Sharp inequalities for Riesz transforms

Adam Osękowski

Adam Osękowski

Studia Mathematica, Tome 223 (2014), p. 1-18 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $f : ℝ^{d} \to ℝ$ , $\int_{A} | R_{j} f (x) | d x \leq C_{p} {| | f | |}_{L^{p} (ℝ^{d})} {| A |}^{1 / q}$ , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = {[2^{q + 2} Γ (q + 1) / π^{q + 1} \sum_{k = 0}^{\infty} {(- 1)}^{k} / {(2 k + 1)}^{q + 1}]}^{1 / q}$ , 1 < p < 2, and $C_{p} = {[4 Γ (q + 1) / π^{q} \sum_{k = 0}^{\infty} 1 / {(2 k + 1)}^{q}]}^{1 / q}$ , 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.

Publié le : 2014-01-01

Zbl 1305.42011

EUDML-ID : urn:eudml:doc:285404

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm222-1-1,
     author = {Adam Os\k ekowski},
     title = {Sharp inequalities for Riesz transforms},
     journal = {Studia Mathematica},
     volume = {223},
     year = {2014},
     pages = {1-18},
     zbl = {1305.42011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm222-1-1}
}

Adam Osękowski. Sharp inequalities for Riesz transforms. Studia Mathematica, Tome 223 (2014) pp. 1-18. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm222-1-1/