Arbeitsgruppe Analysis und Partielle Differentialgleichungen

V5B8: Selected Topics in Analysis - Weighted inequalities

Summer Semester 2017

No lecture on July 28.
Dr. Pavel Zorin-Kranich
Instructor

Lectures

  • Friday 10-12, SR 0.011

Topics

  1. Hardy-Littlewood maximal function and Muckenhoupt weights
  2. Extrapolation and connection with vector-valued estimates
  3. Fefferman-Stein maximal inequality
  4. Sparse operators and sharp weighted estimates for Calderón-Zygmund operators
  5. Classical theory of A_p weights: reverse Hölder inequality, factorization, open property

Prerequisites

Basic concepts from measure theory and functional analysis (Lebesgue integration theory, Lp spaces). Previous knowledge of Calderón-Zygmund theory might be helpful for motivation but is not assumed.

Exam

Oral exams take place on August 4 and August 10.

Lecture notes

The notes posted here are mainly for my own preparation (no attempt at completeness or correctness is claimed), but they should give an idea of what is being covered.
  1. Fefferman-Stein inequality
  2. The Ap condition
  3. Extrapolation, sparse operators
  4. Calderón-Zygmund operators
  5. Calderón-Zygmund operators, continued
  6. May 26: sparse domination for a nontangentional maximal operator
  7. June 2: A and the reverse Hölder inequality
  8. June 23: mixed A estimates
  9. June 30: intrinsic square function, only Section 2 without Lemma 2.2 and with v=1, beta=1.
  10. July 7: Embedding A into Ap; Orlicz spaces
  11. July 14: Fefferman-Stein type inequality for sparse operators, following this article, another proof of sparse domination following this article
  12. July 21: Counterexample to Muckenhoupt-Wheeden conjecture

Literature

  • J. Duoandikoetxea. “Extrapolation of weights revisited: new proofs and sharp bounds”. In: J. Funct. Anal. 260.6 (2011), pp. 1886–1901.
  • C. Fefferman and E. M. Stein. Some Maximal Inequalities, American Journal of Mathematics Vol. 93, No. 1 (Jan., 1971), pp. 107-115
  • J. García-Cuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics, 1985.
  • David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez. Weights, Extrapolation and the Theory of Rubio de Francia, 2011.
  • David Cruz-Uribe, Extrapolation and Factorization, 2017
  • Andrei K. Lerner and Fedor Nazarov. Intuitive dyadic calculus: the basics, 2015
  • Moen. Sharp weighted bounds without testing or extrapolation. Arch. Math. (Basel) 99 (2012), no. 5, 457–466.
  • Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Transactions AMS, 1972