# 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

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topMartín-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

top- [A] K. F. Andersen, Weighted inequalities for maximal functions associated with general measures, Trans. Amer. Math. Soc. 326 (1991), 907-920. Zbl0736.42013
- [AM] K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9-26. Zbl0501.47011
- [AS] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), 547-557. Zbl0664.26002
- [CHS] A. Carbery, E. Hernandez and F. Soria, Estimates for the Kakeya maximal operator and radial functions in ${\mathbb{R}}^{n}$, in: Harmonic Analysis (Sendai, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 41-50.
- [JT] W. Jurkat and J. Troutman, Maximal inequalities related to generalized a.e. continuity, Trans. Amer. Math. Soc. 252 (1979), 49-64. Zbl0441.42023
- [KG] V. Kokilashvili and M. Gabidzashvili, Two weight weak type inequalities for fractional type integrals, Math. Inst. Czech. Acad. Sci. Prague 45 (1989).
- [LT] M. Lorente and A. de la Torre, Weighted inequalities for some one-sided operators, Proc. Amer. Math. Soc. 124 (1996), 839-848. Zbl0895.26002
- [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
- [MPT] F. J. Martín-Reyes, L. Pick and A. de la Torre, ${A}_{\infty}^{+}$ condition, Canad. J. Math. 45 (1993), 1231-1244.
- [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
- [S] E. T. Sawyer, Weighted inequalities for the one sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61. Zbl0627.42009
- [SW] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Pres 1971.

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