Weighted estimates for the averaging integral operator (Q634821)

Under the following monotonicity conditions on weights \(v\) and \(w\): 1) \(v(x)x^{\rho }\) is equivalent to a non-decreasing function for some \( \rho >0,\) and 2) \(\left( w(x)x\right) ^{1/q}\simeq \left( v(x)x\right) ^{1/p}\). It is shown that the weighted \(L^{p}(v)-L^{q}(w)\) boundedness of the average operator \(Af(x):=\frac{1}{x}\int_{0}^{x}f(t)dt,\) (\(0<x<\infty ),\) implies that there exists \(\varepsilon _{0}\in (0,p-1)\) such that \(A\) is bounded from \(L^{p-\varepsilon }(v(x)^{1+\delta }x^{\gamma })\) to \(L^{q-\varepsilon q/p}(w(x)^{1+\delta }x^{\delta (1-q/p)}x^{\gamma q/p}),\) for all \( \varepsilon ,\delta ,\gamma \in [ 0,\varepsilon _{0}),\) and vice versa.