Sharp inequalities for Riesz transforms

Adam Osękowski¹

¹ Faculty of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 222 (2014) no. 1, pp. 1-18

Résumé

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.

Détail

DOI : 10.4064/sm222-1-1

Keywords: establish following sharp local estimate family riesz transforms mathbb borel subset mathbb function mathbb mathbb int leq mathbb quad infty here p harmonic conjugate bigg frac vargamma sum infty frac bigg quad bigg frac vargamma sum infty frac bigg quad leq infty enables determine precise values weak type constants riesz transforms infty proof rests appropriate martingale inequalities which independent interest