Weighted norm inequalities for the Hardy maximal function
Author:
Benjamin Muckenhoupt
Journal:
Trans. Amer. Math. Soc. 165 (1972), 207226
MSC:
Primary 46E30; Secondary 26A86, 42A40
MathSciNet review:
0293384
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Abstract: The principal problem considered is the determination of all nonnegative functions, , for which there is a constant, C, such that where , J is a fixed interval, C is independent of f, and is the Hardy maximal function, The main result is that is such a function if and only if where I is any subinterval of J, denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case when or , a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.
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 C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107115. MR 0284802 (44:2026)
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 F. Forelli, The Marcel Riesz theorem on conjugate functions, Trans. Amer. Math. Soc. 106 (1963), 369390. MR 26 #5340. MR 0147827 (26:5340)
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 G. H. Hardy and J. E. Littlewood, A maximal theorem with functiontheoretic applications, Acta Math. 54 (1930), 81116. MR 1555303
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 , Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243260. MR 40 #3159. MR 0249918 (40:3159)
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 , Mean convergence of Hermite and Laguerre series. II, Trans. Amer. Math. Soc. 147 (1970), 433460. MR 41 #711. MR 0256051 (41:711)
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 A. Zygmund, Trigonometric series. Vols. I, II, 2nd ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0236587 (38:4882)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202933846
PII:
S 00029947(1972)02933846
Keywords:
Hardy maximal function,
mean summability,
Fourier series,
Gegenbauer series,
weighted norm inequalities
Article copyright:
© Copyright 1972 American Mathematical Society
