Książka - szczegóły

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
$||P^{(ε)}u||_{L^{p}(ℝⁿ;X)} ≤ C||u||_{L^{p}(ℝⁿ;X)}^{1/𝓣} ||R_{i₀}u||_{L^{p}(ℝⁿ;X)}^{1 - 1/𝓣}$, (1)
where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type 𝓣 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
$||S_{λ}u||_{L^{p}(ℝⁿ;X)} ≤ C·2^{-λ/𝓒}||u||_{L^{p}(ℝⁿ;X)}$, (2)
where the constant C depends only on n, p, the UMD-constant of X and the Rademacher cotype 𝓒 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 $f(x) ≤ C (1 + ||x||_{Xⁿ})^{p}$.