Sharp estimates of the k-modulus of smoothness of Bessel potentials

0342833 - MÚ 2011 RIV GB eng J - Journal Article
Gogatishvili, Amiran - Neves, J. S. - Opic, Bohumír
Sharp estimates of the k-modulus of smoothness of Bessel potentials.
Journal of the London Mathematical Society. Roč. 81, č. 3 (2010), s. 608-624. ISSN 0024-6107. E-ISSN 1469-7750
R&D Projects: GA ČR GA201/08/0383
Institutional research plan: CEZ:AV0Z10190503
Keywords : embeddings * spaces * optimality * compact
Subject RIV: BA - General Mathematics
Impact factor: 0.828, year: 2010
http://jlms.oxfordjournals.org/content/81/3/608

Let X(n)=X(n, µn) be a rearrangement-invariant Banach function space over the measure space (n, µn), where µn stands for the n-dimensional Lebesgue measure in n. We derive a sharp estimate for the k-modulus of smoothness of the convolution of a function fX(n) with the Bessel potential kernel g, where (0, n). The above estimate is very important in applications. For example, it enables us to derive optimal continuous embeddings of Bessel potential spaces HX(n) in a forthcoming paper, where, in limiting situations, we are able to obtain embeddings into Zygmund-type spaces rather than Hölder-type spaces. In particular, such results show that the Brézis–Wainger embedding of the Sobolev space Wk+1, n/k(n), with k and k<n–1, into the space of ‘almost’ Lipschitz functions, is a consequence of a better embedding which has as its target a Zygmund-type space.
Permanent Link: http://hdl.handle.net/11104/0185456