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

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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.

**[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,*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]**-,*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**to*, 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**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*, Pacific J. Math.**115**(1984), 217-255. MR**762212 (86j:42028)****[W1]**J. M. Wilson,*Weighted inequalities for the dyadic square function without dyadic*, 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]**-,*weighted norm inequalities for the square function*, , Illinois J. Math, (to appear).

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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