Papers

In the present paper, we consider the maximal operator \[ \mathcal M_{\Omega}f(x) := \sup_{r>0}\frac{1}{r^n} \int_{|y|<r} |f(x-y)| \left| \Omega\!\left(\frac{y}{|y|}\right) \right|\,dy. \] A longstanding open conjecture raised by E.M.Stein asks whether the maximal operator $\mathcal M_{\Omega}$ is of weak type $(1,1)$ when $\Omega$ is merely in $L^1(\mathbb S^{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of $\mathcal M_{\Omega}$ with kernel $\Omega\in\mathcal X(\mathbb S^{n-1})$, yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here$\mathcal X(\mathbb S^{n-1})$ is the space related to the Hausdorff–Choquet angular space and $$ L\log^+\!L(\mathbb S^{n-1}) \subsetneq \mathcal X(\mathbb S^{n-1})\subset L^1(\mathbb S^{n-1}). $$ Prior to our work, the condition $\Omega \in L\log^+\!L(\mathbb S^{n-1}) $ remained the best known size restriction guaranteeing the weak-type$ (1,1)$ bounds for all dimensions $n > 2$. To our knowledge, the present paper is the first to relax this threshold for $n>2$. Finally, for the general Hausdorff–Choquet scale $\mathcal H\mathcal C_{\alpha}$, we show that $\alpha=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.

Let $\Omega\in L^1(S^{d-1})$ have vanishing average, and let $T_\Omega^\ast$ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative sparse bounds for $T_\Omega^\ast$. If $\Omega\in L^\infty(S^{d-1})$, then, for every $1<p<\infty$, \[ \|T_\Omega^\ast\|_{(1,p)\text{-}\mathrm{sparse}} \lesssim_d p'\|\Omega\|_{L^\infty(S^{d-1})}. \] For unbounded angular kernels, if $1<q<\infty$ and $\Omega\in L^{q,1}\log L(S^{d-1})$, then the same estimate holds for $q'\leq p<\infty$, with the right-hand side replaced by \[ C_{d,q}p' \|\Omega\|_{L^{q,1}\log L(S^{d-1})}. \] These estimates retain a genuine $L^1$ average in the first entry of the sparse form. In the bounded-kernel case, the upper bound has the same linear growth in $p'$ as the known sparse bound for the nonmaximal operator. The unbounded-kernel estimate includes the critical exponent $p=q'$. As a consequence, we obtain weighted weak-type $(1,1)$ estimates for all $A_1$ weights in the bounded-kernel case and for weights in $A_1\cap RH_{q'}$ in the unbounded-kernel case. The proof combines physical-space linearization and microlocal decomposition with localized sparse testing. An amplitude decomposition of the second input, together with the Rademacher–Menshov inequality, yields the quantitative dependence on $p$.

Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.

Let $K_{a,m}$ and $K_{b,n}$ be missing-digit Cantor sets with initial consecutive digit sets, and write $c_a=m/(a-1)$ and $c_b=n/(b-1)$. We prove that if $c_a+c_b\geq 1$, then the radial projection of $K_{a,m}\times K_{b,n}$ from every observer has nonempty interior; no multiplicative-independence assumption is needed for this implication. Conversely, when the bases are multiplicatively independent and $c_a+c_b<1$, we exhibit explicit unbounded open sets of observers for which the radial image is compact and nowhere dense. Rational observers with the same property are dense in each exterior corner region and occur arbitrarily close to the four corners of the product. Thus $c_a+c_b=1$ is the exact threshold for the all-observers interior property within the multiplicatively independent initial-block family. In particular, this supplies an explicit two-set counterexample to the nonempty-interior conclusions of two conjectures of Yu. We also establish a sufficient modular phase obstruction for affine translates of such sets. Combining it with a fixed-pin positive-measure theorem of Banaji and Yu yields consecutive-block division sets that are compact, perfect, of positive Lebesgue measure, and nowhere dense, with both factor dimensions tending to one. For the same family, Fourier $l^1$-dimension estimates and the incidence argument used in Yu's product theorem imply that the two self-products contain intervals for all sufficiently large $r$ and $s$, while the cross-division set remains nowhere dense.

We study an oracle subspace-perturbation model for 3D localization. The procedure is given an $s$-dimensional subspace $\widetilde{{\mathcal U}}$ and a deterministic error bound \[ \varepsilon_{\mathrm{sub}} = \left\|P_{\widetilde{{\mathcal U}}}-P_{{\mathcal U}}\right\|_2, \] where $\mathcal{U}$ is an $s$-dimensional subspace of far-field patterns with wavenumber $\kappa$, and $P_{\mathcal{V}}$ denotes orthogonal projection onto a subspace $\mathcal{V}$. Under explicit separation and conditioning hypotheses for arbitrary point clouds, we prove that the perturbed spherical MUSIC objective \[ \widetilde{q}({\mathbf z}) = 1-\left\|P_{\widetilde{\mathcal{U}}}\varphi_{{\mathbf z}}\right\|_2^2 \] has a unique strongly convex well in every ball $B_{\gamma/\kappa}(x_j)$ and a uniform value gap outside the union of the certified wells. A fixed-step gradient map with $h\asymp\kappa^{-2}$ leaves each well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an $O(\kappa^{-1})$-mesh, followed by gradient descent and duplicate removal, recovers all relevant minima with localization bound \[ \mathfrak{R}(\varepsilon_{\mathrm{sub}}) \lesssim \frac{\varepsilon_{\mathrm{sub}}}{\kappa}. \] The arbitrary-cloud frame analysis yields the sufficient condition \[ \kappa\delta_X\gtrsim s^{2/3} \] through an absolute coherence row sum and Gershgorin's theorem. We construct lower-frame counterexamples below the $s^{1/6}$ scale, upper-frame counterexamples below the $s^{1/3}$ scale, and examples showing that $2/3$ is optimal for the absolute-row-sum argument. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove \[ \mathfrak{R}(\varepsilon_{\mathrm{sub}}) \sim \frac{\varepsilon_{\mathrm{sub}}}{\kappa}. \]

We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{\alpha_k}, $$ where $c_k, \alpha_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We improve previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $\alpha_k$. For linear combinations of square roots of positive quadratic polynomials on $\mathbb{R}$ we prove the linear bound $2n$, answering a question of N.~Alon. A modification of the argument yields a linear bound for a question of A.~Gabrielov, D.~Novikov, and B.~Shapiro related to Maxwell's conjecture. The article describes two independent approaches: an elementary ODE method in the general case, which also gives a polynomial bound for the number of critical points of one dimensional Gaussian mixtures, and a PDE method for the case of positive quadratic polynomials, which connects the problem to the number of nodal domains of solutions to $\Delta u + \lambda u = 0$ on the punctured hyperbolic plane. As a byproduct of the second approach, we describe a curious relation between axially symmetric harmonic functions on $\mathbb{R}^3\setminus\{(x,0,0)\}$ and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue $1/4$.

We investigate the ratio asymptotics and the asymptotic zero distribution of a sequence of polynomials that satisfy a recurrence relation of order $r+1$ with all recurrence coefficients, except the last one, equal to zero. Such a sequence is part of a system of multiple orthogonal polynomials and it satisfies the symmetry property $P_n(\omega_{r+1} z) = \omega_{r+1}^n P_n(z)$, where $\omega_{r+1}$ is the primitive $(r+1)$th root of unity. We consider the unbounded regime in which the recurrence coefficients exhibit algebraic growth and, after division by $n^{\gamma}$ become asymptotically periodic and bounded. After the appropriate scaling, we establish ratio asymptotics and characterize the limiting ratio as the distinguished solution of an algebraic equation. We then determine the limiting zero distribution through its Stieltjes transform and investigate the associated \(\St\)-transform, which in several cases yields connections with hypergeometric polynomial sequences and distributions arising in free probability. The recurrence is represented by a two-diagonal non-self-adjoint Hessenberg operator, so that the limiting zero measure also admits a natural interpretation as a limiting empirical spectral distribution of its rescaled finite sections. Our analysis is based solely on the positivity and asymptotic behavior of the recurrence coefficients and requires no explicit knowledge of the underlying orthogonality measures.

How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\left({L}/{(N+1)}\right)^2$ when $1\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée–Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.

We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.

We prove sharp both-sided estimates for the $W$-invariant trigonometric Dunkl heat kernel in rank one case, including root systems $A_1$ and $BC_1$. Consequently, we obtain sharp bounds for the non-compact Jacobi heat kernel in the full range of the type parameters.

We improve the currently best known bound for the norm of the embedding of $PW^2$ into $PW^q$ for all $q>2$. We show that our bound is asymptotically optimal as $q\to \infty$. Our approach combines the recently established theorem of optimal Paley-Wiener concentration with a Hardy-Littlewood-Pólya majorization argument.

We study orthogonal polynomials of $d = d_1+d_2$ variables with respect to a wrapped product weight function ${\bm W}({\bm x},{\bm y}) = W_1({\bm x}/\rho({\bm y})) W_2(\bm y)$ for $(\bm x, \bm y) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$, where $\rho$ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For $d =2$, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in $d= d_1+d_2$ variables that have not been studied before if either $d_1>1$ or $d_2 > 1$, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when $d = 3$. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.

In this paper, we establish a homological existence criterion for the discrete $L_p$ Minkowski problem for all $p<0$, where $p=-n$ corresponds to the celebrated centro-affine case. For discrete measures with antipodal support, we obtain a complete existence characterization for arbitrary positive masses. In the even case, a solution can be chosen origin-symmetric.

The aim of this paper is to establish new regions in the frame sets of Hermite functions $h_n$. A classical result of Gröchenig and Lyubarskii shows that the Gabor system $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame for $L^2(\mathbb{R})$ whenever the lattice density exceeds $n+1$. We show that, for every $\eta>0$ and all sufficiently large $n$, the same Gabor system forms a frame whenever $ab\leq n^{-\frac{2}{3}-\eta}$. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters $a$ or $b$ is small. Namely, for every $\delta>0$ and $\rho\in(0,\frac{1}{2})$ and all sufficiently large $n$ we prove that if $\min\{a,b\}\leq n^{-\frac{1}{2}-\delta}$ and $ab\leq \frac{1}{2}-\rho$ then $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame.

We study operators acting on functions defined on the half-space, with methods inspired by sparse domination. Using the specific link between dyadic grids on the boundary and Whitney regions on the half-space, we obtain a very precise form domination with model operators of Hardy type. We introduce mixed-norm off-diagonal estimates that measures both tangential and transversal decay without requiring pointwise control in the transversal variable, substantially weakening the assumptions used in earlier tent space theories. This allows us to prove optimal weighted tent space extrapolation from local boundedness plus off-diagonal decay. The resulting framework yields new bounds for operators arising from non-autonomous elliptic and parabolic PDEs with non-smooth coefficients.

We give a complete characterization of the zero set of the Zak transform of the second-order B-spline $Q_2$. As a consequence, we obtain a new proof of the Lemvig–Nielsen obstructions to the Gabor frame property of $Q_2$, and, along the hyperbola $ab=1/q$, we show these obstructions are sharp: every point they do not exclude belongs to the frame set. This gives the first complete characterization of $\mathcal F(Q_2)$ along an entire curve in this family, resolving a question left open by Lemvig and Nielsen.

We prove Conjecture 1.9 of Gabrielov-Novikov-Shapiro. More precisely, for every $p>0$, every nonconstant restriction to a line of a potential generated by $n$ point charges has at most $2n-1$ critical points. The bound is sharp, even for positive charges. The proof uses a duality with an auxiliary planar potential generated by collinear charges and Morse theory.

We establish higher-order Gaussian upper bounds for the heat semigroups associated with a broad class of sectorial maximally subelliptic quadratic forms on manifolds with boundary. Near non-characteristic boundary points and in the interior, we obtain pointwise bounds for all mixed derivatives of the heat kernel in time and along the Hörmander vector fields, expressed in the associated Carnot–Carathéodory geometry. The results apply to operators of arbitrary even order, systems, nonsymmetric forms, and boundary conditions beyond the Dirichlet case.

We completely solve Colombo's 1928 determinant problem. For distinct real $x_1,\ldots,x_N$, $N\geq 2$, and an integer $D\geq 1$, we prove that $\det[(x_j-x_i)^D]\neq 0$ if and only if $D\geq N-1$ and either $N$ is even or $D$ is even. The even-exponent case follows from Dyn–Goodman–Micchelli (1986); the remaining odd case is proved by a strict Pfaffian sign theorem. The new odd-exponent theorem and its complete proof chain have also been formalized in Lean 4.

We construct explicit translational tiles without spectra in three finite abelian $p$-groups. The first is a $64$-point subset of $\Z_4^4\times\Z_2^2$. The other two are a $512$-point subset of $\F_2^{13}$ and a $2187$-point subset of $\F_3^9$. Consequently, the tile-to-spectral implication fails for finite abelian $p$-groups, and it already fails within the class of elementary abelian groups for both $p=2$ and $p=3$. Two elementary mechanisms organize the examples. A two-layer obstruction turns a spectral non-tile with two suitable tiling complements into a tile without a spectrum. A fiber–clique obstruction converts a family of tiling complements with controlled common Fourier zeros into an elementary abelian counterexample. All coordinate data are included. The finite claims are certified by three short, self-contained programs using exact integer arithmetic and exhaustive searches; the accompanying source files recompute every assertion used in the proofs.

In this paper, we obtain a full characterization of the finite positive Borel measures $\mu$ on $\mathbb D$ for which the embedding $$ \operatorname{id}:\mathcal D_{p-1}^p\longrightarrow L^p(\mu),\qquad p>2, $$ is bounded. More precisely, for any dyadic system $\mathcal D$ on $\mathbb T$, we prove that this embedding is bounded if and only if $$ \mathcal C_{p,\mathcal D}(\mu)+\mathcal H_{p,\mathcal D}(\mu)<\infty, $$ where $\mathcal C_{p,\mathcal D}(\mu)$ and $\mathcal H_{p,\mathcal D}(\mu)$ denote the packing energy and the Haar energy of $\mu$, respectively. This resolves a longstanding characterization problem in the theory of Dirichlet-type spaces that arose from Wu's 1999 conjecture, corresponds to the endpoint case not covered by the work of Arcozzi, Rochberg, and Sawyer in 2002, and remained open after the works of Girela and Peláez in 2006 and Galanopoulos, Girela, and Peláez in 2011. We also construct finite measures showing that the two energy conditions are genuinely distinct. The key ingredient in the proof is a reduction of the Dirichlet embedding to a dyadic Whitney embedding, which allows us to combine weighted Hardy inequalities on trees with probabilistic arguments.

In this work we are concerned with the controllability of time-varying lumped control systems governed by a semilinear differential equation. We consider the semilinear system as a perturbation of a linear system. Assuming the underlying linear system is controllable, and the nonlinear forcing function satisfies a boundedness condition which is adapted to the underlying linear system, we show that the semilinear system is also approximately controllable

We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants $C_{G,L}$ and $\beta_{G,L}$. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carathéodory distance; the leading coefficient $\mathfrak F_{G,K}(Z)$ is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of $\mathrm{SU}(N)$, recovering the classical $\mathrm{SU}(2)$ and CR sphere spectra.

In this paper we prove the sharp weak estimates for the Riesz potentials $$\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\leq \gamma_{n,s} v_n^{\frac{n-s}{n}}\frac{\Gamma(s/2)\Gamma((n+2-s)/2)}{\Gamma(n/2)}\|f\|_{L^1},~~ \text{when}~~0<s<\min\{n,2\}$$ and $$ \|I_sf\|_{L^{\frac{n}{n-s},\infty}} \leq \gamma_{n,s}v_n^{\frac{n-s}{n}}\|f\|_{L^1}, ~~\text{when}~~2\leq s<n,$$ where $\gamma_{n,s}=2^{-s}\pi^{-\frac{n}{2}}\frac{\Gamma(\frac{n-s}{2})}{\Gamma(\frac{s}{2})}$ and $v_n$ is the volume of the unit ball. The isoperimetric inequality for the Riesz capacity and the Newtonian isocapacitary inequality play a crucial role in our approach.

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-\Delta)$ of two-dimensional Schrödinger operators $H=-\Delta+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón–Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind—an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodniĭ is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence $(a_m)$ satisfying $a_m\leq 7m^8$ and a function $f\in L^1(\mathbb T)$ for which $0$ is a Lebesgue point, $f(0)=0$, and the means $m^{-1}\sum_{k=1}^m S_{a_k}f(0)$ are unbounded.

We establish ground-state representations and critical Hardy inequalities in the discrete $L^p$ setting. In particular, we construct critical Hardy weights for the discrete Dirichlet Laplacian on the half-line and the discrete fractional Laplacian on the integers for all $p\in(1,\infty)$. Our approach uses Sobolev–Bregman forms, for which the ground-state representations are exact identities and the corresponding Hardy weights are expressed through linear operators acting on powers of positive superharmonic functions.

We prove the refined Koornwinder–Kostenko–Teschl conjecture. For the normalized weighted Jacobi function and all $n\in\mathbb N\_0$, $\alpha,\beta\ge0$, and $-1\le x\le1$, $$ |g\_n^{(\alpha,\beta)}(x)| \le \left[ \frac{(n+1)(n+\alpha+\beta+1)} {(n+\alpha+1)(n+\beta+1)} \right]^{1/4}\le1. $$ The proof combines a central contour estimate with Sturm–Sonin localization and an exact inverse moment. It also yields an $n$-uniform extreme-lobe theorem and a sharp canonical-product first-lobe principle. Applied to the discrete Laguerre evolution, the estimate gives the optimal positive-parameter decay. For $-1<\alpha\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\|\mathrm{e}^{-\mathrm{i} tH_\alpha}\|_{\ell^1\to\ell^\infty} =(1+t^2)^{-(1+\alpha)/2}$. Thus the large-time decay exponent is $\min\{1,1+\alpha\}$ for every $\alpha>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.

Let $p_n^{(\alpha,\beta)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^\alpha(1+x)^\beta$ on $[-1,1]$, where $\alpha,\beta\ge-1/2$, and put $S=\alpha+\beta+1$. We prove the uniform degree–parameter estimate $$ (1-x)^{\alpha+1/2}(1+x)^{\beta+1/2} \bigl|p_n^{(\alpha,\beta)}(x)\bigr|^2 \le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\bigr\}. $$ This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erdélyi–Magnus–Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor $S^{1/3}$ in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss–Jacobi quadrature weights.

The Hörmander-Bernhardsson constant $\mathscr{C}$ is the sharp constant in $|f(0)|\le \mathscr{C} \|f\|_1$ for entire functions of exponential type $\le \pi$. We prove that $\mathscr{C} = 2\pi \theta_*^{-2}$ where $\theta_*$ is the least positive singularity of the regular solution $v$ with $v(0)=0$ of the cosh-Gordon equation $v_{\theta\theta} + v_\theta/\theta = \cosh(v)$. It is known that $\mathscr{C}$ is a scaling limit in $n$ from the analogous problem for polynomials of degree $\le n$. We reformulate the polynomial problem as a Padé approximation problem at infinity. The associated matrix Riemann-Hilbert problem is analyzed by a Deift-Zhou steepest descent whose local parametrix is built from a Painlevé transcendent.