On Lp–L1 estimates of logarithmic-type in Hardy–Sobolev spaces of the disk and the annulus

A generalization to the Hardy-Sobolev spaces $H^{k,p}$ of an $L^p$-$L^1$ logarithmic type estimate

“…Note that Theorem 2.1 can be extended to any bounded subset of functions in H k,p (G). Note also that this kind of results generalizes those established in [7], [12], [14], [13], [19], [20] and also improves upon [5], Lemma 4.2, since the upper bound in (2.1) has no log-log term in the numerator.…”

“…The purpose of this paper is to establish a logarithmic estimate of optimal-type in the Hardy-Sobolev space H k,p (G); k ∈ N * , 1 p ∞ and G is either the open unit disk D or the annulus G s of radii (s, 1), 0 < s < 1 of the complex space C. More precisely, we study the behavior on the boundary of G with respect to the L p -norm of any function f in the unit ball of the Hardy-Sobolev H k,p (G) starting from its behavior on any open connected subset I ⊂ ∂G of the boundary of G with respect to the L 1 -norm. Our result can be viewed as an extension of those established in [5], [7], [8], [12], [14], [13], [19], [20].…”

“…We recall also the following inequality, linked to [11], Theorem 5.6, and [13], Lemma 3.4, which will be useful for the proof of Lemma 4.3 and Lemma 4.8.…”