A short proof of an inequality of Littlewood and Paley

Pavlović, Miroslav

doi:10.1090/S0002-9939-06-08434-6

A short proof of an inequality of Littlewood and Paley
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by Miroslav Pavlović

Proc. Amer. Math. Soc. 134 (2006), 3625-3627

DOI: https://doi.org/10.1090/S0002-9939-06-08434-6

Published electronically: June 15, 2006

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Abstract:

A very short proof is given of the inequality \begin{equation*} \int _{|z|<1}|\nabla u(z)|^p (1-|z|)^{p-1} dxdy \le C_p\left (\frac 1{2\pi }\int _0^{2\pi } |f(e^{it})|^p dt -|u(0)|^p\right ), \end{equation*} where $p>2,$ and $u$ is the Poisson integral of $f\in L^p(\partial \mathbb D),$ $\mathbb D=\{z: |z|<1\}.$

References

Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
J.E. Littlewood and R.E.A.C. Paley, Theorems on Fourier series and power series. II, Proc. Lond. Math. Soc. 42(1936), 52–89.
Daniel H. Luecking, A new proof of an inequality of Littlewood and Paley, Proc. Amer. Math. Soc. 103 (1988), no. 3, 887–893. MR 947675, DOI 10.1090/S0002-9939-1988-0947675-0
Miroslav Pavlović, A Littlewood-Paley theorem for subharmonic functions, Publ. Inst. Math. (Beograd) (N.S.) 68(82) (2000), 77–82. MR 1826098
P. Stein, On a theorem of M. Riesz, J. London Math. Soc. 8(1933), 52–89.