Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4590115 | Journal of Functional Analysis | 2014 | 19 Pages |
Abstract
We prove a dimension-invariant imbedding estimate for Sobolev spaces of first order into a small Lebesgue space, and we establish the optimality of its fundamental function. Namely, for any 1
p where Yp(0,1) is a rearrangement-invariant Banach function space independent of the dimension n, Bn is the ball in Rn of measure 1 and cp is a constant independent of n, is satisfied by the small Lebesgue space L(p,pâ²/2(0,1). Moreover, we show that the smallest space Yp(0,1) (in the sense of the continuous imbedding) such that (â) is true has the fundamental function equivalent to that of L(p,pâ²/2(0,1). As a byproduct of our results, we get that the space Lp(logL)p/2 is optimal in the framework of the Orlicz spaces satisfying (â).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Alberto Fiorenza, Miroslav Krbec, Hans-Jürgen Schmeisser,