Sharp inequalities for Riesz transforms

Adam Osękowski. "Sharp inequalities for Riesz transforms." Studia Mathematica 222.1 (2014): 1-18. <http://eudml.org/doc/285404>.

@article{AdamOsękowski2014,
abstract = {We establish the following sharp local estimate for the family $\{R_\{j\}\}_\{j=1\}^\{d\}$ of Riesz transforms on $ℝ^\{d\}$. For any Borel subset A of $ℝ^\{d\}$ and any function $f: ℝ^\{d\} → ℝ$, $∫_\{A\} |R_\{j\}f(x)|dx ≤ C_\{p\}||f||_\{L^\{p\}(ℝ^\{d\})\}|A|^\{1/q\}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_\{p\} = [2^\{q+2\}Γ(q+1)/π^\{q+1\} ∑_\{k=0\}^\{∞\} (-1)^\{k\}/(2k+1)^\{q+1\}]^\{1/q\}$, 1 < p < 2, and $C_\{p\}= [4Γ(q+1)/π^\{q\} ∑_\{k=0\}^\{∞\} 1/(2k+1)^\{q\}]^\{1/q\}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.},
author = {Adam Osękowski},
journal = {Studia Mathematica},
keywords = {Riesz transform; Hilbert transform; weak-type inequality; martingale inequality; sharp constant},
language = {eng},
number = {1},
pages = {1-18},
title = {Sharp inequalities for Riesz transforms},
url = {http://eudml.org/doc/285404},
volume = {222},
year = {2014},
}

TY - JOUR
AU - Adam Osękowski
TI - Sharp inequalities for Riesz transforms
JO - Studia Mathematica
PY - 2014
VL - 222
IS - 1
SP - 1
EP - 18
AB - We establish the following sharp local estimate for the family ${R_{j}}_{j=1}^{d}$ of Riesz transforms on $ℝ^{d}$. For any Borel subset A of $ℝ^{d}$ and any function $f: ℝ^{d} → ℝ$, $∫_{A} |R_{j}f(x)|dx ≤ C_{p}||f||_{L^{p}(ℝ^{d})}|A|^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = [2^{q+2}Γ(q+1)/π^{q+1} ∑_{k=0}^{∞} (-1)^{k}/(2k+1)^{q+1}]^{1/q}$, 1 < p < 2, and $C_{p}= [4Γ(q+1)/π^{q} ∑_{k=0}^{∞} 1/(2k+1)^{q}]^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.
LA - eng
KW - Riesz transform; Hilbert transform; weak-type inequality; martingale inequality; sharp constant
UR - http://eudml.org/doc/285404
ER -