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Abstract  

We introduce the generalized fractional integrals (generalized B-fractional integrals) generated by the ΔB Laplace-Bessel differential operator and give some results for them. We obtain O’Neil type inequalities for the B-convolutions and give pointwise rearrangement estimates of the generalized B-fractional integrals. Then we get the L p,γ-boundedness of the generalized B-convolution operator, the generalized B-Riesz potential and the generalized fractional B-maximal function. Finally, we prove a sharp pointwise estimate of the nonincreasing rearrangement of the generalized fractional B-maximal function.

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Abstract  

We consider the generalized shift operator associated with the Laplace-Bessel differential operator

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\Delta _B = \sum\limits_{i = 1}^n {\frac{{\partial ^2 }} {{\partial x_j^2 }}} + \sum\limits_{i = 1}^k {\frac{{\gamma _i }} {{x_i }}\frac{\partial } {{\partial x_i }}}$$ \end{document}
, and study the modified B-Riesz potential Ĩ α, β generated by the generalized shift operator acting in the B-Morrey space in the limiting case. We prove that the operator Ĩ α, β, 0 < α < n + |γ|, is bounded from the B-Morrey space L (n+|γ|−λ)/α,λ,γ(ℝk,+ n) to the B-BMO space BMOγ(ℝk,+ n).

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