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Burkholder inequalities for submartingales, Bessel processes and conformal martingales
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 6, December 2013
- pp. 1675-1698
- 10.1353/ajm.2013.0050
- Article
- Additional Information
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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.