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Elliptic Riesz operators on the weighted special atom spaces. (English) Zbl 0879.42007

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 S. Bloom and 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 \(L(\phi)\).

MSC:

42A99 Harmonic analysis in one variable
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0646.46021
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