Reduction principle for Gaussian KK-inequality

We study interpolation properties of operators (not necessarily linear) which satisfy a specific $K$-inequality corresponding to endpoints defined in terms of Orlicz--Karamata spaces modeled upon the example of the Gaussian--Sobolev embedding. We prove a reduction principle for a fairly wide class of such operators.Comment: 20 page

Finite element approximation of the p(⋅)p(\cdot)-Laplacian

We study a~priori estimates for the Dirichlet problem of the $p(\cdot)$-Laplacian, $$-\mathrm{div}(|\nabla v|^{p(\cdot)-2} \nabla v) = f.$$ We show that the gradients of the finite element approximation with zero boundary data converges with rate $O(h^\alpha)$ if the exponent $p$ is $\alpha$-H\"{o}lder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $|\nabla v|^{\frac{p-2}{2}} \nabla v$

On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces

We consider generalized Orlicz-Morrey spaces M_{\Phi,\varphi}(\Rn) including their weak versions WM_{\Phi,\varphi}(\Rn). In these spaces we prove the boundedness of the Riesz potential from M_{\Phi,\varphi_1}(\Rn) to M_{\Psi,\varphi_2}(\Rn) and from M_{\Phi,\varphi_1}(\Rn) to WM_{\Psi,\varphi_2}(\Rn). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(\varphi_{1},\varphi_{2})$, which do not assume any assumption on monotonicity of $\varphi_{1}(x,r)$, $\varphi_{2}(x,r)$ in r.Comment: 23 pages. J. Funct. Spaces Appl.(to appear