Abstract

The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\Bbb{R}^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0<p<\infty$, what is the optimal value of the constant $C_{p,d}$ in the inequality $$\|Y\|_p\leq C_{p,d}\|X\|_p?$$ We answer this question by considering a more general related problem for nonnegative submartingales. This enables us to study extension of the above inequality to the case when $d>1$ is not an integer, which has further interesting applications to stopped Bessel processes and to the behavior of smooth functions on Euclidean domains. The inequality for conformal martingales, which has its roots on the study of the $L^p$ norms of the Beurling-Ahlfors singular integral operator, extends a recent result of Borichev, Janakiraman, and Volberg.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1675-1698
Launched on MUSE
2013-11-27
Open Access
No
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