Vector-valued singular integrals revisited-with random dyadic cubes (Q2903926)

With the help of random dyadic systems, the author gives a short proof of Figiel's vector-valued \(T1\) theorem.NEWLINENEWLINELet \(1<p<\infty\), and let \(X\) be a UMD space with the unconditionality constant \(\beta_{p,X}\) of martingale differences in \(L^p(\mathbb R;X)\). Let \(T\) be a Calderón-Zygmund operator on \(\mathbb R^n\) which satisfies the standard kernel estimates, the weak boundedness property \(|\langle \chi_I,\,T\chi_I\rangle|\leq C|I|\) for all cubes \(I\), and \(T1=T^*1=0\). Then \(T\) extends to a bounded linear operator on \(L^p(\mathbb R;X)\), and NEWLINE\[NEWLINE\|T\|_{\mathcal L(L^p(\mathbb R;X))}\leq C\,\beta_{p,X}^2.NEWLINE\]NEWLINE He also gives a simpler proof of a certain square function estimate of J. Bourgain for translations of functions with a limited frequency spectrum.NEWLINENEWLINERandom dyadic systems were used effectively by \textit{F. Nazarov, S. Treil} and \textit{A. Volberg} [Int. Math. Res. Not. 1997, No. 15, 703--726 (1997; Zbl 0889.42013); Acta Math. 190, No. 2, 151--239 (2003; Zbl 1065.42014)]. On the space \((\{0,1\}^n)^{\mathbb Z}\), consider the natural probability, which makes the coordinates independent and uniformly distributed over the set \(\{0,1\}^n\). This induces a probability on the family of all dyadic systems \(\mathcal D^\beta\) in the definition below.NEWLINENEWLINELet NEWLINE\[NEWLINE\mathcal D^0:=\bigcup_{j\in\mathbb Z}\mathcal D_j^0,\quad \mathcal D_j^0:=\{2^{-j}([0,1)^n+m);m\in \mathbb Z^n\}NEWLINE\]NEWLINE be the standard system of dyadic cubes. For every NEWLINE\[NEWLINE\beta=(\beta_j)_{j\in\mathbb Z}\in (\{0,1\}^n)^{\mathbb Z},NEWLINE\]NEWLINE let NEWLINE\[NEWLINE\mathcal D^\beta:=\bigcup_{j\in\mathbb Z}\mathcal D_j^\beta,\quad \mathcal D_j^\beta :=D_j^0+\sum_{i>j}2^{-i}\beta_i.NEWLINE\]