Papers

Let $\mathcal H=-\Delta+|x|^2$ be the Hermite operator on $\mathbb R^2$, and let $\Pi_\lambda$ denote the spectral projection corresponding to $\lambda=2N+2$. We prove the sharp log-free endpoint estimate $||\Pi_\lambda||_{L^2(\mathbb R^2)\to L^{10/3}(\mathbb R^2)}\lesssim\lambda^{-1/10}$. The proof uses a spectral decomposition in polar coordinates and combines Koch-Tataru localized spectral projection bounds with a Liouville-Green representation, van der Corput estimates for exponential sums, and a weighted $TT^*$ argument across radial scales.

In this article we consider polynomials orthogonal with respect to the inner product $$\langle P,Q \rangle_S = \int_{\mathbb{R}}P(x)Q(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x + \lambda\int_{\mathbb{R}}P^\prime(x)Q^\prime(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x$$ where $V$ is a polynomial of even degree (at least four) and positive leading coefficient, and $\lambda > 0$ a constant. The above inner product is a special case of the so-called Sobolev inner product. We show how one can represent the associated Sobolev orthogonal polynomials as a species of Type I multiple-orthogonal polynomial. This correspondence makes use of the WKB asymptotics of a certain second order ODE. From this we may write a Riemann–Hilbert problem for the Sobolev orthogonal polynomials, from which one can deduce a Christoffel–Darboux-type formula for the projection kernel.

Let $L =-\Delta_{\mathbb{H}^n} +V$ be the Schödinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian, and $V$ is a nonnegative potential belonging to the reverse Hölder class $RH_q(\mathbb{H}^n)$ for some $q > Q/2$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. In this paper, motivated by the work of De León-Contreras and Torrea \cite{DT}, we introduce the Lipschitz spaces $\Lambda_L^\alpha (\mathbb{H}^n)$, $0< \alpha <2$, adapted to $L$ via a pointwise second-order difference condition involving the critical radius function $\rho$ related to $V$, and also introduce another type of Lipschitz spaces $\Gamma^{\alpha/2}_L(\mathbb{H}^n)$, $0< \alpha <\infty$, adapted to $L$ in terms of the heat semigroup $e^{-tL}$. We show that for $0< \alpha <2-(Q/q)$, $\Lambda_{L}^\alpha (\mathbb{H}^n) =\Gamma_L^{\alpha/2} (\mathbb{H}^n)$ with equivalent norms. Applications of $\Gamma^{\alpha/2}_L(\mathbb{H}^n)$ to the regularity of the fractional powers of the operator $L$ are also given.

We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|\beta|\leq1/2$. For the spectral scale $n+(\alpha+\beta+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.

In this article we study the analog of Kakeya needle problem for $(n-1)$-rectifiable sets in $\mathbb{R}^n$ and construct the related Nikodym type sets. The novelty of our approach lies in combining three ingredients: the two-dimensional Venetian blind-type construction for isometries in $\mathbb{R}^n$; the normal geometry of a $(n-1)$-rectifiable set viewed in the projective setting; and measure estimates of moving $(n-1)$-rectifiable sets by isometries in $\mathbb{R}^n$. Together, these ingredients enable us to move $(n-1)$-rectifiable sets along paths of isometries in $\mathbb{R}^n$ and cover a Lebesgue null set.

Recent work produced a linearly dependent system of twelve time–frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $\alpha=\frac13+10^{-12}\sqrt2$ and $\beta=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $\lambda\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(\alpha,\beta/2)f=\lambda f$, where $W$ denotes the Weyl time–frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb R)$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.

Let $Hf(x)=x^{-1}\int_0^x f(t)\,dt$ be the Hardy operator on $\mathbb R_+$. Korenovskii proved that $$ \Vert Hf\Vert_{\mathrm{BMO}}\geq \frac{e\alpha_0}{4}\Vert f\Vert_{\mathrm{BMO}}, $$ for every nonincreasing and locally integrable $f$, where $\alpha_0$ is defined by the relation $\Vert H\chi_{(0,1)}\Vert_{\mathrm{BMO}} = \alpha_0 \Vert \chi_{(0,1)}\Vert_{\mathrm{BMO}}$, and conjectured that the factor $e/4$ could be removed. We prove this conjecture by showing that every nonincreasing locally integrable $f$ satisfies $$ \Vert Hf\Vert_{\mathrm{BMO}}\geq \alpha_0\Vert f\Vert_{\mathrm{BMO}}. $$ The constant $\alpha_0$ is optimal, with equality for the one-jump functions $\chi_{(0,a)}$.

In this paper, we provide a sufficient condition under which the one-dimensional local zeta function admits a holomorphic continuation to the complex plane cut along the negative real axis.

We study random Bernoulli measures $\mu_{\mathbf{p}}$ in a random environment $\mathbf{p}=(p_1,p_2,\ldots)$, where $p_n$ are independent and identically distributed on $(0,1)$. For almost every environment $\mathbf{p}$, we establish several almost sure properties of $\mu_{\mathbf{p}}$: its local dimension, $L^q$ dimensions, Rajchman property, mutual singularity with any fixed measure, and a dichotomy for normal numbers of its typical points.

We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schrödinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schrödinger equation fails when one allows small perturbations of the paraboloid, which can be viewed as a higher-dimensional extension of the phenomenon proved by Fu, Ren, and Wang. Our approach uses new lower bounds for incidence estimates originally proven by Cairo and Zhang, for which we provide an alternative proof based on homogeneous dynamics. Moreover the estimates are essentially sharp at the decoupling endpoint for the paraboloid $p = \frac{2(d+2)}{d}$.

We exhibit nineteen explicit vectors in $\mathbb{C}^6$, with Gaussian-integer entries, that do phase retrieval: the intensities $|\langle a_j,x\rangle|^2$, $j=1,\dots,19$, determine every $x\in\mathbb{C}^6$ up to a unimodular scalar. This gives $m{\mathbb{C}}(6)\le 19=4d-5$, one below the generic count $4d-4$; combined with the lower bound $m{\mathbb{C}}(6)\ge 18$ of Wang and Xu, the minimal measurement number in dimension six is 18 or 19. The claim is verified by an exact rational computation, archived and rerunnable: after an elementary reduction, phase retrieval is equivalent to the emptiness of the real zero set of a system of five quadratic equations in five unknowns, and the archived eliminant of that system is a squarefree integer polynomial of degree fourteen with no real roots.

We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo–Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo–Nirenberg target spaces are precisely those given by the Calderón–Lozanovskiĭ space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo–Nirenberg inequalities hold, thereby answering an open question posed by K. Leśnik, T. Roskovec, and F. Soudský. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.

Building on the vector-Zak and cohomological framework developed by Faulhuber, Petersen, van Velthoven, and Voigtlaender in their twelve-point counterexample, we give a computer-assisted four-point counterexample with a nonzero complex-valued Schwartz window. Every symplectic triangle determinant of the explicit configuration has absolute value below one, placing it in the intrinsically subcritical regime.

A cyclic $n$-root is an $n$-dimensional complex vector that solves a particular set of multivariate homogeneous polynomial equations. There is a one-to-one correspondence between unimodular cyclic $n$-roots and bi-unimodular vectors (CAZAC sequences) with leading entry one. It was conjectured by Björck and Saffari that the set of cyclic $n$-roots is finite if and only if $n$ is square free. It is known that such a set is infinite if $n$ is not square free, and finite if $n$ is prime. A critical reduction in Haagerup's proof for prime $n$ is to show that infinity of cyclic $n$-roots (for any $n$) implies existence of two vectors with disjoint supports in both the time domain and the frequency domain. In this paper we show that such pair of vectors always exist if $n$ is composite, indicating that the original reduction is not adequate for composite square free cases. A discussion on the existence of a single vector whose support is disjoint with its Fourier transform is also included.

Consider the oscillatory integral operators \begin{equation} T_{\lambda}f(y)=\int_{\mathbb{R}^{2}}e^{i\lambda S\left( x_{1},x_{2}% ,y\right) }\Phi(x_{1},x_{2},y)f(x_{1},x_{2})dx_{1}dx_{2},\nonumber \end{equation} where $\Phi(x_{1},x_{2},y)\in C_{0}^{\infty}\left( \mathbb{R}^{3}\right) $, $S\left( x_{1},x_{2},y\right) \in C_{0}^{\infty}\left( \mathbb{R} ^{3}\right) $ is real valued, and $\lambda$ is a large real number. We prove that, if $S\left( x_{1},x_{2},y\right) =y^{n_{1}}h_{n-n_{1}}\left(x_{1},x_{2}\right) +\cdots+y^{n_{s}}h_{n-n_{s}}\left( x_{1},x_{2}\right) $ is a homogeneous polynomial of degree $n,$ where $0<n_{1}<n_{2}<\cdots <n_{s}<n$, and $h_{n-n_{1}}\left( x_{1},x_{2}\right) $ and $h_{n-n_{s}}\left( x_{1},x_{2}\right) $ are non-degenerate in the sense that there are no multiple factors when they are factored into linear terms over complex numbers, then for $\delta=\max \left( \frac{n}{3},\frac{n-n_{s}}{2},n_{1}\right) >1,$ $\left \Vert T_{\lambda}\right \Vert _{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/\left( 2\delta \right) }\right) $, while in the endpoint case $\delta=1$ the bound becomes $\left \Vert T_{\lambda}\right \Vert_{L^{2}\rightarrow L^{2}}=O\left( \lambda^{-1/2}\log \lambda \right) $. The decay rate is sharp, up to a power of $\log \lambda$ when $\delta=1$. We further show that $\delta$ is exactly the modified Newton distance for the phase function, thus verifies the conjecture of \citet{Greenleaf07} in this case.

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that \[ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha} <1. \] By his variation formula, this inequality is sufficient for every positive zero of $\pi_n$ to move to the right as $c$ increases. For $0<c<1$, a first-crossing argument proves the conjecture throughout $0<\alpha<\beta$. Together with the earlier regions recorded by Gautschi and established by Milovanović, this settles $\beta\geq0$. In the negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, an ensemble Ward bound yields the region $c^2\leq3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\geq1$, $-1<\alpha<\beta$, and $0<c\leq1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.

We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only Hölder regular in space—$C^{\alpha-}(\mathbb{T}^d)$ for some $\alpha \in (0,1)$—thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $\alpha = 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.

Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one $q\times p$ matrices of weights, with arbitrary $p$ and $q$: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at $(x,1)$. For the mixed beta–Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer $G$-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with $p$ subdiagonals and $q$ superdiagonals. All recurrence coefficients are given by finite Gamma–Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss–Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.

We study strong asymptotics of orthogonal polynomials on an interval with varying weights, extending the framework of Totik's theorem on weighted orthogonal polynomials. Our results generalize this theorem in several directions. In particular, we allow the supports of the associated equilibrium measures to converge to a proper subinterval of the original interval of orthogonality or even collapse to a point. These extensions are motivated by applications to multiple orthogonality where such varying measures arise naturally.

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

We prove that the set of time-frequency shifts $\{e^{2\pi i \beta l t} g(t-\alpha k) : k,l \in \mathbb{Z}\}$ with a continuous, integrable totally positive function $g$ and lattice parameters $\alpha,\beta>0$ generates a frame for $L^2(\mathbb{R})$ if and only if $\alpha\beta<1$. This fully settles the so-called frame set problem for the class of totally positive functions. As a closely related result we prove a sharp Kadets-type theorem for every shift-invariant space generated by a continuous totally positive function. The proofs are based on Fredholm theory and limit-operator theory. A formalization of our main result in Lean 4 is also provided.

In this paper, we disprove a conjecture of Liu, Lou, and Zhu concerning Carleson embeddings of $\mathcal Q_p$ spaces into tent spaces for $0<p<1$. More precisely, we construct a finite positive $p$-Carleson measure $\mu$ on the unit disc $\mathbb D$ such that the canonical embedding $$ \operatorname{id}:\mathcal Q_p \longrightarrow \mathcal T_{p,2}^2(\mu) $$ is not bounded. The main ingredient is a new family of $\mathcal Q_p$ test functions that encodes Cantor-type structures on the unit circle $\mathbb T$ into the analytic behavior of functions in $\mathcal Q_p$. This construction is inspired by ideas developed in a recent work of the first and third authors on composition operators on $\mathcal Q_p$ spaces.

We prove that, for every $n\ge1$, the degree-$n$ Bernstein operator on $[0,\pi]$ preserves positive-definiteness on the circle $S^1$. Equivalently, if a continuous function on $[0,\pi]$ defines a positive-definite isotropic kernel on $S^1$, then its Bernstein polynomial approximation of any fixed degree does as well. The proof reduces the problem to the nonnegativity of the cosine coefficients of the Bernstein images $Q_{n,m}=B_n[\cos(mx)]$, which we prove using an explicit coefficient formula and a two-regime positivity argument. We also discuss the higher-dimensional sphere analogue and show that the naive affine Bernstein operator fails to preserve the positive-definite cone already on $S^2$.

On the $q$-quadratic lattices that carry the Askey–Wilson polynomials, the two roles played by a single Bernstein basis on the linear lattice, namely being a nonnegative partition of unity and carrying an orthogonal connection with the classical families, split between two distinct bases. The affine (spectral) basis retains the Bézier properties. A coprimality argument completely classifies the first-order lattice ladder: its solution space has dimensions two and one in degrees one and two, respectively, and is trivial in every degree at least three for every $q\in(0,1)$. The affine factors do not satisfy the weight-shift mechanism that produces the $q$-Racah connection for the generalized-power basis. The generalized-power basis, built from products of two Askey–Wilson monomials, does not furnish a nonnegative partition of unity in a natural positive parameter region, specified in the text; there the unique normalisation summing to unity has sign-changing coefficients. Nevertheless, its connection coefficients with the Askey–Wilson polynomials are identified completely: they are $q$-Racah polynomials $R_m(\mu(k);\,ad/q,\,bc/q,\,q^{-n-1},\,a/b\,|\,q)$ multiplied by fully explicit prefactors. The proof is bispectral, valid for every degree, and rests on the tridiagonal action of the Askey–Wilson operator on the generalized-power basis, with explicit band coefficients. A scaled limit recovers the Wilson/Racah connection, while a second limit gives the $q$-Hahn polynomial part of the normalised $q$-linear connection coefficients. A numerical case study on the NACA 2412 airfoil illustrates the orthogonal machinery.

Bounded ratios of products of minors of positive definite matrices have a long history, starting with Hadamard's inequality in 1893. It states that for every positive semidefinite matrix $A$ $$ \det A \le A_{11} \cdots A_{nn}. $$ This inequality was subsequently generalized by Fisher and then further by Koteljanskii. The latter states that for every positive semidefinite matrix $A$ and any index sets $\alpha_1, \alpha_2 \subseteq \{1,\ldots, n\}$ one has $$ \det A[\alpha_1 \cup \alpha_2] \det A[\alpha_1 \cap \alpha_2] \le \det A[\alpha_1] \det A[\alpha_2], $$ where $A[\alpha]$ denotes the principal submatrix determined by the indexes in $\alpha$. In a manuscript published only on the arXiv in 2008, Hall and Johnson made three conjectures about ratios of products of principal minors of $4\times4$ positive definite matrices, denoted by $R_i$, $i=1,2,3$, see (2) and (3). They hypothesized that the supremum of $R_1$ was $27/16$, while the supremum of the other two ratios was $1$. Such ratios are called absolutely bounded. The conjecture for $R_1$ was affirmed in [17] and it is the only known bounded determinantal ratio with supremum bigger than one. The goal of this paper is to affirm the conjecture for $R_2$ and $R_3$. It is known that the upper bound for the ratios $R_i$, $i=1,2,3$, does not follow from repeated applications of Koteljanskii's inequality. In addition, Hall and Johnson showed that $R_i$ is bounded above by $4$, for $i=1,2,3$.

We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.

We develop a rigorous framework for the kinematics of a material point whose trajectory contains a jump discontinuity, modeled as a smooth curve on an interval punctured at one instant, with finite one-sided limits of position and velocity. A position jump is called an instantaneous teleportation. The event is treated as exogenous to Newtonian dynamics, imposed only before and after the removed instant. Separating the position and velocity discontinuities, we prove that the net linear impulse equals the momentum jump $\mathbf J=m\,\Delta\mathbf v$, carried entirely by the velocity discontinuity, while the position jump contributes none. This yields a momentum-compatibility condition: an instantaneous displacement transfers no net impulse if and only if the velocity is preserved. For rigid bodies it splits into preservation of translational and angular velocity. For the rotating, revolving Earth, a kinematic fact governs the estimates: for two points of the same Earth at the same instant the orbital velocity cancels, bounding the jump by $2\omega R_{\oplus}\approx 9.3\times10^{2}\ \mathrm{m\,s^{-1}}$; the larger scale $6.0\times10^{4}\ \mathrm{m\,s^{-1}}$ arises only between orbital epochs or frames. The internal stress depends on how momentum is delivered, and the implied strains lie outside linear elasticity, so the figures are indicative.

New sharp bounds for the product and the ratios of modified Bessel functions are presented. Most bounds for the product are derived as direct consequences of previously established bounds for the ratios of consecutive orders, except for the lower bound $I_\nu(x)K_\nu(x) > \frac{1}{2}(x^2 + \nu^2 + 1/5)^{-1/2}$, which had been conjectured for $x > 0$ and $\nu > -1$ and we prove in the present paper, showing that the constant $1/5$ can not be lowered. Moreover, very sharp bounds are obtained for the ratios (and consequently for the product) by asymptotically optimizing certain uniparametric inequalities. These optimized bounds are remarkably accurate: they remain extremely sharp for both small and large $x$ with fixed $\nu$, and for large $\nu$ with fixed $x$ or fixed $z = x/\nu$. As a consequence, they provide precise upper and lower estimates across a wide range of parameters.

We give an analytic construction of a tau function for the $q$-difference sixth Painlevé equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.

We study long-time Strichartz estimates for the one-dimensional Schrödinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schrödinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint $L^4$ Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell–Vinogradov systems.