SINGULAR INTEGRALS SUPPORTED BY SUBVARIETIES FOR VECTOR-VALUED FUNCTIONS

In this paper, we show that singular integrals supported by subvarieties are bounded on $L^{p}(\mathbb{R}^{n};\mathbf{X})$ for $1<p<\infty$ and some UMD space $\mathbf{X}$. In the terminology from operator space theory, we prove that singular integrals supported by subvarieties are completely $L^{p}$-bounded.

The Littlewood—Paley—Rubio de Francia property of a Banach space for the case of equal intervals

Let X be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of X-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents p ≥ 2 if and only if the space X is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

Interpolation of compact operators by the methods of Calderón and Gustavsson–Peetre

Let X = (X0, X1) and Y = (Y0, Y1) be Banach couples and suppose T:X→Y is a linear operator such that T:X0→Y0 is compact. We consider the question whether the operator T:[X0, X1]θ→[Y0, Y1]θ is compact and show a positive answer under a variety of conditions. For example it suffices that X0 be a UMD-space or that X0 is reflexive and there is a Banach space so that X0 = [W, X1]α, for some 0<α<1.