Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

M. S. Dzhabrailov, S. Z. Khaligova


In this paper it is proved that anisotropic fractional maximal operator $M_{\a,\sigma}$, $0 \le \a < |\sigma|$ is bounded on anisotropic generalized Morrey spaces $M_{p,\varphi,\sigma}$, where $|\sigma|=\sum_{i=1}^n \sigma_i$ is the homogeneous dimension of $\Rn$. We find the conditions on the pair $(\varphi_1,\varphi_2)$ which ensure the Spanne-Guliyev type boundedness of the operator $M_{\a,\sigma}$ from anisotropic generalized Morrey space $M_{p,\varphi_1,\sigma}$ to $M_{q,\varphi_2,\sigma}$, $1<p\le q<\i$, $1/p-1/q=\a/|\sigma|$, and from the space $M_{1,\varphi_1,\sigma}$ to the weak space $WM_{q,\varphi_2,\sigma}$, $1< q<\i$, $1-1/q=\a/|\sigma|$. We also find conditions on the $\varphi$ which ensure the Adams-Guliyev type boundedness of $M_{\a,\sigma}$ from $M_{p,\varphi^{\frac{1}{p}},\sigma}$ to $M_{q,\varphi^{\frac{1}{q}},\sigma}$ for $1<p<q<\i$ and from $M_{1,\varphi,\sigma}$ to $WM_{q,\varphi^{\frac{1}{q}},\sigma}$ for $1<q<\i$.

As applications, we establish the boundedness of some Sch\"{o}dinger type operators on anisotropic generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse H\"{o}lder class.

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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