The Boundedness of the $B$-Riesz Potential in the $B$-Morrey Spaces
DOI:
https://doi.org/10.5644/SJM.04.1.09Keywords:
$B$-maximal operator, $B$-Riesz potential, $B$-Morrey spaces, Sobolev-Morrey type estimatesAbstract
We consider the generalized shift operator ($B$ shift operator),
generated by the Laplace-Bessel differential operator $\Delta
_{B}=\sum_{i=1}^{k}B_{i}+ \sum_{j=k+1}^{n}\frac {\partial ^2
}{\partial x_j^2},$ $B=\left(B_1,\ldots,B_k \right)$,
$B_i=\frac{\partial ^2}{\partial x_i^2}+ \frac{\gamma_i
}{x_i}\frac\partial {\partial x_i},$ $\gamma_i>0,$
$i=1,\ldots,k$, $|\gamma|=\gamma_1+\dots+\gamma_k$. The
$B$-maximal functions and the $B$-Riesz potentials, generated by
the Laplace-Bessel differential operator $\Delta_{B}$ are
investigated. We study the $B$-Riesz potentials in the $B$-Morrey
spaces. The inequality of Sobolev-Morrey type is established for the $B$-Riesz potentials.
2000 Mathematics Subject Classification. 42B20, 42B25, 42B35
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