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

Abstract

Let X(Rn) = X(Rn, ¥in) be a rearrangement-invariant Banach function space over the measure space (Rn, ¥in), where ¥in stands for the n-dimensional Lebesgue measure in Rn. We derive a sharp estimate for the k-modulus of smoothness of the convolution of a function f io X(Rn) with the Bessel potential kernel g¥o, where ¥o io (0, n). Such an estimate states that if g¥o belongs to the associate space of X, then ¥ok(f . g¥o, t)   tn 0 s¥o/n.1f . (s) ds for all t io (0, 1) and every f io X(Rn) provided that k  [¥o] + 1 (f . denotes the non-increasing rearrangement of f). One of the key steps in the proof of the sharpness of this estimate is the assertion that sgn iOjg¥o iOxj 1 (x) = (.1)j , with ¥o io (0, n) and j io N, for all x in a small circular half-cone which has its vertex at the origin and its axis coincides with the positive part of the x1-axis. The above estimate is very important in applications. For example, it enables us to derive optimal continuous embeddings of Bessel potential spaces H¥oX(Rn) in a forthcoming paper, where, in limiting situations, we are able to obtain embeddings into Zygmund-type spaces rather than Hi§older-type spaces. In particular, such results show that the Br¢¥ezis.Wainger embedding of the Sobolev space Wk+1,n/k(Rn), with k io N and k < n. 1, into the space of i®almosti¯ Lipschitz functions, is a consequence of a better embedding which has as its target a Zygmund-type space.