Vector-Valued Singular Integrals and Littlewood–Paley Theory

Let T be a bounded operator from \(L^{p}(\textbf{R}^n)\) to itself for some \(p\in (1,\infty )\) . One may wonder if a stronger estimate of the form \( \Big \Vert \Big ( \sum _{j=1}^N |T(f_j) |^ s \Big )^{\frac{1}{s}} \Big \Vert _{L^p} \le C \Big \Vert \Big ( \sum _{j=1}^N | f_j |^ q \Big )^{\frac{1}{q}} \Big \Vert _{L^p} \) might hold, where \(f_j\in L^{p}(\textbf{R}^n)\) and \(1\le q,s\le \infty \) (with the obvious modifications when q or s is infinite). Naturally, we would like this estimate to hold with a constant C independent of N, so that we can let \(N\rightarrow \infty \) . We will derive estimates of the form (4.1.1) by introducing operators acting on finite sequences of functions. We fix positive integers M, N and \(1\le q,s\le \infty \) .