Boundedness of a class of vector-valued multiplier operators in \(L_ p(T^{\infty})\). (English. Russian original) Zbl 0608.47032

Bendikov, A. D.; Pavlov, I. V.

Sib. Math. J. 27, 1-7 (1986); translation from Sib. Mat. Zh. 27, No. 1, 3-10 (1986).

Starting from a question of E. M. Stein [Bull. Am. Math. Soc. New Ser. 9, 71-73 (1983; Zbl 0515.42018)], the authors study the boundedness of a class of vector multiplier operators on the infinite dimensional torus \(T^{\infty}\). These results, which have been previously announced [see ”Tez. dokl. Vil’nyus” 3, 26-27 (1981)], are given here in full details.

Reviewer: F.-H.Vasilescu

MSC:

47B38	Linear operators on function spaces (general)
42B15	Multipliers for harmonic analysis in several variables
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Riesz operators; boundedness of a class of vector multiplier operators on the infinite dimensional torus

Citations:

Zbl 0515.42018

References:

[1]	E. M. Stein, ?Some results in harmonic analysis in Rn, for n??,? Bull. Am. Math. Soc.,9, No. 1, 71-73 (1983). · Zbl 0515.42018 · doi:10.1090/S0273-0979-1983-15157-1
[2]	A. D. Bendikov and I. V. Pavlov, ?Spaces Hp and BMO on the infinite dimensional torus,? in: The Third International Vilnius Conference on Theory of Probability and Mathematical Statistics. Abstracts of Reports [in Russian], Vol. III, Vilnius (1981), pp. 26-27.
[3]	I. V. Pavlov, ?Probabilistic-analytic approach to the investigation of hardy spaces and BMO on the infinite-dimensional torus,? Author’s Abstract of Candidate’s Thesis, Physicomathematical Sciences, Vilnius (1982).
[4]	P. A. Meyer, ?Demonstration probabiliste de certaines inégalités de Littlewood-Paley,? in: P. A. Meyer (ed.), Seminaire de Probabilites, X, Lect. Notes in Math., Vol. 511, Springer-Verlag, Berlin-New York (1976), pp. 125-183. · Zbl 0332.60032
[5]	A. D. Bendikov, ?On harmonic functions for a category of Markov processes and projective limits in it,? Usp. Mat. Nauk,31, No. 2(188), 209-210 (1976). · Zbl 0331.60049
[6]	E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970). · Zbl 0207.13501
[7]	A. Zygmund, Trigonometric Series, Vol. 2, Cambridge Univ. Press (1959). · Zbl 0085.05601
[8]	E. M. Stein and G. Weiss, Introduction to Harmonic Analysis on Euclidean Spaces, Princeton Univ. Press (1971). · Zbl 0232.42007