Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts (Q2769336)

Our main objects are paraproduct operators, Carleson measures in the upper half-plane, Hankel operators, \(H^1\) multipliers. All these objects are vector (operator) valued. Our goal is to find to what extent the noncommutative analogs of classical scalar-valued theorems are valid. We do that by finding the sharp dimensional estimates for our objects. The interest in this subject has many reasons, and one of them is the recent development of noncommutative weighted estimates of Calderón-Zygmund operators. The estimates of some Calderón-Zygmund operators in Banach valued function spaces have a long history. Such estimates have been later developed by Burkholder and Bourgain. The difference between the weighted and unweighted case is that in the weighted case the norm is not shift invariant. In the unweighted case, to calculate the norm of a vector function \(f\) we first gauge the vector \(f(t)\) \((t\in\mathbb{R})\) by Minkowski's function of the convex body \(B\), and \(B\) is the same for all \(t\). In the weighted case, we gauge the vector \(f(t)\) \((t\in\mathbb{R})\) by Minkowski's function of the ellipsoid \(B(t)\), and \(B(t)\) are different for different \(t\).NEWLINENEWLINENEWLINEOn the level of estimates the difference of unweighted and weighted cases amounts to the following fundamental feature. If we consider a Calderón-Zygmund operator \(T\) (the usual one, with scalar kernel) such that \(T1= 0\), \(T^*1= 0\), then there is no ``paraproduct operator'' neither in the scalar unweighted nor in the vector unweighted case. Notice that may be the most important example of such a \(T\) is the Hilbert transform \(H\) given by \(Hf(x):= {1\over\pi} \int_{\mathbb{R}} {f(t)\over x-t} dt\) understood in the sense of principal value: \(Hf(x):= \lim_{\varepsilon\to 0} {1\over \pi} \int_{\mathbb{R}\setminus B(x,\varepsilon)}\dots\), where \(B(x,\varepsilon)\) is the disc centered at \(x\) of radius \(\varepsilon\).NEWLINENEWLINENEWLINEBut if we consider the same \(T\) in the weighted case (in both scalar and vector-valued spaces) the ``paraproduct operator'' does appear. It has operator valued symbol, it acts on functions with values in the vector space \(E\), and we need to estimate it in terms of the dimension of \(E\). As our results in the present article show there is no hope to get dimensionless estimates.