Norm inequalities for the Littlewood-Paley function 𝑔*_{𝜆}

Muckenhoupt, Benjamin; Wheeden, Richard L.

doi:10.1090/S0002-9947-1974-0387973-X

Norm inequalities for the Littlewood-Paley function $g^{\ast } _{\lambda }$
HTML articles powered by AMS MathViewer

by Benjamin Muckenhoupt and Richard L. Wheeden PDF

Trans. Amer. Math. Soc. 191 (1974), 95-111 Request permission

Abstract:

Weighted norm inequalities for ${L^p}$ and ${H^p}$ are derived for the Littlewood-Paley function $g_\lambda ^ \ast$. New results concerning the boundedness of this function are obtained, by a different method of proof, even in the unweighted case. The proof exhibits a connection between $g_\lambda ^\ast$ and a maximal function for harmonic functions which was introduced by C. Fefferman and E. M. Stein. A new and simpler way to determine the behavior of this maximal function is given.

References

R. R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838–2839. MR 303226, DOI 10.1073/pnas.69.10.2838
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
R. F. Gundy and R. L. Wheeden, Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series, Studia Math. 49 (1973/74), 107–124. MR 352854, DOI 10.4064/sm-49-2-107-124
Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
Benjamin Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1973/74), 101–106. MR 350297, DOI 10.4064/sm-49-2-101-106
Benjamin Muckenhoupt and Richard L. Wheeden, Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249–258. MR 285938, DOI 10.1090/S0002-9947-1971-0285938-7
Marvin Rosenblum, Summability of Fourier series in $L^{p}(d\mu )$, Trans. Amer. Math. Soc. 105 (1962), 32–42. MR 160073, DOI 10.1090/S0002-9947-1962-0160073-1
C. Segovia and R. L. Wheeden, On the function $g_{\lambda }^{\ast }$ and the heat equation, Studia Math. 37 (1970), 57–93. MR 285907, DOI 10.4064/sm-37-1-57-93
Elias M. Stein, A maximal function with applications to Fourier series, Ann. of Math. (2) 68 (1958), 584–603. MR 100197, DOI 10.2307/1970157
E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. MR 125392, DOI 10.2307/1970308
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
A. Zygmund, On the Littlewood-Paley function $g^*(\theta )$, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 208–212. MR 77700, DOI 10.1073/pnas.42.4.208