Papers

We study Brownian motion killed upon exiting arbitrary open sets $\Omega\subset\mathbb R^n$, $n\in[3,\infty)\cap\mathbb N$. Let $\alpha\in(0,1]$. The infimal radius $d_\alpha(x)$ at which the absorber near $x$ carries an $\alpha$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_n\alpha$ within a dimension-dependent multiple of the diffusive time scale $d_\alpha(x)^2$. Independently, if $p\in (0,1)$ and $\tau_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \begin{align*} \int_\Omega\frac{|u\left(x\right)|^2}{\tau_p\left(x\right)}\,\mathrm d x \le\frac{2}{p^2}\int_\Omega|\nabla u\left(x\right)|^2\,\mathrm d x \end{align*} for every $u\in C_{\mathrm{c}}^\infty(\Omega)$. Together these statements give a capacitary Hardy inequality with constant $C_n\alpha^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $\alpha^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya–Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.

We develop a whole-trajectory relaxation framework for dissipative evolutions with non-convex, state- and history-dependent kinetics. The framework arises naturally from a game-theoretical formulation of energy–dissipation evolution. State–rate consistency is eliminated by causal Volterra reconstruction, reducing the problem to a diagonal equilibrium condition on complete rate trajectories. We establish a direct method based on vanishing-defect approximability, compactness, and complementary semicontinuity. The relaxation mechanism is computed exactly for a reduced simple-shear model with a non-convex kinetic potential and a local memory variable. The enriched state is a Young measure on complete local histories. We identify explicitly the sequential lower-semicontinuous envelope of the pure diagonal defect, characterize its zero set, construct pure recovery sequences for every relaxed zero, and prove existence through a frozen-best-response time discretization. The resulting coexistence of distinct rate populations with distinct memories provides a reduced mechanism for kinetic shear banding and connects naturally with negative strain-rate sensitivity and dynamic strain aging such as underlie the Portevin–Le Chatelier effect.

It is known that $q$-Painlevé equations admit the symmetry of the extended affine Weyl groups. Kajiwara, Noumi and Yamada developed a unified description of the symmetry by introducing representations of the extended affine Weyl groups. On the other hand, a parallel translation associated with the extended affine Weyl groups describes the time evolution in Sakai's theory of discrete Painlevé equations. In this paper, we introduce the equivalences to the representations of Kajiwara, Noumi and Yamada. The equivalences imply reducibility of the representations, and they are related to the expressions of the parallel translations of the $q$-Painlevé equations.

Point-mass perturbations and finite-difference operations alter the finite-degree structure of discrete orthogonal polynomials, but their combined effect at the Mehler–Heine scale is not immediate. We study the monic Charlier and Meixner families under a pure Uvarov modification by a fixed mass $A\delta_{0}$ at the endpoint of the support and determine the asymptotic behaviour of their forward and backward differences of arbitrary fixed order. Using a rank-one connection formula, we show that the perturbation coefficient decays factorially in the Charlier case and exponentially, with an algebraic prefactor, in the Meixner case. In both families this decay is faster than every algebraic power of $n^{-1}$ and therefore suppresses the growth produced by any fixed number of finite differences. Consequently, for every fixed $A\geq0$ and $k\in\Nzero$, the mass-modified and classical families have the same locally uniform Mehler–Heine limits in $\C$. Forward differences preserve the reciprocal-Gamma profile up to the factor $(-1)^k$, whereas backward differences translate the limiting argument by $k$, shifting the limiting zero lattice from $\Nzero$ to $k+\Nzero$. We also derive a first-order shift equation for the common forward limit and characterize its entire solution space. Complex-plane portraits and real-axis computations illustrate the predicted limiting profiles, the two zero lattices, and the asymptotic disappearance of the fixed endpoint mass. These results identify a scale-separation mechanism governing the fixed-order asymptotic stability of the Charlier and Meixner families under rank-one endpoint perturbations.

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d \gamma)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.

We prove optimality of the differentiability bound for isotropic positive definite functions on every even-dimensional sphere. If the even continuation of such a function on the $d$-dimensional sphere is $2k$ times differentiable at zero, then the function has $2k+\lfloor(d-1)/2\rfloor$ continuous interior derivatives; previously, optimality was known only in odd dimensions. We construct a function on the two-dimensional sphere whose first derivative does not exist at the equator and transfer it to all even dimensions by turning bands and spherical montée. The resulting examples are strictly positive definite, have $2k$ but not $2k+2$ derivatives at zero, and are not positive definite in the next dimension.

Lacey used sparse domination to study the sharp weighted norm estimate of the maximal function of predictable multipliers in discrete time filtration spaces. Domelevo, Petermichl, and Škreb developed the self similarity argument known as sparse domination in an abstract martingale setting with a continuous time parameter. In our investigation, we establish sparse domination for discrete-time martingale transforms, introducing the novel concept of conditional sparsity as a core property of our approach. The conditional sparsity framework enables derivation of sharp weighted estimates and a mixed-norm estimate \( A_p^\alpha A_r^\beta \) that improves upon known sharp \( L^p \) bounds. Moreover, we develop dedicated sparse domination specifically for Doob's maximal operator, recovering the sharp bound as a direct application. Finally, we focus on the application of sparse theory to quantitative two-weight estimates.

In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^{\alpha}_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^{\alpha}_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an admissible range of exponents, with the required smoothness parameter $\alpha$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.

For every odd integer $d\geq3$, a continuous function $\varphi\colon[0,\infty)\to\mathbb R$ supported in $[0,\pi]$ and isotropic positive definite on $\mathbb R^d$ remains so on $\mathbb S^d$. In even dimensions, recent work shows that this transfer fails under every prescribed positive upper bound on the support. We prove an exact-support refinement with a construction uniform in the prescribed radius. More precisely, for each $d=2m\geq2$ and $R\in(0,\pi]$, we construct a function $\varphi$ whose radial extension belongs to $C_c^\infty(\mathbb R^d)$ and has support radius exactly $R$, such that $\varphi(\|\mathbf{x}-\mathbf{y}\|_2)$ is strictly positive definite on $\mathbb R^d$, whereas $\varphi(\rho(\mathbf{x},\mathbf{y}))$ is not positive definite on $\mathbb S^d$. Thus every admissible support radius is attained by a smooth, strictly Euclidean positive-definite counterexample.

A sharpened version of Carlson's integral inequality is established, which features a remainder term involving a relative distance to the set of extremal functions. A similar approach leads to an alternative derivation of Carlson's and Landau's discrete inequalities and provides another characterization of the optimal constant in Carlson's inequality for finite sums.

The large-degree phase of the Jacobi polynomial $P_n^{(\alpha,\beta)}(\cos\theta)$ contains, at each order in $\Nn^{-1}$, $\Nn=n+(\alpha+\beta+1)/2$, an additive constant $\kappa_r$ that the phase equation does not determine. These constants are antisymmetric in $(\alpha^2,\beta^2)$ and vanish for $\alpha=\beta$. We obtain three results. First, on the diagonal $\alpha=\beta$ the logarithm of the midpoint amplitude is computed in closed form: if $L_n$ denotes the coefficient of $(2\Nn)^{-n}$ in that logarithm, then $\Upsilon_A(t)=\sum_{n\ge1}L_nt^n/(n-1)!$ equals \[ \Upsilon_A(t)=\frac{e^{t}\bigl(\cosh2\alpha t-\cosh t\bigr)}{\sinh 2t}. \] The proof is algebraic: the identity is established for half-odd-integer $\alpha$, where the expansion terminates, by means of two classical recurrences for Bessel polynomials, and is extended to all $\alpha$ by a degree argument. Second, the derivative of the logarithm of the amplitude with respect to $\alpha^2$ at the Legendre point $\alpha=\beta=0$ is computed in closed form to all orders. An exact reduction of an Appell $F_3$ function to a Gauss function shows that it is a parameter derivative of a zero-balanced ${}_2F_1$ at argument $2$; the real part of its expansion is a trigamma series, and the imaginary part is the unique formal solution of an explicit central-difference equation. Third, we deduce that the coefficients $w_m$ of an explicit even power series attached to the imaginary part satisfy $\nu_2(w_m)=-2\nu_2((2m)!)$ for every $m\ge0$ ($\nu_2$ the $2$-adic valuation). This determines, for every $r$, the exact power of $2$ in the denominator of the coefficient of $\alpha^2$ in the polynomial that represents $\kappa_r$. We also give a factorisation of the amplitude into products of pairs of Gauss functions, further valuation formulas, and a numerical illustration for the zeros.

We show Lagrange and Hermite interpolation are possible using Stieltjes polynomials, linear combinations of iterated Stieltjes integrals of a constant function. We introduce divided differences via Newton interpolation and provide an explicit error formula of Peano kernel type via a new Taylor formula. Finally, we use the properties the space of Stieltjes polynomials has to recover two fundamental approximating properties: the uniqueness of the best uniform polynomial approximant and the Chebyshev equioscillation theorem.

For $N$ lattices in $\R^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $\Omega(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained open since the work of Kolountzakis and Wolff \cite{kolwolff-1999Mathematika}. We close this gap by constructing, for every $d\geq 2$ and all sufficiently large $N$, lattice families satisfying the same volume and intersection conditions that admit a nonnegative common tiling function with support diameter $O(N^{1/d})$, thereby also answering Question 1 of Kolountzakis and Papageorgiou \cite{kolPapageorgiou-functions-2022jfaa}. We also obtain the optimal $O(\sqrt N)$ upper bound by constructing, for any prescribed family of plane lattices whose volumes lie in a fixed bounded set independent of $N$, a pairwise trivially intersecting family with the same respective volumes and with bases arbitrarily close to suitable bases of the prescribed lattices.

We study pointwise convergence of polynomial ergodic averages over short intervals whose left endpoints tend to infinity. For a polynomial orbit of degree $d\geq2$ and doubly lacunary starting times, we prove $L^p$ variational estimates, and hence almost-everywhere convergence, for $1<p<\infty$ in the range $c>(d-1)/d$. This gives the first pointwise ergodic theorem for polynomial orbits along short intervals. We also show that the endpoint $L^1$ fails along every infinite subsequence. In a different direction, we prove that substantially denser sequences of starting times exhibit the strong sweeping-out property.

We prove that sparse domination of a bilinear form implies convex body domination. More precisely, if a bilinear form admits an $(r,s)$-sparse bound, then its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound. The proof relies on a randomization argument. We establish the result both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application, we deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.

In this paper, we consider a class of first-order abstract retarded functional differential equations in Banach spaces, incorporating a time-dependent infinite delay governed by a regulated function. We establish the existence of mild solutions for the nonlinear equation and show that the family of solution maps for the linear equation forms a well-defined evolution family of bounded linear operators on an appropriate phase space. Furthermore, we leverage this evolution family to prove a variation of constants formula for the inhomogeneous linear problem.

We study the connection between AAA (triple A) algorithm and multi-point Padé approximation when the approximated functions belong to either the Nevanlinna-Pick class or are Pólya frequency series.

In this paper, we establish a connection between joint moments of characteristic polynomials and their derivatives in the Circular Jacobi Ensemble, a generalisation of the Circular Unitary Ensemble, and solutions of nonlinear Painlevé equations. For finite $N$, we show that these joint moments are characterised by a solution of the $\sigma$-Painlevé V equation for all real moment exponents in their admissible range. Under an appropriate large-$N$ scaling limit, we further prove that the limiting joint moments admit a representation in terms of a solution of the $\sigma$-Painlevé III$'$ equation for a certain range of moment exponents. As applications, we answer a question posed by Assiotis et al. in [Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7] by showing that the characteristic function of a distinguished random variable is connected with the $\sigma$-Painlevé III$'$ equation for complex parameters. Furthermore, we investigate joint moments involving higher-order derivatives of characteristic polynomials in the Circular Jacobi Ensemble. As a consequence, we extend a result of Assiotis et al. in [Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11] concerning the joint moments of a sequence of random variables arising from the ergodic decomposition of Hua-Pickrell measures, from real parameters to complex parameters.

We give a roadmap for the study of critical fractional Hardy inequalities for Dirichlet and Sobolev–Bregman forms. We focus on homogeneous forms on the real line, in particular the form of the fractional Laplacian and the Servadei–Valdinoci form. We prove criticality results in both the Dirichlet and Sobolev–Bregman settings.

We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.

Let $d \ge 2$ and $1 \le k \le d-1$. We show that if $f:\mathbb{R}^d\to\mathbb{R}$ is nonnegative and measurable and $0<m\le M <\infty$ then it is impossible that on almost all affine $k$-planes $P$ in $\mathbb{R}^d$ the integral of $f$ on $P$ lies between $m$ and $M$. Let $G = \mathbb{Z}^k \times \{0\}^{d-k}$. Using $m=M=1$ and $f$ being the indicator function of a measurable set in $\mathbb{R}^d$ this then implies that there is no measurable Steinhaus set for the group $G$ in $\mathbb{R}^d$. In other words there is no measurable set $S \subseteq \mathbb{R}^d$ such that $S$ tiles $\mathbb{R}^d$ with $T(G)$, for all $T \in O(d)$. We also generalize our impossibility results concerning the size of line-integrals of functions to integrals on strips in the plane.

The classical Borell–Brascamp–Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ \lambda = ( \lambda_1 , \dots , \lambda_m ) $, we set $ z_\lambda ( x ) := \sum_{ i = 1 }^m \lambda_i x_i $ and denote by $ M_p^\lambda $ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell–Brascamp–Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa–Leindler inequality of Cordero-Erausquin–Maurey.

Using Gaussian heat-kernel estimates, we classify planar Dunkl Newton kernels in weighted Euclidean BMO and obtain the corresponding local classification for unit-ball Green kernels. For atomic Newton potentials supported on a regular reflection orbit, orbit BMO detects exactly whether the coefficients are constant along the orbit. We also prove an intrinsic Dunkl–CLMS theorem in the heat-semigroup Hardy space and derive intrinsic and Euclidean-source Newton–Wente estimates, extending the BMO–Hardy-space method of Chanillo and Li to the Dunkl setting. Further consequences include sharp local atomic estimates, same-domain criteria, a localized $L^1$-to-BMO bound, and a Brezis–Merle-type estimate.

Although the Heil–Ramanathan–Topiwala conjecture is false in full generality, its failure makes the classification of positive geometric and arithmetic regimes more urgent. We settle the critical mixed-arithmetic regime for four time-frequency shifts. For \(z=(x,\omega)\) and \(w=(y,\eta)\), set \(\sigma(z,w)=x\eta-y\omega\). Let \(u,v\in\R^2\) satisfy \(\lvert\sigma(u,v)\rvert=1\), write \(\nu=\alpha u+\beta v\), and assume that \(0,u,v,\nu\) are distinct. If \(\dim_{\Q}\operatorname{span}_{\Q}\{1,\alpha,\beta\}=2\) then, for every nonzero \(f\in L^2(\R)\), the vectors \(f,\pi(u)f,\pi(v)f,\pi(\nu)f\) are linearly independent. This includes both four-point configurations in Chris Heil's Conjecture~9.2. The proof first converts a putative dependence into a scalar cocycle over an irrational rotation. On a positive-measure family of zero-free fibres, winding and continued-fraction returns force the periodic holonomy to be constant. The return multiplier is Laurent polynomial, however, so its ungauged holonomy is algebraic; the constant-holonomy law simultaneously makes it an irrational character. These conclusions are incompatible. The principal theorem and its complete formal dependency chain have been fully certified end-to-end in Lean~4, including the physical all-nonzero case, the complete three-point HRT theorem, the coefficient-reduction step, and the final four-point conclusion. The accompanying \href{https://www.dropbox.com/scl/fi/jd9xeqqera147wdc82r8k/3f260460-07d5-45f5-8d1c-ee1d2879c0af-aristotle-32-.tar.gz?rlkey=a24njadfds5ty9ez473lk03lv&e=1&dl=0} {Lean~4 certification archive} contains reproducible source, pinned build instructions, and transitive axiom audits.

This monograph investigates the Lebesgue boundedness of planar directional maximal operators $D_{\Omega}$. These are maximal averages of functions over line segments in $\mathbb R^2$ whose slopes lie in a specified set $\Omega\subseteq\mathbb R$. A large body of work has identified a geometric property of $\Omega$, called finite-order lacunarity, as a key factor in ensuring that $D_{\Omega}$ is Lebesgue bounded. While several variations of this notion exist, they all centre on the distribution of gaps in $\Omega$. Building on earlier work, an article of Bateman(2009) asserted a dichotomy for such operators. Namely, $D_{\Omega}$ is bounded on $L^p$ for all $p\in (1,\infty)$ precisely when the slope set $\Omega$ is finite-order lacunary, or equivalently, when $\Omega$ does not admit Kakeya-type sets. Conversely, sublacunary direction sets $\Omega$ admit Kakeya-like phenomena, implying that $D_{\Omega}$ is unbounded on $L^p$ for all $p\in [1,\infty)$. Recent work of Hagelstein, Radillo-Murguia, and Stokolos(2024) identified a gap in the proof of this assertion and produced counterexamples for which the separation mechanism underlying that proof fails, demonstrating the need for a corrected framework. We establish the corrected characterization by introducing a new notion of admissible finite-order lacunarity that faithfully reflects the combinatorial structure of the direction set. This leads to a tree-theoretic characterization in terms of finite splitting number and provides the foundation for new geometric and probabilistic constructions establishing the equivalence between finite-order lacunarity, the absence of Kakeya-type sets, and the boundedness of directional maximal operators. The resulting framework not only resolves the gap in the earlier proof, but also identifies admissible finite-order lacunarity as the structural invariant governing these phenomena.

We prove a Fefferman–Stein good-$\lambda$ inequality for the Dunkl Poisson semigroup associated with a finite reflection group and a non-negative multiplicity function. For arbitrary complex-valued $f\in C_c^\infty(\mathbb R^N)$, with no $G$-invariance assumption, it compares the orbit-conical non-tangential maximal function $\mathcal N_P^\beta f$ with the area function $\mathcal S_Pf$ formed from the full space-time Dunkl carré du champ, including its reflection-difference energy. The main obstruction is that a general cut-off creates wall differences not controlled by the local Euclidean-gradient product identity. The good set $E=\{x:\mathcal N_P^\beta f(x)\le\lambda\}$ is $G$-invariant; by the equivariance of the Poisson semigroup, so is $a=\varphi(P_t \mathbf 1_E)$, and hence all reflection differences of the cut-off vanish. Poisson maximal and tail estimates, together with the $L^2$ Littlewood–Paley estimate for $P_t \mathbf 1_{E^c}$, then yield the desired distribution inequality. Its integrated form gives maximal-to-area estimates for every $0<p<2$ and endpoint $H^1$-to-$L^1$ bounds for the orbit-conical and Euclidean-conical intrinsic area functions. For chamber lifts of globally smooth data, the inequality has an equivalent formulation on a fundamental chamber, where orbit cones become Euclidean cones and the reflection energy becomes a finite wall coupling. Combined with the known semigroup square-function characterization, these bounds characterize the Dunkl Poisson maximal Hardy space among $L^1(d\omega)$ data.

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\ll(\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki–Kwaśnicki–Małecki–Stós and Kwaśnicki, and confirms the conjectural $O_\alpha(n^{-2})$ remainder suggested by the numerical simulations of Kaleta–Kwaśnicki–Małecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $\alpha$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.

We use Morse Theory to prove that the Burkholder function is quasiconvex on symmetric matrices, which implies several sharp inequalities for the Beurling–Ahlfors transform when acting on real-valued functions. More generally, we prove that the separating functions introduced by Székelyhidi are quasiconvex on symmetric matrices.

We define the notion of regular parabolic (bi-)Lipschitz images as the parabolic (bi-)Lipschitz maps from $n$-dimensional space time which, up to translation, fix the $t$ variable and whose spatial components are each regular parabolic Lipschitz functions. We show that any parabolic Ahlfors-David regular is parabolic uniformly rectifiable if and only if it has big pieces of parabolic Lipschitz images of $n$-dimensional space time if and only if it has big pieces of parabolic bi-Lipschitz images of $n$-dimensional space time. This further extends the David-Semmes theory to the parabolic setting. Our proof combines the ideas of the first authors previous work [BH12,BHH+22] and some ideas of Azzam and Schul [AS12]. The proof easily adapts (and is far less complicated) to the Euclidean case to give an alternative proof of the analogous fact.