This course will cover some of the basic tools
of analysis that are extremely useful in many areas of
mathematics, including PDE's, stochastic analysis,
harmonic analysis and complex analysis. Specific topics
covered in the course include: Geometric lemmas (Vitali,
Wiener, etc.) and geometric decomposition theorems
(Whitney, etc.) and their applications to
differentiation theory and to the Hardy-Littlewood
maximal function; convolutions; approximations to the
identity and their applications to boundary value
problems in R^{d} with L^{p}-data; the
Fourier transform and its basic properties on L^{1}
and L^{2} (including Plancherel's theorem);
interpolation theorems for linear operators
(Marcinkiewicz, Riesz-Thorin); the basic Calderón-Zygmund
singular integral theory and some of its applications;
the Hardy-Littlewood-Sobolev inequalities for fractional
integration and powers of the Laplacian and other
elliptic operators; the inequalities of Nash and Sobolev
viewed from the point of the heat semigroup. The
basics properties of Lusin and Littlewood-Paley square
functions and applications to Hormander's multiplier
theorem.
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