Optimal logarithmic estimates in Hardy–Sobolev spaces Hk,∞

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

“…Further remarks concerning Theorem 1.1. As a consequence of Theorem 1.1, we find again for α = 1 the results of the authors [5], Theorem 2.6, when G = D and those of the second author [13], Theorem 3.7, when G = G s . The following consequences are immediate.…”

“…Then, referring to [3], Lemma 4.1, [14], Lemma 2.4, and [5], Lemma 2.3, we get the following two estimates of the function f and its radial primitive F t in the case of the unit disk D.…”

“…In the present paper we establish some optimal logarithmic estimates in the Hölder-Hardy spaces H k,α (G), extending thus the earlier cases [3], [5], [18], [20]. Our main result is the following: Theorem 1.1.…”

“…More recently, a (1/ log k )-type optimal estimates with respect to the L ∞ norm have been established by the authors in H k,∞ (D) ( [5]). For the annulus G s , a (1/ log)-type estimate with respect to the L 2 norm has been proved by J.…”

“…We also prove a variant of the Hardy-Landau-Littlewood inequality amid the class of Hölder functions on a finite interval. Section 3 is devoted to the proof of our main results and to recovering the results of [5], [13] as a consequence of Theorem 1.1. As applications, in the last part of the paper we study the stability of Cauchy's problem for the Laplace operator in the class of Hölder solutions, and the stability of the Robin inverse problem in the case of a flux ϕ ∈ W 1,2 0 (I).…”