\(L_p\)-maximal regularity on Banach spaces with a Schauder basis (Q1614939)

Let \(1 \leq p < \infty\). A Banach space \(X\) has type \(p\) if there exists a constant \(C > 0\) such that for each finite sequence \((x_k)_{k=1}^{K}\) in \(X\) the following inequality holds: \[ \left(\int_{0}^{1} \|\sum_{k=1}^{K} \varepsilon_{k}(t) x_k\|^2 dt\right)^{1\over 2} \leq C\left(\sum\limits_{k=1}^{K} \|x_k\|^p \right)^{1\over p},\tag{1} \] where \((\varepsilon_{k})_{k=1}^{\infty}\) is the standard sequence of Rademacher functions on \([0, 1]\). The authors prove the following Theorem. Let \(X\) be a Banach space of type \(p > 1\) having the Schauder decomposition \((E_n)_{n=1}^{\infty}\). If \(X\) has maximal regularity property, then there exists a constant \(C > 0\) such that for any block basic sequence \((u_k)_{k=1}^{N}\) with respect to the decomposition \((E_n)_{n=1}^{\infty}\), the following inequality holds: \[ \frac{1}{C} \sum\limits_{k=1}^{N} \|u_k\|^2 \leq \int_{0}^{1} \Biggl\|\sum_{k=1}^{K} \varepsilon_{k}(t) x_k\Biggr\|^2 dt \leq C \sum_{k=1}^{N} \|u_k\|^2. \] An application of this result to the maximal regularity property of certain classes of operators is given. Analogous results are discussed for \(L_{r}\)-regularity in \(L_{s}\) spaces.