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## Studia Mathematica

1997 | 122 | 1 | 1-14
Tytuł artykułu

### Some weighted inequalities for general one-sided maximal operators

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
1-14
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-06-06
poprawiono
1996-06-15
Twórcy
autor
autor
Bibliografia
• [A] K. F. Andersen, Weighted inequalities for maximal functions associated with general measures, Trans. Amer. Math. Soc. 326 (1991), 907-920.
• [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.
• [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.
• [CHS] A. Carbery, E. Hernandez and F. Soria, Estimates for the Kakeya maximal operator and radial functions in $ℝ^n$, 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.
• [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.
• [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.
• [MPT] F. J. Martín-Reyes, L. Pick and A. de la Torre, $A_∞^+$ 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.
• [S] E. T. Sawyer, Weighted inequalities for the one sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61.
• [SW] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Pres 1971.
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Bibliografia
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