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''fractional Sobolev space, Liouville space''
 
''fractional Sobolev space, Liouville space''
  
A [[Banach space|Banach space]] of integrable functions or distributions on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201701.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201702.png" />, which generalizes the ordinary [[Sobolev space|Sobolev space]] of functions whose derivatives belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201703.png" />-classes, and their duals. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201704.png" /> denotes the [[Laplace operator|Laplace operator]], the Bessel potential space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201707.png" />, can be defined as the space of functions (or distributions) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201708.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b1201709.png" /> belongs to the [[Lebesgue space|Lebesgue space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017010.png" />, normed by the corresponding Lebesgue norm. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017011.png" />, which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017012.png" /> is a kind of fractional differentiation (cf. also [[Fractional integration and differentiation|Fractional integration and differentiation]]), is most easily defined by means of the [[Fourier transform|Fourier transform]]. It corresponds, in fact, to multiplication of the Fourier transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017013.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017014.png" />. The operator clearly has the group properties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017016.png" />.
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A [[Banach space|Banach space]] of integrable functions or distributions on the $n$-dimensional Euclidean space ${\bf R} ^ { n }$, which generalizes the ordinary [[Sobolev space|Sobolev space]] of functions whose derivatives belong to $L ^ { p }$-classes, and their duals. If $\Delta$ denotes the [[Laplace operator|Laplace operator]], the Bessel potential space $L _ { \alpha } ^ { p }$, $1 < p < \infty$, $- \infty < \alpha < \infty$, can be defined as the space of functions (or distributions) $f$ such that $( I - \Delta ) ^ { \alpha / 2 } f$ belongs to the [[Lebesgue space|Lebesgue space]] $L ^ { p }$, normed by the corresponding Lebesgue norm. The operator $( I - \Delta ) ^ { \alpha / 2 } = \mathcal{G}_{ - \alpha}$, which for $\alpha &gt; 0$ is a kind of fractional differentiation (cf. also [[Fractional integration and differentiation|Fractional integration and differentiation]]), is most easily defined by means of the [[Fourier transform|Fourier transform]]. It corresponds, in fact, to multiplication of the Fourier transform of $f$ by $( 1 + | \xi | ^ { 2 } ) ^ { \alpha / 2 }$. The operator clearly has the group properties $\mathcal{G} _ { \alpha } \mathcal{G} _ { \beta } = \mathcal{G} _ { \alpha + \beta }$, and $\mathcal{G} _ { \alpha } ^ { - 1 } = \mathcal{G} _ { - \alpha }$.
  
It is a theorem of A.P. Calderón that for positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017018.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017019.png" /> coincides (with equivalence of norms) with the Sobolev space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017020.png" /> of functions all of whose derivatives (in the distributional, or weak sense) of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017021.png" /> are functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017022.png" />.
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It is a theorem of A.P. Calderón that for positive integers $\alpha$ and $1 < p < \infty$ the space $L _ { \alpha } ^ { p }$ coincides (with equivalence of norms) with the Sobolev space $W _ { \alpha } ^ { p }$ of functions all of whose derivatives (in the distributional, or weak sense) of order at most $\alpha$ are functions in $L ^ { p }$.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017023.png" /> the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017024.png" /> are themselves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017025.png" />-functions, which can be represented as Bessel potentials of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017026.png" />-functions. In fact, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017027.png" /> is then the Fourier transform of an integrable function, the Bessel kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017028.png" />, and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017029.png" /> can be represented by a convolution with this kernel. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017032.png" />, if and only if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017034.png" />, where the integral is taken over all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017035.png" /> with respect to the Lebesgue measure.
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For $\alpha &gt; 0$ the elements of $L _ { \alpha } ^ { p }$ are themselves $L ^ { p }$-functions, which can be represented as Bessel potentials of $L ^ { p }$-functions. In fact, the function $( 1 + | \xi | ^ { 2 } ) ^ { - \alpha / 2 }$ is then the Fourier transform of an integrable function, the Bessel kernel $G _ { \alpha } ( x )$, and the operator ${\cal G} _ { \alpha }$ can be represented by a convolution with this kernel. In other words, $f \in L _ { \alpha } ^ { p }$, $1 < p < \infty$, $\alpha > 0$, if and only if there is a $g \in L ^ { p }$ such that $f ( x ) = \mathcal{G} _ { \alpha } g ( x ) = \int G _ { \alpha } ( x - y ) g ( y ) d y$, where the integral is taken over all of ${\bf R} ^ { n }$ with respect to the Lebesgue measure.
  
The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017036.png" /> can be expressed explicitly by means of a modified Bessel function of the third kind (cf. also [[Bessel functions|Bessel functions]]), also known as a Macdonald function, and for this reason the Bessel potentials were given their name by N. Aronszajn and K.T. Smith in 1961. More important than the exact expression for the kernel is the fact that it is a suitable modification of the (Marcel) Riesz kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017038.png" />, whose Fourier transform is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017039.png" />. The Bessel kernel has the same properties as the Riesz kernel for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017040.png" />, but thanks to the fact that its Fourier transform behaves nicely at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017041.png" />, it decays exponentially at infinity. In contrast to the Riesz kernel it is therefore an integrable function, and this is its main advantage.
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The kernel $G_{\alpha}$ can be expressed explicitly by means of a modified Bessel function of the third kind (cf. also [[Bessel functions|Bessel functions]]), also known as a Macdonald function, and for this reason the Bessel potentials were given their name by N. Aronszajn and K.T. Smith in 1961. More important than the exact expression for the kernel is the fact that it is a suitable modification of the (Marcel) Riesz kernel $I_{\alpha} (x) = c_{ \alpha } |x|^{\alpha-n}$, $0 < \alpha < n$, whose Fourier transform is $|\xi|^{-\alpha}$. The Bessel kernel has the same properties as the Riesz kernel for small $x$, but thanks to the fact that its Fourier transform behaves nicely at $0$, it decays exponentially at infinity. In contrast to the Riesz kernel it is therefore an integrable function, and this is its main advantage.
  
The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017042.png" /> appear naturally as interpolation spaces that are obtained from Sobolev spaces by means of the complex interpolation method (cf. also [[Interpolation of operators|Interpolation of operators]]). They are included in the more general scale of Lizorkin–Triebel spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017043.png" />; in fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017044.png" /> (with equivalence of norms) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017046.png" />. This equivalence is a highly non-trivial result of so-called Littlewood–Paley type. Related to this are very useful representations by means of atoms.
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The spaces $L_{\alpha}^{p}$ appear naturally as interpolation spaces that are obtained from Sobolev spaces by means of the complex interpolation method (cf. also [[Interpolation of operators]]). They are included in the more general scale of Lizorkin–Triebel spaces $F_{\alpha}^{p, q}$; in fact, $L_ {\alpha}^{p} = F_{\alpha}^{p, 2}$ (with equivalence of norms) for $-\infty < \alpha < \infty$ and $1 < p < \infty$. This equivalence is a highly non-trivial result of so-called Littlewood–Paley type. Related to this are very useful representations by means of atoms.
  
The [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017047.png" />, also known as the Dirichlet space, and its generalizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017048.png" /> are intimately related to classical [[Potential theory|potential theory]]. The study of more general non-Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120170/b12017050.png" />, motivated by investigations of non-linear partial differential equations, has lead to the creation of a new non-linear potential theory, and many of the results and concepts of the classical theory have been extended to the non-linear setting, sometimes in unexpected ways.
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The [[Hilbert space]] $W_{1}^{2}$, also known as the Dirichlet space, and its generalizations $L_{\alpha}^{2}$ are intimately related to classical [[potential theory]]. The study of more general non-Hilbert spaces $L_{\alpha}^{p}$ and $W_{\alpha}^{p}$, motivated by investigations of non-linear partial differential equations, has lead to the creation of a new non-linear potential theory, and many of the results and concepts of the classical theory have been extended to the non-linear setting, sometimes in unexpected ways.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.R. Adams,   L.I. Hedberg,   "Function spaces and potential theory" , Springer  (1996)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Triebel,   "Theory of function spaces II" , Birkhäuser  (1992)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top"> D.R. Adams, L.I. Hedberg, "Function spaces and potential theory" , Springer  (1996)</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top"> H. Triebel, "Theory of function spaces II" , Birkhäuser  (1992)</td></tr>
 +
</table>

Latest revision as of 20:14, 19 November 2023

fractional Sobolev space, Liouville space

A Banach space of integrable functions or distributions on the $n$-dimensional Euclidean space ${\bf R} ^ { n }$, which generalizes the ordinary Sobolev space of functions whose derivatives belong to $L ^ { p }$-classes, and their duals. If $\Delta$ denotes the Laplace operator, the Bessel potential space $L _ { \alpha } ^ { p }$, $1 < p < \infty$, $- \infty < \alpha < \infty$, can be defined as the space of functions (or distributions) $f$ such that $( I - \Delta ) ^ { \alpha / 2 } f$ belongs to the Lebesgue space $L ^ { p }$, normed by the corresponding Lebesgue norm. The operator $( I - \Delta ) ^ { \alpha / 2 } = \mathcal{G}_{ - \alpha}$, which for $\alpha > 0$ is a kind of fractional differentiation (cf. also Fractional integration and differentiation), is most easily defined by means of the Fourier transform. It corresponds, in fact, to multiplication of the Fourier transform of $f$ by $( 1 + | \xi | ^ { 2 } ) ^ { \alpha / 2 }$. The operator clearly has the group properties $\mathcal{G} _ { \alpha } \mathcal{G} _ { \beta } = \mathcal{G} _ { \alpha + \beta }$, and $\mathcal{G} _ { \alpha } ^ { - 1 } = \mathcal{G} _ { - \alpha }$.

It is a theorem of A.P. Calderón that for positive integers $\alpha$ and $1 < p < \infty$ the space $L _ { \alpha } ^ { p }$ coincides (with equivalence of norms) with the Sobolev space $W _ { \alpha } ^ { p }$ of functions all of whose derivatives (in the distributional, or weak sense) of order at most $\alpha$ are functions in $L ^ { p }$.

For $\alpha > 0$ the elements of $L _ { \alpha } ^ { p }$ are themselves $L ^ { p }$-functions, which can be represented as Bessel potentials of $L ^ { p }$-functions. In fact, the function $( 1 + | \xi | ^ { 2 } ) ^ { - \alpha / 2 }$ is then the Fourier transform of an integrable function, the Bessel kernel $G _ { \alpha } ( x )$, and the operator ${\cal G} _ { \alpha }$ can be represented by a convolution with this kernel. In other words, $f \in L _ { \alpha } ^ { p }$, $1 < p < \infty$, $\alpha > 0$, if and only if there is a $g \in L ^ { p }$ such that $f ( x ) = \mathcal{G} _ { \alpha } g ( x ) = \int G _ { \alpha } ( x - y ) g ( y ) d y$, where the integral is taken over all of ${\bf R} ^ { n }$ with respect to the Lebesgue measure.

The kernel $G_{\alpha}$ can be expressed explicitly by means of a modified Bessel function of the third kind (cf. also Bessel functions), also known as a Macdonald function, and for this reason the Bessel potentials were given their name by N. Aronszajn and K.T. Smith in 1961. More important than the exact expression for the kernel is the fact that it is a suitable modification of the (Marcel) Riesz kernel $I_{\alpha} (x) = c_{ \alpha } |x|^{\alpha-n}$, $0 < \alpha < n$, whose Fourier transform is $|\xi|^{-\alpha}$. The Bessel kernel has the same properties as the Riesz kernel for small $x$, but thanks to the fact that its Fourier transform behaves nicely at $0$, it decays exponentially at infinity. In contrast to the Riesz kernel it is therefore an integrable function, and this is its main advantage.

The spaces $L_{\alpha}^{p}$ appear naturally as interpolation spaces that are obtained from Sobolev spaces by means of the complex interpolation method (cf. also Interpolation of operators). They are included in the more general scale of Lizorkin–Triebel spaces $F_{\alpha}^{p, q}$; in fact, $L_ {\alpha}^{p} = F_{\alpha}^{p, 2}$ (with equivalence of norms) for $-\infty < \alpha < \infty$ and $1 < p < \infty$. This equivalence is a highly non-trivial result of so-called Littlewood–Paley type. Related to this are very useful representations by means of atoms.

The Hilbert space $W_{1}^{2}$, also known as the Dirichlet space, and its generalizations $L_{\alpha}^{2}$ are intimately related to classical potential theory. The study of more general non-Hilbert spaces $L_{\alpha}^{p}$ and $W_{\alpha}^{p}$, motivated by investigations of non-linear partial differential equations, has lead to the creation of a new non-linear potential theory, and many of the results and concepts of the classical theory have been extended to the non-linear setting, sometimes in unexpected ways.

References

[a1] D.R. Adams, L.I. Hedberg, "Function spaces and potential theory" , Springer (1996)
[a2] H. Triebel, "Theory of function spaces II" , Birkhäuser (1992)
How to Cite This Entry:
Bessel potential space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_potential_space&oldid=16844
This article was adapted from an original article by L.I. Hedberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article