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

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 with a constant c_p, related to the imbedding of W_0^{1,p}(B_n) into Y_p(0,1), where Yp(0,1) is a rearrangement-invariant Banach function space independent of the dimension n, B_n is the ball in R^n of measure 1 and c_p 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 (log L)p/2 is optimal in the framework of the Orlicz spaces satisfying the imbedding inequality.