非倍测度空间上的插值定理和多线性分数次积分算子
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摘要
本论文主要研究非倍测度空间上的插值定理和多线性分数次积分算子及其交换子的有界性。全文共分两章,第一章致力于研究伴随非倍测度μ的Hardy空间上的插值定理,其中μ为满足某种增长性条件的非负Radon测度。我们建立了一个新的插值定理,该定理改进了Tolsa在[25]的插值定理。第二章,我们研究一类由多线性分数次积分和RBMO(μ)函数生成的交换子,借助于Sharp极大函数估计,建立了该类算子在赋予测度μ的Lebesgue乘积空间上的有界性,推广了谌稳固和Sawyer在[3]的结果。
This dissertation is devoted to the study of the interpolation theorem related to Hardy spaces and the boundedness of multilinear fractional integral operator and its commutators in non-doubling measure spaces. It consists of two chapters.
The first chapter is concerning with the interpolation theorem on Hardy space associated toμ, whereμis the nonnegative Radon measure satisfying some growth condition. We establish a new interpolation theorem which improves the interpolation theorem of Tolsa in [25].
Chapter 2 deals with a class of commutators generated by multilinear fractional integrals and RBMO(μ) functions. The boundedness of such operators on product of Lebesgue spaces withμare established, which extends the result of Chen and Sawyer in [3].
This dissertation is devoted to the study of the interpolation theorem related to Hardy spaces and the boundedness of multilinear fractional integral operator and its commutators in non-doubling measure spaces. It consists of two chapters.
The first chapter is concerning with the interpolation theorem on Hardy space associated toμ, whereμis the nonnegative Radon measure satisfying some growth condition. We establish a new interpolation theorem which improves the interpolation theorem of Tolsa in [25].
Chapter 2 deals with a class of commutators generated by multilinear fractional integrals and RBMO(μ) functions. The boundedness of such operators on product of Lebesgue spaces withμare established, which extends the result of Chen and Sawyer in [3].
引文
1. S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31(1)(1982), 7-16
2. M. Bramanti, M. Cerutti, Commutators of singular integrals on homogeneous spaces, Boll. Un. Mat. Ital, 10(1996), 843-883.
3. W. Chen and E. T. Sawyer, A note on commutators of fractional integrals with RBMO(μ) functions, Illinois Math. J. 46 (4) (2002), 1287-1298.
4. Y. Ding, S. Lu, P. Zhang, Weak type estimate for the commutator of fractional integral, Sci. in china (Ser A), 44(7)(2001), 289-299.
5. J. Garcia-Cuerva and A. E. Gatto, Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162 (3) (2004), 245-261.
6. J. Garacia and J. Martell, Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces, Indiana Univ. Math. J. 50 (3) (2001), 1241-1280.
7. L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Prentice Hall, 2004.
8. L. Grafakos and R. Torres. Multilinear Caldcron-Zygmund Theory, Adv. Math. 165 (2002), 124-164.
9. E. Harboure, C. Segovia, J. L. Torrea, Boundedness of commutators of fractional and singular integrals for the extreme values of p, Illinois J. Math., 41(4)(1997),676-700.
10. S. Lu, Q. Wu, D. Yang, Boundedness of commutators on Hardy type spaces, Sci. in China (Ser. A), 45(8)(2002), 984-997.
11. G. Hu, Y. Meng and D. Yang, Multilinear commutators for fractional integrals in non-homogeneous spaces, Publ. Mat. 48 (2004), 335-367.
12. G. Hu, Y. Meng and D. Yang, Multilinear commutators for singular integrals with non doubling measures, Integr. Equa. Oper. Theory 51 (2005), 235-255.
13. G. Hu, Y. Meng and D. Yang, New atomic characterization of H~1 space with non-doubling measures and applications, Math. Proc. Camb. Phil. Soc, 138(2005), 151-171.
14. G. Hu and D. Yang, Weighted norm inequalities for maximal singular integral operators with non doubling measures, Submitted.
15. C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1-15.
16. J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for non-doubling measures, Duke Math. J., 102 (2000), 533-565.
17. Y. Meng, D. Yang, Boundedness of commutators with Lips-chitz functions in non-homogeneous spaces, Taiwanese J. Math., 10(4), 2006.
18. F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calderon-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 15 (1997), 703-726.
19. F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderon-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 9 (1998), 463-487.
20. C. Perez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal 128(1995), 163-185.
21. C. Perez and R. Torres, Sharp maximal function estimates for multilinear singular integrals, Contemp. Math. 320 (2003), 323-331.
22. C. Perez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc. 65 (2002), 672-692.
23. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
24. X. Tolsa, A proof of weak (1,1) inequality for singular integrals with non doubling measures based on a Calderon-Zygmund decomposition, Publ. Mat, 45 (2001), 163-174.
25. X. Tolsa, BMO, H~1 and Calderon-Zygmund operators for non doubling measures, Math. Ann., 319 (2001), 89-149.
26. X. Tolsa, The space H~1 for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc, 355 (2003), 315-348.
27. X. Tolsa, Plainves problem and the semiadditivity of analytic capacity, Acta Math., 190(1)(2003), 105-149.
28. X. Tolsa, Cotlar's inequality without the doubling condition and existence of principal values for the cauchy integral of measures, J. Reine Angew. Math., 502(1998), 199-235.
29. X. Tolsa, A T(1) theorem for non-doubling measures with atoms, Proc. London Math. Soc, 82(3)(2001), 195-228.
30. X. Tolsa, Littlewood-paley theory and the T(1) theorem with non-doubling measures, Adv. Math., 164(1)(2001), 57-116.
31. J. Verdera, The fall of the doubling condition in Calderon-Zygmund theory, Publ. Mat.. Extra (2002), 275-292.
32. J. Xu, Boundedness of multilinear singular integrals for non-doubling measures, J. Math. Anal. Appl. 307 (2007), 471-480.
33. J. Xu, Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures, Science in China: Ser. A Math. 50 (2007), 361-376.
2. M. Bramanti, M. Cerutti, Commutators of singular integrals on homogeneous spaces, Boll. Un. Mat. Ital, 10(1996), 843-883.
3. W. Chen and E. T. Sawyer, A note on commutators of fractional integrals with RBMO(μ) functions, Illinois Math. J. 46 (4) (2002), 1287-1298.
4. Y. Ding, S. Lu, P. Zhang, Weak type estimate for the commutator of fractional integral, Sci. in china (Ser A), 44(7)(2001), 289-299.
5. J. Garcia-Cuerva and A. E. Gatto, Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162 (3) (2004), 245-261.
6. J. Garacia and J. Martell, Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces, Indiana Univ. Math. J. 50 (3) (2001), 1241-1280.
7. L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Prentice Hall, 2004.
8. L. Grafakos and R. Torres. Multilinear Caldcron-Zygmund Theory, Adv. Math. 165 (2002), 124-164.
9. E. Harboure, C. Segovia, J. L. Torrea, Boundedness of commutators of fractional and singular integrals for the extreme values of p, Illinois J. Math., 41(4)(1997),676-700.
10. S. Lu, Q. Wu, D. Yang, Boundedness of commutators on Hardy type spaces, Sci. in China (Ser. A), 45(8)(2002), 984-997.
11. G. Hu, Y. Meng and D. Yang, Multilinear commutators for fractional integrals in non-homogeneous spaces, Publ. Mat. 48 (2004), 335-367.
12. G. Hu, Y. Meng and D. Yang, Multilinear commutators for singular integrals with non doubling measures, Integr. Equa. Oper. Theory 51 (2005), 235-255.
13. G. Hu, Y. Meng and D. Yang, New atomic characterization of H~1 space with non-doubling measures and applications, Math. Proc. Camb. Phil. Soc, 138(2005), 151-171.
14. G. Hu and D. Yang, Weighted norm inequalities for maximal singular integral operators with non doubling measures, Submitted.
15. C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1-15.
16. J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for non-doubling measures, Duke Math. J., 102 (2000), 533-565.
17. Y. Meng, D. Yang, Boundedness of commutators with Lips-chitz functions in non-homogeneous spaces, Taiwanese J. Math., 10(4), 2006.
18. F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calderon-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 15 (1997), 703-726.
19. F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderon-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 9 (1998), 463-487.
20. C. Perez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal 128(1995), 163-185.
21. C. Perez and R. Torres, Sharp maximal function estimates for multilinear singular integrals, Contemp. Math. 320 (2003), 323-331.
22. C. Perez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc. 65 (2002), 672-692.
23. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
24. X. Tolsa, A proof of weak (1,1) inequality for singular integrals with non doubling measures based on a Calderon-Zygmund decomposition, Publ. Mat, 45 (2001), 163-174.
25. X. Tolsa, BMO, H~1 and Calderon-Zygmund operators for non doubling measures, Math. Ann., 319 (2001), 89-149.
26. X. Tolsa, The space H~1 for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc, 355 (2003), 315-348.
27. X. Tolsa, Plainves problem and the semiadditivity of analytic capacity, Acta Math., 190(1)(2003), 105-149.
28. X. Tolsa, Cotlar's inequality without the doubling condition and existence of principal values for the cauchy integral of measures, J. Reine Angew. Math., 502(1998), 199-235.
29. X. Tolsa, A T(1) theorem for non-doubling measures with atoms, Proc. London Math. Soc, 82(3)(2001), 195-228.
30. X. Tolsa, Littlewood-paley theory and the T(1) theorem with non-doubling measures, Adv. Math., 164(1)(2001), 57-116.
31. J. Verdera, The fall of the doubling condition in Calderon-Zygmund theory, Publ. Mat.. Extra (2002), 275-292.
32. J. Xu, Boundedness of multilinear singular integrals for non-doubling measures, J. Math. Anal. Appl. 307 (2007), 471-480.
33. J. Xu, Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures, Science in China: Ser. A Math. 50 (2007), 361-376.