25.1.6 Fourier analysis and complex-line incidence342F8D0FF9AE

For Fourier inversion on the Schwartz space and Plancherel's theorem (lemma 2.9.2, theorem 2.9.4) compare [Exercises 38–42, formula (11), Theorem 43 and Exercise 44, Tao09Fourier]. Our Fourier transform uses the character $\mathrm e^{-2\pi \mathrm i x\cdot \xi }$. Gaussian integration gives the exact constant in the Fourier expression of the Riesz energy, theorem 2.9.6; the kernel there is $|x-y|^{-s}$, without the normalization of a fundamental solution.

For the dyadic decomposition compare [Proposition 4.3 and proof, pp. 23–24, Tao06Fourier3]; for the weak type $(1,1)$ estimate and its consequences by interpolation and duality compare [Lemma 2.7, Corollaries 2.9–2.10 and proofs, pp. 7–9, Tao06Fourier4]. We treat only the Hessian multipliers $-\xi _ a\xi _ b/|\xi |^2$ (lemma 2.9.10), whose kernels are obtained by differentiating the heat kernel. Their weak type $(1,1)$ bound and their $L^ p$ bounds are lemma 2.9.11, theorem 2.9.12; the exponents $p>2$ are obtained by duality. The interpolation inequality lemma 2.9.15 and Morrey's inequality lemma 2.9.16 are used for metric Laplacians in section 16.10.

The star-shaped annular sets, the projections onto complex hyperplanes and the incidence argument follow [Lemmas 1.3–1.5, pp. 2–4, PSHD]; this manuscript carries no author's name and no publication data. The Borel selection required by the argument is lemma 2.9.21: the sets of high values on circles are closed and, by the submean inequality, have positive length, so a median with respect to arc length gives a Borel choice, and no general measurable selection theorem for compact-valued maps is needed. The unitarily invariant measures are constructed from the Gaussian measure on matrices, and volumes in projective space are controlled by an explicit estimate for projective balls instead of volume charts. In corollary 2.9.28 the function is assumed to be Borel; in the proof of theorem 5.8.4 this measurability is established before the corollary is applied.