Operator-valued Fourier multiplier theorems on \(L_{p}\)(\(X\)) and geometry of Banach spaces.
The paper under review considers the extension of scalar-valued multiplier theorems to operator-valued multiplier theorems. Let \(X\), \(Y\) be Banach spaces and \({\mathcal B}(X,Y)\) the space of bounded linear operators from \(X\) to \(Y\). A bounded measurable function \(m\colon\mathbb{R}^N\to {\mathcal B}(X,Y)\) is a Fourier multiplier from \(L_q(\mathbb{R}^N,X)\) to \(L_q(\mathbb{R}^N,Y)\) if the map \(f\in{\mathcal S}(\mathbb{R}^N,X)\mapsto {\mathcal F}^{-1}(m\cdot {\mathcal F} f)\in L_\infty(\mathbb{R}^N,Y)\), where \({\mathcal F}\) is the Fourier transform and \({\mathcal S}\) the Schwartz class, extends to a bounded operator from \(L_q(\mathbb{R}^N,X)\) to \(L_q(\mathbb{R}^N,Y)\). Since multiplier theorems in general do not extend from the scalar case to the Banach space-valued case, some geometrical conditions on the Banach spaces are needed. A Banach space \(X\) has the unconditional martingale difference (UMD) property if all \(X\)-valued martingale difference sequences are unconditional in \(L_p(X)\), \((1<p<\infty)\). Results of \textit{J. Bourgain} [Ark. Mat. 21, 163--168 (1983; Zbl 0533.46008)] and \textit{D. L. Burkholder} [Conference on harmonic analysis in honor of Antoni Zygmund, Vol. 1, 270--286 (1983; Zbl 0532.42001)] show that this property is equivalent to the boundedness on \(L_p(X)\) of vector-valued Hilbert transform. The main result of the paper gives a condition on a bounded measurable function \(m\colon\mathbb{R}^N\setminus\{0\}\to {\mathcal B}(X,Y)\) so that it is a Fourier multiplier from \(L_q(\mathbb{R}^N,X)\) to \(L_q(\mathbb{R}^N,Y)\) \((1<q<\infty)\). The condition depends on the Fourier type of \(X\) and \(Y\) (\(X\), \(Y\) have non-trivial Fourier type since they have the UMD property). The condition is rather technical and uses vector-valued Besov spaces and the \(R\)-boundedness of a certain map induced by \(m\). The concept of \(R\)-boundedness strengthens boundedness in norm: A collection of operators \(\tau\subset B(X,Y)\) is \(R\)-bounded if there exists a constant \(C>0\) such that \(\int_0^1 \| \sum_{k=1}^nr_k(s)T_k(x_k)\| ds\leq C\int_0^1\| \sum_{k=1}^nr_k(s)x_k\| ds\), for any \(n\in \mathbb{N}\), \(T_1,\dots ,T_n\in\tau\) and \(x_1,\dots ,x_n\in X\). In a series of corollaries, the authors obtain vector-valued generalizations of classical multiplier theorems, and give simpler conditions for \(m\) being a Fourier multiplier on different function spaces such as vector fractional Sobolev spaces, \(H^1(X)\) and \(BMO(X)\).
scientific article; zbMATH DE number 2037117
scientific article