# Daesung Kim

### Research Interest

Probability and stochastic analysis

Singular integrals and Fourier multipliers, and extremal problems for nonlocal operators

Functional and geometric inequalities

Optimal transport theory

### Papers

1. Martingale transforms and the Hardy–Littlewood–Sobolev inequality for semigroups, Potential Anal. 45 (2016), no. 4, 795–807. [article] [arXiv] [abstract]

We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev(HLS) inequality based on this representation. The proof rests on a new inequality for a fractional Littlewood–Paley $g$-function.

2. (with Rodrigo Bañuelos) On square functions and Fourier multipliers for nonlocal operators , submitted, arXiv preprint (2017) [arXiv] [abstract]

Using Ito's formula for processes with jumps, we give a simple direct proof of the Hardy-Stein identity proved in [Banuelos et at 2017]. We extend the proof given in that paper to non-symmetric Levy-Fourier multipliers.

3. (with Emanuel Indrei) Deficit Estimates for the Logarithmic Sobolev Inequality, submitted, arXiv preprint (2018), [arXiv] [abstract]

We identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. The techniques are based on optimal transport theory and Fourier analysis. We also discuss a probabilistic approach.

4. Instability Results for the Logarithmic Sobolev Inequality and its Application to the Beckner–Hirschman Inequality, submitted, arXiv preprint (2018) [arXiv] [abstract]

We provide an example to show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}$ distance for $p>1$. The results imply that the stability bounds for the logarithmic Sobolev inequality with respect to $W_{1}$, $W_{2}$, and $L^{1}$ in the space of probability measures with bounded second moments are best possible. As an application of the example, we prove instability results for the Beckner--Hirschman inequality in terms of $L^{p}$ distances with specific measures and range of $p$.

### Talks

1. Functional inequalities in Probability, University of Connecticut, November 2018 (short talk session)
2. Recent trends in continuous and discrete probability, Georgia Institute of Technology, June 2018 (short talk session) [slide]
3. Probability seminar, Seoul National University, March 2018 [slide]
4. Probability seminar, Purdue University, September 2017 [slide]
5. Probability seminar, University of Illinois at Urbana–Champaign, March 2017 [slide]
6. Seminar on Stochastic processes, University of Maryland, March 2016 (short talk session)
7. Probability seminar, Purdue University, February 2016 [slide]