Weighted inequalities for multilinear singular integrals, spring 2016

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News

       •  There is no class on April 25-26 due to Prof. Olevskii's course 

• Recall that there is no class on April, 19 (next Tuesday).
  
  
• Second part of the course starting on 11.04
• First part notes now available. 
• Classes start on Monday 14 March: everybody is welcome!

Teaching schedule

Weeks 11-18, Monday and Tuesday 14-16 in room C122.

Easter holiday 24.-30.3. 

Contents

Part I: Sharp weighted bounds for the multilinear maximal function and sparse operators

Part II: Sharp bounds for multilinear Calderón-Zygmund operators

Purpose

The purpose of this course is to give a short but detailed introduction to multilinear weighted inequalities and the usual techniques of proof in the area.

On one hand, we start describing the main object in this area, the multilinear maximal function, and how it controls the class of multilinear Calderón-Zygmund operators and allow us to define the right class of multiple weights. We also prove the generalization of Muckenhoupt's one and two-weight problems for the multilinear maximal function as well as some multiple (sharp) weighted inequalities for multilinear maximal functions and sparse operators.

On the other hand, we give a pointwise control of multilinear Calderón-Zygmund operators of Dini type by sparse operators. As a consequence of this result and using some mixed weighted bounds for a general class of sparse operators, we will be able to show similar bounds for several multilinear operators such us Calderón-Zygmund operators, their commutators with BMO functions, square functions and Fourier multipliers.

Course material

Part I: Sharp weighted bounds for the multilinear maximal function and sparse operators (W. Damián)

• Updated version of the notes (Last update: 02.05.2016) 
• Class slides (extra material not contained in the lecture notes, i.e., preliminaries, detailed proofs):
  • Class 1 (Introduction)
    
    • Information about the course and basic introduction to linear weighted inequalities (motivation)
      
  • Class 2 (Muckenhoupt weights)
    • Basic definitions not included in the lecture notes.
      
  • Class 3 (Characterization)
    • A more detailed proof of the characterization of the multiple $A_{\vec{P}}$ classes in terms of the Muckenhoupt $A_p$ classes of weights.
  • Class 4, 5 and 6 (see notes).
  • Class 7 (Auxiliary results for 2nd part).

Part II: Sharp bounds for multilinear Calderón-Zygmund operators (K. Li)

• Class  1:     Domination theorem of bilinear Calderón-Zygmund operators.
• Class  2-4:  $A_p$-$A_\infty$ bound for general bilinear sparse operators.
• Class  5:     Applications to commutators and multilinear square functions.

Registration

No registration needed, just come to the first lecture.

Exercises

Exercises will be collected and then graded by Wendolín Damián and Kangwei Li.

Passing the course (if you want the 5 credits) is based on doing the exercises.

Assignments

• Set 1: Deadline 11.04.16
  Set1.pdf
• Set 2: Deadline 02.05.16

Course feedback

Course feedback can be given at any point during the course. Click here.