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Keywords:
variable exponent; fractional maximal function; Riesz potential; Sobolev's inequality; weighted Morrey space; double phase functional
variable exponent; fractional maximal function; Riesz potential; Sobolev's inequality; weighted Morrey space; double phase functional
Summary:
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 {\mathbb H} $, and $b(\cdot )$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.
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