Lectures on Harmonic Analysis

By T. Wolff. 2003

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Arithmetic Progressions in Sets of Fractional Dimension (2007, background): “[10], [22], [33]) asserts that if E ⊂ R is a compact set then dimH(E) = sup{α ≥ 0 : ∃ a probabilistic measure μ supported on E such that (A) holds for some C1 = C1(α)}.”
THE CAUCHY PROBLEM FOR SEMILINEAR PARABOLIC EQUATIONS IN BESOV SPACES (2004, methodology): “By the Leibniz rule for the fractional-order derivatives (using the Littlewood-Paley decomposition theory; see [2, 31] for details), one easily obtains that u ∈ C([ ̄, T ∗);Wmδ,p) for any m ∈ N.”
A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral (2013, methodology): “, Chapter 8 in [16], ) means that for any > 0 there exists C > 0 such that if Bδ is a ball of radius δ centered at the origin, then (3.”
Fourier transforms of Gibbs measures for the Gauss map (2013, background): “The main idea of bounding the L2 norm is to study the distribution of the differences Ta(x) − Tb(x) for different n-regular words a and b and exploit stationary phase type inequalities (see Section 6 of [34]) to bound oscillative Lebesgue integrals in the terms of derivatives of the oscillations tha”
On the multilinear restriction and Kakeya conjectures (2005, background): “See [37] for a detailed account of these facts.”