Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving A-Laplacian ?Δ _A u = ?divA(?u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset \({\Omega \subseteq \mathbb{R}^n}\) . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the form $$\int_\Omega F_{\bar{A}}(|\xi|) \mu_1(dx) \leq \int_\Omega \bar{A}(|\nabla \xi|)\mu_2(dx),$$ where \({\bar{A}(t)}\) is a Young function related to A and satisfying Δ′-condition, while \({F_{\bar{A}}(t) = 1/(\bar{A}(1/t))}\) . Examples involving \({\bar{A}(t) = t^p{\rm log}^\alpha(2+t), p \geq 1, \alpha \geq 0}\) are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30–50, 2013), where we dealt with inequality ?Δ _p u ≥ Φ, leading to Hardy and Hardy–Poincaré inequalities with the best constants.     

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J.?Fourier Anal Appl. 1:311?C353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.