Harmonic Analysis

Analysis in general tends to revolve around the study of general classes of functions (often real-valued or complex-valued) and operators (which take one or more functions as input, and return some other function as output). Harmonic analysis 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)| dx ≤ A. A typical question in harmonic analysis might then be the following: if a function f : R → 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?