Bessel potential operator

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

with symbol

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

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

The set of functions

is known as the Bessel potential space.

extends to an isomorphism between the Bessel potential spaces: [a1], [a2], [a5], and even between more general Besov–Triebel–Lizorkin spaces [a6].

Now, let be a special Lipschitz domain. A linear operator is said to be a Bessel potential operator of order for (briefly written as ) if [a3]:

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

being -multipliers (cf. also Multiplier theory) and [a3].

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].

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