Bessel potentials with logarithmic components and Sobolev-type embeddings. (English) Zbl 0973.46021

Opic, Bohumír; Trebels, Walter

Anal. Math. 26, No. 4, 299-319 (2000).

This very technical and well written paper treats the problem of existence of continuous embeddings of the Sobolev space \(W_k^P({\mathcal R}^n)\) of \(L^p({\mathcal R}^n)\) integrable functions whose (weak) derivatives of order \(\leq k\) (\(k\in{\mathcal N}\) fixed) are also integrable.
Introducing the spaces of Bessel potentials \[ H^{\sigma}(L^p)({\mathcal R}^n)=\{u: u=g_{\sigma}\ast f, f\in L^p\},\;1\leq p\leq \infty, \] and \[ L^p_{\sigma,\alpha}({\mathcal R}^n) =H^{\sigma,\alpha}(L^p)({\mathcal R}^n)= \{u: u=g_{\sigma,\alpha}\ast f, f\in L^p\},\;||u||_{\sigma,\alpha}:=||f||_p, \] where the Bessel kernels are given by their Fourier transforms \[ {\hat g}_{\sigma}(\xi)=(1+|\xi|^2)^{-\sigma/2}, \sigma>0; \] \[ {\hat g}_{\sigma,\alpha}(\xi)=(1+|\xi|^2)^{-\sigma/2}(1+ \log{(1+|\xi|^2)})^{-\alpha}, \sigma\geq 0, \alpha\in{\mathcal R}, \] the author improves on the known embedding (using \(g_{\sigma}\)) \[ L_{\sigma^{*}}^{n/\sigma}({\mathcal R}^n) \hookrightarrow L^{\infty}({\mathcal R}^n), \sigma^{*}>0, 0<\sigma<n \] with the aid of the smoother kernel \(g_{\sigma,\alpha}\).
The paper is written compactly and contains quite a number of results (including the case where either the target space is near \(L^{\infty}\) or where the source space is near to \(L^1\)).

Reviewer: Marcel G.de Bruin (Delft)

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38	Linear operators on function spaces (general)
47G10	Integral operators
42B35	Function spaces arising in harmonic analysis
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

continuous embedding; (quasi) normed spaces; Sobolev spaces; Lipschitz continuity; Orlicz spaces