ИСТИНА

Two linear operators play a crusial role in much of the modern analysis: the Fourier transform and the Hilbert transform (the latter relates the boundary values of real and imaginary parts of an analytic function in a half-plane). Both are unitary in L2. In a series of three talks I will review classical inequalities (Hilbert's, Hardy's, etc.) concerning L^p estimates, consider discrete versions (with integrals replaced by sums) and truncated versions, where certain limit laws (Szego-type theorems) are known. These talks can be considered as an introduction to the now well established techniques of harmonic analysis. The final goal of the first talk is to prove a seemingly new estimate for the Laplace transform observed along a curve in the complex plane. This is a joint work with Dr.Anatoli Merzon.