Bessel potential operator
A classical Bessel potential operator is a generalized convolution operator (or a pseudo-differential operator)
where is the Laplace operator, and are, respectively, the Fourier transform and its inverse, and is a generalized kernel (cf. also Kernel of an integral operator). If , the kernel is the modified Bessel function of the third kind (cf. also Bessel functions) and
The set of functions
is known as the Bessel potential space.
i) is translation invariant: with , ;
ii) there exists a continuous and invertible extension for all , ;
iii) and its inverse preserve supports within : , provided and (here, stands for the closure of ).
is said to be a Bessel potential operator for (briefly, ) if and if it generates an additive group , , , [a3].
The following assertions are basic for Bessel potential operators.
1) For a special Lipschitz domain the inclusion holds if and only if
is a generalized convolution, with
The group of can be generated as follows: for [a3].
2) Let , and be as in 1). There exists a generalized kernel such that for all ; if , then .
If is another special Lipschitz domain and , , then for all [a3].
3) Let be as in 1). Any operator arranges an isomorphism of the Bessel potential spaces of functions vanishing at the boundary
(the same for the -spaces).
4) Let be as in 1) and let, further, be the restriction and let be one of its right inverses, for . Then the restricted adjoint operator arranges an isomorphism, where . The isomorphism is independent of the choice of a right inverse (the same for the -spaces).
5) For all and any general Lipschitz domain (even for a manifold with a Lipschitz boundary) there exist pseudo-differential operators and such that and will be isomorphisms (the same for the - and -spaces). is independent of the choice of . If is the principal symbol of (cf. also Symbol of an operator), then will be the principal symbol of . can be chosen, among others, with principal symbols from the Hörmander class [a3], [a4].
6) The operators and from the above assertion can be applied to the lifting of pseudo-differential operators (i.e. to reduction of the order): if is a pseudo-differential operator with principal symbol , then will be an equivalent pseudo-differential operator, with principal symbol [a3], [a4].
|[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|
Bessel potential operator. R. Duduchava (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bessel_potential_operator&oldid=14142