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Harmonic Analysis: Course Outline (MAT 622)

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms.

Course Outline

Review: Norm spaces, measure spaces, measurable and continuous functions, analytic functions.  Harmonic Functions and Dirchlet Problem. Poisson Kernels.

Interpolation of Operators:  Interpolation of Linear operators, Real Method and Complex Method. Riesz--Thorin interpolation Theorem, Marcinkiewicz Interpolation Theorem, Stein Interpolation Theorem. Applications of Interpolation results.

Maximal Functions: Locally Integrable function, Vitalli Covering Lemma, Hardy-Littlewood Maximal Operators, Muckhenhoupt weights and its properties.

Integral Operators: Minkowski’s Integral inequality, integral operators with homogeneous kernels, Hilbert’s inequality, Hardy Operator and Hardy’s Inequality. Riesz Representation Theorem, Adjoint integral operators. Compact Integral Operators and Arzela Ascoli’s Theorem. L2 spaces and Radon-Nikodym Theorem.

Convolutions and Potentials:  Convolutions, support of convolutions, Riesz Potential operator, Potentials in Lebesgue spaces, Hedberg’s Inequality and consequences

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