Czechoslovak Mathematical Journal, Vol. 73, No. 4, pp. 1201-1217, 2023
Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces
Yoshihiro Mizuta, Tetsu Shimomura
Received October 4, 2022. Published online May 24, 2023.
Abstract: 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, $p(x)<q(x)$ for $x \in\h$, and $b(\cdot)$ is nonnegative and Hölder continuous of order $\theta\in(0,1]$.
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Affiliations: Yoshihiro Mizuta, Department of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8521, Japan, e-mail: yomizuta@hiroshima-u.ac.jp; Tetsu Shimomura (corresponding author), Department of Mathematics, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima 739-8524, Japan, e-mail: tshimo@hiroshima-u.ac.jp
