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In this paper, the necessary and sufficient conditions are found for the boundedness of the rough Bfractional integral operators from the Lorentz spaces L p,s,γ to L q,r,γ , 1 < p < q < ∞, 1 r s ∞, and from L 1,r,γ to L q,∞,γ ≡ W L q,γ , 1 < q < ∞, 1 r ∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.

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In this study, the boundedness of the high order Riesz-Bessel transformations generated by generalized shift operator in weighted L p,ω,γ -spaces with general weights is proved.

In this paper, the mean value formula depends on the Bessel-generalized shift operator corresponding to the solutions of the boundary value problem related to the multidimensional Bessel operator are studied. In addition, Riesz transforms R B related to the multidimensional Bessel operators are studied. Since a Bessel-generalized shift operator is a translation operator corresponding to the multidimensional Bessel operator, we construct a family of R B by using a Bessel-generalized shift operator. Finally, we analyze weighted inequalities involving R B . MSC: Primary 31B05; 35G10; secondary 34B30; 47F05