Geodesic
Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 4, pp. 1201-1217
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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)
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 $\theta \in (0,1]$.
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 
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     title = {Sobolev type inequalities for fractional maximal functions and {Riesz} potentials in {Morrey} spaces of variable exponent on half spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1201--1217},
     year = {2023},
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Mizuta, Yoshihiro; Shimomura, Tetsu. Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 4, pp. 1201-1217. doi: 10.21136/CMJ.2023.0442-22

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