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## Studia Mathematica

1997 | 126 | 1 | 67-94
Tytuł artykułu

### Two-weight norm inequalities for maximal functions on homogeneous spaces and boundary estimates

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let D be an open subset of a homogeneous space(X,d,μ). Consider the maximal function $M_φ f(x) = sup1/φ(B) ʃ_{B∩∂D} |f|dν$, x∈ D, where the supremum is taken over all balls of the form B = B(a(x),r) with r > t(x) = d(x,∂D), a(x)∈ ∂D is such that d(a(x),x) < 3/2 t(x)$and φ is a nonnegative set function defined for all Borel sets of X satisfying the quasi-monotonicity and doubling properties. We give a necessary and sufficient condition on the weights w and v for the weighted norm inequality (0.1)$(ʃ_D [M_φ(f)]^q wdμ)^{1/q} ≤ c(ʃ_{∂D} |f|^p vdν)^{1/p}$to hold when 1 < p < q < ∞,$σdν = v^{1-p'}dν$is a doubling weight, and dν is a doubling measure, and give a sufficient condition for (0.1) when 1 < p ≤ q < ∞ without assuming that σ is a doubling weight but with an extra assumption on φ. Another characterization for (0.1) is also provided for 1 < p ≤ q < ∞ and D of the form Y×(0,∞), where Y is a homogeneous space with group structure. These results generalize some known theorems in the case when$M_φ$is the fractional maximal function in$ℝ^{n+1}_+$, that is, when$M_φ f(x,t) = M_γ f(x,t) = sup_{r>t} 1/(ν(B(x,r))^{1-γ}) ʃ_{B(x,r)} |f|dν$, where$(x,t) ∈ ℝ^{n+1}_+$, 0 < γ < 1, and ν is a doubling measure in$ℝ^n\$.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
67-94
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-11-14
poprawiono
1997-02-12
Twórcy
autor
• Departamento de Matemática, Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, Caixa Postal 668, São Carlos, SP 13.560-970, Brazil
Bibliografia
• [C] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601-628.
• [MS] R. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257-270.
• [S1] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545.
• [SW1] E. T. Sawyer and R. L. Wheeden, Carleson conditions for the Poisson integral, Indiana Univ. Math. J. 40 (1991), 639-676.
• [SW2] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874.
• [SWZ] E. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal. 5 (1996), 523-580.
• [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
• [W1] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 257-272.
• [W2] R. L. Wheeden, Norm inequalities for off-centered maximal operators, Publ. Mat. 37 (1993), 429-441.
Typ dokumentu
Bibliografia
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