Harmonic Analysis

Fourier’s transformation: Fourier’s basic L¹ transformation theorem, L² theorem and Plancherels theorem; the class of tempered distributions. Basic Concepts of Real Variables Theorem: maximal function; proximal behavior to general points in measurable sets; decomposition in Rⁿ open set cubes; an interpolation theorem for Lp; Interpolation of Operators: M. Riesz convexity theorem and interpolation of operators in L^p spaces; Marcinkiewicz interpolation theorem; L(p,q) spaces; interpolation of analytical groups of operators. Singular Integrals: Hilbert transformation; singular integral operators with odd nucleus; singular integral operators with even nucleus; singular integral operators that commute with dilation; analogous to vector values. Riesz transformation: Poisson integrals and Harmonic Spheres: Riesz transformation; Poisson integrals and identity approximations; Riesz transformation of higher orders and harmonic spheres. Multiple Fourier Series: elementary properties; Poisson summation formula; multiplying transformations. Littlewood-Paley theorem and Multipliers. Littlewood-Paley g function; The g function (lick); multipliers; application of partial sum operators; dyadic decomposition; Marcinkiewicz multiplier theorem. Hardy spaces: maximal characterization of H^p; atomic decomposition of H^p ; singular integrals. H^p and BMO: the space of limited medium oscillation functions; the sharp function; an elementary approach and a dyadic version; other properties of BMO; an interpolation theorem.

References:
DUOANDIKOETXEA, J. – Fourier Analysis. Graduate Studies in Mathematics, 29, AMS, Providence, RI 2001.
STEIN, E. – Harmonic Analysis, Princeton University Press, Princeton, New Jersey, 1993.
STEIN, E. – Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.
STEIN, E., WEISS, G. – Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.

* Standard program. The teacher has the autonomy to make any changes