Sufficiency conditions for 𝐿^{𝑝}-multipliers with power weights

Muckenhoupt, Benjamin; Wheeden, Richard L.; Young, Wo-Sang

doi:10.1090/S0002-9947-1987-0876461-9

Sufficiency conditions for $L^ p$-multipliers with power weights
HTML articles powered by AMS MathViewer

by Benjamin Muckenhoupt, Richard L. Wheeden and Wo-Sang Young

Trans. Amer. Math. Soc. 300 (1987), 433-461

DOI: https://doi.org/10.1090/S0002-9947-1987-0876461-9

PDF | Request permission

Abstract:

Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function of Hörmander type. The operators are initially defined on the space ${\mathcal {S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use weight functions of the form ${\left | x \right |^\alpha }$ for $\alpha$ larger than usually considered. For these weight functions, if $(\alpha + 1)/p$ is not an integer, a strict inequality on $\alpha$ is shown to be sufficient for a norm inequality to hold. A sequel to this paper shows that the weak version of this inequality is necessary.

References

A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution. II, Advances in Math. 24 (1977), no. 2, 101–171. MR 450888, DOI 10.1016/S0001-8708(77)80016-9
A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225
A. Carbery, G. Gasper, and W. Trebels, On localized potential spaces, J. Approx. Theory 48 (1986), no. 3, 251–261. MR 864749, DOI 10.1016/0021-9045(86)90049-3
W. C. Connett and A. L. Schwartz, The theory of ultraspherical multipliers, Mem. Amer. Math. Soc. 9 (1977), no. 183, iv+92. MR 435708, DOI 10.1090/memo/0183
George Gasper and Walter Trebels, A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math. 65 (1979), no. 3, 243–278. MR 567079, DOI 10.4064/sm-65-3-243-278
I. I. Hirschman, The decomposition of Walsh and Fourier series, Mem. Amer. Math. Soc. 15 (1955), 65. MR 72269
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
Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235–254. MR 561835, DOI 10.1090/S0002-9947-1980-0561835-X
Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, DOI 10.4064/sm-44-1-31-38

Necessity conditions for

multipliers with power weights

300

Benjamin Muckenhoupt, Richard L. Wheeden, and Wo-Sang Young, $L^{2}$ multipliers with power weights, Adv. in Math. 49 (1983), no. 2, 170–216. MR 714588, DOI 10.1016/0001-8708(83)90072-5
Benjamin Muckenhoupt, Richard L. Wheeden, and Wo-Sang Young, Sufficiency conditions for $L^p$-multipliers with power weights, Trans. Amer. Math. Soc. 300 (1987), no. 2, 433–461. MR 876461, DOI 10.1090/S0002-9947-1987-0876461-9
Benjamin Muckenhoupt and Wo Sang Young, $L^{p}$ multipliers with weight $x^{kp-1}$, Trans. Amer. Math. Soc. 275 (1983), no. 2, 623–639. MR 682722, DOI 10.1090/S0002-9947-1983-0682722-5
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
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972