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.)

Zbl 0646.46021
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