Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weighted norm inequalities for the continuous square function


Author: J. Michael Wilson
Journal: Trans. Amer. Math. Soc. 314 (1989), 661-692
MSC: Primary 42B20
DOI: https://doi.org/10.1090/S0002-9947-1989-0972707-9
Erratum: Trans. Amer. Math. Soc. 321 (1990), null.
MathSciNet review: 972707
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove new weighted norm inequalities for real-variable analogues of the Lusin area function. We apply our results to obtain new: (i) weighted norm inequalities for singular integral operators; (ii) weighted Sobolev inequalities; (iii) eigenvalue estimates for degenerate Schrödinger operators.


References [Enhancements On Off] (What's this?)

  • [CWW] S. Y. A. Chang, J. M. Wilson and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), 217-246. MR 800004 (87d:42027)
  • [CW1] S. Chanillo and R. L. Weeden, $ {L^p}$ estimates for fractional integrals and Sobolev inequalities, with applications to Schrödinger operators, preprint (1985).
  • [CW2] -, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), 277-294. MR 891775 (88e:42036)
  • [F] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129-206. MR 707957 (85f:35001)
  • [FS1] C. L. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 92 (1971), 107-115. MR 0284802 (44:2026)
  • [FS2] -, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [RF] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. 126 (1987), 109-130. MR 898053 (90e:42030)
  • [G] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. MR 628971 (83g:30037)
  • [GJ] J. B. Garnett and P. W. Jones, The distance in $ {\text{BMO}}$ to $ {L^\infty }$, Ann. of Math. 108 (1978), 373-393. MR 506992 (80h:46037)
  • [KS] R. Kerman and E. T. Sawyer, Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms and Carleson measures, preprint (1984).
  • [K] D. Krutz, Littlewood-Paley and multiplier theorems on weighted $ {L^p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), 235-254. MR 561835 (80f:42013)
  • [M] B. Muckenhoupt, Weighted norm inequalities for classical operators, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R.I., 1979, pp. 69-84. MR 545240 (80i:42015)
  • [Sch] M. Schechter, The spectrum of the Schrödinger operator, preprint (1987).
  • [St] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton N.J., 1970. MR 0290095 (44:7280)
  • [U] A. Uchiyama, The Fefferman-Stein decomposition of smooth functions and its applications to $ {H^p}({{\mathbf{R}}^n})$, Pacific J. Math. 115 (1984), 217-255. MR 762212 (86j:42028)
  • [W1] J. M. Wilson, Weighted inequalities for the dyadic square function without dyadic $ {A_\infty }$, Duke Math. J. 55 (1987), 19-49. MR 883661 (88d:42034)
  • [W2] -, A sharp inequality for the square function, Duke Math. J. 55 (1987), 879-887. MR 916125 (89a:42029)
  • [W3] -, $ {L^p}$ weighted norm inequalities for the square function, $ 0 < p < 2$, Illinois J. Math, (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B20

Retrieve articles in all journals with MSC: 42B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0972707-9
Keywords: Lusin area function, weighted norm inequality, Calderón-Zygmund operator, Sobolev inequality, Schrödinger operator
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society