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<title>Pseudodifferentialoperator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Pseudodifferentialoperator</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Ein Pseudodifferentialoperator ist eine mathematische Funktion, die eine Erweiterung des Konzepts des <a href="Differentialoperator" title="Differentialoperator">Differentialoperators</a> darstellt. Sie sind ein wichtiger Bestandteil der Theorie der <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">partiellen Differentialgleichungen</a> sowie der <a href="Mikrolokale_Analysis" title="Mikrolokale Analysis">mikrolokalen Analysis</a>. Die Grundlagen der Theorie stammen von <a href="Lars_H%C3%B6rmander" title="Lars Hörmander">Lars Hörmander</a>. Eingeführt wurden sie 1965 durch <a href="Joseph_Kohn" title="Joseph Kohn">Joseph Kohn</a> und <a href="Louis_Nirenberg" title="Louis Nirenberg">Louis Nirenberg</a>.

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Lineare_Differentialoperatoren_mit_konstanten_Koeffizienten">Lineare Differentialoperatoren mit konstanten Koeffizienten</h3></div>

Man betrachte den linearen Differentialoperator mit konstanten Koeffizienten

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<annotation encoding="application/x-tex">{\displaystyle p(D):=\sum _{\alpha }a_{\alpha }\,D^{\alpha }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8d4b56496413eb2fb63cadc52046cf3ebf26fb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:18.589ex; height:5.509ex;" alt="{\displaystyle p(D):=\sum _{\alpha }a_{\alpha }\,D^{\alpha }}" loading="lazy"></dd></dl>

der auf dem Raum der glatten Funktionen mit <a href="Kompakter_Raum" title="Kompakter Raum">kompaktem</a> <a href="Tr%C3%A4ger_(Mathematik)" title="Träger (Mathematik)">Träger</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> operiert. Er kann als Komposition einer <a href="Kontinuierliche_Fourier-Transformation" class="mw-redirect" title="Kontinuierliche Fourier-Transformation">Fouriertransformation</a>, einer einfachen Multiplikation mit dem Polynom (dem sogenannten Symbol)

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<annotation encoding="application/x-tex">{\displaystyle p(\xi )=\sum _{\alpha }a_{\alpha }\,\xi ^{\alpha }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6fed05a1a9f70deebbc3f37e1392a4481142338.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:16.158ex; height:5.509ex;" alt="{\displaystyle p(\xi )=\sum _{\alpha }a_{\alpha }\,\xi ^{\alpha }}" loading="lazy"></dd></dl>

und der inversen Fouriertransformation:

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<annotation encoding="application/x-tex">{\displaystyle (1)\quad p(D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }p(\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/751eea4190d14827af24f9605bfd1f5b5e3d30f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.25ex; height:6.009ex;" alt="{\displaystyle (1)\quad p(D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }p(\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }" loading="lazy"></dd></dl>

geschrieben werden. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} _{0}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} _{0}^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5faee3f16b785f2f545df1a4790bbada882b2c6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.559ex; height:2.843ex;" alt="{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} _{0}^{n}}" loading="lazy">

ein <a href="Multiindex" title="Multiindex">Multiindex</a>,

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<annotation encoding="application/x-tex">{\displaystyle D^{\alpha }=(-i\partial _{1})^{\alpha _{1}}\dots (-i\partial _{n})^{\alpha _{n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5be9871da6a1172ecb86d588876423c68db173a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.751ex; height:2.843ex;" alt="{\displaystyle D^{\alpha }=(-i\partial _{1})^{\alpha _{1}}\dots (-i\partial _{n})^{\alpha _{n}}}" loading="lazy">

ein Differentialoperator,

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<annotation encoding="application/x-tex">{\displaystyle \partial _{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57023ce0c60b39ddd06853b983b3f1dcc04db740.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.144ex; height:2.843ex;" alt="{\displaystyle \partial _{j}}" loading="lazy">

steht für Ableitung nach der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">-ten Komponente und

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<annotation encoding="application/x-tex">{\displaystyle a_{\alpha }\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab65fe582d11fcb0691ca0549dca6d02bb85e943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.901ex; height:2.009ex;" alt="{\displaystyle a_{\alpha }\,}" loading="lazy"> sind komplexe Zahlen.

Analog ist ein Pseudodifferentialoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> mit Symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x,\xi )}">

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<annotation encoding="application/x-tex">{\displaystyle p(x,\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/657d9ea5d51566bab68b4df5eb1c2c68167a851e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle p(x,\xi )}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> ein Operator der Form

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<annotation encoding="application/x-tex">{\displaystyle (2)\quad Pu(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }p(x,\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/896536f58d69de1705f9d14b62f9411bd9fdbe48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.456ex; height:6.009ex;" alt="{\displaystyle (2)\quad Pu(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }p(x,\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }" loading="lazy">,</dd></dl>

mit einer allgemeineren Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> im Integranden, wie unten weiter ausgeführt wird.

<dl><dt>Herleitung von Formel (1)</dt></dl>

Die Fouriertransformation einer glatten Funktion

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy">

mit kompaktem Träger in

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> ist

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<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}(\xi ):=\int \mathrm {e} ^{-\mathrm {i} y\xi }u(y)\,\mathrm {d} y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aecbd5bc8729beca514698d2d571841cce4ce803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.171ex; height:5.676ex;" alt="{\displaystyle {\hat {u}}(\xi ):=\int \mathrm {e} ^{-\mathrm {i} y\xi }u(y)\,\mathrm {d} y}" loading="lazy"></dd></dl>

und inverse Fouriertransformation ergibt

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<annotation encoding="application/x-tex">{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \mathrm {e} ^{\mathrm {i} x\xi }{\hat {u}}(\xi )\,\mathrm {d} \xi ={\frac {1}{(2\pi )^{n}}}\int \int \mathrm {e} ^{\mathrm {i} (x-y)\xi }u(y)\,\mathrm {d} y\,\mathrm {d} \xi \,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82104d22c15c622bff73022fff6840a2d2d34190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.807ex; height:6.009ex;" alt="{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \mathrm {e} ^{\mathrm {i} x\xi }{\hat {u}}(\xi )\,\mathrm {d} \xi ={\frac {1}{(2\pi )^{n}}}\int \int \mathrm {e} ^{\mathrm {i} (x-y)\xi }u(y)\,\mathrm {d} y\,\mathrm {d} \xi \,.}" loading="lazy"></dd></dl>

Wendet man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(D)}">

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<annotation encoding="application/x-tex">{\displaystyle p(D)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d3c836183257f935db4d8d46fae3750b21dbc97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.992ex; height:2.843ex;" alt="{\displaystyle p(D)}" loading="lazy"> auf diese Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> an und benutzt

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<annotation encoding="application/x-tex">{\displaystyle p(D)\,\mathrm {e} ^{\mathrm {i} (x-y)\xi }=\mathrm {e} ^{\mathrm {i} (x-y)\xi }\,p(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c79645b7e9e161da595d9641d59b53844bfa23f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:26.405ex; height:3.343ex;" alt="{\displaystyle p(D)\,\mathrm {e} ^{\mathrm {i} (x-y)\xi }=\mathrm {e} ^{\mathrm {i} (x-y)\xi }\,p(\xi )}" loading="lazy">,</dd></dl>

erhält man (1).

<div class="mw-heading mw-heading3"><h3 id="Darstellung_von_Lösungen_von_partiellen_Differentialgleichungen">Darstellung von Lösungen von partiellen Differentialgleichungen</h3></div>

Um eine partielle Differentialgleichung

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<annotation encoding="application/x-tex">{\displaystyle p(D)\,u=f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b70edd56e46be26666e37071651bf88c8dee7009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.086ex; height:2.843ex;" alt="{\displaystyle p(D)\,u=f}" loading="lazy"></dd></dl>

zu lösen, werden beide Seiten (formal) fouriertransformiert, wobei sich algebraische Gleichungen ergeben:

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<annotation encoding="application/x-tex">{\displaystyle p(\xi )\,{\hat {u}}(\xi )={\hat {f}}(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65335b88923895034a192c179620b02c6dc1d3e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.291ex; height:3.343ex;" alt="{\displaystyle p(\xi )\,{\hat {u}}(\xi )={\hat {f}}(\xi )}" loading="lazy">.</dd></dl>

Falls das Symbol

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<annotation encoding="application/x-tex">{\displaystyle p(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/722a5061e1a6b168527e738b8419a82fa7a770cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.098ex; height:2.843ex;" alt="{\displaystyle p(\xi )}" loading="lazy">

immer ungleich Null ist für

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<annotation encoding="application/x-tex">{\displaystyle \xi \in \mathbb {R} ^{n}}</annotation>

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kann man durch

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<annotation encoding="application/x-tex">{\displaystyle p(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/722a5061e1a6b168527e738b8419a82fa7a770cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.098ex; height:2.843ex;" alt="{\displaystyle p(\xi )}" loading="lazy"> dividieren:

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<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}(\xi )={\frac {1}{p(\xi )}}{\hat {f}}(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de494f71b061310e244d0b03544872b2ab8dd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.651ex; height:6.009ex;" alt="{\displaystyle {\hat {u}}(\xi )={\frac {1}{p(\xi )}}{\hat {f}}(\xi )}" loading="lazy"></dd></dl>

Die Lösung lautet dann mit Anwendung der umgekehrten Fouriertransformation:

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<annotation encoding="application/x-tex">{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \mathrm {e} ^{\mathrm {i} x\xi }{\frac {1}{p(\xi )}}{\hat {f}}(\xi )\,\mathrm {d} \xi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/007e6978e5648980318218a833bb9cd9583cbc56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.377ex; height:6.009ex;" alt="{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \mathrm {e} ^{\mathrm {i} x\xi }{\frac {1}{p(\xi )}}{\hat {f}}(\xi )\,\mathrm {d} \xi }" loading="lazy">.</dd></dl>

Dabei wird folgendes vorausgesetzt:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(D)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d3c836183257f935db4d8d46fae3750b21dbc97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.992ex; height:2.843ex;" alt="{\displaystyle p(D)}" loading="lazy"> ist ein linearer Differentialoperator mit konstanten Koeffizienten,</li>

<li>sein Symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\xi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/722a5061e1a6b168527e738b8419a82fa7a770cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.098ex; height:2.843ex;" alt="{\displaystyle p(\xi )}" loading="lazy"> ist niemals Null für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8a718e9c68da4acb51d12eb9d9474bd5aed8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.767ex; height:2.676ex;" alt="{\displaystyle \xi \in \mathbb {R} ^{n}}" loading="lazy">,</li>

<li>sowohl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> als auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> haben wohldefinierte Fouriertransformierte.</li></ol>

Die letzte Annahme kann mit der Theorie der <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">Distributionen</a> abgeschwächt werden. Die ersten beiden Annahmen können wie folgt abgeschwächt werden:

Man setze in der letzten Formel die Fouriertransformation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ein:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \int \mathrm {e} ^{\mathrm {i} (x-y)\xi }{\frac {1}{p(\xi )}}f(y)\,\mathrm {d} y\,\mathrm {d} \xi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd716eb007008e064ef7dcc8d0ef85a91d2d94d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.873ex; height:6.009ex;" alt="{\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \int \mathrm {e} ^{\mathrm {i} (x-y)\xi }{\frac {1}{p(\xi )}}f(y)\,\mathrm {d} y\,\mathrm {d} \xi }" loading="lazy">.</dd></dl>

Das ist ähnlich Formel (1), nur dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{p(\xi )}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{p(\xi )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d01a700526a6683ac32112f8e54b6727c55a0202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.671ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{p(\xi )}}}" loading="lazy"> kein Polynom ist, sondern eine Funktion allgemeinerer Art.

<div class="mw-heading mw-heading2"><h2 id="Definition_des_Pseudodifferentialoperators">Definition des Pseudodifferentialoperators</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Die_Symbolklasse">Die Symbolklasse</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Symbolklasse" title="Symbolklasse">Symbolklasse</a></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(x,\xi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(x,\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7ddd6243ebed602789695c98ba65be0debddd90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.433ex; height:2.843ex;" alt="{\displaystyle a(x,\xi )}" loading="lazy"> eine <a href="Glatte_Funktion" title="Glatte Funktion">unendlich oft differenzierbare Funktion</a> auf

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \times \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6bf9f1c88504a47081e93f1dd169aca6166782b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.415ex; height:2.343ex;" alt="{\displaystyle \Omega \times \mathbb {R} ^{n}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> offen, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b9aa033deb5e523e2d98098c1b1418f97b33e5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {R} }" loading="lazy">, mit

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\partial _{\xi }^{\alpha }\partial _{x}^{\beta }a(x,\xi )|\leq C_{\alpha ,\beta ,K}\,(1+|\xi |)^{m-|\alpha |}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/337566db2bf3799c63182a9f9a9cb3dbbb5e4188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.676ex; height:3.843ex;" alt="{\displaystyle |\partial _{\xi }^{\alpha }\partial _{x}^{\beta }a(x,\xi )|\leq C_{\alpha ,\beta ,K}\,(1+|\xi |)^{m-|\alpha |}}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9075ef2df706bf2764f76182a1ddf55ff231a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle x\in K}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subset \Omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/685a8b4e07d58637ae0e38517fc1009c614677b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K\subset \Omega }" loading="lazy"> kompakt ist, für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy">, alle Multiindizes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4b46b57cfa0011b643037751809904d915c1b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }" loading="lazy">, eine Konstante

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\alpha ,\beta ,K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42bb8501757d342de6d27f40de285e6199316a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.263ex; height:2.843ex;" alt="{\displaystyle C_{\alpha ,\beta ,K}}" loading="lazy">, so gehört <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> zur Symbolklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{m}(\Omega \times \mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{m}(\Omega \times \mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2f087836285ccff8609ab7eb5f0d67147a4c983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.421ex; height:2.843ex;" alt="{\displaystyle S^{m}(\Omega \times \mathbb {R} ^{n})}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Pseudodifferentialoperator">Pseudodifferentialoperator</h3></div>

Sei wieder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> eine glatte Funktion aus der Symbolklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{m}(X\times \mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{m}(X\times \mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8651ac72f6b887ae2e8e5718312bf6cdc2148ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.723ex; height:2.843ex;" alt="{\displaystyle S^{m}(X\times \mathbb {R} ^{n})}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23a6b446fb9736703b3fe09ff010de5ef2e75f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.975ex; height:2.343ex;" alt="{\displaystyle X\subset \mathbb {R} ^{n}}" loading="lazy">. Ein Pseudodifferentialoperator der Ordnung m ist gewöhnlicherweise eine Abbildung

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}(X)\to {\mathcal {E}}(X)\quad {\text{bzw.}}\quad {\mathcal {S}}(X)\to {\mathcal {S}}(X),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd717ee1c17c4129b971ae6a28630680ed605d1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.415ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(X)\to {\mathcal {E}}(X)\quad {\text{bzw.}}\quad {\mathcal {S}}(X)\to {\mathcal {S}}(X),}" loading="lazy"></dd></dl>

welche durch

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Pu)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }a(x,\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05b82d2d8cd9b27d8f95781adbc82603eeb7861a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.032ex; height:6.009ex;" alt="{\displaystyle (Pu)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }a(x,\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }" loading="lazy"></dd></dl>

definiert ist. Der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"> ist der Raum der <a href="Testfunktion" title="Testfunktion">Testfunktionen</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"> ist der Raum der <a href="Glatte_Funktion" title="Glatte Funktion">glatten Funktionen</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> ist der <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Raum</a>.

<div class="mw-heading mw-heading3"><h3 id="Eigentlich_getragener_Pseudodifferentialoperator">Eigentlich getragener Pseudodifferentialoperator</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> ein Pseudodifferentialoperator. Im Folgenden sei

<msub>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{P}(x,y):=\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }a(x,\xi )\,\mathrm {d} \xi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88240cf58f86956fc0fbacef516b0be3f04222d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.485ex; height:5.676ex;" alt="{\displaystyle K_{P}(x,y):=\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }a(x,\xi )\,\mathrm {d} \xi }" loading="lazy"></dd></dl>

der <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</a> des Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. Der Pseudodifferentialoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> heißt eigentlich getragen, falls die Projektionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1},\pi _{2}:\operatorname {supp} (K_{P})\to X}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \pi _{1},\pi _{2}:\operatorname {supp} (K_{P})\to X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cd8949c434a74acaec9fff744c644bcd597befd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.366ex; height:2.843ex;" alt="{\displaystyle \pi _{1},\pi _{2}:\operatorname {supp} (K_{P})\to X}" loading="lazy"> <a href="Eigentliche_Abbildung" title="Eigentliche Abbildung">eigentlich</a> sind.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<ul><li>Lineare Differentialoperatoren der Ordnung m mit glatten, beschränkten Koeffizienten können als Pseudodifferentialoperatoren der Ordnung m aufgefasst werden.</li>

<li>Der <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle K(x,y):=\int _{\mathbb {R} ^{n}}^{OS}\mathrm {e} ^{\mathrm {i} \langle x-y,\xi \rangle }a(x,\xi )\,\mathrm {d} \xi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c87e8a58e904a364ab6b05f6e8264d4be6630c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.903ex; height:6.343ex;" alt="{\displaystyle K(x,y):=\int _{\mathbb {R} ^{n}}^{OS}\mathrm {e} ^{\mathrm {i} \langle x-y,\xi \rangle }a(x,\xi )\,\mathrm {d} \xi }" loading="lazy"></dd></dl></dd>

<dd>ist außer auf der Diagonalen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,y)|x=y\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x,y)|x=y\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62d835bafdc89a6d9d74064bb1fbc9471e4a90d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.884ex; height:2.843ex;" alt="{\displaystyle \{(x,y)|x=y\}}" loading="lazy"> ein glatter <a href="Schwartz-Kern" class="mw-redirect" title="Schwartz-Kern">Schwartz-Kern</a>.</dd></dl>

<ul><li>Die Transponierte eines Pseudodifferentialoperators ist ebenfalls wieder ein Pseudodifferentialoperator.</li>

<li>Falls ein linearer Differentialoperator der Ordnung m <a href="Elliptischer_Differentialoperator" class="mw-redirect" title="Elliptischer Differentialoperator">elliptisch</a> ist, ist sein Inverses ein Pseudodifferentialoperator der Ordnung −m. Man kann also lineare, elliptische Differentialgleichungen mehr oder weniger explizit mit Hilfe der Theorie der Pseudodifferentialoperatoren lösen.</li>

<li>Differentialoperatoren sind lokal. Das bedeutet, dass man nur den Wert einer Funktion in der Umgebung eines Punktes zu kennen braucht, um die Wirkung des Operators zu bestimmen. Pseudodifferentialoperatoren sind pseudolokal, das bedeutet, dass diese den singulären Träger einer Distribution nicht vergrößern. Es gilt also

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sing\,supp} (Au)\subset \operatorname {sing\,supp} (u)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9355deb3e0c59656cf142a68657e0d1e033646cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:29.518ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle \operatorname {sing\,supp} (Au)\subset \operatorname {sing\,supp} (u)}" loading="lazy">.</dd></dl></li>

<li>Da der Schwartz-Raum <a href="Dichte_Teilmenge" title="Dichte Teilmenge">dicht</a> im Raum der <a href="Quadratintegrierbar" class="mw-redirect" title="Quadratintegrierbar">quadratintegrierbaren Funktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"> liegt, ist es möglich mittels Stetigkeitsargumenten einen Pseudodifferentialoperator auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"> <a href="Einschr%C3%A4nkung_(Mathematik)" class="mw-redirect" title="Einschränkung (Mathematik)">fortzusetzen</a>. Gilt außerdem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Psi ^{0}(\mathbb {R} ^{n}\times \mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle A\in \Psi ^{0}(\mathbb {R} ^{n}\times \mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/738e2f126684424f6cf4a17f48debd9fff9c5a64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.889ex; height:3.176ex;" alt="{\displaystyle A\in \Psi ^{0}(\mathbb {R} ^{n}\times \mathbb {R} ^{n})}" loading="lazy"> dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle A\colon L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8b47824d2148a5a259908800844c07174327523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.077ex; height:3.176ex;" alt="{\displaystyle A\colon L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}" loading="lazy"> ein <a href="Beschr%C3%A4nkter_Operator" title="Beschränkter Operator">beschränkter</a> also stetiger Operator.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Komposition_von_Pseudodifferentialoperatoren">Komposition von Pseudodifferentialoperatoren</h3></div>

Pseudodifferentialoperatoren mit dem <a href="Schwartz-Raum" title="Schwartz-Raum">Schwartz-Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0078d18e4675b6e7e2acb6c2c25c65294193e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"> als Definitionsbereich bilden diesen in sich selbst ab. Sie sind sogar ein Isomorphismus auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0078d18e4675b6e7e2acb6c2c25c65294193e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy">.<a href="#cite_note-1">[1]</a> Außerdem bilden eigentlich getragene Pseudodifferentialoperatoren den Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/260741c735b326a154f2737ed63e97f6cb6d7540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.379ex; height:2.843ex;" alt="{\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}" loading="lazy"> in sich ab. Daher ist es möglich für solche Operatoren die <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Komposition</a> zweier Pseudodifferentialoperatoren zu betrachten, was wieder einen Pseudodifferentialoperatoren ergibt.

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in S^{m_{1}}(X\times \mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle a\in S^{m_{1}}(X\times \mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5fbdeca996ab7f2f12ac207627aad852217622e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.625ex; height:2.843ex;" alt="{\displaystyle a\in S^{m_{1}}(X\times \mathbb {R} ^{n})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in S^{m_{2}}(X\times \mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle b\in S^{m_{2}}(X\times \mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/005e0ee2b4d1fba30768f289ea249d9fd27555f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.393ex; height:2.843ex;" alt="{\displaystyle b\in S^{m_{2}}(X\times \mathbb {R} ^{n})}" loading="lazy"> zwei Symbole und seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{a}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5830e67f78d703f1bc6ff6d691691cba661ef48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.594ex; height:2.509ex;" alt="{\displaystyle P_{a}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{b}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{b}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92b684b99147aa175db98057461b2dc2653f704e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.43ex; height:2.509ex;" alt="{\displaystyle P_{b}}" loading="lazy"> die entsprechenden Pseudodifferentialoperatoren, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}\circ P_{b}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{a}\circ P_{b}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dab8fad88569c1d21aed52b5738ea83893acc61a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.219ex; height:2.509ex;" alt="{\displaystyle P_{a}\circ P_{b}}" loading="lazy"> wieder ein Pseudodifferentialoperator. Das Symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> des Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}\circ P_{b}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{a}\circ P_{b}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dab8fad88569c1d21aed52b5738ea83893acc61a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.219ex; height:2.509ex;" alt="{\displaystyle P_{a}\circ P_{b}}" loading="lazy"> ist ein Element des Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{m_{1}+m_{2}}(X\times \mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle S^{m_{1}+m_{2}}(X\times \mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dba2dadd208df38a59a1ae15d83609f2a053fc49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.107ex; height:3.009ex;" alt="{\displaystyle S^{m_{1}+m_{2}}(X\times \mathbb {R} ^{n})}" loading="lazy"> und es hat die <a href="Symbolklasse#Asymptotische_Entwicklung_eines_Symbols" title="Symbolklasse">asymptotische Entwicklung</a>

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<annotation encoding="application/x-tex">{\displaystyle c\sim \sum _{\mu =0}^{\infty }{\frac {(-i)^{|\mu |}}{\mu !}}{\frac {\partial ^{\mu }a}{\partial \xi ^{\mu }}}(x,\xi ){\frac {\partial ^{\mu }b}{\partial x^{\mu }}}(x,\xi )\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9e3c0657757328ab0f41d1f3e259d5b0cbf41df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.021ex; height:7.509ex;" alt="{\displaystyle c\sim \sum _{\mu =0}^{\infty }{\frac {(-i)^{|\mu |}}{\mu !}}{\frac {\partial ^{\mu }a}{\partial \xi ^{\mu }}}(x,\xi ){\frac {\partial ^{\mu }b}{\partial x^{\mu }}}(x,\xi )\,,}" loading="lazy"></dd></dl>

was

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<annotation encoding="application/x-tex">{\displaystyle c-\sum _{\mu <N}{\frac {(-i)^{|\mu |}}{\mu !}}{\frac {\partial ^{\mu }a}{\partial \xi ^{\mu }}}(x,\xi ){\frac {\partial ^{\mu }b}{\partial x^{\mu }}}(x,\xi )\in S^{m_{1}+m_{2}-N}(X\times \mathbb {R} ^{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/481773c9b866380dcd4e04b7cd7ab084ece9c4a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.789ex; height:7.509ex;" alt="{\displaystyle c-\sum _{\mu <N}{\frac {(-i)^{|\mu |}}{\mu !}}{\frac {\partial ^{\mu }a}{\partial \xi ^{\mu }}}(x,\xi ){\frac {\partial ^{\mu }b}{\partial x^{\mu }}}(x,\xi )\in S^{m_{1}+m_{2}-N}(X\times \mathbb {R} ^{n})}" loading="lazy"></dd></dl>

bedeutet.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Adjungierter_Operator">Adjungierter Operator</h3></div>

Für jedes Paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ,\,\psi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi ,\,\psi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/083f06247e07a6d49eed245f4274862730a96775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.32ex; height:2.509ex;" alt="{\displaystyle \phi ,\,\psi }" loading="lazy"> von Schwartz-Funktionen sei

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<annotation encoding="application/x-tex">{\displaystyle (\phi ,\psi )=\int _{X}\phi (x){\overline {\psi (x)}}\mathrm {d} x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58511ba9060543882b1f4a3729e092a964268756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.066ex; height:5.676ex;" alt="{\displaystyle (\phi ,\psi )=\int _{X}\phi (x){\overline {\psi (x)}}\mathrm {d} x}" loading="lazy"></dd></dl>

eine <a href="Bilinearform" title="Bilinearform">Bilinearform</a> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\colon {\mathcal {S}}(X)\to {\mathcal {S}}(X)}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon {\mathcal {S}}(X)\to {\mathcal {S}}(X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7593f8a48b2274c387328d6be98c32f8d35513bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.957ex; height:2.843ex;" alt="{\displaystyle P\colon {\mathcal {S}}(X)\to {\mathcal {S}}(X)}" loading="lazy"> ein Pseudodifferentialoperator mit Symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in S^{m}(X\times \mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in S^{m}(X\times \mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49220e51fa65e3ad9bf12d0b375bd509a70b0168.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.793ex; height:2.843ex;" alt="{\displaystyle a\in S^{m}(X\times \mathbb {R} ^{n})}" loading="lazy">. Dann ist der <a href="Adjungierter_Operator" title="Adjungierter Operator">formal adjungierte Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24635cb75ac6b04bbdb6054de6df7eca458d3ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.876ex; height:2.343ex;" alt="{\displaystyle P^{*}}" loading="lazy"> bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ,\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fc515c912925128800226dd0b017be508069e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}" loading="lazy"> wieder ein Pseudodifferentialoperator und sein Symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e8ebbdafec4421aba60c540bcd665fe9aaa5c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.343ex;" alt="{\displaystyle a^{*}}" loading="lazy"> ist ein Element des Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{m}(X\times \mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{m}(X\times \mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8651ac72f6b887ae2e8e5718312bf6cdc2148ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.723ex; height:2.843ex;" alt="{\displaystyle S^{m}(X\times \mathbb {R} ^{n})}" loading="lazy"> und es hat die asymptotische Entwicklung

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{*}\sim \sum _{\mu =0}^{\infty }{\frac {(-i)^{|\mu |}}{\mu !}}\,{\frac {\partial ^{\mu }}{\partial x^{\mu }}}{\frac {\partial ^{\mu }b}{\partial \xi ^{\mu }}}\,{\overline {a(x,\xi )}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5181d6ce1c9f0c6530d961f76bf9b1376553dcb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.993ex; height:7.509ex;" alt="{\displaystyle a^{*}\sim \sum _{\mu =0}^{\infty }{\frac {(-i)^{|\mu |}}{\mu !}}\,{\frac {\partial ^{\mu }}{\partial x^{\mu }}}{\frac {\partial ^{\mu }b}{\partial \xi ^{\mu }}}\,{\overline {a(x,\xi )}}\,.}" loading="lazy"><a href="#cite_note-3">[3]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Pseudodifferentialoperatoren_auf_Distributionenräumen">Pseudodifferentialoperatoren auf Distributionenräumen</h2></div>

Mit Hilfe des formal adjungierten Operators ist es möglich Pseudodifferentialoperatoren auf <a href="Distribution_(Mathematik)" title="Distribution (Mathematik)">Distributionenräumen</a> zu definieren. Dazu betrachtet man statt der Bilinearform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ,\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fc515c912925128800226dd0b017be508069e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}" loading="lazy"> die <a href="Duale_Paarung" title="Duale Paarung">duale Paarung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ,\cdot )_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}}">

<msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359d4021bb9e6ab426a1b56fa0ae236330997b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.169ex; height:3.343ex;" alt="{\displaystyle (\cdot ,\cdot )_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}}" loading="lazy"> zwischen dem Schwartz-Raum und seinem <a href="Dualraum" title="Dualraum">Dualraum</a>. Die duale Paarung kann als <a href="Stetige_Fortsetzung" class="mw-redirect" title="Stetige Fortsetzung">stetige Fortsetzung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ,\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fc515c912925128800226dd0b017be508069e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}" loading="lazy"> verstanden werden. Daher ist es möglich Pseudodifferentialoperatoren auf dem Dualraum des Schwartz-Raum also dem Raum der <a href="Temperierte_Distribution" title="Temperierte Distribution">temperierten Distributionen</a> zu definieren.

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\colon {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2fa3c93bbbbff6b46fcd19f47e64f3c9594667b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.79ex; height:2.843ex;" alt="{\displaystyle P\colon {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"> ein Pseudodifferentialoperator und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathcal {S}}'(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in {\mathcal {S}}'(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/002f32e103345e665cf34a609b5dc6d901ae6351.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.078ex; height:3.009ex;" alt="{\displaystyle u\in {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"> eine temperierte Distribution. Dann ist der fortgesetzte Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {P}}\colon {\mathcal {S}}'(\mathbb {R} ^{n})\to {\mathcal {S}}'(\mathbb {R} ^{n})}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {P}}\colon {\mathcal {S}}'(\mathbb {R} ^{n})\to {\mathcal {S}}'(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5d9cb52ae0e504e2ccc06990c3eb64f47790af6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.276ex; height:3.176ex;" alt="{\displaystyle {\tilde {P}}\colon {\mathcal {S}}'(\mathbb {R} ^{n})\to {\mathcal {S}}'(\mathbb {R} ^{n})}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in {\mathcal {S}}(\mathbb {R} ^{n})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in {\mathcal {S}}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d24603731948f00fbad05857abff0fddb2f94d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.166ex; height:2.843ex;" alt="{\displaystyle v\in {\mathcal {S}}(\mathbb {R} ^{n})}" loading="lazy"> definiert durch

<mover>

</mover>

</mrow>

<msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\tilde {P}}u,v)_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}:=(u,P^{*}v)_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09355227fb8241d17a531c09c1755bf638f40b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:42.133ex; height:3.676ex;" alt="{\displaystyle ({\tilde {P}}u,v)_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}:=(u,P^{*}v)_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}\,.}" loading="lazy"></dd></dl>

Für Pseudodifferentialoperatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\colon {\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {E}}(\mathbb {R} ^{n})}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon {\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {E}}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5bb8708752a3c21a99c3736e8de5ea48585691d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.908ex; height:2.843ex;" alt="{\displaystyle P\colon {\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {E}}(\mathbb {R} ^{n})}" loading="lazy"> gilt Analoges. Der bezüglich der Bilinearform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot ,\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fc515c912925128800226dd0b017be508069e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}" loading="lazy"> adjungierte Operator ist ein Pseudodifferentialoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{*}\colon {\mathcal {E}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P^{*}\colon {\mathcal {E}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7371353da73f7efff4ee9e768d0d1681359ebfab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.038ex; height:2.843ex;" alt="{\displaystyle P^{*}\colon {\mathcal {E}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n})}" loading="lazy"> und diesen kann man ebenfalls analog zu einem Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {P}}\colon {\mathcal {E}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {P}}\colon {\mathcal {E}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/534835ef74a5056a0111645da2042f37b47b046a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.369ex; height:3.176ex;" alt="{\displaystyle {\tilde {P}}\colon {\mathcal {E}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})}" loading="lazy"> stetig fortsetzen. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}'(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}'(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/963053e27b2c2698296922a9a79e6171c2a2a7a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.182ex; height:3.009ex;" alt="{\displaystyle {\mathcal {D}}'(\mathbb {R} ^{n})}" loading="lazy"> der Raum der Distributionen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6b31e955bf24d690800eff9e33f552adcef3311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.727ex; height:3.009ex;" alt="{\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}" loading="lazy"> der Raum der Distributionen mit kompaktem Träger.

<div class="mw-heading mw-heading2"><h2 id="Pseudodifferentialoperatoren_auf_Mannigfaltigkeiten">Pseudodifferentialoperatoren auf Mannigfaltigkeiten</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{c}^{\infty }(X)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{c}^{\infty }(X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af90942f33f95f8a96827b9a01e5dbb52d5b6514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.463ex; height:2.843ex;" alt="{\displaystyle C_{c}^{\infty }(X)}" loading="lazy"> der Raum der <a href="Testfunktion" title="Testfunktion">Testfunktionen</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subset \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23a6b446fb9736703b3fe09ff010de5ef2e75f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.975ex; height:2.343ex;" alt="{\displaystyle X\subset \mathbb {R} ^{n}}" loading="lazy">, sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine <a href="Kompakter_Raum" title="Kompakter Raum">kompakte</a> <a href="Glatte_Mannigfaltigkeit" class="mw-redirect" title="Glatte Mannigfaltigkeit">glatte Mannigfaltigkeit</a> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U_{j},\phi _{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U_{j},\phi _{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e24b6618981c6b08b5bd660ca93238f1a88f63a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.526ex; height:3.009ex;" alt="{\displaystyle (U_{j},\phi _{i})}" loading="lazy"> eine <a href="Atlas_(Mathematik)" title="Atlas (Mathematik)">Karte</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Eine stetige Abbildung

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon C^{\infty }(M)\to C^{\infty }(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37d2ba72fb5755bb90dcdc3ff57e127850d5fde5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.243ex; height:2.843ex;" alt="{\displaystyle P\colon C^{\infty }(M)\to C^{\infty }(M)}" loading="lazy"></dd></dl>

ist ein Pseudodifferentialoperator, falls er lokal in jeder Karte wie ein Pseudodifferentialoperator in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> dargestellt werden kann. Konkret heißt dies, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> ist ein Pseudodifferentialoperator, falls für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0},\,\psi _{1}\in C_{c}^{\infty }(U_{j})}">

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0},\,\psi _{1}\in C_{c}^{\infty }(U_{j})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53f111016f4c37628e1f50a52093f9934a840d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.376ex; height:3.009ex;" alt="{\displaystyle \psi _{0},\,\psi _{1}\in C_{c}^{\infty }(U_{j})}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}=1}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/185d8a9e5d91f60349f4025399ba449a33417bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.828ex; height:2.509ex;" alt="{\displaystyle \psi _{1}=1}" loading="lazy"> in einer Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {supp} (\psi _{0})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {supp} (\psi _{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f84b1b9348b629c3e4fe0de190a5e232402fce93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle \operatorname {supp} (\psi _{0})}" loading="lazy"> der <a href="Linearer_Operator" title="Linearer Operator">Operator</a>

<msub>

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {P}}_{i}(u)(y):=\psi _{0}(x)\cdot P(\psi _{1}\cdot u\circ \phi _{i})(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9d5ba4b77724eab7b446ddc6dcd5905c73ef968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.496ex; height:3.176ex;" alt="{\displaystyle {\tilde {P}}_{i}(u)(y):=\psi _{0}(x)\cdot P(\psi _{1}\cdot u\circ \phi _{i})(x)}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\phi _{i}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y=\phi _{i}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/544bea8284088438ae80a0546b97db3b0b941486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.578ex; height:2.843ex;" alt="{\displaystyle y=\phi _{i}(x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in C^{\infty }(\phi _{i}(\Omega _{i}))}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle u\in C^{\infty }(\phi _{i}(\Omega _{i}))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29d4413b28ec573d2a044ab228c69e5a4cbad1d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.125ex; height:2.843ex;" alt="{\displaystyle u\in C^{\infty }(\phi _{i}(\Omega _{i}))}" loading="lazy"> ein Pseudodifferentialoperator ist.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>José García-Cuerva: Fourier Analysis and Partial Differential Equations. CRC Press, Boca Raton FL u. a. 1995, ISBN 0-8493-7877-X.</li>

<li>Lars Hörmander: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators (= Grundlehren der mathematischen Wissenschaften. Bd. 274). Springer, Berlin 1985, ISBN 3-540-13828-5.</li>

<li>Michail A. Shubin: Pseudodifferential Operators and Spectral Theory. 2nd edition. Springer, Berlin u. a. 2001, ISBN 3-540-41195-X.</li>

<li><a href="Michael_E._Taylor" title="Michael E. Taylor">Michael E. Taylor</a>: Pseudodifferential Operators (= Princeton Mathematical Series. Bd. 34). Princeton University Press, Princeton NJ 1981, ISBN 0-691-08282-0.</li>

<li>Michael E. Taylor: Partial differential equations. Band 1–2. Springer, New York u. a. 1996, ISBN 0-387-94653-5 (Bd. 1), ISBN 0-387-94651-9 (Bd. 2).</li>

<li><a href="Fran%C3%A7ois_Treves" title="François Treves">François Treves</a>: Introduction to Pseudo Differential and Fourier Integral Operators. 2 Bände. Plenum Press, New York NY u. a. 1980;

<ul><li>Band 1: Pseudodifferential Operators. ISBN 0-306-40403-6;</li>

<li>Band 2: Fourier Integral Operators. ISBN 0-306-40404-4.</li></ul></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblink">Weblink</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9906155">Mark Joshi, Vorlesungen über Pseudodifferentialoperatoren, englisch</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Man Wah Wong: <cite style="font-style:italic">An introduction to pseudo-differential operator</cite>. 2. Auflage. World Scientific, River Edge NJ 1999, ISBN 981-02-3813-4, S. 31–33.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Pseudodifferentialoperator&rft.au=Man+Wah+Wong&rft.btitle=An+introduction+to+pseudo-differential+operator&rft.date=1999&rft.edition=2.&rft.genre=book&rft.isbn=9810238134&rft.pages=31-33&rft.place=River+Edge+NJ&rft.pub=World+Scientific" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Man Wah Wong: <cite style="font-style:italic">An introduction to pseudo-differential operator</cite>. 2. Auflage. World Scientific, River Edge NJ 1999, ISBN 981-02-3813-4, S. 54–60.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Pseudodifferentialoperator&rft.au=Man+Wah+Wong&rft.btitle=An+introduction+to+pseudo-differential+operator&rft.date=1999&rft.edition=2.&rft.genre=book&rft.isbn=9810238134&rft.pages=54-60&rft.place=River+Edge+NJ&rft.pub=World+Scientific" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Man Wah Wong: <cite style="font-style:italic">An introduction to pseudo-differential operator</cite>. 2. Auflage. World Scientific, River Edge NJ 1999, ISBN 981-02-3813-4, S. 62–69.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Pseudodifferentialoperator&rft.au=Man+Wah+Wong&rft.btitle=An+introduction+to+pseudo-differential+operator&rft.date=1999&rft.edition=2.&rft.genre=book&rft.isbn=9810238134&rft.pages=62-69&rft.place=River+Edge+NJ&rft.pub=World+Scientific" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Christopher D. Sogge: Fourier Integrals in Classical Analysis. (= Cambridge Tracts in Mathematics. Bd. 105). Digitally printed version. Cambridge University Press, Cambridge u. a. 2008, ISBN 978-0-521-06097-4, S. 106.

</li>

</ol>

<div>

Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4047640-6">4047640-6</a> </div>

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