Daesung Kim

Research Interest

Functional inequalities: the Hardy-Littlewood-Sobolev inequality and log Sobolev inequality.

Martingale transforms and Stochastic analysis, Lévy processes, the fractional Laplacian, non-local operators and related PDEs.

Papers

  1. Instability Results for the Logarithmic Sobolev Inequality and its Application to the Beckner–Hirschman Inequality, 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 an instability result for the Beckner--Hirschman inequality in terms of $L^{p}$, for $p>2$.

  2. Deficit Estimates for the Logarithmic Sobolev Inequality (with Emanuel Indrei), summitted, 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), summitted, 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. Stability results for Logarithmic Sobolev inequality, Probability seminar, Seoul Nat'l University, March 2018 [slide]
  2. Stability results for Logarithmic Sobolev inequality, Probability seminar, Purdue University, September 2017 [slide]
  3. Hardy-Stein identity and Square functions, Probability seminar, UIUC, March 2017 [slide]
  4. The expected exit time of the stable processes, Seminar on Stochastic processes, University of Maryland, March 2016 (short talk session)
  5. Martingale transforms and the Hardy-Littlewood-Sobolev inequality, Probability seminar, Purdue University, February 2016 [slide]