Shorter proof yields dimension-free Lp bounds for maximal Riesz transforms
New argument uses Stein's spherical maximal inequality to eliminate dimension dependence
· “Shorter proof of dimension-free L^p estimates for maximal Riesz transforms”
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Two criteria spot curves that inscribe every rectangle
New conditions using finite variation and conformal tails show which Jordan curves admit continuous Legendrian lifts.
This paper investigates which functions maximize mixed-norm Strichartz inequalities for…
In low dimensions 1-5, Gaussian functions are stable local maximizers of mixed-norm Strichartz inequalities only when the spatial exponent…
· “Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schr\"odinger Equation”
Small delays cannot destroy a system's exponential dichotomy
Two weighted solution-space pairs fix the exact stability exponent, proving hyperbolicity survives delay perturbations.
Two-parameter variation of torus averages hits sharp bounds
Variation along both radii of torus averages in 3D gets Lp–Lq bounds that one-parameter theory cannot see.
· “Two-parameter variational estimates for averages over tori”
Cyclic-group Bohnenblust–Hille constants grow polynomially
For each fixed group size, a dimension-free power bound replaces exponential growth.
· “Polynomial Bohnenblust--Hille bounds for product of cyclic groups”
One dimension-free bound settles the old L2 lattice-ball question
One p-independent bound controls the ball maximal average in every dimension and radius, for every p>1.
· “Full-radius dimension-free maximal inequalities for discrete Euclidean balls”
Analytic proof settles Mockenhaupt's three-term majorant conjecture
A single argument covers every integer k≥4, resolving all previously open cases and completing the proof for all k.
· “Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture”
Dimension-free ball maximal bound proved for all exponents p>1
A 30-year-old question of Stein on discrete Euclidean averages gets an affirmative answer for every dimension and every p>1.
· “Dimension-free estimates for the full discrete Euclidean ball maximal function”
Loomis-Whitney inequalities now cover every Reiter-Heisenberg group
An explicit bound in every dimension, plus Sobolev and isoperimetric consequences.
Sharp weak constant for Riesz potentials with 0 < s < 2
Isoperimetric inequality for Riesz capacity yields an explicit formula in terms of gamma functions and ball volume.
· “Sharp constants for weak estimates of the Riesz Potentials”
Bessel positivity boundary is strictly convex
The curve separating sign-definite Bessel integrals from sign-changing ones has no inflection on its whole closed range.
All coordinates of Chenevier's critical vectors are positive
The proof hinges on strictly negative Verblunsky coefficients and the Delsarte–Genin transformation, yielding positivity in every degree.
· “Positivity and Asymptotics for Chenevier's Orthogonal Polynomials”
Hermite polynomials are the only extremizers of the finite free Stam inequality
Proof uses a matrix defect identity and turns classification into rigidity of root trajectories under convolution.
Jump sets are always rectifiable in sub-Riemannian spaces
A new Whitney extension proves every L¹ function's jump set is covered by C¹ hypersurfaces.
Gowers U^k uncertainty conjecture fails for k≥6
A simple two-point signal on Z_3 produces a counterexample, exposing the role of coset structure.
· “A counterexample to the Gowers U^k uncertainty conjecture for kge 6”
Every uncountable Borel set yields a zero-measure Kakeya set
Countability is the sole obstruction: a compact set of measure zero contains a unit segment in every direction with centre in any given…
Multilinear Hilbert transforms along curves: bounded for all r>1/3
The n-linear bound now covers every r>1/3, pushing maximal operators from r≥1 into the quasi-Banach range.
· “Boundedness of Multilinear Hilbert Transforms Along Moment Curves”
Fractional and logarithmic Helmholtz equations are equivalent for all nonnegative…
New proof using abstract operator theory extends the equivalence to any nonnegative linear operator.
· “The equivalence of solutions to fractional and logarithmic Helmholtz equations”
Bidiagonal factorization tames Pi├▒eiro orthogonal polynomials
Recurrence matrix for mixed orthogonal polynomials on the step line factors into explicit bidiagonal matrices.
· “Pi\~neiro mixed orthogonal polynomials and bidiagonal factorization”
Complex Gaussian matrix: average singular value bounded for all d
The two bounds differ by less than 1/(16πd^4), sharpening unitary-Grothendieck guarantees.
· “Non-asymptotic bounds for the average singular value of a complex Gaussian matrix”
Optimal covering theorem for balls resolves Solyanik conjectures
A new angular separation argument achieves the sharp Solyanik estimate (1-α)^{2/(n+1)} and optimal constants for maximal operators.
· “A sharp covering theorem and Solyanik estimates for Euclidean balls”
Solvability of singular DAE boundary problems reduced to two checks
Existence becomes a range condition and consistency identities—no regular-pair assumption needed.
Unified confluence scheme ties ten multiple orthogonal families to a Hahn-like ancestor
Bernstein transform on Jacobi-like weights yields a discrete ancestor that limits to Kravchuk, two Meixners, two Charlier, Jacobi, two…
· “An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality”
Critical Hardy weights for discrete Laplacians for all L^p exponents
Sobolev–Bregman forms give exact identities, extending criticality theory from p=2 to every p>1
s-波共振让二维波算子端点有界
完整分类揭示阈值奇点类型决定L^1和L^∞行为，与高维直觉相反
· “Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schr\"odinger Operators”
Symmetric Laguerre zero products prove Hermite conjecture
Strict monotonicity with degree extends to all pairs and a one-parameter deformation, settling two conjectures.
Weak-type bound for Dunkl–Riesz transform depends on reflection count
Obstacle method yields dimension-free endpoint estimate; constant 2 for reflection-invariant functions.
· “Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform”
Cantor-dust–circle sum has interior iff λ>1/4
Settles Simon–Taylor problem, and refines thresholds for infinite, continuum, and positive-dimensional intersections.
· “Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle”
Exact relaxation proved: shear-band memories survive the limit
The relaxed state of a shear-banding material is a history measure, not a convexified rate law.
Capacitary distance yields sharp Hardy inequality for all domains
Solves Maz'ya's Problem 8: on any open set, ∫|u|^2/d_α^2 ≤ (C_n/α^2)∫|∇u|^2, and the exponent 2 is optimal over all domains.
Explicit optimal Hardy weight for all fractional orders on the half-line
A sharp inequality for the discrete fractional Laplacian matching the continuous constant and extending the Birman inequality.
· “Optimal fractional discrete Hardy inequalities on the half-line”
New bases disprove the Calderón-sum and wavelet-set conjectures
The Calderón sum stays ≤ 1/3 and no wavelet set exists, yet the system is an orthonormal basis, even for determinant one
Rough damping still stabilizes the wave equation on the torus
Observability from thin annuli yields exponentially decaying energy under H^{1/2} damping satisfying GGCC.
Every uniformly perfect set gets an optimal polynomial mesh
A Green-sublevel argument removes the need for boundary regularity, covering Cantor sets, fat Cantor sets, and real Julia Cantor sets.
· “Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels”
One denominator dictates all higher Taylor coefficients
This minimal order is stable under coordinate changes and flags exact cancellations before normal-form steps.
· “Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields”
Refined decoupling estimate found false
Counterexample and non-circular repair keep the 5/4 Falconer result intact
· “Repairing the refined-decoupling proof of the 5/4 planar pinned Falconer theorem”
New functional identity generalizes Jacobi to n terms
Pair of functions obeying a three-term law yields an n-term identity that drives a general finite-sum transformation.
· “Notices on a new functional identity and allied series transformations”
Capacity gauge solves open problem of quasisymmetric composition on $Q_\alpha$ spaces
A long-standing problem asking which quasisymmetric maps preserve $Q_\alpha$ spaces is answered with a new intrinsic geometric quantity…
Critical chaos on the circle is Rajchman
Fourier coefficients of the canonical critical Gaussian multiplicative chaos vanish almost surely, solving a problem of Garban and Vargas.
· “Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman”
Dimension-free lattice counts for every integer q ≥ 2
Number of integer points inside a small ℓ^q ball grows like (C(q)+o(1))^d, completing earlier q=1,2 results.
· “Lattice points in high-dimensional ell^q balls with small radii”
Geronimus transformation unifies three orthogonal polynomial types
Generalized Hermite polynomials serve as the primary example; the framework yields recurrence relations, zero interlacing, and…
· “A unified approach via Geronimus transformation to various types of orthogonal polynomials”
Chebyshev-Fourier limits obey forced degree-mesh window
Under order-below-two conditions, degree r_k and mesh d_k obey d_k^{-1}=o(r_k) and r_k=o(d_k^{-2}).
· “Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants”
Gowers norms get a strictly better log-convexity bound
New L^p autocorrelation estimates force the U^d norm below the product of neighboring Gowers norms.
Positive curved trace ensures Minkowski sum threshold despite zero Fourier decay
A strictly convex Lipschitz graph can lack all pointwise arclength decay yet force positive measure for every translate set of dimension >1.
· “Minkowski sums with convex curves without pointwise Fourier decay”
One condition guarantees unique bridge deflection and fast iteration
The classical Melan suspension-bridge equation has a unique solution under realistic loads, computable by a simple proven algorithm.
Bounded Hajłasz–Sobolev extension domains have null boundaries
In Q‑doubling metric spaces, a quasiadditive set function argument forces the boundary measure to be zero when 0<q<p<q*_Q.
· “On the measure of the boundary of a Hajl{}asz-Sobolev (M^(1,p),M^(1,q))-extension domain”
Critical L^p bound for Born-Jordan distribution proved
Resolves the critical boundedness problem; p* = 2d/(d-2) is the optimal exponent.
Minimum variance equals the closest Laguerre zero-gap
At degree m, the optimal repeated-pole kernel deletes the closest adjacent pair of zeros of L_{m+2}.
· “Least Variability in a Polynomial-Square Class of Rational Kernels”
Firm tangential paths block harmonic limit convergence
New Littlewood-type theorem for half-spaces covers sequential approach regions, settling a question raised by Nagel–Stein.
· “A Littlewood-Type Theorem for Harmonic Functions in Euclidean Half-Spaces”
Two-weight condition fails for one-sided maximal operator in 3D+
A logarithmic obstruction shows that the natural Muckenhoupt condition is not sufficient for dimensions three and higher.
· “On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions”
Linear q-difference systems get explicit monodromy formulas
Canonical solutions, isomonodromic deformations, and a q→1 limit that recovers the classical theory.
· “Monodromy and Isomonodromy for Linear q-Difference Systems with Coefficient Matrix Ax+B”
AI finds verified counterexamples to 18 open conjectures
Every refutation comes with a certificate in exact arithmetic that anyone can rerun from source.
Error exponent for curve lattice-point counts drops to n/(2n-1)
In every dimension n ≥ 4 the new bound beats the previous best and closes most of the gap to the conjectured optimum.
· “Counting integral points near a curve in the n-dimensional Euclidean space”
Ergodic ×p-measure keeps dimension under every linear projection
Sharp tube estimate defies algebraic resonances that normally cause dimension loss in rational directions.
· “Ergodic times p-invariant measures on mathbb{T}² with no dimension dropping projections”
Beta derivative equals infinite product at odd integers
A Wallis-type product with polynomial exponents gives the Dirichlet beta function's derivative at negative odd integers explicitly.
Counterexamples set Schrödinger-type convergence threshold at 1/2
For a broad class of polynomial phase symbols, convergence fails below Sobolev regularity 1/2 and holds above it.
· “Counterexamples for generalizations of the non-elliptic Schr\"{o}dinger maximal operator”
Positivity of coefficients in Hurwitz-type matrix polynomials holds for deg≤3
Explicit formulas and a counterexample resolve a conjecture; the property fails for degree four.
0.5409288219: the sharp constant is set by one Painlevé singularity
Hörmander–Bernhardsson equals 2π/θ∗², with θ∗ the first singularity of cosh-Gordon, matching 100 digits.
· “A Painlev\'e equation for the H\"ormander-Bernhardsson constant”
Bi-Lipschitz transformations preserve rectifiability square-function test
Composing density differences with a bi-Lipschitz function leaves the Carleson condition unchanged, making the logarithmic version a…
· “Square Functions and Rectifiability under Monotone Transformations of the Density”
Uniform Jacobi bound proves two long-standing conjectures
A three-regime envelope for orthonormal Jacobi polynomials settles the Erdélyi–Magnus–Nevai and Krasikov conjectures with explicit constant.
· “The Erd\'elyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials”
Sharp Jacobi bound proved
The weighted Jacobi bound settles a long-open conjecture and pins the Laguerre decay exponent at min{1, 1+α}.
An analysis preprint claims to refute a 1997 conjecture by constructing an integrable…
The paper asserts an explicit convex sequence with a_m <= 7m^8 and an L1 function whose Cesaro Fourier means are unbounded at a Lebesgue…
· “A Counterexample to Belinsky's Conjecture on Ces\`aro Means at Lebesgue Points”
Tiles without spectra exist in finite p-groups
The counterexamples hold even in elementary abelian groups for p=2 and 3, not just in exponent-4 groups.
· “Translational tiles without spectra in finite abelian p-groups”
Maximal rough singular integrals admit a pure $(1,p)$-sparse bound
$(1,p)$-sparse bound with $L^1$ average; linear $p'$ growth for bounded kernels, critical $p=q'$ for $L^{q,1}\log L$.
Two constants control sub-Laplacian heat trace and zeta determinant
For simply connected two-step pairs, the zeta-regularized determinant decomposes exactly into local, lattice, and spectral terms.
· “Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants”
Bounded nonlinear term preserves approximate controllability
Even with time-varying linear dynamics, a sufficiently bounded nonlinear perturbation does not destroy the ability to steer the state…
· “Controllability of time-varying lumped semilinear systems”
This paper characterizes the boundedness of the Dirichlet Carleson embedding for p>2
Finite positive Borel measures mu on the disk make the Dirichlet-to-L^p embedding bounded if and only if the packing energy and Haar energy…
Digit-restricted prime gaps: exact counts for pairs and triples
A single localized Fourier exponent decides which digit-restricted prime gaps obey exact asymptotic formulae.
One finite test decides which fractal frequency sets form bases
The test checks integer points of a fractal attractor, so spectrum verification becomes a finite computation.
· “Characterization on spectra of self-similar measures and the dual spectral set conjecture”
Every C¹ bump's gradient range equals the closure of its interior
Settles a quarter-century-old open problem in every dimension, via rigidity from symplectic topology.
· “Regular closed gradient ranges and asymptotic gradient values”
Unit lemniscate areas shrink to zero for every capacity-one set
Answering Erdős Problem 1040: at diameter one or above, unit lemniscate area shrinks to zero, no smoothness needed.
Finite convolution transform for hypergeometric series
Divided differences yield a new identity that expresses an r+2F_{r+1}(1) series as a finite sum of r+1F_r(1) series.
· “Terminating Zero-Balanced Hypergeometric Series Using Divided Differences”
2^d times the Nikolskii constant tends to π/2
The proof squeezes the sharp constant between explicit test functions and a Bessel-model zero comparison.
· “Exact Asymptotics of the Multidimensional Nikolskii Constant”
Curved Kakeya lifts to linear
Dense Hörmander-type curve families would gain full Hausdorff dimension; quadratic curves get exact integer dimensions.
Sharp Gaussian bounds extend to non-characteristic boundaries
Controls every time and vector-field derivative, for all even orders, systems, and non-Dirichlet boundary conditions.
· “Higher-order Gaussian bounds for maximally subelliptic boundary value problems”
Charge potential on a line: at most 2n−1 critical points
Settles conjecture of Gabrielov–Novikov–Shapiro; bound is sharp
Q2 frames on ab=1/q when |F(b)|>1/(2q)
Complete characterization of a B-spline frame set along an entire curve, resolving a question left open by Lemvig and Nielsen
Flat tori fail negative type at every power
A facet-by-facet Fourier signature destabilizes Haar measure and yields exact maximizers on rectangular and hexagonal tori.
Hermite Gabor frames proven safe down to lattice area n^{-2/3}
Near the axes, the frame property survives up to product 1/2 once one step shrinks faster than n^{-1/2}.
· “Asymptotic safety regions for Gabor frames generated by Hermite functions”
