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

Let X (ℝ n ) = X (ℝ n , μ n ) be a rearrangement‐invariant Banach function space over the measure space (ℝ n , μ n ), where μ n stands for the n ‐dimensional Lebesgue measure in ℝ n . We derive a sharp estimate for the k ‐modulus of smoothness of the convolution of a function f ∈ X (ℝ n ) with the Bessel potential kernel g σ , where σ ∈ (0, n ). Such an estimate states that if g σ belongs to the associate space of X , thenω k ( f * g σ , t ) ≺ ˜∫ 0 t ns σ / n − 1f * ( s ) d s for   all   t ∈ ( 0 , 1 )   and   every   f ∈ X ( ℝ 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 ⁡∂ j g ω∂ x 1 j( x ) = ( − 1 ) j , with σ ∈ (0, n ) and j ∈ 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 x 1 ‐axis. The above estimate is very important in applications. For example, it enables us to derive optimal continuous embeddings of Bessel potential spaces H σ X (ℝ n ) in a forthcoming paper, where, in limiting situations, we are able to obtain embeddings into Zygmund‐type spaces rather than Hölder‐type spaces. In particular, such results show that the Brézis–Wainger embedding of the Sobolev space W k +1, n / k (ℝ n ), with k ∈ ℕ 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.