B L m p , ν estimates for the Riesz transforms associated with Laplace-Bessel operator

In this paper, we introduce higher order Riesz-Bessel transforms which we can express partial derivatives of order α of I m , ν f for f ∈ L p , ν . In addition, we establish relationship between Riesz potential with higher order Riesz-Bessel transform related to generalized shift operator. By using this relationship, we make some improvements of integral estimates for I m , ν f and higher order Riesz-Bessel transform R m ν in the Beppo Levi space B L m p , ν . We prove an estimate for the singular integral operator with convolution type generated by generalized shift operator in the Beppo Levi spaces.