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 $f(x) ≤ C (1 + ||x||_{Xⁿ})^{p}$.
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We obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.
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In the context of spaces of homogeneous type, we develop a method to deterministically construct dyadic grids, specifically adapted to a given combinatorial situation. This method is used to estimate vector-valued operators rearranging martingale difference sequences such as the Haar system.
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