Let X(?~n) = X(?~n, μ_n) be a rearrangement-invariant Banach function space over the measure space (?~n, μ_n), where μ_n 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 f ∈ X(?~n) with the Bessel potential kernel gσ, where σ ∈ (0, n). Such an estimate states that if gσ belongs to the associate space of X, then ωk(f * g_σ, t) t_n. s~(σ-1)f*(s) ds for all t ∈ (0, 1) and every f ∈ X(?n) provided that k ≥ [σ] + 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 ?jgσ/ ?x~j_1(x) = (-1)~j with σ ∈ (0, n) and j ∈ ? 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~σX(?~n) in a forthcoming paper, where, in limiting situations, we are able to obtain embeddings into Zygmund-type spaces rather than H?lder-type spaces. In particular, such results show that the Brézis-Wainger embedding of the Sobolev space W~(k+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.
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