Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Mizuta, Yoshihiro ; Shimomura, Tetsu

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 4, pp. 1201-1217

Résumé

Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{\mathbb H,\nu }f$ and Riesz potentials $I_{\mathbb H,\alpha }f$ in weighted Morrey spaces of variable exponent on the half space $\mathbb H$. We also obtain Sobolev type inequalities for a $C^1$ function on $\mathbb H$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t) = t^{p(x)} + (b(x) t)^{q(x)}$, where $p(\cdot )$ and $q(\cdot )$ satisfy log-Hölder conditions, $p(x)$ for $x \in {\mathbb H} $, and $b(\cdot )$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.

Détail

DOI : 10.21136/CMJ.2023.0442-22

Classification : 31B15, 42B25, 46E30
Keywords: variable exponent; fractional maximal function; Riesz potential; Sobolev's inequality; weighted Morrey space; double phase functional