Analysis
16 results · 26 papers · 10 with a Lean formalization
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 circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.
2 papersLeanProves 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 \(B_{\mathcal S}(t)=|t|-1\) for t ≤ −2. A strict bound \(B_b(-1)\lt 1/4\) for bounded univalent functions disproves Kraetzer's prediction at that parameter.
2 papersLeanResolves 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 measure.
1 paperLeanResolves 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 \(O_\varepsilon(\delta^{-\varepsilon})\) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
2 papersProves 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 at the \(L\log L\) scale.
1 paperLeanConstructs 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 polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.
3 papersLeanProves 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 extension operator is bounded from \(L^p(\Sigma)\) to \(L^p(\mathbb R^3)\) for every p > 3.
2 papersResolves 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 \(\delta\gt \max\{3|1/p-1/2|-1/2,0\}\).
1 paperResolves 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 \(\max\{0,1-3/p\}\). In particular, the critical L3 estimate holds with every positive Sobolev loss.
1 paperProves 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 Schrödinger convergence problem, including the planar endpoint H1/3.
2 papersResolves 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 Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.
1 paperLeanProves 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 gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric \(L^3\times L^3\times L^3\) point.
3 papersLeanProves 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 uniform over inner truncations and yields an L2-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.
1 paperLeanFor 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. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.
2 papersLeanResolves 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 \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) with an absolute constant. It belongs locally to \(W^{1,1}\) and has a globally integrable weak gradient.
1 paperLeanProves 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 conjecture for slopes 1, 2, 3.
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