Some weighted inequalities for general one-sided maximal operators

F. Martín-Reyes; A. de la Torre

Studia Mathematica (1997)

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

Abstract

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We characterize the pairs of weights on ℝ for which the operators M h , k + f ( x ) = s u p c > x h ( x , c ) ʃ x c f ( s ) k ( x , s , c ) d s 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 = s u p c > x 1 / ( c - x ) β ʃ x c f ( s ) / ( c - s ) 1 - α d s . 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.

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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  8. [MOT] F. J. Martín-Reyes, P. Ortega Salvador and A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319 (1990), 517-534. Zbl0696.42013
  9. [MPT] F. J. Martín-Reyes, L. Pick and A. de la Torre, A ∞ + condition, Canad. J. Math. 45 (1993), 1231-1244. 
  10. [MT] F. J. Martín-Reyes and A. de la Torre, Two weight norm inequalities for fractional one-sided maximal operators, Proc. Amer. Math. Soc. 117 (1993), 483-489. Zbl0769.42010
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