Atomic Decomposition of Function Spaces and Fractional Integral and Differential Operators

The method ot atomic decomposition of Besov and Lizorkin-Triebel function spaces is combined with basic ideas from the theory of singular integral operators and applied to the study of fractional integral and diierential operators (FIDO), of which the Riesz potential is considered in detail as a model example. The main new results are: characterization of the Riesz potential by an innnite-dimensional matrix with certain speciic properties; generalization of the \lifting" property of the Riesz potential in Besov spaces, for the case of diierent metric indices; generalization of Sobolev's theorem about boundedness of the Riesz potential between Lebesgue spaces, for the case of Lizorkin-Triebel spaces. Our results can be extended to other FIDO, e.g., the Bessel potential, the Riemann-Liouville, Caputo and Gr unwald-Letnikov FIDO, etc. Other possible applications of the atomic decomposition of function spaces via (bi)-orthonormal wavelets to the theory of FIDO are also discussed in brief.