An improvement of dimension-free Sobolev imbeddings in r.i. spaces

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<∞, the inequality(⁎)‖f⁎‖Yp(0,1)≤cp‖∇f‖Lp(Bn)∀f∈W01,p(Bn),∀n>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,p′/2(0,1). Moreover, we show that the smallest space Yp(0,1) (in the sense of the continuous imbedding) such that (⁎) is true has the fundamental function equivalent to that of L(p,p′/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 (⁎).