Molecular decomposition and Fourier multipliers for holomorphic Besov and Triebel-Lizorkin spaces
A generalization of the transform approach which was useful in the decomposition for holomorphic Besov and Triebel-Lizorkin spaces is the so-called molecular decomposition. Molecules are special families of functions that allow for corresponding stable decompositions to form bounded operators between specified classes of functions. In this paper the notion of molecules is introduced in the setting of holomorphic Besov and Triebel-Lizorkin spaces on the disk. After a necessary background which contains the notion of an $\alpha$-Weyl derivative $J^\alpha$, holomorphic Besov and Triebel-Lizorkin spaces, distributions, holomorphic Hardy and Hardy-Sobolev spaces, wavelet bases, and discrete decompositions of the holomorphic Besov and Triebel-Lizorkin spaces, an extension is given to the families of smooth molecules, using almost diagonal operators. So, a smooth molecular synthesis (Theorem 3.8) and smooth molecular analysis (Theorem 3.12) are obtained. Using the obtained molecular decompositions, sufficient conditions for the boundedness of Fourier multipliers on holomorphic Besov and Triebel-Lizorkin spaces are proved (Theorem 4.2). As applications of the obtained results, a characterization of the holomorphic Besov and Triebel-Lizorkin spaces is given by an equivalent norm characterization.
scientific article; zbMATH DE number 7031254
scientific article