Some weighted inequalities for general one-sided maximal operators

We characterize the pairs of weights on ℝ for which the operators ${\displaystyle M_{h,k}^{+}f\left(x\right)=\underset{\text{sup}}{c>x}h(x,c){\int }_x^{c}f\left(s\right)k(x,s,c)ds}$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on ${\displaystyle \{(x,c):x<c\}}$, while k is defined on ${\displaystyle \{(x,s,c):x<s<c\}}$. If ${\displaystyle h(x,c)={(c-x)}^{-\beta }}$, ${\displaystyle k(x,s,c)={(c-s)}^{\alpha -1}}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator ${\displaystyle M_{\alpha ,\beta }^{+}f=\underset{c>x}{\text{sup}}\frac{1}{{(c-x)}^{\beta }}{\int }_x^c\frac{f\left(s\right)}{{(c-s)}^{1-\alpha }}ds}$. For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal weights and 1 < p < ∞, weak and strong type are equivalent. If we take α = β we characterize the strong type weights for the operator ${\displaystyle M_{\alpha ,\alpha }^{+}}$ introduced by W. Jurkat and J. Troutman in the study of ${\displaystyle C_{\alpha }}$ differentiation of the integral.