Fourier Analysis

elective monographs

I would like to present a part of the Fourier analysis appearing currently in the applied mathematics.

The main point will be the Paley-Littlewood decomposition, defining structure of functions. This point of view in a natural way introduces us Besov B^s_{p,q} and Triebel F^s_{p,q} function spaces -- being a fractional generalization of the classical Sobolev spaces. To understand the properties of this approach we recall the theory of Fourier multipliers -- the Marcinkiewicz theorem, which extends the elementary features of the L_2- on L_p-spaces. This part of the theory can be relatively easily extended on the nonlinear problems. We introduce the paraproducts to control the multiplication beyond the classical point of view. We plan to consider applications to concrete problems from PDEs, too.

I would like to present a part of the Fourier analysis appearing currently in the applied mathematics. The main point will be the Paley-Littlewood decomposition, defining structure of functions. This point of view in a natural way introduces us Besov B^s_{p,q} and Triebel F^s_{p,q} function spaces -- being a fractional generalization of the classical Sobolev spaces. To understand the properties of this approach we recall the theory of Fourier multipliers

-- the Marcinkiewicz theorem, which extends the elementary features of the L_2- on L_p-spaces. This part of the theory can be relatively easily extended on the nonlinear problems. We introduce the paraproducts to control the multiplication beyond the classical point of view. We plan to consider applications to concrete problems from PDEs, too.

The lecture schedule:

1. Elementary properties of functions;

2. Besov and Triebel spaces;

3. Maximal function and singular operators;

4. A_p;

5. L_p=F^0_{p,2};

6. The Marcinkiewicz theorem;

7. BMO and Hardy spaces.

8. Paraproducts and imbeddings theorems.

1. J. Duoandikoetxea, Fourier analysis. AMS, Providence, RI, 2001.

2. M.E. Taylor, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. AMS, Providence, RI, 2000.