Sharp inequalities for Riesz transforms

• Autorzy: Adam Osękowski
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 60G44: Martingales with continuous parameter
• 42B20: Singular and oscillatory integrals (Calder\'on-Zygmund, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2014
• Tom: 222
• Numer: 1
• Strony: 1-18

Daty

wydano

2014

Twórcy

autor

Adam Osękowski

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

Identyfikatory

DOI

10.4064/sm222-1-1