Lecture Notes 4 for 247a
نویسنده
چکیده
In this set of notes we begin the theory of singular integral operators operators which are almost integral operators, except that their kernel K(x, y) just barely fails to be integrable near the diagonal x = y. (This is in contrast to, say, fractional integral operators such as Tf(y) := ∫ Rd 1 |x−y|d−s f(x) dx, whose kernels go to infinity at the diagonal but remain locally integrable.) These operators arise naturally in complex analysis, Fourier analysis, and also in PDE (in particular, they include the zeroth order pseudodifferential operators as a special case, about which more will be said later). It is here for the first time that we must make essential use of cancellation: cruder tools such as Schur’s test which do not take advantage of the sign of the kernel will not be effective here.
منابع مشابه
Lecture Notes 5 for 247a
for various pairs of weight functions w1 : X → R + and w2 : Y → R . Roughly speaking, these weighted estimates are a variant of the usual L estimates which assert that if f avoids the regions where w1 is large, then Tf avoids the regions where w2 is large. If X = Y and one can establish the above type of estimate for many pairs w1, w2, with w1 “looking similar to” w2, then these estimates begin...
متن کاملLecture Notes 2 for 247a
The idea is to use the three lines lemma from last week’s notes. The problem is that the inequality is not complex analytic in θ as stated. However we can fix this as follows. Observe that if fθ is a simple function with L pθ (X) norm 1, then we can factorise fθ = F 1−θ 0 F θ 1 a where F0, F1 are non-negative simple functions with L 0(X) and L1(X) norms respectively equal to 1, and a is a simpl...
متن کاملLecture Notes 1 for 247a
The aim of this course is to introduce the basic tools and theory of real-variable harmonic analysis very roughly speaking, the art of estimating the size of an output function in terms of the size of an input function, when a known transformation (linear, multilinear, or nonlinear) is applied. In particular we shall focus on the classical Calderón-Zygmund-Stein theory, in which we study such o...
متن کاملFactors affecting students tendency of Univercity students to Lecture Notes
Introduction: Many studies detected factors contributing to the students’ tendency to lecture notes. This study aimed at evaluating the factors affecting students tendency to lecture notes in Hormozgan University of Medical Sciences. Methods: In this descriptive study, 179 students from medicine, nursing & midwifery, health, and Paramedicine schools were selected through stratified random...
متن کاملLecture Notes 3 for 247a
1. The Hardy-Littlewood maximal inequality Let us work in Euclidean space R d with Lebesgue measure; we write |E| instead of µ(E) for the Lebesgue measure of a set E. For any x ∈ R d and r > 0 let B(x, r) := {y ∈ R d : |x − y| < r} denote the open ball of radius r centred at x. Thus for instance |B(x, r)| ∼ d r d. For any c > 0, we use cB(x, r) = B(x, cr) to denote the dilate of B(x, r) around ...
متن کامل