On the rearrangement estimates and the boundedness of the generalized fractional integrals associated with the Laplace-Bessel differential operator

We introduce the generalized fractional integrals (generalized B-fractional integrals) generated by the Δ _B 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. V. S. Guliyev was partially supported by the grant of INTAS (Nr. 05-1000008-8157). Z. V. Safarov was partially supported by INTAS YS Post Doctoral Fellowship (Nr. 05-113-4671).