Extrapolation results on variable exponent grand Lebesgue space with B p ( · ) $B_{p(\cdot)}$ weights

In this paper, we study Rubio de Francia extrapolation theorems in the framework of the variable grand Lebesgue spaces with $B_{p(·)}$ weights. As an application of the extrapolation theorems, we prove the boundedness of the Hardy averaging operator and the fractional Riemann Liouville transform for nonnegative and nonincreasing measurable functions. Some structural properties of the weighted grand Lebesgue spaces with variable exponent are also investigated.