Bessel potential space

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)