Convolution on homogeneous groups
Using methods from wavelet theory due to \textit{M. Frazier} and \textit{B. Jawerth} [Indiana Univ. Math. J. 34, 777--799 (1985; Zbl 0551.46018)] the authors give interesting characterizations of convolutions on homogeneous groups. While the presentation in the paper is (necessarily) rather technical, the results are natural! Hence the authors are right to expect more results for operators on homogeneous groups proved along these lines. For those who know some of the analysis on homogeneous groups, here a bit more. On \(G\) the homogeneous norm is taken and the basic space for working with the Fourier transform is the Schwartz space on \(G\) with functions having all moments vanishing. Consider functions whose derivatives decay as those of homogeneous functions of degree \(j\). Take the inverse Fourier transform of these functions. Convolution of two elements in these spaces each of ``homogeneity'' degree \(j_1\) and \(j_2\), respectively, yields a function of ``homogeneity degree'' of degree \(j_1+ j_2\).
scientific article; zbMATH DE number 5757969