A multiplier related to symmetric stable processes

Year 2017, Volume: 46 Issue: 2, 217 - 228, 01.04.2017

Deniz Karlı

Abstract

In two recent papers [5] and [6], we generalized some classical results of Harmonic Analysis using probabilistic approach by means of a $d$-
dimensional rotationally symmetric stable process. These results allow one to discuss some boundedness conditions with weaker hypotheses.
In this paper, we study a multiplier theorem using these more general results. We consider a product process consisting of a $d$-dimensional
symmetric stable process and a 1-dimensional Brownian motion, and use properties of jump processes to obtain bounds on jump terms and
the $L^p(\mathbb{R}^d)$-norm of a new operator.

Keywords

symmetric stable process , Fourier multiplier , harmonic extension , bounded operator

References

• Applebaum, D. Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics), 2nd ed., Cambridge University Press, 2009.
• Bass, R. F. Probabilistic Techniques in Analysis. Springer, New York,1995.
• Bass, R. F. Stochastic Processes (Cambridge Series in Statistical and Probabilistic Mathematics), 1 ed., Cambridge University Press, 2011.
• Bouleau, N. and Lamberton, D. Théorie de Littlewood-Paley-Stein et processus stables, Sémin. Probab. (Strasbourg) 20 (1986), 162185.
• Karlı, D. Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion, Potential Analysis 38 (2011), no. 1, 95117. (DOI:10.1007/s11118-011- 9265-6) arXiv:1010.4904v2.
• Karlı, D. An Extension of a Boundedness Result for Singular Integral Operators, Colloquium Mathematicum 145 (2016), no. 1, 1533. (DOI: 10.4064/cm6722-1-2016) arXiv:1501.05164.
• Meyer, P.A. Démonstration Probabiliste de Certaines Inégalites de Littlewood-Paley, Sémin. Probab. (Strasbourg) 10 (1976), 164174.
• Meyer, P.A. Démonstration probabiliste de certaines inégalites de Littlewood-Paley. Exposé IV : semi-groupes de convolution symétriques. Séminaire de probabilités (Strasbourg) 10, (1976), 175-183.
• Meyer, P.A. Retour sur la theorie de Littlewood-Paley. Séminaire de probabilités (Strasbourg) 15, (1981), 151-166.
• Sato, K.-I. Lévy Processes and Innitely Divisible Distributions (Cambridge Studies in Advanced Mathematics), Cambridge University Press, 1999.