Abstract
Let μ be a Radon measure on Rd which may be non–doubling. The only condition satisfied by μ is that μ(B(x, r)) ≤ Cr n for all x ∈ ℝd, r > 0 and some fixed 0 < n ≤ d. In this paper, the authors prove that the boundedness from H 1(μ) into L 1,∞(μ) of a singular integral operator T with Calderón–Zygmund kernel of Hörmander type implies its L 2(μ)–boundedness.
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Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar’s inequalities for Calderón–Zygmund operators on non–homogeneous spaces. Internat Math. Res. Notices, 9, 463–487 (1998)
Nazarov, F., Treil, S., Volberg, A.: Accretive system Tb–theorems on nonhomogeneous spaces. Duke Math. J., 113, 259–312 (2002)
Nazarov, F., Treil, S., Volberg, A.: The Tb–theorem on non–homogeneous spaces. Acta Math., 190, 151–239 (2003)
Orobitg, J., Pérez, C.: Ap weights for nondoubling measures in ℝn and applications. Trans. Amer. Math. Soc., 354, 2013–2033 (2002)
Tolsa, X.: A T(1) theorem for non–doubling measures with atoms. Proc. London Math. Soc., 82(3), 195–228 (2001)
Tolsa, X.: BMO, H 1 and Calderón–Zygmund operators for non doubling measures. Math. Ann., 319, 89–149 (2001)
Tolsa, X.: Littlewood–Paley theory and the T(1) theorem with non–doubling measures. Adv. Math., 164, 57–116 (2001)
Tolsa, X.: A proof of the weak (1, 1) inequality for singular integrals with non doubling measures based on a Calderón–Zygmund decomposition. Publ. Mat., 45, 163–174 (2001)
Tolsa, X.: The space H 1 for nondoubling measures in terms of a grand maximal operator. Trans. Amer. Math. Soc., 355, 315–348 (2003)
Tolsa, X.: Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math., 190, 105–149 (2003)
Verdera, J.: The fall of the doubling condition in Calderón–Zygmund theory. Publ. Mat., Vol. Extra, 275–292 (2002)
Hu, G., Meng, Y., Yang D.: Boundedness of Riesz potentials in non–homogeneous spaces. Submitted (2004)
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This project is supported by NNSF (No. 10271015) of China, and the third (corresponding) author is also supported by RFDP (No. 20020027004) of China
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Fu, X.L., Hu, G.E. & Yang, D.C. A Remark on the Boundedness of Calderón–Zygmund Operators in Nony–homogeneous Spaces. Acta Math Sinica 23, 449–456 (2007). https://doi.org/10.1007/s10114-005-0723-1
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DOI: https://doi.org/10.1007/s10114-005-0723-1