On the LittlewoodPaley theory for mixed norm spaces
Author:
John A. Gosselin
Journal:
Trans. Amer. Math. Soc. 256 (1979), 113124
MSC:
Primary 42C10
MathSciNet review:
546910
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Abstract: An inequality of LittlewoodPaley type is proved for the mixed norm spaces , , , on the interval . 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 WalshFourier series on is established. This result can be viewed as an extension of a similar result for the HardyLittlewood maximal function due to C. Fefferman and E. M. Stein.
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 A. Benedeck and R. Panzone, The spaces with mixed norm, Duke Math. J. 29 (1961), 301324.
 [2]
 A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), 97101. MR 0420115 (54:8132)
 [3]
 C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107115. 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 WalshFourier series of functions, Proc. Internat. Congress Math. (Nice, 1970), vol. 2, 1971, pp. 655661. MR 0511000 (58:23330)
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 R. A. Hunt and W. S. Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274277. MR 0338655 (49:3419)
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 R. E. A. C. Paley, A remarkable series of orthogonal functions. I, Proc. London Math. Soc. 34 (1932), 241264.
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 D. Pankratz, Almost everywhere convergence of lacunary partial sums of WalshFourier series, Doctoral Dissertation, Purdue University, Lafayette, Ind., 1974.
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 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
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 , Topic i harmoni analysi relate t the LittlewoodPaley theory, Ann. of Math. Studies, no. 63, Princeton Univ. Press, Princeton, N. J., 1970.
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 W. S. Young, Some maximal inequalities for martingales, part of Doctoral Dissertation, Purdue University, Lafayette, Ind., 1973. MR 1824594 (2002b:53102)
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 A. Zygmund, Trigonometric series, Vols. I, II, 2nd ed., Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719790546910X
PII:
S 00029947(1979)0546910X
Keywords:
Mixed norm spaces,
LittlewoodPaley function,
Khinchine's inequality,
Rademacher functions,
WalshFourier series,
HardyLittlewood maximal function,
lacunary maximal partial sum operator
Article copyright:
© Copyright 1979
American Mathematical Society
