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)

 
 

 

On the Littlewood-Paley theory for mixed norm spaces


Author: John A. Gosselin
Journal: Trans. Amer. Math. Soc. 256 (1979), 113-124
MSC: Primary 42C10
DOI: https://doi.org/10.1090/S0002-9947-1979-0546910-X
MathSciNet review: 546910
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An inequality of Littlewood-Paley type is proved for the mixed norm spaces $ {L_P}({l_r})$, $ 1 < p$, $ r < \infty $, on the interval $ [0,1]$. This result makes use of recent work by C. Fefferman and A. Cordoba on the boundedness of singular integrals on these spaces. As an application of this inequality, boundedness of the lacunary maximal partial sum operator for Walsh-Fourier series on $ {L_p}({l_r})$ is established. This result can be viewed as an extension of a similar result for the Hardy-Littlewood maximal function due to C. Fefferman and E. M. Stein.


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

  • [1] A. Benedeck and R. Panzone, The spaces $ {L^p}$ with mixed norm, Duke Math. J. 29 (1961), 301-324.
  • [2] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), 97-101. MR 0420115 (54:8132)
  • [3] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • [4] R. John, Weighted norm inequalities for singular and hypersingular integrals, Doctoral Dissertation, Rutgers University, New Brunswick, N. J., 1975.
  • [5] R. A. Hunt, Almost everywhere convergence of Walsh-Fourier series of $ {L^2}$ functions, Proc. Internat. Congress Math. (Nice, 1970), vol. 2, 1971, pp. 655-661. MR 0511000 (58:23330)
  • [6] R. A. Hunt and W. S. Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274-277. MR 0338655 (49:3419)
  • [7] R. E. A. C. Paley, A remarkable series of orthogonal functions. I, Proc. London Math. Soc. 34 (1932), 241-264.
  • [8] D. Pankratz, Almost everywhere convergence of lacunary partial sums of Walsh-Fourier series, Doctoral Dissertation, Purdue University, Lafayette, Ind., 1974.
  • [9] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
  • [10] -, Topic i harmoni analysi relate t the Littlewood-Paley theory, Ann. of Math. Studies, no. 63, Princeton Univ. Press, Princeton, N. J., 1970.
  • [11] W. S. Young, Some maximal inequalities for martingales, part of Doctoral Dissertation, Purdue University, Lafayette, Ind., 1973. MR 1824594 (2002b:53102)
  • [12] A. Zygmund, Trigonometric series, Vols. I, II, 2nd ed., Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42C10

Retrieve articles in all journals with MSC: 42C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0546910-X
Keywords: Mixed norm spaces, Littlewood-Paley function, Khinchine's inequality, Rademacher functions, Walsh-Fourier series, Hardy-Littlewood maximal function, lacunary maximal partial sum operator
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society