An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

GDML_Books, (2014), p.

Résumé

We prove an interpolatory estimate linking the directional Haar projection $P^{(ε)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^{p}(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If $ε_{i₀} = 1$, the result is the inequality $||P^{(ε)}u||_{L^{p}(ℝⁿ;X)} ≤ C||u||_{L^{p}(ℝⁿ;X)}^{1/} ||R_{i₀}u||_{L^{p}(ℝⁿ;X)}^{1 - 1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^{p}(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{λ}$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection. The main result on stripe operators is the estimate $||S_{λ}u||_{L^{p}(ℝⁿ;X)} ≤ C·2^{-λ/}||u||_{L^{p}(ℝⁿ;X)}$, (2) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher cotype of $L^{p}(ℝⁿ;X)$. The proof of (2) relies on a uniform bound for the shift operators Tₘ, $0 ≤ m < 2^{λ}$, acting on the image of $S_{λ}$. Mainly based upon inequality (1), we prove a vector-valued result on sequential weak lower semicontinuity of integrals of the form u ↦ ∫ f(u)dx, where f: Xⁿ → ℝ⁺ is separately convex satisfying $f(x) ≤ C (1 + ||x||_{Xⁿ})^{p}$.

Détail

Zbl 1321.46042

EUDML-ID : urn:eudml:doc:286038