# Daesung Kim

### Research Interest

Stability of functional and geometric inequalities

Martingale transforms and Stochastic analysis

Optimal transport

### Papers

1. 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 $p$.

2. Deficit Estimates for the Logarithmic Sobolev Inequality (with Emanuel Indrei), 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.

3. On square functions and Fourier multipliers for nonlocal operators (with Rodrigo Bañuelos), 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.

4. Martingale transforms and the Hardy–Littlewood–Sobolev inequality for semigroups, Potential Anal. (2016), pp.1–13 [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.

### Talks

1. Recent trends in continuous and discrete probability, Georgia Tech, June 2018 (short talk session) [slide]
2. Probability seminar, Seoul Nat'l University, March 2018 [slide]
3. Probability seminar, Purdue University, September 2017 [slide]
4. Probability seminar, UIUC, March 2017 [slide]
5. Seminar on Stochastic processes, University of Maryland, March 2016 (short talk session)
6. Probability seminar, Purdue University, February 2016 [slide]