Hyunwoo Kwon (Will)

Learning seminars on Harmonic Analysis related to PDEs

Goal

The goal of this learning seminar is to study harmonic analysis tools that are closely related to the theory of PDEs.

Time/Venue

We will meet Tuesday from 4 PM – 5:30 PM at 205 Kassar House.

Schedule

Date Topic Refs Speakers
Sep/20 Organizational meeting    
Sep/27 Calderon-Zygmund decomposition and $L_p$-boundedness of singular integrals 1,7,11 Will
Oct/04 Application of Calderon-Zygmund decomposition, John-Nirenberg theorem 1,12 Will
Oct/11 Introduction to sparse operators(guest lecture)   Jose
Oct/18 $A_p$-weights (Reverse Holder inequality, Singular integrals) 5,12 Tainara
Oct/25 $A_p$-weights (Rubio de Francia extrapolation) 2,3 Nathan
Nov/1 Littlewood-Paley square function theorem 5,9 Hanye
Nov/8 No seminar (Election day)    
Nov/15 Commutator estimates (Coifman-Rosenberg-Weiss) 6,8 Nathan
Nov/22 Paraproducts and Coifman-Meyer theorem 6,10 Will
Nov/29 Strichartz estimates for Schrodinger equations 12 Haram
Dec/6 Strichartz estimates for wave equations 12 Bruno

Notes

  • Week 1 file
  • Week 2 file
  • Paraproduct and Coifman-Meyer theorem (upon request)

References

  1. L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), no.1, 1–21.
  2. H. Dong and D. Kim, On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights, Trans. Amer. Math. Soc. 370 (2018), no. 7, 5081–5130.
  3. J. Duoandikoetxea, Fourier analysis, GSM, vol 29, AMS, Providence, RI, 2001.
  4. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, Vol. 105, Princeton University Press, Princeton, NJ, 1983.
  5. L. Grafakos, Classical Fourier analysis, 3rd Edition, GTM, vol. 249, Springer, New York, 2014.
  6. L. Grafakos, Modern Fourier analysis, 3rd Edition, GTM, vol. 249, Springer, New York, 2014.
  7. N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, GSM, vol. 96, AMS, Providence, RI, 2008.
  8. D. Li, On Kato-Ponce and fractional Leibniz, Rov. Math. Iberoam. 35 (2019), 23–100.
  9. C. Muscalu and W. Schlag, Classical and multilinear harmonic analysis, Vol. I, Cambridge University Press, Cambridge, 2013.
  10. C. Muscalu and W. Schlag, Classical and multilinear harmonic analysis, Vol. II, Cambridge University Press, Cambridge, 2013.
  11. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30., Princeton University Press, Princeton, N.J., 1970.
  12. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, andPrinceton Mathematical Series, vol.~43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.