On the Boundedness of the Generalized Translation Operator on Variable Exponent Lebesgue Spaces

In this paper we are deal with the generalized translation operator generated by the Bessel operator in variable exponent Lebesgue spaces. The behavior of this generalized translation operator is well known on weighted Lebesgue spaces. But, there are some differences in the behavior of these operators on the variable exponent Lebesgue spaces. For example, the generalized translation operator is bounded in the variable exponent Lebesgue space L-p(+),L-gamma(R-+(n)) if and only if the exponent is constant. The aim of this paper is to give some the regularity conditions which ens(u)re the boundedness of generalized translation operator T-y on variable exponent Lebesgue spaces if p is nonconstant.