Papers

The unfair $0$–$1$ polynomial conjecture asks whether a factorization \[C(x)=A(x)B(x),\] with $A$ and $B$ monic and having nonnegative real coefficients, and $C$ a polynomial with all the coefficients 0 and 1, must already be a factorization into $0$–$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that \[1+a x^2+x^k\] divides a $0$–$1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case $k=5$. We prove that no such factorization exists for any odd $k\ge 341$. The intermediate cases (5 < k < 341) are treated in the companion paper.

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function, set $s(t)=\Phi(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later restated the assertion as Open Problem~4.14. We prove the conjecture by two complementary methods, sharing only a local certificate near the modular fixed point. The first proof is a direct theta-series argument: a directed-rounding Taylor certificate near $t=0$ is joined to a dominant-first-summand estimate with rigorous theta-tail bounds. The second proof uses Jacobi's nonlinear third-order differential equation for $\theta_3$ to obtain a three-dimensional autonomous phase space, a sharp elliptic monotonicity theorem, and an exact quartic reduction of the target inequality. The quartic boundary is a polynomial shear of the quadratic cone $XY=Z^2$. A directed interval certificate proves that every possible cone contact on the only remaining compact interval points strictly into the desired region; beyond that interval a pointwise monotonicity theorem closes the argument. The finite certificates and exact symbolic checks are supplied as reproducible scripts. The result implies the associated double Turán inequalities through the theorem of Csordas–Dimitrov, but no assertion of the Riemann Hypothesis is made.

The goal of this paper is to prove an upper bound for the $L^4$ norm of a trigonometric polynomial whose spectrum is a nontrivial strictly monotone polynomial with integer coefficients of degree three and higher, via its $L^2$ norm. Our condition on the coefficients of the trigonometric polynomial in question is that they form a complex sequence whose modulus is decreasing. Second, we obtain a similar result in the case where the spectrum is formed by perfect squares, under a more strict condition on the coefficients. Our results strengthen and complement those by S. Bochkarev and A. Córdoba. We also give an answer to a conjecture of Eceizabarrena and Da Rocha and determine the sharp order of growth of the $L^4$ norm of the trigonometric polynomial in this conjecture.

We develop additive-energy estimates and weighted Young inequalities for finite sets on bounded-degree real algebraic varieties. For an irreducible $m$-dimensional variety $V$, let $\sigma(V)=2m-\dim\overline{V-V}^{\mathrm{Zar}}$ and $\alpha(V)=\max{2,1+\frac{2\sigma(V)}{m}}$. For every $a\in[\alpha(V),3)$ we define a finite-degree translation-partition flag parameter $\Lambda_{a,R}(X;V)$ and prove $E(X)\ll \Lambda_{a,R}(X;V)^{3-a}|X|^{a+\varepsilon}$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}^3$. For codimension-two quadratic threefolds ${(u,Q_1(u),Q_2(u))\in\mathbb{R}^3}\subset\mathbb{R}^5$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|^{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L^4$ restriction bounds and off-diagonal Young inequalities; at the near-diagonal threshold the sharp region is $1\le p,q\le 2$ and $p^{-1}+q^{-1}\ge 1$. We also prove a sharp turning-complexity extension of the Cushman-Demeter-Wu theorem: $J_3(P)\ll_{\varepsilon}\kappa(P)^2|P|^{3+\varepsilon}$, with matching examples at every power scale.

Erdős Problem 1154 asks whether every number in $[0,1]$ occurs as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$. Motivated by this problem, Liang Yu asked whether the reals of an inner model can have Hausdorff dimension strictly between zero and one when their dimension is computed in an outer model. We prove a stronger result: if $\mathcal I \subseteq 2^\omega$ is any Turing ideal, then $\dim_{\mathrm H} \mathcal I \in \{0,1\}$. Equivalently, the real-closed field whose reals have Turing degrees in $\mathcal I$ has Hausdorff dimension zero or one. The proof combines digit interleaving with the Furstenberg-set theorem of Orponen and Shmerkin.

Representations of the Kohno-Drinfeld Lie algebra associated with several classical multivariate distributions have played an important role in recent developments in the theory of quantum superintegrable systems. In this work, we analyze the corresponding Gaudin models for the multivariate Hahn, Dirichlet, multinomial, and negative multinomial distributions. More precisely, using tools from representation theory, we construct multivariate orthogonal polynomials with respect to these distributions as common eigenfunctions of Gaudin operators. The polynomials are parametrized by solutions to the Bethe ansatz equations or, equivalently, by the roots of Heine-Stieltjes polynomials.

Maximal-density factoring must often make one parent object carry several forms of information at once: child mass, two multiplicity levels, a density estimate, and the ability to survive later geometric refinements. These properties are not stable under arbitrary deletion. We introduce a cellular factoring method that places positive child mass on a weighted bipartite graph of labelled parents and half-open spatial cells. After regularizing edge weights and cell degrees, one edge set defines both the child shading and the parent shading. Compatibility and pointwise multiplicity bounds are then exact, while any later selection of whole parent-cell incidences lifts quantitatively back to the children. The combinatorial core does not require Euclidean geometry: it extends to finite measurable atomic partitions with comparable atom measures and admits an explicit finite-iteration loss. We combine this joint construction with a weighted planar incidence estimate, a resolution-limited cellwise angular selection, a labelled slab estimate, and a global synchronization of tubelet and fine-tube mass. The resulting rectangular parent estimate has shading exponent $2-\sigma$, which is sharp uniformly over the class considered here. With the stated GWZ factoring data and corresponding $K_F(\beta)$ or $K_{KT}(\beta)$, we obtain the two small-middle interfaces used in the reduction. For the very-non-sticky branch, assuming both $K_{KT}(\beta)$ and $K_F(\beta)$, the stated GWZ Lemma 9.1 hypotheses together with the source-uniformity and refinement interface (I1)–(I4) yield the terminal gain for the represented fine-tube input; this conditional application supplies the corresponding input to the surrounding reduction, while the sharp-convex-parent problem lies outside its scope.

We consider evolution of piecewise linear curves with boundary by linear hyperbolic curvature flows. Given appropriate boundary behaviours we find corresponding self-similar solutions. More generally, given any specific boundary trajectories we write down general solutions using finite Fourier expansions. Finally we consider the problem of evolving one piecewise linear curve to another using nonhomogeneous linear hyperbolic flow.

Multiplicity factoring is usually formulated for child families at comparable scales. For children of mixed geometry, thickening at the shortest parent scale produces nonuniform inflation ratios, and a single worst-case replacement does not preserve the natural density normalization. We prove a multiplicity-factoring theorem for finite indexed convex parent–child families in $\mathbb{R}^4$ that accommodates arbitrary child shapes, scales, orientations, aspect ratios, and repetitions. The local geometry of each assigned family is encoded by a thickening-weighted Frostman coefficient and a mean-normalized inflation efficiency, both determined by the base family before any shading refinement. The resulting coarse density satisfies $\mathcal{L}^{-A_\varepsilon}(w_1/w_4)^\varepsilon \mathfrak{P}_\varepsilon\lambda^{K_\varepsilon}$, up to the stated parameter-dependent constant, where $\mathfrak{P}_\varepsilon$ is an explicit profile of the parent loads and local efficiencies. The proof thickens arbitrary measurable shadings, projects along a shortest parent direction, establishes an indexed three-dimensional convex-union estimate in the presence of collisions, and lifts the resulting density response back to four dimensions. A weighted Hölder inequality then assembles the nonuniform parent data, while a common cellular refinement regularizes the fine and coarse multiplicities and yields the stated parentwise multiplicity-product estimate. Under a relative convex Frostman hypothesis, the profile is expressed explicitly in the minimum, mean, and maximum inflation ratios. The comparable-scale regime follows as a specialization after a preliminary load selection. Thus the theorem supplies a structural factoring input for four-dimensional overlap arguments; deriving new Kakeya maximal or Hausdorff-dimension estimates would require additional analytic ingredients.

Let $N\ge1$, $p\in[1,\infty)$, $\gamma\in(0,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any $\lambda\in(0,\infty)$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{\lambda,p,\gamma}(u;\Omega) :=\lambda\iint_{\Omega\times\Omega} \mathbf 1_{\left\{(x,y)\in\Omega\times\Omega:\ x\neq y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+\gamma}}\geq\lambda\right\}} |x-y|^{\gamma-N}\,dx\,dy. \end{align*} In this article, we prove that, as $\lambda\to\infty$, the family $G_{\lambda,p,\gamma}$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega)$, to the functional \begin{align*} \Psi_{p,\gamma}^{\mathrm{cell}}(u;\Omega):= \begin{cases} C_{N,p,\gamma}^{\mathrm{cell}}\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^{\mathrm{cell}}|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[1mm] \infty,&\hbox{otherwise}, \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma}^{\mathrm{cell}}$ are independent of $\Omega$ and characterized by a cell formula. This gives an affirmative answer to the problem posed by Brezis [Open Problem~9.3, Rend. Lincei Mat. Appl. 2023].

In this note, we revisit one of Erdélyi's product formulas for the univariate confluent hypergeometric function ${}_{1}F_{1}$ and extend it to the multivariate confluent hypergeometric functions $\Phi_2^{(k)}$, $\Psi_2^{(k)}$, as well as to a certain matrix-variate confluent hypergeometric function. A useful connection related to fractional calculus is also given.

For non-integer $\theta > 3$ and $\kappa \geq 1$, we show that the smallest $r_0$ such that the mean value estimate \[ \int_{-\kappa}^{\kappa} \Big| \sum_{X < x \leq 2X} e(\alpha x^{\theta}) \Big|^{2r} d\alpha \ll_{\epsilon} \kappa X^{2r - \theta+\epsilon} \] holds for all integers $r \geq r_0$ satisfies $2r_0 \leq \theta^2(1+O(\theta^{-1/2}))$. This is an improvement over the previous bound by Poulias of $2r_0 \leq (\lfloor 2\theta \rfloor + 1)(\lfloor 2\theta \rfloor + 2)$. As a consequence, the bound on the asymptotic order of the minimum number of variables required to prove the expected asymptotic formula for the number $R_{s,\theta}(N)$ of solutions $(x_1,\ldots,x_s) \in \mathbb{N}^s$ to the Diophantine equation \[ \lfloor x_1^{\theta} \rfloor+\cdots+\lfloor x_s^{\theta} \rfloor = N \] is improved by a factor of $4$. We also discuss a certain Diophantine system which arises naturally from our proof, which may have applications to other counting problems and may be of independent interest.

We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.

Christ [Chr01] looked at a certain boundedness problem for trilinear operators. In this paper we set up a framework of mixed-norm Brascamp-Lieb inequalities and identify a new interesting regime not covered by the study of classical Brascamp-Lieb. Positive results in [Chr01] can be viewed as the first nontrivial progress in this new regime. We will also prove another mixed-norm Brascamp-Lieb inequality, showcasing how one can use a tensor product trick to slightly sharpen Christ's argument and also obtain the endpoint case. We then discuss the connections between mixed-norm Brascamp-Lieb and the Kakeya Conjectures, as well as unique new difficulties for these mixed-norm Brascamp-Lieb inequalities compared to the classical setting.

In this paper, through the introduction of Lorentz–Muckenhoupt classes, we systematically investigate the boundedness of maximal operators on multiplier weighted Lorentz spaces. As applications, we give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe. Second, Hardy inequalities in Lorentz spaces are established, with the critical case $p=d$, in which the classical Hardy inequality fails. Finally, we apply the Lorentz estimates to fractional Schrödinger equations with singular potentials.

In this paper we study the behavior of the density ratios of harmonic measure at points with vanishing density. Given an arbitrary open set $\Omega\subset\mathbb R^{n+1}$ with harmonic measure $\omega$, we show that at $\omega$-almost every point $x\in\partial\Omega$ where $\liminf_{r\to0}\frac{\omega(B(x,r))}{r^n}=0$, the density ratio $\frac{\omega(B(x,r))}{r^n}$ is not almost monotone with respect to the radius $r$, and therefore exhibits arbitrarily large oscillations at small scales.

Let $\|\cdot\|_p$ denote the Schatten $p$-norm and let $|A|=(A^*A)^{1/2}$. For $2\leq p<\infty$, let $x_{p,m}>1$ be the unique solution of $x_{p,m}^p=2x_{p,m}+m-1$, and set \[ C_{p,m}=\frac{\sqrt{x_{p,m}(x_{p,m}+m-1)}}{(x_{p,m}^p+m-1)^{1/p}}. \] We prove the sharp inequality \[ \|A_1+\cdots+A_m\|_p\leq C_{p,m}\bigl\||A_1|+\cdots+|A_m|\bigr\|_p \] for arbitrary complex matrices of arbitrary size. Equivalently, if $q=p/(p-1)$ and $R,X_1,\cdots,+X_m$ are positive semidefinite, then \[ \|RX_1\|_1+\cdots\|RX_m\|_1 \leq C_{p,m}\|R\|_q\|X_1+\cdots X_m\|_p. \] For $1<p<2$, we also show that the formula proposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.oposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.

Let $\tilde{\Gamma}(x)$ denote the modified gamma function recently introduced by the authors as an alternating analogue of the classical Euler gamma function. In this paper, we establish a complete alternating generalization of Mizuno's celebrated product formula: \begin{equation*} \prod_{m=0}^{\infty}\left(\prod_{j=1}^{n}(m+x_{j})^{(-1)^{m}}\right) =\frac{\left(\sqrt{\frac{\pi}{2}}\right)^n}{\prod_{j=1}^{n}\tilde{\Gamma}(x_{j})} =\prod_{j=1}^{n}\left(\prod_{m=0}^{\infty}(m+x_{j})^{(-1)^{m}}\right). \end{equation*} This identity, which we refer to as the alternating Mizuno formula, replaces the classical gamma function $\Gamma$ and the constant $\sqrt{2\pi}$ by their natural alternating counterparts $\tilde{\Gamma}$ and $\sqrt{\pi/2}$. As immediate consequences, we recover the alternating Lerch formula and, by specializing to $x=2$ and multiplying by $2$, a remarkably concise derivation of Wallis' famous product \begin{equation*} \frac{2\cdot2}{1\cdot3}\cdot\frac{4\cdot4}{3\cdot5}\cdot\frac{6\cdot6}{5\cdot7}\cdot\cdots=\frac{\pi}{2}. \end{equation*} More generally, by exploiting polynomial factorizations, we derive Kurokawa–Wakayama type formulas for alternating products over cyclotomic fields. Beyond these product identities, we investigate the multiple alternating gamma functions $\bar\Gamma_N(x)$, obtaining explicit factorizations in terms of Barnes' multiple gamma functions, closed-form evaluations involving the Glaisher–Kinkelin constant, and a Gauss–Legendre type multiplication formula. Finally, we introduce an alternating analogue of Shintani's double sine function and establish a clean arithmetic dichotomy: for algebraic $\tau$, its special values at integer points are algebraic precisely when $\tau$ is rational, and transcendental otherwise.

We prove a four-way Szegő theorem for $L^p$ extremal polynomials on compact supports $K=K_0\cup X\subset\mathbb R$, where $K_0$ is a regular compact set and $X$ is a finite or countable set of isolated points. For every $2\le p\le\infty$, including the weighted Chebyshev case $p=\infty$ under the corresponding assumptions on the weight, any three of the Parreau–Widom condition for $K_0$, the Blaschke condition for $X$, the Szegő condition for the weight, and the Widom condition $0<\limsup_{n\rightarrow\infty}W_{p,n}<\infty$ imply the fourth. As a consequence, for every regular compact set $K\subset\mathbb R$ of positive capacity, the Parreau–Widom condition is equivalent both to boundedness of the unweighted Chebyshev Widom factors and to boundedness of the equilibrium-measure $L^2$ Widom factors. For every $0<p\le\infty$, we also prove upper and lower bounds for the Widom factors in which the contributions of the weight, the isolated points, and the gaps of $K_0$ appear separately. Finally, we give examples illustrating the sharpness of our results. For $p=2$, we realize every combination of the following five properties that is not excluded by the implications proved in this work: the Parreau–Widom condition, the Blaschke condition, the Szegő condition, boundedness of the Widom factors from above, and boundedness of the Widom factors away from zero.

We investigate the behavior of lattice points in 1-symmetric convex bodies–those invariant under both coordinate permutations and sign changes. In this setting we introduce a discrete analogue of the isotropic constant and establish concentration of mass properties for lattice points that parallel classical mass concentration results for convex bodies in isotropic position. These geometric estimates are applied to prove dimension-free $\ell^p$ bounds ($p \ge 2$) for discrete dyadic maximal operators and for maximal functions in the small-scale regime. Furthermore, we demonstrate that coordinate permutation invariance is a necessary condition by constructing 1-unconditional convex bodies in isotropic position for which these dimension-free estimates fail. Our results provide a framework that generalizes several recent works on dimension-free estimates for discrete maximal functions, including those by Bourgain, Mirek, Stein, and Wróbel.

We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $\beta_K=-K$ and $\alpha_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(\alpha,\beta)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_n^{\alpha,\beta}$ from the constant polynomial. We solve in triangular form the connection problem with the $\alpha=0$ family, identified with the Gegenbauer polynomials, and transfer the ladder relations to the $(n,\ell)$ lattice. Under $x=\cot\chi$ the in-level operators depend on $\ell$ alone. The dressed functions carry intrinsic lowest weight $\mathrm{su}(1,1)$ modules along columns of fixed $\ell$; rows need a boundary prescription and a rescaling. The deformed functions carry the same modules through higher order operators, which yield relations between Romanovski polynomials with different parameter pairs. Algebraically the family lies in the Jacobi class, and we indicate throughout which statements are inherited from that framework by transformation or specialization and which are not.

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil–Ramanathan–Topiwala conjecture by constructing twelve linearly dependent time–frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand–Shilov class $\mathcal S_1^1(\mathbb R)$; their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in $\mathcal S_1^1(\mathbb R)$. Moreover, if $0\ne f\in\mathcal S_s^\sigma(\mathbb R)$ and $s<1$ or $\sigma<1$, then every finite system of distinct time–frequency shifts of $f$ is linearly independent. Thus four is the least possible cardinality and $(1,1)$ is the sharp coordinatewise threshold within the Roumieu Gelfand–Shilov scale.

The notion of a regular point is extended to that of a weak regular point, and so the notion of a strong natural boundary may be described locally as the absence of weak regular points, just as a natural boundary is locally the absence of regular points. The main result is a sufficient condition for a strong natural boundary, which is the conjunction of the unboundedness of the coefficients of the underlying power series with either of two specific gap density conditions imposed on the exponents that yields a lacunary sequence.

We prove probabilistic pointwise convergence to the initial datum for the 2D periodic quintic NLS for data in $H^s(\mathbb T^2)$ with $s > 0$. This is an improvement with respect to the deterministic setting, in which convergence is known to fail if $s < 1/3$. The proof is based on a nonlinear maximal characterization for convergence, Bourgain's linear-nonlinear decomposition and the corresponding nonlinear smoothing for which we employ random tensor estimates.

There are several non-equivalent notions of infinitesimal symmetry in the literature of second order linear differential equations: Lie point symmetries, vertical (gauge) symmetries, operator symmetries, infinitesimal contact symmetries, and Lie–Bäcklund operators. We construct an explicit correspondence among the first three, We then describe the Lie algebra $\mathcal L_\Omega(\mathbb U)$ of infinitesimal contact transformations of the contact system $\Omega=\langle dy-y'\,dx\rangle$, with coefficients in a differential field $\U$ of functions of $x$ and $y$. We obtain a canonical decomposition $\mathcal L_\Omega(\mathbb U)=\prod_{k\ge0}\mathcal L_\Omega^k\mathbb U$ into $\mathbb C$-vector spaces, each parametrized by $\mathbb U$ (by $\mathbb U\oplus\mathbb U$ for $k=0$), and we compute the algebraic differential formulae for the Lie bracket in these coordinates. Applied to the symmetry problem, we prove that a contact vector field with generating function $W$ is a symmetry of if and only if $A^{2}W=aW+b\,AW$, where $A$ is the vector field in the jet space corresponding to the equation; equivalently, if and only if $W=F_1(u_1,u_2)\phi_1+F_2(u_1,u_2)\phi_2$ for arbitrary functions $F_1,F_2$ of the two first integrals of $A$ and a basis $\phi_1,\phi_2$ of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of $\Lom(\U)$ never captures the whole symmetry algebra: for $a\neq0$ the graded part reduces to the point symmetries, while for $y''=0$ it is an infinite dimensional but still proper subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law $W\mapsto\mu^{-1}(W\circ\varphi)$ for characteristics under a contact transformation with conformal factor $\mu$, which governs the transport of evolutionary representatives.

We prove some conjectures of K. C. Au concerning the descents of cyclotomic levels in certain sums over multiple polylogarithms. Via answers to a question of Au in some particular cases, we also confirm Broadhurst's conjecture on honorary multiple zeta values at even weights.

We study properties of the calculus $\varphi\mapsto\varphi(A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten–von Neumann class $\boldsymbol{S}_p$. It turns out that this calculus defined on the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ in the case $p\le2$ admits a commutator Lipschitz estimate and is multiplicative modulo $\boldsymbol{S}_p$. On the other hand in the case $p>2$ there is no commutator Lipschitz estimate. There is no commutator Lipschitz estimate in the operator norm as well.

We give a characterization of the existence of Lipschitz surjections between dust-like self-similar sets of the same dimension under the assumption that the similarity ratios are commensurable. Our results apply more generally to graph-directed fractals.

In this paper, we study a Durrmeyer variant of the positive linear operators based on two-variable Hermite polynomials recently introduced by G. Krech (2016). Our main objective is to establish a complete asymptotic expansion for these operators as n tends to infinity for locally integrable functions of polynomial growth. The coefficients of the expansion are explicitly expressed in terms of the derivatives of the function. As a corollary, a Voronovskaja-type formula is obtained.

Let $\alpha\in\mathbb{R}$, $p\in(0,\infty)$, and $q\in(0,\infty]$. In this article, we develop a theory of matrix-weighted anisotropic Besov spaces associated with an expansive matrix $A$ and an $\mathcal A_{p,\infty}$-matrix weight $W$. We first introduce the homogeneous spaces $\dot B_{p,q}^{\alpha}(A,W)$ and establish their $\varphi$-transform characterization. Then we construct counterexamples to show that the assumption $W\in\mathcal A_{p,\infty}$ in this characterization cannot be relaxed to $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$. The same counterexamples also show that this weaker condition $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$ is insufficient to ensure the well-definedness of $\dot B_{p,q}^{\alpha}(A,W)$. Next we characterize $\mathcal A_{p,\infty}$-matrix weights via the rescaled maximal operator, which leads naturally to a new concept of the critical rescaling index that quantitatively captures the self-improving behavior of matrix weights. In terms of this index, we obtain optimal boundedness for almost diagonal operators on the associated sequence spaces $\dot b_{p,q}^{\alpha}(A,W)$. Based on these, we further establish the molecular characterization of $\dot B_{p,q}^{\alpha}(A,W)$ and some sharp boundedness results for pseudo-differential operators on these spaces.