Abstract In the paper we prove weighted Hardy type inequality $$\begin{aligned} \int _{{{\mathbb {R}}}^N}V\varphi ^2 \mu (x)dx\le _{\mathbb {R}^N}|\nabla \varphi |^2\mu (x)dx +K\int {R}^N}\varphi ^2\mu (x)dx, \end{aligned}$$ ∫ R </mml:mrow...

see for instance [29]. Notice that the exponent p or q could be larger than N+2 N−2 . Hence the usual Sobolev space H 0 (Ω)×H1 0 (Ω) is not suitable to handle the problem. To study the problem (1.2) under the condition (1.3), a key observation was done by Hulshof and Van de Vorst [16], De Figueiredo and Felmer [9]. In order to solve this problem, the main idea is to destroy the symmetry between...

and Applied Analysis 3 Now we recall definitions and some properties of the Smirnov class, the Privalov class, the Bergman-Privalov class, and the Zygmund F-algebra on Bn or D. The space of all holomorphic functions on X Bn or D is denoted by H X . For each 0 < p ≤ ∞, the Hardy space is denoted by H X with the norm ‖ · ‖p. 2.1. Smirnov Class N∗ X Let X ∈ {Bn, Dn}. The Nevanlinna class N X on X ...

We consider a more general class than the class of Schur multipliers namely the Abel-Schur multipliers, which in turn coincide with the bounded linear operators on `2 preserving the diagonals. We extend to the matrix framework Theorem 2.4 (a) of a paper of Anderson, Clunie, and Pommerenke published in 1974, and as an application of this theorem we obtain a new proof of the necessity of an old t...

Associated to the class of restricted-weak type weights for the Hardy operator Rp, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal functi...