%0 Thesis %A Kim, Daesung %D 2019 %T Stability for functional and geometric inequalities and a stochastic representation of fractional integrals and nonlocal operators %U https://hammer.purdue.edu/articles/thesis/Stability_for_functional_and_geometric_inequalities_and_a_stochastic_representation_of_fractional_integrals_and_nonlocal_operators/8282051 %R 10.25394/PGS.8282051.v1 %2 https://hammer.purdue.edu/ndownloader/files/15516749 %K stability %K functional and geometric inequalities %K stochastic analysis %K Probability %K Stochastic Analysis and Modelling %K Calculus of Variations, Systems Theory and Control Theory %X
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.

The second topic of the thesis is a stochastic representation of fractional integrals and nonlocal operators. 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. We also prove the Hardy-Stein identity for non-symmetric pure jump Levy processes and the L^p boundedness of a certain class of Fourier multiplier operators arising from non-symmetric pure jump Levy processes. The proof is based on Ito's formula for general jump processes and the symmetrization of Levy processes.
%I Purdue University Graduate School