Elliptic Riesz operators on the weighted special atom spaces (Q1907693)

Although the author states the concept of elliptic Riesz operators for multiple Fourier series he considers only the one-dimensional case. But in this case elliptic Riesz operator is just the original generalized Riesz operator. The concept of weighted special atom space \(B(\omega)\) with respect to a weight \(\omega\) can be found in the paper of \textit{S. Bloom} and \textit{G. Soares de Souza} [Ill. J. Math. 33, No. 2, 181-209 (1989; Zbl 0646.46021)]. A Banach space \(L(\phi)\) concerning the decreasing rearrangement of functions is defined with respect to the function \(\phi(t)={\omega(t)\over t}\), which becomes a Lorentz space under certain conditions. The author proves that, in the one-dimensional case, the maximal elliptic Riesz operators (which are just generalized Riesz operators for single Fourier series) of positive order in both non-conjugate and conjugate cases are bounded from \(B(\omega)\) to \(L(\phi)\).