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## Colloquium Mathematicum

1996 | 69 | 1 | 95-115
Tytuł artykułu

### On weighted inequalities for operators of potential type

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, we discuss a class of weighted inequalities for operators of potential type on homogeneous spaces. We give sufficient conditions for the weak and strong type weighted inequalities sup_{λ>0} λ|{x ∈ X : |T(fdσ)(x)|>λ }|_{ω}^{1/q} ≤ C (∫_{X} |f|^{p}dσ)^{1/p} and (∫_{X} |T(fdσ)|^{q}dω )^{1/q} ≤ C (∫_X |f|^{p}dσ )^{1/p} in the cases of 0 < q < p ≤ ∞ and 1 ≤ q < p < ∞, respectively, where T is an operator of potential type, and ω and σ are Borel measures on the homogeneous space X. We show that under certain restrictions on the measures those sufficient conditions are also necessary. A consequence is given for the fractional integrals in Euclidean spaces.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
95-115
Opis fizyczny
Daty
wydano
1996
otrzymano
1994-02-23
poprawiono
1994-08-19
Twórcy
autor
• Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121, U.S.A.
Bibliografia
• [1] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.
• [2] I. Genebashvili, A. Gogatishvili and V. Kokilashvili, Criteria of general weak type inequalities for integral transforms with positive kernels, Proc. Georgian Acad. Sci. (Math.) 1 (1993), 11-34.
• [3] R. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), 257-270.
• [4] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-275.
• [5] C. Pérez, Two weighted norm inequalities for Riesz potentials and uniform $L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), 31-44.
• [6] E. T. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), 339-345.
• [7] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, ibid. 308 (1988), 533-545.
• [8] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874.
• [9] E. T. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal., to appear.
• [10] I. E. Verbitsky, Weight norm inequalities for maximal operators and Pisicr's theorem on factorization through $L^{p∞}$, Integral Equations Oper. Theory 15 (1992), 124-153.
• [11] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 257-272.
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Bibliografia
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