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1. CMB 2010 (vol 54 pp. 113)
On the Norm of the Beurling-Ahlfors Operator in Several Dimensions
The generalized Beurling-Ahlfors operator $S$ on
$L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the
exterior algebra with its natural Hilbert space norm, satisfies the
estimate
$$\|S\|_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*-1),\quad
p^*:=\max\{p,p'\}$$
This improves on earlier results in all dimensions $n\geq 3$. The
proof is based on the heat extension and relies at the bottom on
Burkholder's sharp inequality for martingale transforms.
Categories:42B20, 60G46 |
2. CMB 1999 (vol 42 pp. 321)
Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces We shall study some connection between averaging operators and
martingale inequalities in rearrangement invariant function spaces.
In Section~2 the equivalence between Shimogaki's theorem and some
martingale inequalities will be established, and in Section~3 the
equivalence between Boyd's theorem and martingale inequalities with
change of probability measure will be established.
Keywords:martingale inequalities, rearrangement invariant function spaces Categories:60G44, 60G46, 46E30 |
3. CMB 1999 (vol 42 pp. 221)
Boundedness of the $q$-Mean-Square Operator on Vector-Valued Analytic Martingales We study boundedness properties of the $q$-mean-square operator
$S^{(q)}$ on $E$-valued analytic martingales, where $E$ is a
complex quasi-Banach space and $2 \leq q < \infty$. We establish
that a.s. finiteness of $S^{(q)}$ for every bounded $E$-valued
analytic martingale implies strong $(p,p)$-type estimates for
$S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new
characterizations (in terms of analytic and stochastic properties
of the function $S^{(q)}$) of the complex spaces $E$ that admit an
equivalent $q$-uniformly PL-convex quasi-norm. We also obtain a
vector-valued extension (and a characterization) of part of an
observation due to Bourgain and Davis concerning the
$L^p$-boundedness of the usual square-function on scalar-valued
analytic martingales.
Categories:46B20, 60G46 |