Fourier multipliers on weighted 𝐿^{𝑝}-spaces

Quek, T.

doi:10.1090/S0002-9939-99-04812-1

Fourier multipliers on weighted $L^p$-spaces
HTML articles powered by AMS MathViewer

by T. S. Quek PDF

Proc. Amer. Math. Soc. 127 (1999), 2343-2351 Request permission

Abstract:

In his 1986 paper in the Rev. Mat. Iberoamericana, A. Carbery proved that a singular integral operator is of weak type $(p,p)$ on $L^{p}(\mathbb {R}^{n})$ if its lacunary pieces satisfy a certain regularity condition. In this paper we prove that Carbery’s result is sharp in a certain sense. We also obtain a weighted analogue of Carbery’s result. Some applications of our results are also given.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
A. Baernstein II and E. T. Sawyer, Embedding and multiplier theorems for $H^P(\textbf {R}^n)$, Mem. Amer. Math. Soc. 53 (1985), no. 318, iv+82. MR 776176, DOI 10.1090/memo/0318
Anthony Carbery, Variants of the Calderón-Zygmund theory for $L^p$-spaces, Rev. Mat. Iberoamericana 2 (1986), no. 4, 381–396. MR 913694, DOI 10.4171/RMI/40
Alessandro Figà-Talamanca and Garth I. Gaudry, Multipliers of $L^{p}$ which vanish at infinity, J. Functional Analysis 7 (1971), 475–486. MR 0276689, DOI 10.1016/0022-1236(71)90029-2
C. S. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968/69), 283–323. MR 0438109, DOI 10.1512/iumj.1969.18.18024
Karel de Leeuw, On $L_{p}$ multipliers, Ann. of Math. (2) 81 (1965), 364–379. MR 174937, DOI 10.2307/1970621
C. W. Onneweer, On absolutely convergent Fourier series, Ark. Mat. 12 (1974), 51–58. MR 352852, DOI 10.1007/BF02384746
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673, DOI 10.1007/BFb0091154