Classical Analysis and ODEs
Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
sort pith recommended most recent
Boundary recovery is more involved than for fractional powers, yet works under reverse Hölder potentials.
· “Extension theorems for logarithmic Schr\"odinger and discrete Laplacian operators”
The weight is null-critical and fixes the sharp constant for n^{-2σ}.
· “An optimal fractional Hardy inequality on the discrete half-line”
Quantitative closeness at every scale and point forces local equivalence to a cone over a simplicial complex.
· “A generalization of Reifenberg's theorem in R^N for flat cones”
For decreasing weights and several concrete weight sequences, the classical constant is not best possible.
Entrywise transforms of Lorentz–Gram matrices reduce to one integral family, completing the curvature trilogy.
A fractional obstacle decomposition plus a Lewy-Stampacchia estimate settles a 1986 problem from the International Congress of…
· “A dimension-free weak-type (1,1) bound for the vector Riesz transform on mathbb{R}^n”
A closed ray of Schwartz functions yields the 196560 ratio through Poisson summation.
· “The 196560 auxiliary-function conjecture for the Leech lattice”
This settles two open questions on time-lag classes and yields a John--Nirenberg inequality.
A 1969 prediction, revised by Christiansen–Simon–Zinchenko, is proven for every analytic Jordan arc.
Settles two 2009 conjectures and gives finite-degree bounds in Bessel zeros for spherical cubature.
· “Affine Scaling of Jacobi Zeros: Sharp Orderings Beyond Gautschi's Conjectures”
Gabor shifts form a frame exactly when lattice density is below 1, closing the irrational case.
· “Gabor Frames of Totally Positive Functions: A Complete Characterization”
The probability-flow ODE yields a unique regular Lagrangian flow transporting p0 onto the diffusion marginals under Sobolev/BV score…
· “Mimicking diffusion processes with differential equations”
Removes the small-γ restriction from earlier estimates and supplies an explicit constant that works for every x > 0.
· “Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions”
Twelve standard forms are obtained, including new quartic and quintic examples, for use in the Painlevé equivalence problem.
· “Modified Painlev\'e systems with meromorphic solutions for polynomial Hamiltonians of all degrees”
A new integral for the area of half-Holder curves makes this work even for the simplest vector-valued case.
· “Area of H\"older curves and coarea formula on the Heisenberg group”
Degree-mismatch criterion rules out all higher-degree symmetric polynomials for nonconstant continuous F on the positive reals.
Lacunary hopping weights make wave packets flatten like |t|^{-N} for every N, with no universal speed limit.
· “Arbitrarily Fast Quantum Dispersion in Long-Range Crystals”
Quantitative strengthening of the norm-convergence theorem with computable constants, verified in Lean 4.
A Legendre-series seed with a nondifferentiable equator closes the last gap in the spherical regularity theorem.
· “Optimal differentiability of isotropic positive definite functions on even-dimensional spheres”
A conditional-sparsity argument yields sharp weighted L^p estimates on every filtered probability space.
· “Weighted Estimation by Discrete-time Sparse Domination on Martingale Spaces”
The paper proves the corrected dichotomy: Kakeya-type sets, unbounded Lp operators, and sublacunarity coincide.
New proof needs no reflection-group invariance; yields H^1-to-L^1 endpoint bounds and the Hardy-space characterization.
· “A Fefferman--Stein inequality for the Dunkl Poisson semigroup and its chamber-lifted formulation”
A new proof gives explicit remainder terms and uniform eigenfunction bounds for every fractional order between 0 and 2.
Sharp weak-type and Lp bounds for the Beurling–Ahlfors transform follow from one integral inequality.
· “Quasiconvexity of the Burkholder function on symmetric matrices”
Every uniformly rectifiable space-time set contains large smooth graph patches at all scales, and conversely.
For parabolic Lipschitz graphs, a finite square Dini-beta integral is exactly absolute continuity between surface and caloric measure.
· “The parabolic Dini-β condition and absolute continuity of surface and caloric measure”
Every coefficient carries exactly the predicted power of 2; one p-adic series decides the whole sharp law.
· “The dyadic denominator law for the phase constants of the Jacobi zeros”
Complete indeterminacy of a matrix Nevanlinna-Pick problem is equivalent to convergence of a matrix series built from rational functions of…
· “The Hamburger Criterion for Matrix Nevanlinna--Pick Interpolation”
The radial extremal function splits into two half-type factors, so the sharp constant follows from a d+1-parameter ODE.
Mass-to-geometry ratio plus short endpoint runs yields the exponent and an exact dimension-distortion law.
· “Local Scaling and Dimension Distortion of Generalized Cantor Functions”
Criterion: once the un-averaged operator is L2-bounded with the right constant, the averaged operator is weak type (1,1).
· “A criterion on weak type (1,1) bound of rough singular integrals”
At Fresnel phases 3π/4 and 7π/4 the quarter-arcs glue smoothly; waves split off from circles.
A single parameter shift transfers monodromy, spectra, and finite-gap results across the whole component.
· “Componentwise Geometry and Monodromy of Generalized Lam\'e Equations”
Needles in every direction must fill nearly full volume, unlocking harmonic analysis breakthroughs.
· “Rotating needles in space: the road to the Kakeya conjecture, and why it matters”
Interpolation over the extremizer's zeros yields the equation and closes the even-dimension gap.
· “The H\"ormander--Bernhardsson function in higher dimensions”
A Euclidean positive definite function can turn negative on the sphere even with arbitrarily small support.
· “Isotropic Positive Definite Functions on Spheres: Even-Dimensional Gneiting's Problem”
Longitudinal plus transverse beam integrals invert by Fourier-Mellin, extending tensor tomography to Gaussian beams.
· “Gaussian beam Radon transform for tensor fields in mathbb{R}²”
Restricted-type Schatten bound settles the long-open endpoint conjecture and its transport analogue.
Closed-form coefficients follow; the difference operator has order ℓ, confirming the predicted order equality.
· “Finite-term recurrences in a generalized Bochner--Krall family”
Almost surely, the count matches the tube area times Q^3, except at one critical thickness.
The paper settles the attainability half of Arnold's 1988 problem with explicit Hölder-continuous space-filling surfaces.
· “Space-filling surfaces: sharp H\"older continuous parameterizations from squares to cubes”
Bourgain–Brezis L1 inequalities survive when coordinates scale differently, with Lorentz and Besov refinements.
The same symbol class, as VMO, gives compactness; operator norm equals the Bloom oscillation.
· “Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights”
A 3×3 Riemann–Hilbert analysis gives explicit prefactors and O(1/n) uniform error everywhere, including near endpoints.
· “Strong asymptotics for Jacobi-Pi\~neiro orthogonal polynomials”
All self-similar motions fall into four classes; any curve can evolve to any other.
As the threshold λ→∞, the nonlocal energy lands on a universal constant times the local Sobolev or total-variation energy.
· “Gamma-Convergence of Weak-Type Nonlocal Functionals on Bounded Domains”
A tensor-product transference upgrades discrete counting bounds to a sharp analytic inequality, with Kakeya consequences.
They reach critical power weights that classical Muckenhoupt classes miss, enabling new endpoint estimates.
· “Properties and applications of Lorentz--Muckenhoupt classes”
Products of multivariate and matrix hypergeometric functions become one integral, opening convolution identities.
· “Product formulas for multivariate and matrix-variate confluent hypergeometric functions”
For 1≤p≤2, φ(A,B) is controlled by input commutators; for p>2 even finite-rank counterexamples block any such bound.
· “Commutator estimates for functions of noncommuting self-adjoint operators”
A dyadic seminorm on each cube controls the v-weighted oscillation by the w-weighted gradient in L^p, for p<q and p>q.
· “Generalization for Poincar\'e--Sobolev inequalities with local weights”
It improves the previous best bound for every d ≥ 52 divisible by 4 and matches the exact optimum at d = 12.
· “Positive quasimodular forms and the sign uncertainty principle”
Functions cannot concentrate on fractal position and frequency sets at once under the Hankel transform.
· “Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform”
The exact jump, from 6 to 4 at ε = 3−2√2, shows self-similar geometry controls progression scales.
No doubled terms means one measure; reducible values and triple repeats spoil uniqueness.
· “A characterization of uniqueness of purely atomic finite measures with central Cantor set range”
A new theorem decides when every mix of two weighted exponential systems stays complete in L²[0,1].
The test replaces arc measure with extremal-function measures, making the old sufficient condition a special case.
· “Extremal functions and zero sets for the Dirichlet space”
The number of refined directions holding λ points of E is at most q^{2n-1}|E|λ^{-2n}, and the full exponent diagram is settled.
· “Sharp refined-direction Kakeya estimates in finite Heisenberg groups”
The (C, α) means along a_k = P(k) reproduce f(x) at every Lebesgue point, answering both 1936 questions.
· “Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points”
Below the Lipschitz endpoint a vertical fiber bound suffices; at the endpoint, Dini–Zygmund summability takes over.
The construction settles an open question about when Fourier interpolation breaks down.
An Orlicz-type A1 condition on the weights yields the correct logarithmic loss at every commutator order.
A rotation-number comparison proves infinite relative periodic orbits on every compact regular energy surface, including hip-hop and…
· “Relative Periodic Orbits in the Gutzwiller-type Anisotropic Kepler Problem and n-body Problem”
The proof constructs quasisymmetric metrics that reduce any finite-dimensional measure's Hausdorff dimension at will
· “Conformal Dimension of Measures and Quasisymmetric Dimension Reduction”
The bound is sharp: no conformal map can push a line's preimage longer than π², and examples approach it.
A symmetric measure lift turns prediction errors, Verblunsky coefficients, and Toeplitz determinants into evaluations at x=2.
· “Lommel polynomials and explicitly solvable prediction problems on the unit circle”
An explicit limiting counterexample forces any valid constant above pi, answering a 1974 question.
· “On the optimal constant in the Montgomery-Vaughan weighted Hilbert inequality”
It settles an open extension problem with the optimal pointwise bound.
Matrix Corona plus a symmetry average yields bounded slice solutions for any n≥2.
· “The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem”
It minimizes Wehrl entropy and proves spherical caps maximize all Toeplitz eigenvalue sums.
· “Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere”
Inside a critical radius the density is an explicit integral; outside, the constraint saturates.
The rank is the only part that cannot be compressed; the transverse directions shrink by half.
· “Kakeya sets and dimension compression in compact Lie groups”
Derivatives of the matching map stay smooth even when nonlinear terms grow without bound.
Unitary gap jumps extend steepest-descent asymptotics to matrix weights on several intervals.
· “Matrix SzegH{o} Function and Matrix Orthogonal Polynomials for Multiple Cuts”
A double sum of factorial ratios equals one trigonometric series, yielding a restricted sum formula.
For bounded lower oscillation, no larger exponential rate and no smaller prefactor can hold.
Two gaps in the key theorem leave the general case of Conjecture A open.
New hypotheses let Federer's projection theorem pass through critical sets, settling the rectifiable endpoint.
· “Finite Good Witnesses for Generalized Curve Projections at the Rectifiable Endpoint”
Federer's projection theorem is extended to distance maps, radial views, and circles, with small exceptional sets.
· “Applications of Nonlinear Projections to Rectifiable 1-sets”
The classification now covers Guinand–Weil formulas for the Selberg class, via log-derivative generating functions.
· “Classification of Fourier summation formulas on a horizontal strip”