Riesz fractional integrals in grand lebesgue spaces on ℝ n

Abstract

Abstract We introduce conditions on the construction of grand Lebesgue spaces on ℝ n which imply the validity of the Sobolev theorem for the Riesz fractional integrals I α and the boundedness of the maximal operator, in such spaces. We also give an inversion of the operator I α by means of hypersingular integrals, within the frameworks of the introduced spaces. We also proof the denseness of $C_0^\infty {(\mathbb R)^n}$ C 0 ∞ ( ℝ ) n in a subspace of the considered grand space.