Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces
The author of this paper considers vector-valued analytic functions of bounded mean oscillation (BMO). He states: The paper is divided into five sections. We start with a section of preliminary character to recall the notions on geometry of Banach spaces to be used throughout the paper and which also contains the basic properties of vector-valued analytic functions of bounded mean oscillation and functions in the vector valued Bloch space. Section 2 is devoted to give some sufficient conditions on the derivative of the function or on the Taylor coefficients of the function to assure that the function belongs to \(\text{BMOA} (X)\). It is shown that one has \(M_p(f',r)= O((1-r)^{-\frac{1}{p'}})\) for some \(1\leq p<\infty\) or \(\| x_n\|= O(\frac 1n)\) (in the case of \(B\)-convex spaces) implies that \(f(z)= \sum_{n=0}^\infty x_nz^n\in \text{BMOA} (X)\). Section 3 deals with multipliers between spaces of vector valued functions defined on different Banach spaces \(X\) and \(Y\). This is done by looking at functions with values in the space of operators \({\mathcal L}(X,Y)\) and considering the natural convolution mentioned above. We also introduce two new notions based on the vector-valued formulations of the Hardy-Littlewood inequalities previously pointed out, called the (HL)-property and the (HL)*-property respectively. It is shown that under the assumptions of (HL)-property on \(X\) and (HL)*-property on \(Y\) one has \((H^1(X), \text{BMOA} (Y))={\mathcal B}({\mathcal L}(X,Y))\). Section 4 is devoted to the study of these properties. It is shown that the spaces having the so-called (HL) and (HL)* must satisfy the Paley property (see definition in Section 1) and have type 2 respectively. It is also shown that the natural duality between them holds for UMD spaces. We investigate Lebesgue spaces \(L^p\) and Schatten classes \(\sigma_p\) having such properties. The tools to deal with Schatten classes are the use of certain factorization and interpolation results holding for functions in Hardy spaces with values in them. Finally Section 5 is devoted to present several applications of different nature of the previous results.
vector-valued analytic function of bounded mean oscillation
scientific article; zbMATH DE number 1182210