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A classical Bessel potential operator is a generalized convolution operator (or a [[Pseudo-differential operator|pseudo-differential operator]])
 
A classical Bessel potential operator is a generalized convolution operator (or a [[Pseudo-differential operator|pseudo-differential operator]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104201.png" /></td> </tr></table>
+
$$
 +
( { \mathop{\rm Id} } - \Delta )  ^  \nu  \varphi = {\mathcal F} ^ {- 1 } \lambda  ^  \nu  {\mathcal F} \varphi = k _  \nu  * \varphi,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104202.png" /></td> </tr></table>
+
$$
 +
\varphi \in S ( \mathbf R  ^ {n} ) ,  \nu \in \mathbf R,
 +
$$
  
 
with symbol
 
with symbol
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b1104203.png" /></td> </tr></table>
+
$$
 +
\lambda  ^  \nu  ( \xi ) = ( 1 + \left | \xi \right |  ^ {2} ) ^ { {\nu / 2 } } = {\mathcal F} k _  \nu  ( \xi ) ,  \xi \in \mathbf R  ^ {n} ,
 +
$$
  
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
+
where $  \Delta = \sum _ {j =1 }  ^ {n} { {\partial  ^ {2} } / {\partial  x _ {j}  ^ {2} } } $
 +
is the [[Laplace operator|Laplace operator]], $  {\mathcal F} $
 +
and $  {\mathcal F} ^ {-1 } $
 +
are, respectively, the [[Fourier transform|Fourier transform]] and its inverse, and $  k _  \nu  ( x ) $
 +
is a generalized kernel (cf. also [[Kernel of an integral operator|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|Bessel functions]]) and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042010.png" /></td> </tr></table>
+
$$
 +
k _  \nu  * \varphi ( x ) = \int\limits _ {\mathbf R  ^ {n} } {k _  \nu  ( x - y ) \varphi ( y ) }  {dy }
 +
$$
  
 
is an ordinary [[Convolution of functions|convolution of functions]] [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]].
 
is an ordinary [[Convolution of functions|convolution of functions]] [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]].
Line 17: Line 43:
 
The set of functions
 
The set of functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042011.png" /></td> </tr></table>
+
$$
 +
H _ {p}  ^ {s} ( \mathbf R  ^ {n} ) = ( { \mathop{\rm Id} } - \Delta ) ^ {- s } L _ {p} ( \mathbf R  ^ {n} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042012.png" /></td> </tr></table>
+
$$
 +
=  
 +
\left \{ {u = ( { \mathop{\rm Id} } - \Delta ) ^ {- s } \varphi } : {\varphi \in L _ {p} ( \mathbf R  ^ {n} ) } \right \} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042013.png" /></td> </tr></table>
+
$$
 +
1 < p < \infty,  s \in \mathbf R,
 +
$$
  
 
is known as the Bessel potential space.
 
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]]].
+
$  ( { \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} ) } $[[#References|[a1]]], [[#References|[a2]]], [[#References|[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} ) $[[#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]]]:
+
Now, let $  \Omega \subset  \mathbf R  ^ {n} $
 +
be a special Lipschitz domain. A [[Linear operator|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 [[#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" />;
+
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 <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" />;
+
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) <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" />).
+
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 $).
  
<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]]].
+
$  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} } $[[#References|[a3]]].
  
 
The following assertions are basic for Bessel potential operators.
 
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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042048.png" /></td> </tr></table>
+
$$
 +
B _ {0} ^ {\pm  1 } u = {\mathcal F} ^ {- 1 } {\mathcal B} _ {0} ^ {\pm  1 } {\mathcal F} u = k _ {B _ {0}  } * u
 +
$$
  
 
is a generalized convolution, with
 
is a generalized convolution, with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042049.png" /></td> </tr></table>
+
$$
 +
[ ( 1 + \left | \xi \right | \mid  ^ {2} ) ^ {- {\nu / 2 } } {\mathcal B} _ {0} ( \xi ) ] ^ {\pm  1 }
 +
$$
  
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]]].
+
being $  L _ {p} $-
 +
multipliers (cf. also [[Multiplier theory|Multiplier theory]]) and $  \Omega +  \supp  k _ {B} \subset  {\overline \Omega \; } $[[#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]]].
+
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 $[[#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" />.
+
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 <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]]].
+
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 $[[#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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042072.png" /></td> </tr></table>
+
$$
 +
{H \widetilde{ {}}  } _ {p}  ^ {s} ( \Omega ) = \left \{ {\varphi \in H  ^ {s} _ {p} ( \mathbf R _ {n} ) } : { \supp  \varphi \subset  {\overline \Omega \; } } \right \}
 +
$$
  
(the same for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110420/b11042073.png" />-spaces).
+
(the same for the $  {F \widetilde{ {}}  } _ {p,q }  ^ {s} ( \Omega ) $-
 +
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).
+
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 <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]]].
+
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|Symbol of an operator]]), then $  [ { {b ( x, \xi ) } bar } ]  ^  \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} ) $[[#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]]].
+
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 $  [ { {b ( x, \xi ) } bar } ] ^ {s - r } a ( x, \xi ) [ b ( x, \xi ) ] ^ {- s } $[[#References|[a3]]], [[#References|[a4]]].
  
 
====References====
 
====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>
 
<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>

Revision as of 10:58, 29 May 2020


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 $ [ { {b ( x, \xi ) } bar } ] ^ \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 $ [ { {b ( x, \xi ) } bar } ] ^ {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=46037
This article was adapted from an original article by R. Duduchava (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article