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Abstract

We prove an interpolatory estimate linking the directional Haar projection $P^{(\varepsilon)}$ to the Riesz transform in the context of Bochner–Lebesgue spaces $L^p(\mathbb R^n;X)$, $1 < p < \infty$, provided $X$ is a ${\rm UMD}$-space. If $\varepsilon_{i_0} = 1$, the result is the inequality \begin{equation}\label{eqn:abstract:main_result} \|P^{(\varepsilon)} u\|_{L^p(\mathbb R^n;X)} \leq C \|u\|_{L^p(\mathbb R^n;X)}^{1/\mathcal T} \|R_{i_0} u\|_{L^p(\mathbb R^n;X)}^{1 - 1/\mathcal T},\tag{1} \end{equation} where the constant $C$ depends only on $n$, $p$, the ${\rm UMD}$-constant of $X$ and the Rademacher type $\mathcal T$ of $L^p(\mathbb R^n;X)$.

In order to obtain the interpolatory result (1) we analyze stripe operators $S_\lambda$, $\lambda \geq 0$, which are used as basic building blocks to dominate the directional Haar projection. The main result on stripe operators is the estimate \begin{equation}\label{eqn:abstract:stripe_operator_estimate} \| S_\lambda u \|_{L^p(\mathbb R^n;X)} \leq C\cdot 2^{-\lambda/\mathcal C} \|u\|_{L^p(\mathbb R^n;X)},\tag{2} \end{equation} where the constant $C$ depends only on $n$, $p$, the ${\rm UMD}$-constant of $X$ and the Rademacher cotype $\mathcal C$ of $L^p(\mathbb R^n;X)$. The proof of (2) relies on a uniform bound for the shift operators $T_m$, $0 \leq m < 2^\lambda$, acting on the image of $S_\lambda$.

Mainly based upon inequality (1), we prove a vector-valued result on sequential weak lower semicontinuity of integrals of the form \begin{equation*} u \mapsto \int f (u)\, dx, \end{equation*} where $f : X^n \rightarrow \mathbb R^+$ is separately convex satisfying $f(x) \leq C (1 + \|x\|_{X^n})^p$.