HARMONIC ANALYSIS - UCLA Department of （谐波分析-加州大学洛杉矶分校的）.pdf

HARMONIC ANALYSIS TERENCE TAO Analysis in general tends to revolve around the study of general classes of func- tions (often real-valued or complex-valued) and operators (which take one or more functions as input, and return some other function as output). Harmonic analysis1 focuses in particular on the quantitative properties of such functions, and how these quantitative properties change when apply various (often quite explicit) operators. A good example of a quantitative property is for a function f (x) being uniformly bounded in magnitude by an explicit upper bound M , or perhaps being square integrable with some bound A, thus |f (x)|2 dx ≤ A. A typical question in har- monic analysis might then be the following: if a function f : Rn → R is square integrable, and its gradient ∇f exists and is also square integrable, does this imply that f is uniformly bounded? (The answer is yes when n = 1, no when n 2, and just barely no when n = 2; this is a special case of the Sobolev embedding theorem, which is of fundamental importance in the analysis of PDE.) If so, what are the precise bounds one can obtain? Real and complex functions, such as a real-valued function f (x) of one real variable x ∈ R, are of course very familiar in mathematics, starting back in high school. In many cases one deals primarily with special functions - polynomials, exponentials, trigonometric functions, and other very explicit and concrete functions. Such func- tions typically have a very rich algebraic and geometric structure, and there are many techniques from those fields of mathematics that can be used to give exact solutions to many questions concerning these functions. In contrast, analysis is more concerned with general classes of functions - functions which may have some