Classical Analysis and ODEs
Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
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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”
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”
A long-standing problem asking which quasisymmetric maps preserve $Q_\alpha$ spaces is answered with a new intrinsic geometric quantity…
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”
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”
New L^p autocorrelation estimates force the U^d norm below the product of neighboring Gowers norms.
The proof squeezes the sharp constant between explicit test functions and a Bessel-model zero comparison.
· “Exact Asymptotics of the Multidimensional Nikolskii Constant”
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”
A 1969 Morse-Cairns conjecture is proved; the number of equilibrium points is bounded by a function exponential in the number of charges.
· “The finiteness conjecture for equilibria of electric fields generated by point charges of one sign”
The logarithmic blowup survives even on a geometric modulation sequence, so no sub-(t log t) bound exists.
· “Endpoint Estimates for Stein's Purely Quadratic Carleson Operator”
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”
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”
New conditions using finite variation and conformal tails show which Jordan curves admit continuous Legendrian lifts.
New argument uses Stein's spherical maximal inequality to eliminate dimension dependence
· “Shorter proof of dimension-free L^p estimates for maximal Riesz transforms”
For random α-Hölder maps, images achieve full dimension m exactly when αm ≤ D(μ), a scale-invariant parameter extracted from the measure.
· “A threshold for full packing dimension of H\"{o}lder images of sets and measures”
A new Whitney extension proves every L¹ function's jump set is covered by C¹ hypersurfaces.
Proof uses a matrix defect identity and turns classification into rigidity of root trajectories under convolution.
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”
The curve separating sign-definite Bessel integrals from sign-changing ones has no inflection on its whole closed range.
An explicit bound in every dimension, plus Sobolev and isoperimetric consequences.
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”
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”
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”
Resolves the critical boundedness problem; p* = 2d/(d-2) is the optimal exponent.
The classical Melan suspension-bridge equation has a unique solution under realistic loads, computable by a simple proven algorithm.
The Calderón sum stays ≤ 1/3 and no wavelet set exists, yet the system is an orthonormal basis, even for determinant one
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”
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”
Strict monotonicity with degree extends to all pairs and a one-parameter deformation, settling two conjectures.
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”
Existence becomes a range condition and consistency identities—no regular-pair assumption needed.
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”
For each fixed group size, a dimension-free power bound replaces exponential growth.
· “Polynomial Bohnenblust--Hille bounds for product of cyclic groups”
Recurrence matrix for mixed orthogonal polynomials on the step line factors into explicit bidiagonal matrices.
· “Pi\~neiro mixed orthogonal polynomials and bidiagonal factorization”
New proof using abstract operator theory extends the equivalence to any nonnegative linear operator.
· “The equivalence of solutions to fractional and logarithmic Helmholtz equations”
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”
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”
Two weighted solution-space pairs fix the exact stability exponent, proving hyperbolicity survives delay perturbations.
Countability is the sole obstruction: a compact set of measure zero contains a unit segment in every direction with centre in any given…
Dense Hörmander-type curve families would gain full Hausdorff dimension; quadratic curves get exact integer dimensions.
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”
Answering Erdős Problem 1040: at diameter one or above, unit lemniscate area shrinks to zero, no smoothness needed.
Settles a quarter-century-old open problem in every dimension, via rigidity from symplectic topology.
· “Regular closed gradient ranges and asymptotic gradient values”
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”
A single localized Fourier exponent decides which digit-restricted prime gaps obey exact asymptotic formulae.
A facet-by-facet Fourier signature destabilizes Haar measure and yields exact maximizers on rectangular and hexagonal tori.
Exact averaging of asymmetric boundaries replaces multiplicative amplification, eliminating subpolynomial errors.
For sequences with fixed ratio n/k, the eigenvalue converges to a formula involving α, unifying known cases
· “Asymptotics for the k-Hessian Eigenvalue on the Unit Ball”
If truly zero-free, not even two exponentials can be orthogonal, so the measure cannot be spectral.
· “On the spectrality of the non-homogeneous golden-mean self-similar measure”
Settles the completeness classification for primes in a class of self-similar fractal measures.
· “Primes are Complete for a Class of N-Bernoulli Convolutions Spectral Pair”
A modern exposition that splits circles into transverse and tangent families, then uses separate analytic and geometric tools.
· “Bourgain's proof of the circular maximal theorem: an exposition”
Hausdorff–Choquet angular space captures geometric concentration, extends Christ–Rubio de Francia, and gives sharp dimension exponent.
· “Hausdorff--Choquet Angular Spaces and Weak-Type (1,1) Bounds for Rough Maximal Operators”
$(1,p)$-sparse bound with $L^1$ average; linear $p'$ growth for bounded kernels, critical $p=q'$ for $L^{q,1}\log L$.
The sharp bound 2 follows from a contractive symmetrization, even in infinite-dimensional Hilbert spaces.
A sharp linear bound, answering Alon's question, plus polynomial bounds for general radicals via ODE and PDE methods
· “Estimating the number of real zeros of linear combinations of radicals of polynomials”
Algebraically growing coefficients with periodic modulation determine the limiting zero density without orthogonality measures.
· “Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials”
For bandlimited spherical operators that exactly reproduce degree‑L polynomials, the minimal unavoidable positivity loss is exactly of…
· “Positivity loss in bandlimited spectral reproduction on spheres”
The only exceptions below critical density are the rational densities (q-1)/q, settling a decade-old conjecture.
Precise two-sided estimates proved with a comparison argument, covering A1 and BC1 root systems
· “Sharp estimates for the Jacobi and trigonometric Dunkl heat kernels”
New bound combines optimal concentration and rearrangement to settle leading-order growth for large q.
Two new wrapped-product weights gain explicit second-order operators on ball-simplex and ball-Laguerre domains.
· “Orthogonal polynomials which are eigenfunctions of a partial differential operator”
Nonvanishing homology of a simplicial complex of directions guarantees polytopes for antipodal and even discrete measures.
Complete characterization of a B-spline frame set along an entire curve, resolving a question left open by Lemvig and Nielsen
Settles conjecture of Gabrielov–Novikov–Shapiro; bound is sharp
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”
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…
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”
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”
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”
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”
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”
Sobolev–Bregman forms give exact identities, extending criticality theory from p=2 to every p>1
The weighted Jacobi bound settles a long-open conjecture and pins the Laguerre decay exponent at min{1, 1+α}.
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”
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”
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”
Explicit formulas and a counterexample resolve a conjecture; the property fails for degree four.
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”
A Wallis-type product with polynomial exponents gives the Dirichlet beta function's derivative at negative odd integers explicitly.
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”