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The ρ-variation of the heat semigroup in the Hermitian setting: behaviour in L∞

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  14 June 2011

J. J. Betancor ,
R. Crescimbeni  and
J. L. Torrea
J. J. Betancor
Affiliation:
Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n 38271 La Laguna (Sta Cruz de Tenerife), Spain ([email protected])
R. Crescimbeni
Affiliation:
Departamento de Matemática, Universidad Nacional de Comahue, Buenos Aires 1400, 8300 Neuquén, Argentina ([email protected])
J. L. Torrea
Affiliation:
Departamento de Matemáticas and ICMAT-CSIC-UAM-UCM-UC3M, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain ([email protected])
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Abstract

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Let , ρ > 2, be the ρ-variation of the heat semigroup associated to the harmonic oscillator H = ½(−Δ + |x|2). We show that if f ∈ L∞ (ℝ), the (f)(x) < ∞, a.e. x ∈ ℝ. However, we find a function G ∈ L∞ (ℝ), such that (G)(x) ∉ L∞ (ℝ). We also analyse the local behaviour in L∞ of the operator . We find that its growth is smaller than that of a standard singular integral operator. As a by-product of our work we obtain an L∞ (ℝ) function F, such that the square function

a.e. x ∈ ℝ, where is the classical Poisson kernal in ℝ.

Keywords

Hermite operatorρ-variationheat semigroupPoisson semigroup

MSC classification

Secondary: 42C05: Orthogonal functions and polynomials, general theory 42C15: General harmonic expansions, frames
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 569 - 585
Copyright
Copyright © Edinburgh Mathematical Society 2011

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