Littlewood-Paley and multiplier theorems on weighted 𝐿^{𝑝} spaces

Kurtz, Douglas S.

doi:10.1090/S0002-9947-1980-0561835-X

Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces
HTML articles powered by AMS MathViewer

by Douglas S. Kurtz PDF

Trans. Amer. Math. Soc. 259 (1980), 235-254 Request permission

Abstract:

The Littlewood-Paley operator $\gamma (f)$, for functions f defined on ${{\textbf {R}}^n}$, is shown to be a bounded operator on certain weighted ${L^p}$ spaces. The weights satisfy an ${A_p}$ condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander’s Multiplier Theorem.

References

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
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
I. I. Hirschman, The decomposition of Walsh and Fourier series, Mem. Amer. Math. Soc. 15 (1955), 65. MR 72269
I. I. Hirschman Jr., A note on orthogonal systems, Pacific J. Math. 6 (1956), 47–56. MR 79129, DOI 10.2140/pjm.1956.6.47
Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
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

Weighted norm inequalities for singular and hypersingular integrals

Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362. MR 542885, DOI 10.1090/S0002-9947-1979-0542885-8
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 and Richard L. Wheeden, Norm inequalities for the Littlewood-Paley function $g^{\ast } _{\lambda }$, Trans. Amer. Math. Soc. 191 (1974), 95–111. MR 387973, DOI 10.1090/S0002-9947-1974-0387973-X
Benjamin Muckenhoupt and Richard L. Wheeden, On the dual of weighted $H^{1}$ of the half-space, Studia Math. 63 (1978), no. 1, 57–79. MR 508882, DOI 10.4064/sm-63-1-57-79
H. R. Pitt, Theorems on Fourier series and power series, Duke Math. J. 3 (1937), no. 4, 747–755. MR 1546029, DOI 10.1215/S0012-7094-37-00363-6
N. M. Rivière, Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278. MR 440268, DOI 10.1007/BF02383650
N. M. Rivière and Y. Sagher, On two theorems of Paley, Proc. Amer. Math. Soc. 42 (1974), 238–242. MR 344779, DOI 10.1090/S0002-9939-1974-0344779-0
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
Carlos Segovia and Richard L. Wheeden, On weighted norm inequalities for the Lusin area integral, Trans. Amer. Math. Soc. 176 (1973), 103–123. MR 311921, DOI 10.1090/S0002-9947-1973-0311921-0

Classes

multiplicateurs et fonctions de Littlewood-Paley

263

Elias M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492. MR 82586, DOI 10.1090/S0002-9947-1956-0082586-0
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Richard L. Wheeden, A boundary value characterization of weighted $H^{1}$, Enseign. Math. (2) 22 (1976), no. 1-2, 121–134. MR 417672
Kôsaku Yosida, Functional analysis, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York, Inc., New York, 1968. MR 0239384, DOI 10.1007/978-3-662-11791-0
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776