Weighted criteria for the hardy transform under the B p condition in grand lebesgue spaces and some applications

Abstract

We show that the Hardy operator $$ Hf(x) = \frac{1}{x}\mathop {\int }\limits_0^x f(t)dt $$ from $$ L_{{\text{dec}},w}^{p),\theta } $$ (I) to $$ L_w^{p),\theta } $$ (I), 0 < p < ∞, θ > 0, I = (0, 1), is bounded if and only if the weight w belongs to the well–known class Bp restricted to the interval I. This result is applied to derive a similar assertion for the Riemann–Liouville fractional integral operator and to establish criteria for the boundedness of the Hardy–Littlewood maximal operator in the weighted grand Lorentz space $$ \Lambda_w^{p),\theta } $$ . Bibliography: 23 titles.