Abstract
We consider conditional two-weight estimates for singular and strongly singular integral operators. The conditions governing two-weight estimates shall be simultaneously necessary and sufficient for quite a large class of singular integrals.
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K. Andersen and B. Muckenhoupt, Weighted weak type inequalities with applications to Hilbert transforms and maximal functions, Studia Math., 72 (1982), 9–26.
J. Bradley, Hardy inequality with mixed norms, Canad. Math. Bull., 21 (1978), 405–408.
S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc., 281 (1984), 77–107.
S. Chanillo and M. Christ, Weak (1, 1) bounds for oscillatory singular integrals, Duke Math. J., 55 (1987), 141–155.
S. Chanillo, D. Kurtz, and G. Sampson, Weighted weak (1, 1) and weighted L p estimates for oscillatory kernels, Trans. Amer. Math. Soc., 295 (1986), 127–145.
S. Chanillo and A. Torchinsky, Sharp function and weighted L p estimates for a class of pseudodifferential operators, Ark. Mat., 24 (1986), 1–25.
R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241–249.
D. Edmunds and V. Kokilashvili, Two-weight inequalities for singular integrals, Canadian Math. Bull., 38 (1995), 119–125.
D. Edmunds, V. Kokilashvili, and A. Meskhi, Two-weight estimates for singular integrals defined on spaces of homogeneous type, Canadian J. Math., 52 (2000), 468–502.
D. Edmunds, Bounded and Compact Integral Operators, Kluwer (Dordrecht, Boston, London, 2002).
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), 9–36.
C. Fefferman, L p bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413–417.
J. Garcia-Cuerza and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland (Amsterdam, New York, Oxford, 1985).
I. Genebashvili, A. Gegatishvili, V. Kokilashvili and M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Pitman Monographs and Surveys in Pure and Applied Mathematics, 92, Longman (Harlow, 1992).
V. Guliev, Two-weight L p inequality for singular integral operators on Heisenberg groups, Georgian Math. J., 1 (1994), 367–376.
E. Gusseinov, Singular integrals in the space of functions summable with monotone weight (Russian), Mat. Sb., 132(174) (1977), 28–44.
I. I. Hirschman, Multiplier transforms I, Duke Math. J., 26 (1956), 222–242.
S. Hoffman, Singular integrals with power weights, Proc. Amer. Math. Soc., 110 (1990), 343–353.
Y. Hu, A Weighted Norm Inequality for Oscillatory Singular Integrals, Harmonic Analysis (Tianjin, 1988), Lecture Notes in Math., Springer Verlag (Berlin, 1994).
Y. Hu and Y. Pan, Boundedness of oscillatory singular integrals on Hardy spaces, Ark. Mat., 30 (1992), 311–320.
R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the congugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), 227–251.
V. Kokilashvili, On Hardy’s inequalities in weighted spaces, Soobsch. Akad. Nauk Gruz. SSR, 96 (1979), 37–40 (in Russian).
V. Kokilashvili and A. Meskhi, Two-weight inequalities for singular integrals de-ned on homogeneous groups, Proc. A. Razmzdze Math. Inst., 112 (1997), 57–90.
V. Kokilashvili and A. Meskhi, Two-weight inequalities for singular integrals defined on homogeneous groups, in: Function Spaces V, Proceedings of the Conference, Poznań, Poland, M. Mudzik and L. Skrzypczak, eds., Lecture Notes in Pure and Applied Mathematics, 213, Marcel Dekker (2000).
N. Lyall, A class of strongly singular Radon transforms on the Heisenberg group, to appear in Proc. of Edin. Math. Soc.
V. Maz’ya, Sobolev Spaces, Springer (Berlin, 1985).
B. Muckenhoupt, Hardy’s inequality with weights, Studia Math., 44 (1972), 31–38.
B. Muckenhoupt and R. Wheeden, Two-weight function norm inequalities for the Hardy-Littlewood maximal function and Hilbert transform, Studia Math., 55 (1976), 279–294.
Y. Pan, Oscillatory singular integrals on L p and Hardy spaces, Proc. Amer. Math. Soc., 124 (1996), 2821–2825.
F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals, J. Funct. Anal., 73 (1987), 179–194.
S. Sato, Weighted weak type (1, 1) estimates for oscillatory singular integrals, Studia Math., 141 (2000), 1–24.
F. Soria and G. Weiss, A remark on singular integrals and power weights, Indiana Univ. Math. J., 93 (1994), 187–204.
E. M. Stein, A note on singular integrals, Proc. Amer. Math. Soc., 8 (1957), 250–254.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press (Princeton, 1993).
J. O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer Verlag (Berlin, 1989).
S. Wainger, Special Trigonometric Series in k Dimensions, Memoirs of the AMS 59, American Math. Soc. (1965).
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The second author was supported by the European Commission through the IHP Network HARP 2002–2006.
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Kokilashvili, V., Lyall, N. & Meskhi, A. Two-weight estimates for singular and strongly singular integral operators. Acta Math Hung 116, 1–25 (2007). https://doi.org/10.1007/s10474-007-5265-9
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DOI: https://doi.org/10.1007/s10474-007-5265-9