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