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 |
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