Classical Analysis and ODEs
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Paper Annular variation and dyadic absolute bounds for the triangular Hilbert transform
Proves maximal and annular r-variation bounds, for every r > 2, from complex L^3(\mathbb R^2)\times L^3(\mathbb R^2) to L^{3/2}(\mathbb R^2). The maximal estimate controls both hard truncation endpoints and…
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Paper Real ultraflat Littlewood polynomials and unbounded binary merit factors
Constructs polynomials with N consecutive coefficients in \{-1,1\} whose modulus is (1+o(1))\sqrt N uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood…
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Paper Endpoint Sobolev regularity of centered disk averages
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real f\in W^{1,1}(\mathbb R^2), the centered disk maximal function satisfies $`\|\nabla Mf\|1\le C\|\nabla…
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Paper Riesz transforms and rectifiability in higher codimension
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2\le n\le d-2, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional…
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Paper Brennan's conjecture and the integral-means spectrum
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |\phi'|^s is area-integrable for 4/3\lt s\lt 4. The sharp universal integral-means identity is…
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Paper The L\log L Fourier-convergence conjecture
Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in L\log L(\mathbb T) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture…
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Paper The Falconer distance conjecture
Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E\subset\mathbb R^d with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue…
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