TWO-WEIGHT ESTIMATES FOR STRONG FRACTIONAL MAXIMAL FUNCTIONS AND POTENTIALS WITH MULTIPLE KERNELS

DOI QR코드

DOI QR Code

Kokilashvili, Vakhtang;Meskhi, Alexander

  • 발행 : 2009.05.01

초록

In the paper two-weight inequalities of various type for strong fractional maximal functions and potentials with multiple kernels defined on $\mathbb{R}^2$ are established.

키워드

strong fractional maximal functions;potentials with multiple kernels;two-weight inequality;trace inequality

참고문헌

  1. D. R. Adams, A trace inequality for generalized potentials, Studia Math. 48 (1973), 99–105
  2. D. R. Adams, Lectures on $L^p$-potential theory, Umea Univ. Reports, No.2, 1981
  3. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996
  4. J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405–408
  5. S. Chanillo, D. K. Watson, and R. L. Wheeden, Some integral and maximal operators related to starlike sets, Studia Math. 107 (1993), no. 3, 223–255
  6. R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250
  7. D. E. Edmunds, V. Kokilashvili, and A. Meskhi, On Fourier multipliers in weighted Triebel-Lizorkin spaces, J. Inequal. Appl. 7 (2002), no. 4, 555–591
  8. D. E. Edmunds, Bounded and Compact Integral Operators, Mathematics and its Applications, Vol. 543, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002
  9. R. Fefferman, Multiparameter Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984), 47–130, Ann. of Math. Stud., 112, Princeton Univ. Press, Princeton, NJ, 1986
  10. M. Gabidzashvili, Weighted inequalities for anisotropic potentials, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 82 (1986), 25–36
  11. M. Gabidzashvili, I. Genebashvili, and V. Kokilashvili, Two-weight inequalities for generalized potentials, Trudy Mat. Inst. Steklov. 194 (1992)
  12. Issled. po Teor. Differ. Funktsii Mnogikh Peremen. i ee Prilozh. 14, 89–96; translation in Proc. Steklov Inst. Math. 194 (1993), no. 4, 91–99
  13. J. Garcia–Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116, Mathematical Notes, 104, North-Holland Publishing Co., Amsterdam, 1985
  14. I. Genebashvili, A. Gogatishvili, 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, 1998
  15. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, 1934
  16. L. Hormander, L^p$ estimates for (pluri-) subharmonic functions, Math. Scand. 20(1967), 65–78
  17. R. Hunt, B. Muckenhoupt, and R. L. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251 https://doi.org/10.2307/1996205
  18. E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545
  19. V. M. Kokilashvili, Weighted Lizorkin-Triebel spaces. Singular integrals, multipliers, imbedding theorems, Trudy Mat. Inst. Steklov. 161 (1983), 125–149
  20. V. M. Kokilashvili, Bisingular integral operators in weighted spaces, Soobshch. Akad. Nauk Gruzin. SSR 101 (1981), no. 2, 289–292
  21. V. M. Kokilashvili, On Hardy's inequalities in weighted spaces, Soobshch. Akad. Nauk Gruzin. SSR 96 (1979), no. 1, 37–40
  22. V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,World Scientific Publishing Co., Inc., River Edge, NJ, 1991
  23. V. Kokilashvili and A. Meskhi, On one-sided potentials with multiple kernels, Integral Transforms Spec. Funct. 16 (2005), no. 8, 669–683 https://doi.org/10.1080/10652460500106105
  24. V. Kokilashvili and A. Meskhi, On two-weight estimates for strong fractional maximal functions and potentials with multiple kernels, Proc. A. Razmadze Math. Inst. 137 (2005), 135–140
  25. V. Kokilashvili and A. Meskhi, Two-weighted criteria for integral transforms with multiple kernels, Approximation and probability, 119–140, Banach Center Publ., 72, Polish Acad. Sci., Warsaw, 2006 https://doi.org/10.4064/bc72-0-9
  26. V. G. Maz'ya, Sobolev Spaces, Springer, Berlin, 1985
  27. A. Meskhi, A note on two-weight inequalities for multiple Hardy-type operators, J. Funct. Spaces Appl. 3 (2005), no. 3, 223–237
  28. B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31–38
  29. B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274 https://doi.org/10.2307/1996833
  30. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivative, Theory and Applications, Gordon and Breach Sci. Publishers, 1993
  31. E. T. Sawyer, Weighted norm inequalities for fractional maximal operators, in: Seminar on Harmonic Analysis,Montreal, Que., 1980, pp. 283–309, CMS Conf. Proc., 1, Amer. Math. Soc.Providence, R.I., 1981
  32. E. T. Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), no. 1, 339–345
  33. E. T. Sawyer, Multipliers of Besov and power-weighted $L^2$ spaces, Indiana Univ. Math. J. 33 (1984), no. 3, 353–366
  34. E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874
  35. E. T. Sawyer, R. L. Wheeden, and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal. 5 (1996), no. 6, 523–580
  36. J. O. Stromberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., Vol. 1381, Springer Verlag, Berlin, 1989
  37. K. Tachizawa, On weighted dyadic Carleson's inequalities, J. Inequal. Appl. 6 (2001), no. 4, 415–433 https://doi.org/10.1155/S102558340100025X
  38. R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), no. 3, 257–272
  39. L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930 https://doi.org/10.2307/2372840
  40. E. M. Dynkin and B. P. Osilenker, Weighted estimates for singular integrals and their applications,Mathematical analysis, Vol. 21, 42–129, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983
  41. C. Fefferman and E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115 https://doi.org/10.2307/2373450
  42. M. Gabidzashvili and V. Kokilashvili, Two weight weak type inequalities for fractional type integrals, Preprint, No. 45, Mathematical Institute Czech Acad. Sci., Prague, 1989
  43. A. Gogatishvili and V. Kokilashvili, Criteria of strong type two-weighted inequalities for fractional maximal functions, Georgian Math. J. 3 (1996), no. 5, 423–446 https://doi.org/10.1007/BF02259772
  44. V. Kokilashvili and A. Meskhi, On a trace inequality for one-sided potentials with multiple kernels, Fract. Calc. Appl. Anal. 6 (2003), no. 4, 461–472
  45. V. G. Maz'ya and I. E. Verbitsky, Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), no. 1, 81–115 https://doi.org/10.1007/BF02559606
  46. E. T. Sawyer, Two weight norm inequalities for certain maximal and integral operators, Harmonic analysis (Minneapolis, Minn., 1981), pp. 102–127, Lecture Notes in Math., 908, Springer, Berlin-New York, 1982 https://doi.org/10.1007/BFb0093283
  47. I. E. Verbitsky and R. L. Wheeden, Weighted norm inequalities for integral operators, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3371–3391

피인용 문헌

  1. 1. Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces vol.18, pp.5, 2015, doi:10.4134/JKMS.2009.46.3.523
  2. 2. Potential Operators on Cones of Nonincreasing Functions vol.2012, 2012, doi:10.4134/JKMS.2009.46.3.523
  3. 3. A characterization of the two-weight inequality for Riesz potentials on cones of radially decreasing functions vol.2014, pp.1, 2014, doi:10.4134/JKMS.2009.46.3.523