Martingale transforms and complex uniform convexity

Bourgain, J.; Davis, W. J.

doi:10.1090/S0002-9947-1986-0825718-5

Martingale transforms and complex uniform convexity
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by J. Bourgain and W. J. Davis

Trans. Amer. Math. Soc. 294 (1986), 501-515

DOI: https://doi.org/10.1090/S0002-9947-1986-0825718-5

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Abstract:

Martingale transforms and Calderon-Zygmund singular integral operators are bounded as operators from ${L_2}({L_1})$ to ${L_2}({L_q})$ when $0 < q < 1$. If $Y$ is a reflexive subspace of ${L_1}$ then ${L_1}/Y$ can be renormed to be $2$-complex uniformly convex. A new proof of the cotype 2 property of ${L_1}/{H_1}$ is given.

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