Fourier multipliers and spectral measures in Banach function spaces
Let \(G\) be a locally compact abelian group and denote its dual group by \(\Gamma\). In the paper under review \(G\) will always be infinite. Haar measure on the Borel \(\sigma\)-algebra \(\mathcal{B}(G)\) is denoted by \(\lambda\). Let \(L^0(G)\) denote the space of all \(\mathbb{C}\)-valued, \(\mathcal{B}(G)\)-measurable functions. The space \(L^0(G)\) is a complex Dedekind complete vector lattice for the \(\lambda\)-a.e. pointwise order on the real valued functions. A Banach function space on \(G\) is an ideal \(E \subseteq L^0(G)\) equipped with a norm \(\| \cdot \|_E\) with respect to which \(E\) is a Banach lattice. The authors only consider Banach function spaces whose carrier is equal to \(G\), that is, for every \(A \in \mathcal{B}(G)\) with \(\lambda(A) > 0\) there exists \(B \in \mathcal{B}(G)\) with \(B \subseteq A\) and \(\lambda(B) > 0\) such that \(\chi_B \in E\). For \(f \in L^0(G)\) and \(g \in G\), the translate \(\tau_g f\) of \(f\) by \(g\) is defined by \((\tau_g f)(u) = f(u-g), u \in G\). A Banach function space \(E \subseteq L^0(G)\) is called translation invariant whenever \(\tau_g f \in E\) and \(\| \tau_g f \|_E = \| f \|_E\) for all \(f \in E\) and \(g \in G\). For a Banach function space \(E\) on a locally compact abelian group \(G\), the space of all bounded linear operators on \(E\) will be denoted by \(\mathcal{L}(E)\); when equipped with the strong operator topology it will be denoted by \(\mathcal{L}_s(E)\). For non-compact \(G\) and \(E\) a Banach function space over \((G, \mathcal{B}(G), \lambda)\) we assume that \(E \cap L^2(G)\) is dense in \(E\). Then a function \(\varphi \in L^{\infty}(\Gamma)\) is called an \(E\)-multiplier if there exists an operator \(R_{\varphi} \in \mathcal{L}(E)\) mapping \(E \cap L^2(G)\) into itself and satisfying \[ (R_{\varphi} f)^{\wedge} = \varphi \cdot \widehat{f}, f \in E \cap L^2(G), \] where the above equality is as elements of \(L^2(\Gamma)\). If \(E\) is a Banach function space over \((\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)\), then we say that the Fourier transform is bounded on \(E\) if \(\mathcal{F}\) maps \(E \cap L^2(\mathbb{R})\) into \(E\) and there exists a constant \(C > 0\) such that \(\| \mathcal{F}f\|_E \leq C \| f \|_E\) for all \(f \in E \cap L^2(\mathbb{R})\); here \(\mathcal{F} : L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})\) is the usual Fourier transform operator. In this paper the authors prove the following: Let \(E\) be a Banach function space over \((\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)\) such that \(L^1 \cap L^{\infty}(\mathbb{R}) \subseteq E\) and \(E \cap L^2(\mathbb{R})\) is dense in \(E\). The following are equivalent. {\parindent=5mm \begin{itemize}\item[1.] The Fourier transform is bounded on \(E\). \item[2.] Every \(\varphi \in L^{\infty} (\Gamma)\) is an \(E\)-multiplier. \item[3.]For every bounded interval \(J \subseteq \mathbb{R}\) the function \(\chi_J\) is an \(E\)-multiplier and there exists a constant \(C > 0\) such that \(\| R_{\chi_A} \|_{\mathcal{L}(E)} \leq C\) for every \(A \subseteq \mathbb{R}\) which is a finite union of bounded intervals. \item[4.] \(E = L^2(\mathbb{R})\) with equivalence of norms. \end{itemize}} Moreover assuming, in addition, that \(E\) is translation invariant, all of the above statements are equivalent to: {\parindent=5mm \begin{itemize}\item[5.] There exists a regular spectral measure \(P : \mathcal{B} (\mathbb{R}) \rightarrow \mathcal{L}_s(E)\) satisfying \[ \tau_t = \int_{\mathbb{R}} e^{-its}dP(s), t \in \mathbb{R}. \] \end{itemize}} Similar results for \(G = \mathbb{T}\) and \(G = \mathbb{Z}\) are also given. A proof is given for the case \(G = \mathbb{T}\) and the proof for \(G = \mathbb{Z}\) is left for the reader since it is similar to the case \(G = \mathbb{R}\). The authors remark that similar results for \(G = \mathbb{R}^n, \mathbb{Z}^n (n \in \mathbb{N})\) are easily obtained by adjusting the proofs for the groups \(G = \mathbb{R}, \mathbb{Z}\). This is a well written paper.
scientific article; zbMATH DE number 5534431
scientific article