Stability for Functional and Geometric Inequalities and a Stochastic Representation of Fractional Integrals and Nonlocal Operators

Daesung Kim, Purdue University

Abstract

The dissertation consists of two research topics. The first research direction is to study stability of functional and geometric inequalities. Stability problem is to estimate the deficit of a functional or geometric inequality in terms of the distance from the class of optimizers or a functional that identifies the optimizers. In particular, we investigate the logarithmic Sobolev inequality, the Beckner–Hirschman inequality (the entropic uncertainty principle), and isoperimetric type inequalities for the expected lifetime of Brownian motion. In Chapter 3, we derive several types of stability estimates of the logarithmic Sobolev inequality in terms of the Wasserstein distance, Lp distances, and the Kolmogorov distance. We consider the spaces of probability measures satisfying different conditions on the second moments, the lower bounds of the density, and some integrability of the density. To obtain these results, we employ the optimal transport technique, Fourier analysis, and probability theoretic approach. In Chapter 4, we construct an example to understand the conditions on the space and the distance under which stability of the logarithmic Sobolev inequality does not hold. As an application, we show that stability of the Beckner–Hirschman inequality does not hold for the normalized Lp distance with some weighted measures in Chapter 5. In Chapter 6, we study quantitative improvements of the inequalities for the expected lifetime of Brownian motion, which state that the Lp-norms of the expected lifetime in a bounded domain for 1 ≤ p ≤ ∞, are maximized when the region is a ball with the same volume. Since the inequalities also hold for a general class of Levy processes, it is interesting to see if the quantitative improvement can be extended to general L´evy processes. We discuss the related open problems in that direction. The second topic of the thesis is a stochastic representation of fractional integrals and nonlocal operators. In Chapter 7, we extend the Hardy–Littlewood–Sobolev inequality to symmetric Markov semigroups. To this end, we construct a stochastic representation of the fractional integral using the background radiation process. The inequality follows from a new inequality for the fractional Littlewood–Paley square function. In Chapter 8, we prove the Hardy–Stein identity for non-symmetric pure jump L´evy processes and the Lp boundedness of a certain class of Fourier multiplier operators arising from non-symmetric pure jump L´evy processes. The proof is based on Itˆo’s formula for general jump processes and the symmetrization of L´evy processes.

Degree

Ph.D.

Subject Area

Mathematics

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