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 The geometric case of the Erdős similarity conjecture
For every fixed q\in(0,1), constructs compact subsets of [0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of \{q^n:n\ge1\}, with dilations of either sign.…
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Paper An L^{3} bound for the trilinear Hilbert transform
Proves that the principal-value trilinear Hilbert transform with shifts x-t, x-2t, x-3t is bounded from L^3(\mathbb R)^3 to L^1(\mathbb R). This resolves the L3 exponent case of the standard…
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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 Hilbert transforms along Lipschitz directions
Proves a uniform strong L2 bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is…
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Paper The three-dimensional Bochner–Riesz conjecture
Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on L^p(\mathbb R^3) for every 1\le p\le\infty whenever $`\delta\gt…
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Paper Local smoothing in three dimensions
Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2\lt p\lt \infty, with Sobolev regularity above…
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Paper The exact Sobolev endpoint for Schrödinger convergence
Proves almost-everywhere convergence e^{it\Delta}f\to f as t\downarrow0 for every f\in H^{n/(2(n+1))}(\mathbb R^n) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's…
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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 Kakeya in three and four dimensions
Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps L^3(\mathbb R^3) to L^3(S^2) with norm…
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Paper Fourier restriction for positively curved surfaces
Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface \Sigma\subset\mathbb R^3, including surfaces with boundary, the…
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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 Koebe’s circle-domain conjecture
Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that…
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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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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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