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## Open Mathematics

2014 | 12 | 4 | 636-647
Tytuł artykułu

### Weighted inequalities for some integral operators with rough kernels

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EN
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EN
In this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.
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EN
Kategorie tematyczne
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Czasopismo
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Tom
Numer
Strony
636-647
Opis fizyczny
Daty
wydano
2014-04-01
online
2014-01-17
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autor
autor
Bibliografia
• [1] Bernardis A.L., Lorente M., Riveros M.S., Weighted inequalities for fractional integral operators with kernel satisfying Hörmander type conditions, Math. Inequal. Appl., 2011, 14(4), 881–895
• [2] Ding Y., Lu S., Weighted norm inequalities for fractional integral operators with rough kernel, Canad. J. Math., 1998, 50(1), 29–39 http://dx.doi.org/10.4153/CJM-1998-003-1
• [3] Duoandikoetxea J., Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc., 1993, 336(2), 869–880 http://dx.doi.org/10.1090/S0002-9947-1993-1089418-5
• [4] Duoandikoetxea J., Fourier Analysis, Grad. Stud. Math., 29, American Mathematical Society, Providence, 2001
• [5] García-Cuerva J., Rubio de Francia J.L., Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., 104, North-Holland, Amsterdam, 1985
• [6] Godoy T., Urciuolo M., On certain integral operators of fractional type, Acta Math. Hungar., 1999, 82(1–2), 99–105 http://dx.doi.org/10.1023/A:1026437621978
• [7] Grafakos L., Classical Fourier Analysis, 2nd ed., Grad. Texts in Math., 249, Springer, New York, 2008
• [8] Kurtz D.S., Wheeden R.L., Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc., 1979, 255, 343–362 http://dx.doi.org/10.1090/S0002-9947-1979-0542885-8
• [9] Muckenhoupt B., Wheeden R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 1974, 192, 261–274 http://dx.doi.org/10.1090/S0002-9947-1974-0340523-6
• [10] Riveros M.S., Urciuolo M., Weighted inequalities for integral operators with some homogeneous kernels, Czechoslovak Math. J., 2005, 55(130)(2), 423–432 http://dx.doi.org/10.1007/s10587-005-0032-y
• [11] Riveros M.S., Urciuolo M., Weighted inequalities for fractional type operators with some homogeneous kernels, Acta Math. Sin. (Engl. Ser.), 2013, 29(3), 449–460 http://dx.doi.org/10.1007/s10114-013-1639-9
• [12] Rocha P., Urciuolo M., On the H p-L q boundedness of some fractional integral operators, Czechoslovak Math. J., 2012, 62(137)(3), 625–635 http://dx.doi.org/10.1007/s10587-012-0054-1
• [13] Watson D.K., Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J., 1990, 60(2), 389–399 http://dx.doi.org/10.1215/S0012-7094-90-06015-6
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