Sharp inequalities for Riesz transforms
Volume 222 / 2014
Abstract
We establish the following sharp local estimate for the family $\{R_j\}_{j=1}^d$ of Riesz transforms on $\mathbb R^d$. For any Borel subset $A$ of $\mathbb R^d$ and any function $f:\mathbb R^d\to \mathbb R$, $$ \int_A |R_jf(x)|\,d x\leq C_p\|f\|_{L^p(\mathbb R^d)}|A|^{1/q},\quad\ 1 < p < \infty.$$ Here $q=p/(p-1)$ is the harmonic conjugate to $p$, $$ C_p=\bigg[\frac{2^{q+2}\varGamma(q+1)}{\pi^{q+1}}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{q+1}}\bigg]^{1/q},\quad\ 1 < p < 2,$$ and $$ C_p=\bigg[\frac{4\varGamma(q+1)}{\pi^{q}}\sum_{k=0}^\infty \frac{1}{(2k+1)^{q}}\bigg]^{1/q},\quad\ 2\leq p < \infty.$$ This enables us to determine the precise values of the weak-type constants for Riesz transforms for $1 < p < \infty$. The proof rests on appropriate martingale inequalities, which are of independent interest.
