Lower and upper bounds for the norm of multipliers of multiple trigonometric Fourier series in Lebesgue spaces. (English. Russian original) Zbl 0965.42008
Funct. Anal. Appl. 34, No. 2, 151-153 (2000); translation from Funkts. Anal. Prilozh. 34, No. 2, 86-88 (2000).
Three theorems are announced without proofs. The first of them is a refinement of Hörmander’s multiplier theorem [L. Hörmander, Acta Math. 104, 93-140 (1960; Zbl 0093.11402)], which gives an upper bound for the norm of multipliers. The second theorem provides a lower bound in terms of harmonic intervals in \(\mathbb{Z}^n\). The third theorem asserts both lower (in terms of the Shapiro sequence) and upper bounds for the norm of multipliers.
Reviewer: Ferenc Móricz (Szeged)
MSC:
42B15 | Multipliers for harmonic analysis in several variables |
42B05 | Fourier series and coefficients in several variables |
Citations:
Zbl 0093.11402References:
[1] | E. D. Nursultanov, Mat. Sb.,189, No. 3, 83–102 (1998). |
[2] | E. D. Nursultanov, East J. App., No. 3, 243–275 (1998). |
[3] | L. Hörmander, Acta Math.,104, 93–140 (1960). · Zbl 0093.11402 · doi:10.1007/BF02547187 |
[4] | J. Brillhart and L. Carlitz, Proc. Amer. Math. Soc.,25, 114–118 (1970). · doi:10.1090/S0002-9939-1970-0260955-6 |
[5] | E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970. · Zbl 0207.13501 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.