# Some weighted inequalities for general one-sided maximal operators

Studia Mathematica (1997)

• Volume: 122, Issue: 1, page 1-14
• ISSN: 0039-3223

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## Abstract

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We characterize the pairs of weights on ℝ for which the operators ${M}_{h,k}^{+}f\left(x\right)=su{p}_{c>x}h\left(x,c\right){ʃ}_{x}^{c}f\left(s\right)k\left(x,s,c\right)ds$ 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 $\left(x,c\right):x, while k is defined on $\left(x,s,c\right):x. If $h\left(x,c\right)={\left(c-x\right)}^{-\beta }$, $k\left(x,s,c\right)={\left(c-s\right)}^{\alpha -1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator ${M}_{\alpha ,\beta }^{+}f=su{p}_{c>x}1/{\left(c-x\right)}^{\beta }{ʃ}_{x}^{c}f\left(s\right)/{\left(c-s\right)}^{1-\alpha }ds$. 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 ${M}_{\alpha ,\alpha }^{+}$ introduced by W. Jurkat and J. Troutman in the study of ${C}_{\alpha }$ differentiation of the integral.

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