An example of a Hilbert transform

Samotij, Krzysztof

doi:10.1090/S0002-9939-1991-1065092-2

An example of a Hilbert transform
HTML articles powered by AMS MathViewer

by Krzysztof Samotij PDF

Proc. Amer. Math. Soc. 112 (1991), 965-972 Request permission

Abstract:

We construct a nonnegative integrable function on the real line $\mathbf {R}$ whose Hilbert transform cannot be almost everywhere dominated by the HardyLittlewood maximal function of any finite measure on $\mathbf {R}$.

References

Joaquim Bruna and Boris Korenblum, On Kolmogorov’s theorem, the Hardy-Littlewood maximal function and the radial maximal function, J. Analyse Math. 50 (1988), 225–239. MR 942830, DOI 10.1007/BF02796124
Joaquim Bruna and Boris Korenblum, A note on Calderón-Zygmund singular integral convolution operators, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 2, 271–273. MR 876962, DOI 10.1090/S0273-0979-1987-15515-7
D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class $H^{p}$, Trans. Amer. Math. Soc. 157 (1971), 137–153. MR 274767, DOI 10.1090/S0002-9947-1971-0274767-6
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Alan V. Noell and Thomas H. Wolff, On peak sets for Lip $\alpha$ classes, J. Funct. Anal. 86 (1989), no. 1, 136–179. MR 1013937, DOI 10.1016/0022-1236(89)90068-2