Weighted norm inequalities for the Hardy maximal function. (English) Zbl 0236.26016

The principal problem considered is the determination of all nonnegative functions, \(U(x)\), for which there is a constant, \(C\), such that
\[ \int_J [f^*(x)]^p U(x)\,dx \leq C\int_J | f(x)|^p U(x)\,dx, \] where \(1 < p < \infty\), \(J\) is a fixed interval, \(C\) is independent of \(f\), and \(f^*\) is the Hardy maximal function,
\[ f^*(x) = \sup_{y \neq x;\;y \in J} \frac{1}{y - x}\int_x^y | f(t)| \,dt. \] The main result is that \(U(x)\) is such a function if and only if
\[ \left[\int_I U(x)\,dx\right]\left[\int_I [U(x)]^{-1/(p - 1)}\,dx\right]^{p-1} \leq K| I|^p \] where \(I\) is any subinterval of \(J\), \( | I|\) 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 \(p=1\) or \(p=\infty\), 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.


42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
26D15 Inequalities for sums, series and integrals
42A24 Summability and absolute summability of Fourier and trigonometric series
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[1] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107 – 115. · Zbl 0222.26019 · doi:10.2307/2373450
[2] Frank Forelli, The Marcel Riesz theorem on conjugate functions, Trans. Amer. Math. Soc. 106 (1963), 369 – 390. · Zbl 0121.09802
[3] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), no. 1, 81 – 116. · JFM 56.0264.02 · doi:10.1007/BF02547518
[4] Henry Helson and Gabor Szegö, A problem in prediction theory, Ann. Mat. Pura Appl. (4) 51 (1960), 107 – 138. · Zbl 0178.50002 · doi:10.1007/BF02410947
[5] Benjamin Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231 – 242. · Zbl 0175.12602
[6] Benjamin Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243 – 260. · Zbl 0175.12701
[7] Benjamin Muckenhoupt, Mean convergence of Hermite and Laguerre series. I, II, Trans. Amer. Math. Soc. 147 (1970), 419-431; ibid. 147 (1970), 433 – 460. · Zbl 0191.07602
[8] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17 – 92. · Zbl 0139.29002
[9] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[10] -, On certain operators on \( {L_p}\) spaces, Doctoral Dissertation, University of Chicago, Chicago, Ill., 1955.
[11] G. N. Watson, Notes on generating functions of polynomials. III: Polynomials of Legendre and Gegenbauer, J. London Math. Soc. 8 (1933), 289-292. · Zbl 0007.41101
[12] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968.
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