Bessel potential operator

A classical Bessel potential operator is a generalized convolution operator (or a pseudo-differential operator)

$$( { \mathop{\rm Id} } - \Delta ) ^ \nu \varphi = {\mathcal F} ^ {- 1 } \lambda ^ \nu {\mathcal F} \varphi = k _ \nu * \varphi,$$

$$\varphi \in S ( \mathbf R ^ {n} ) , \nu \in \mathbf R,$$

with symbol

$$\lambda ^ \nu ( \xi ) = ( 1 + \left | \xi \right | ^ {2} ) ^ { {\nu / 2 } } = {\mathcal F} k _ \nu ( \xi ) , \xi \in \mathbf R ^ {n} ,$$

where $\Delta = \sum _ {j =1 } ^ {n} { {\partial ^ {2} } / {\partial x _ {j} ^ {2} } }$ is the Laplace operator, ${\mathcal F}$ and ${\mathcal F} ^ {-1 }$ are, respectively, the Fourier transform and its inverse, and $k _ \nu ( x )$ is a generalized kernel (cf. also Kernel of an integral operator). If $\nu < 0$, the kernel $k _ \nu$ is the modified Bessel function of the third kind (cf. also Bessel functions) and

$$k _ \nu * \varphi ( x ) = \int\limits _ {\mathbf R ^ {n} } {k _ \nu ( x - y ) \varphi ( y ) } {dy }$$

is an ordinary convolution of functions [a1], [a2], [a5].

The set of functions

$$H _ {p} ^ {s} ( \mathbf R ^ {n} ) = ( { \mathop{\rm Id} } - \Delta ) ^ {- s } L _ {p} ( \mathbf R ^ {n} ) =$$

$$= \left \{ {u = ( { \mathop{\rm Id} } - \Delta ) ^ {- s } \varphi } : {\varphi \in L _ {p} ( \mathbf R ^ {n} ) } \right \} ,$$

$$1 < p < \infty, s \in \mathbf R,$$

is known as the Bessel potential space.

$( { \mathop{\rm Id} } - \Delta ) ^ \nu$ extends to an isomorphism between the Bessel potential spaces: ${( { \mathop{\rm Id} } - \Delta ) ^ \nu } : {H ^ {s} _ {p} ( \mathbf R _ {n} ) } \rightarrow {H ^ {s - \nu } _ {p} ( \mathbf R _ {n} ) }$[a1], [a2], [a5], and even between more general Besov–Triebel–Lizorkin spaces $F ^ {s} _ {p,q } ( \mathbf R _ {n} ) \rightarrow F ^ {s - \nu } _ {p,q } ( \mathbf R ^ {n} )$[a6].

Now, let $\Omega \subset \mathbf R ^ {n}$ be a special Lipschitz domain. A linear operator $B : {S ( \mathbf R _ {n} ) } \rightarrow {S ^ \prime ( \mathbf R _ {n} ) }$ is said to be a Bessel potential operator of order $\nu \in \mathbf R$ for $\Omega$( briefly written as $B \in { \mathop{\rm BPO} } ( \nu, \Omega )$) if [a3]:

i) $B$ is translation invariant: $BV _ {h} = V _ {h} B$ with $V _ {h} \varphi ( x ) = \varphi ( x - h )$, $x,h \in \mathbf R ^ {n}$;

ii) there exists a continuous and invertible extension $B : {H ^ {s} _ {p} ( \mathbf R _ {n} ) } \rightarrow {H ^ {s - r } _ {p} ( \mathbf R _ {n} ) }$ for all $s \in \mathbf R$, $1 < p < \infty$;

iii) $B$ and its inverse $B ^ {- 1 }$ preserve supports within ${\overline \Omega \; }$: $\supp B ^ {\pm 1 } \varphi \subset {\overline \Omega \; }$, provided $\varphi \in C _ {0} ^ \infty ( \mathbf R _ {n} )$ and $\supp \varphi \in {\overline \Omega \; }$( here, ${\overline \Omega \; }$ stands for the closure of $\Omega$).

$B$ is said to be a Bessel potential operator for $\Omega$( briefly, $B \in { \mathop{\rm BPO} } ( \Omega )$) if $B \in { \mathop{\rm BPO} } ( 1, \Omega )$ and if it generates an additive group $\{ B ^ \nu \} _ {\nu \in \mathbf R }$, $B ^ \nu \in { \mathop{\rm BPO} } ( \nu; \Omega )$, $B ^ {r} B ^ \nu = B ^ {r + \nu }$, $B ^ {0} = { \mathop{\rm Id} }$[a3].

The following assertions are basic for Bessel potential operators.

1) For a special Lipschitz domain $\Omega \subset \mathbf R ^ {n}$ the inclusion $B _ {0} \in { \mathop{\rm BPO} } ( \nu, \Omega )$ holds if and only if

$$B _ {0} ^ {\pm 1 } u = {\mathcal F} ^ {- 1 } {\mathcal B} _ {0} ^ {\pm 1 } {\mathcal F} u = k _ {B _ {0} } * u$$

is a generalized convolution, with

$$[ ( 1 + \left | \xi \right | \mid ^ {2} ) ^ {- {\nu / 2 } } {\mathcal B} _ {0} ( \xi ) ] ^ {\pm 1 }$$

being $L _ {p}$- multipliers (cf. also Multiplier theory) and $\Omega + \supp k _ {B} \subset {\overline \Omega \; }$[a3].

The group of $B \in { \mathop{\rm BPO} } ( \Omega )$ can be generated as follows: $B ^ \mu = {\mathcal F} ^ {- 1 } {\mathcal B} _ {0} ^ {\mu / \nu } {\mathcal F}$ for $\mu \in \mathbf R$[a3].

2) Let $\Omega$, $\nu$ and $B$ be as in 1). There exists a generalized kernel $k _ {B} \in S ^ \prime ( \mathbf R ^ {n} )$ such that $Bu = k _ {B} * u$ for all $u \in C _ {0} ^ \infty ( \mathbf R ^ {n} )$; if $0 \in {\overline \Omega \; }$, then $\supp k _ {B} \subset {\overline \Omega \; }$.

If $\Omega ^ \prime$ is another special Lipschitz domain and $0 \in {\overline{ {\Omega ^ \prime }}\; }$, $\Omega + \Omega ^ \prime \subset {\overline \Omega \; }$, then ${ \mathop{\rm BPO} } ( \nu; \Omega ^ \prime ) \subset { \mathop{\rm BPO} } ( \nu; \Omega )$ for all $\nu \in \mathbf R$[a3].

3) Let $\nu, \Omega$ be as in 1). Any operator $B ^ \nu \in { \mathop{\rm BPO} } ( \nu, \Omega )$ arranges an isomorphism ${B ^ \nu } : { {H \widetilde{ {}} } _ {p} ^ {s} ( \Omega ) } \rightarrow { {H \widetilde{ {}} } _ {p} ^ {s - \nu } ( \Omega ) }$ of the Bessel potential spaces of functions vanishing at the boundary

$${H \widetilde{ {}} } _ {p} ^ {s} ( \Omega ) = \left \{ {\varphi \in H ^ {s} _ {p} ( \mathbf R _ {n} ) } : { \supp \varphi \subset {\overline \Omega \; } } \right \}$$

(the same for the ${F \widetilde{ {}} } _ {p,q } ^ {s} ( \Omega )$- spaces).

4) Let $\nu, \Omega$ be as in 1) and let, further, ${r _ \Omega } : {S ^ \prime ( \mathbf R ^ {n} ) } \rightarrow {S ^ \prime ( \Omega ) }$ be the restriction and let ${\mathcal l} _ \Omega$ be one of its right inverses, $r _ \Omega {\mathcal l} _ \Omega \varphi = \varphi$ for $\varphi \in S ^ \prime ( \Omega )$. Then the restricted adjoint operator ${ {B \overline{ {}}\; } ^ \nu = r _ \Omega ( B ^ \nu ) ^ {*} {\mathcal l} _ \Omega } : {H _ {p} ^ {s} ( \Omega ) } \rightarrow {H _ {p} ^ {s - \nu } ( \Omega ) }$ arranges an isomorphism, where $H _ {p} ^ {s} ( \Omega ) = r _ \Omega H _ {p} ^ {s} ( \mathbf R ^ {n} )$. The isomorphism is independent of the choice of a right inverse ${\mathcal l} _ \Omega$( the same for the $F _ {p,q } ^ {s} ( \Omega )$- spaces).

5) For all $\nu \in \mathbf R$ and any general Lipschitz domain $\Omega \subset \mathbf R ^ {n}$( even for a manifold $\Omega$ with a Lipschitz boundary) there exist pseudo-differential operators $B ^ \nu$ and ${B \overline{ {}}\; } ^ \nu$ such that ${B ^ \nu } : { {H \widetilde{ {}} } _ {p} ^ {s} ( \Omega ) } \rightarrow { {H \widetilde{ {}} } _ {p} ^ {s - \nu } ( \Omega ) }$ and ${ {B \overline{ {}}\; } ^ \nu } : {H _ {p} ^ {s} ( \Omega ) } \rightarrow {H _ {p} ^ {s - \nu } ( \Omega ) }$ will be isomorphisms (the same for the ${F \widetilde{ {}} } _ {p,q } ^ {s} ( \Omega )$- and $F _ {p,q } ^ {s} ( \Omega )$- spaces). $r _ \Omega ( B ^ {*} ) ^ \nu {\mathcal l} _ \Omega$ is independent of the choice of ${\mathcal l} _ \Omega$. If $[ b ( x, \xi ) ] ^ \nu$ is the principal symbol of $B ^ \nu$( cf. also Symbol of an operator), then $[ {\overline{ {b ( x, \xi ) }}\; } ] ^ \mu$ will be the principal symbol of ${B \overline{ {}}\; } ^ \mu$. $B ^ \nu$ can be chosen, among others, with principal symbols from the Hörmander class $S ^ \nu ( \mathbf R _ {n} )$[a3], [a4].

6) The operators $B ^ \nu$ and ${B \overline{ {}}\; } ^ \mu$ from the above assertion can be applied to the lifting of pseudo-differential operators (i.e. to reduction of the order): if ${a ( x, {\mathcal D} ) } : { {H \widetilde{ {}} } _ {p} ^ {s} ( \Omega ) } \rightarrow {H ^ {s - r } _ {p} ( \Omega ) }$ is a pseudo-differential operator with principal symbol $a ( x, \xi )$, then ${ {B \overline{ {}}\; } ^ {s - r } a ( x, {\mathcal D} ) B ^ {- s } } : {L _ {p} ( \Omega ) } \rightarrow {L _ {p} ( \Omega ) }$ will be an equivalent pseudo-differential operator, with principal symbol $[ {\overline{ {b ( x, \xi ) }}\; } ] ^ {s - r } a ( x, \xi ) [ b ( x, \xi ) ] ^ {- s }$[a3], [a4].

References

 [a1] N. Aronszajn, K. Smith, "Theory of Bessel potentials, Part 1" Ann. Inst. Fourier , 11 (1961) pp. 385–475 [a2] A.P. Calderón, "Lebesque spaces of differentiable functions and distributions" C.B. Morrey (ed.) , Partial Differential Equations , Amer. Math. Soc. (1961) pp. 33–49 [a3] R. Duduchava, F.-O. Speck, "Pseudo-differential operators on compact manifolds with Lipschitz boundary" Math. Nachr. , 160 (1993) pp. 149–191 [a4] R. Schneider, "Bessel potential operators for canonical Lipschitz domains" Math. Nachr. , 150 (1991) pp. 277–299 [a5] E. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970) [a6] H. Triebel, "Interpolation theory, function spaces, differential operators" , North-Holland (1978) [a7] R. Schneider, "Reduction of order for pseudodifferential operators on Lipschitz domains" Comm. Partial Diff. Eq. , 18 (1991) pp. 1263–1286
How to Cite This Entry:
Bessel potential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_potential_operator&oldid=46213
This article was adapted from an original article by R. Duduchava (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article