View source for Bessel potential operator

A classical Bessel potential operator is a generalized convolution operator (or a [[Pseudo-differential operator|pseudo-differential operator]])

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with symbol

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104204.png" /> is the [[Laplace operator|Laplace operator]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104206.png" /> are, respectively, the [[Fourier transform|Fourier transform]] and its inverse, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104207.png" /> is a generalized kernel (cf. also [[Kernel of an integral operator|Kernel of an integral operator]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104208.png" />, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104209.png" /> is the modified Bessel function of the third kind (cf. also [[Bessel functions|Bessel functions]]) and

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is an ordinary [[Convolution of functions|convolution of functions]] [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]].

The set of functions

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is known as the Bessel potential space.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042014.png" /> extends to an isomorphism between the Bessel potential spaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042015.png" /> [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]], and even between more general Besov–Triebel–Lizorkin spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042016.png" /> [[#References|[a6]]].

Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042017.png" /> be a special Lipschitz domain. A [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042018.png" /> is said to be a Bessel potential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042020.png" /> (briefly written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042022.png" />) if [[#References|[a3]]]:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042023.png" /> is translation invariant: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042024.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042026.png" />;

ii) there exists a continuous and invertible extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042029.png" />;

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042030.png" /> and its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042031.png" /> preserve supports within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042032.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042033.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042035.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042036.png" /> stands for the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042037.png" />).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042038.png" /> is said to be a Bessel potential operator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042039.png" /> (briefly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042040.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042041.png" /> and if it generates an additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042045.png" /> [[#References|[a3]]].

The following assertions are basic for Bessel potential operators.

1) For a special Lipschitz domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042046.png" /> the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042047.png" /> holds if and only if

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is a generalized convolution, with

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being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042051.png" />-multipliers (cf. also [[Multiplier theory|Multiplier theory]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042052.png" /> [[#References|[a3]]].

The group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042053.png" /> can be generated as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042054.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042055.png" /> [[#References|[a3]]].

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042058.png" /> be as in 1). There exists a generalized kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042060.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042061.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042062.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042063.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042064.png" /> is another special Lipschitz domain and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042067.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042068.png" /> [[#References|[a3]]].

3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042069.png" /> be as in 1). Any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042070.png" /> arranges an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042071.png" /> of the Bessel potential spaces of functions vanishing at the boundary

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(the same for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042073.png" />-spaces).

4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042074.png" /> be as in 1) and let, further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042075.png" /> be the restriction and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042076.png" /> be one of its right inverses, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042077.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042078.png" />. Then the restricted adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042079.png" /> arranges an isomorphism, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042080.png" />. The isomorphism is independent of the choice of a right inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042081.png" /> (the same for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042082.png" />-spaces).

5) For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042083.png" /> and any general Lipschitz domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042084.png" /> (even for a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042085.png" /> with a Lipschitz boundary) there exist pseudo-differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042087.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042089.png" /> will be isomorphisms (the same for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042090.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042091.png" />-spaces). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042092.png" /> is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042093.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042094.png" /> is the principal symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042095.png" /> (cf. also [[Symbol of an operator|Symbol of an operator]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042096.png" /> will be the principal symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042097.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042098.png" /> can be chosen, among others, with principal symbols from the Hörmander class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042099.png" /> [[#References|[a3]]], [[#References|[a4]]].

6) The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b110420100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b110420101.png" /> from the above assertion can be applied to the lifting of pseudo-differential operators (i.e. to reduction of the order): if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b110420102.png" /> is a pseudo-differential operator with principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b110420103.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b110420104.png" /> will be an equivalent pseudo-differential operator, with principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b110420105.png" /> [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Aronszajn,   K. Smith,   "Theory of Bessel potentials, Part 1"  ''Ann. Inst. Fourier'' , '''11'''  (1961)  pp. 385–475</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.P. Calderón,   "Lebesque spaces of differentiable functions and distributions"  C.B. Morrey (ed.) , ''Partial Differential Equations'' , Amer. Math. Soc.  (1961)  pp. 33–49</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Duduchava,   F.-O. Speck,   "Pseudo-differential operators on compact manifolds with Lipschitz boundary"  ''Math. Nachr.'' , '''160'''  (1993)  pp. 149–191</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Schneider,   "Bessel potential operators for canonical Lipschitz domains"  ''Math. Nachr.'' , '''150'''  (1991)  pp. 277–299</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Stein,   "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Triebel,   "Interpolation theory, function spaces, differential operators" , North-Holland  (1978)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Schneider,   "Reduction of order for pseudodifferential operators on Lipschitz domains"  ''Comm. Partial Diff. Eq.'' , '''18'''  (1991)  pp. 1263–1286</TD></TR></table>