Harmonic Analysis and Its Applications

In these lectures, we concentrate on the motivations, development and applications of the Calderon-Zygmund operator theory. Lecture 1. The differential operators with constant coefficients and the first generation of Calderon-Zygmund operators Consider the following differential operator with constant coefficients: Lu(x) = α a α ∂ α u ∂x α. By taking the Fourier transform, (Lu)(ξ) = α a α (−2πiξ) α ˆ u(ξ). (1.2) This suggests one to consider the following more general Fourier multiplier: Definition 1.3: An operator T is said to be the Fourier multiplier if (T f)(ξ) = m(ξ) ˆ f (ξ). (1.4) (1.2) shows that any classical differential operator is a Fourier multiplier. Example 1: Suppose f ∈ L 2 (R) and F is an analytic extension of f on R 2 + given by F (x + iy) = − i π 1 x + iy − t f (t)dt = − i π (x − t) − iy (x − t) 2 + y 2 f (t)dt = 1 π y (x − t) 2 + y 2 f (t)dt − i π (x − t) (x − t) 2 + y 2 f (t)dt. 1 π y (x−t) 2 +y 2 f (t)dt → f (x) for a. e. x, and, in general, the second term above has no limit. However, one can show p.v 1 x−t f (t)dt exists for a. e. x. Thus lim y→0 F (x + iy) = f (x) + iH(f)(x) (1.6) where H is called the Hilbert transform defined by H(f)(x) = − 1 π f (t) x − t dt.