# 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.

## How to cite

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Martín-Reyes, F., and de la Torre, A.. "Some weighted inequalities for general one-sided maximal operators." Studia Mathematica 122.1 (1997): 1-14. <http://eudml.org/doc/216358>.

@article{Martín1997,
abstract = {We characterize the pairs of weights on ℝ for which the operators $M^\{+\}_\{h,k\}f(x) = sup_\{c>x\}h(x,c) ʃ_\{x\}^\{c\} f(s)k(x,s,c)ds$ 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 $\{(x,c): x < c\}$, while k is defined on $\{(x,s,c): x < s < c\}$. If $h(x,c) = (c-x)^\{-β\}$, $k(x,s,c) = (c-s)^\{α-1\}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M^\{+\}_\{α,β\}f = sup_\{c>x\} 1/(c-x)^\{β\} ʃ_\{x\}^\{c\} f(s)/(c-s)^\{1-α\} ds$. 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 $M^\{+\}_\{α,α\}$ introduced by W. Jurkat and J. Troutman in the study of $C_α$ differentiation of the integral.},
author = {Martín-Reyes, F., de la Torre, A.},
journal = {Studia Mathematica},
keywords = {one-sided maximal operators; Cesàro averages; weights; weighted inequalities; weak and strong type},
language = {eng},
number = {1},
pages = {1-14},
title = {Some weighted inequalities for general one-sided maximal operators},
url = {http://eudml.org/doc/216358},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Martín-Reyes, F.
AU - de la Torre, A.
TI - Some weighted inequalities for general one-sided maximal operators
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 1
SP - 1
EP - 14
AB - We characterize the pairs of weights on ℝ for which the operators $M^{+}_{h,k}f(x) = sup_{c>x}h(x,c) ʃ_{x}^{c} f(s)k(x,s,c)ds$ 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 ${(x,c): x < c}$, while k is defined on ${(x,s,c): x < s < c}$. If $h(x,c) = (c-x)^{-β}$, $k(x,s,c) = (c-s)^{α-1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M^{+}_{α,β}f = sup_{c>x} 1/(c-x)^{β} ʃ_{x}^{c} f(s)/(c-s)^{1-α} ds$. 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 $M^{+}_{α,α}$ introduced by W. Jurkat and J. Troutman in the study of $C_α$ differentiation of the integral.
LA - eng
KW - one-sided maximal operators; Cesàro averages; weights; weighted inequalities; weak and strong type
UR - http://eudml.org/doc/216358
ER -

## References

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