Papers

For a meromorphic function $f$ in the complex plane, we determine when $f$ and $f^{(k)}$ are algebraically dependent. This happens precisely when $f$ belongs to the class $W$ of elliptic functions and their degenerates, or satisfies a linear differential equation $f^{(k)} + af + b = 0$ with constant coefficients. The case of functions having a pole was settled previously. We complete the proof by treating entire solutions.

Given $0\le m<n$ and a positive Radon measure $\mu$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal D(\mu):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}\mu}|G_R^\mu(x)-1|\Bigr],$$ where $G_R^\mu(x):=(4\pi R^2)^{-m/2}\int_{\mathbb R^n} e^{-|z-x|^2/(4R^2)}\,d\mu(z)$. Provided that $\mu$ admits a flat blow-down $\mathcal H^m\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\le\delta<\infty$, called the width of $\mu$, such that $$\mathrm{spt}\mu\subseteq \bar B_\delta(V+x_0),\quad \delta^2\le C(m,n)\mathcal{D}(\mu),$$ where $\bar B_\delta(V+x_0)$ denotes the closed $\delta$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal D(\mu)=0$, the support of $\mu$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $\mu$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $\mu$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.

Let $\Omega=\Omega^+\subset \mathbb{R}^{n+1}$ be a bounded open set with $n$-Ahlfors regular boundary and let $\Omega^-=\mathbb{R}^{n+1}\setminus\overline{\Omega^+}$. For $x\in\partial\Omega$, $r>0$, define \begin{equation*} \varepsilon_{n,\Omega}(x,r):=\frac{1}{r^n} \inf_{H^+}\mathcal{H}^n\left(((\partial B(x,r)\cap H^+)\setminus \Omega^+) \cup ((\partial B(x,r)\cap H^-)\setminus \Omega^-)\right), \end{equation*} where $\mathcal{H}^n$ stands for $n$-dimensional Hausdorff measure and the infimum is taken over all open affine half-spaces $H^+$ such that $x\in \partial H^+$. Here $H^-=\mathbb{R}^{n+1}\setminus \overline{H^+}$. In this work we prove that $\Omega$ is a $C^{1,\alpha}$ open set, with $0<\alpha<1$, if and only if there exists some constant $C>0$ such that $\varepsilon_{n,\Omega}(x,r)\leq C\,r^\alpha$ for all $x\in\partial\Omega$ and all $0<r\leq \mathrm{diam}(\partial\Omega)$.

We identify a normalization defect in a determinantal construction of fourth-order Hahn difference equations and compare its scalar equation with Foupouagnigni's 1998 elimination and its published 2001 counterpart. On an explicit positive orthogonal family, all five prescribed coefficients vanish: the formula gives \(0=0\) and imposes no condition. Along an explicit regular perturbation, division by \(\varepsilon^2\) before specialization recovers a nonzero equation on that family, the same scalar equation as the correctly executed old shift-and-eliminate procedure. On the common nondegenerate domain specified below, both constructions give the same scalar operator up to a nonzero rational factor. The first-order degree bounds follow directly from valid identities of 1998; preserving cancellations gives a bound that can be arbitrarily sharper on explicit admissible inputs, together with exact top coefficients and cancellation criteria. Explicit coefficient formulas do not by themselves yield a new scalar equation on this domain. Our comparison does not exclude corrective or expository contributions; additional computational gain or exceptional coverage must be demonstrated.

Two theorems of Loureiro and Van Assche on ratio and zero limits provided part of the motivation for this note. Their elementary nature permits direct proofs; Appendix B gives complete proofs of both for the canonical initial conditions, using earlier continued-fraction identities. We prove ratio limits and weak limits of normalized zero measures for varying higher-order three-term recurrences with arbitrary coefficient periods and independent limiting functions for their phases. Joint local coefficient limits give ratio convergence without uniform coefficient bounds. For zero measures, convergence in measure to finite measurable profiles suffices, without boundedness or integrability; the limiting measures may have noncompact support. Coefficients may vanish, and real diagonal terms are allowed for ordinary three-term recurrences. The proofs use interlacing, partial fractions with nonnegative residues, local computations of moments, and stability of zero counts under changes of recurrence coefficients. Truncation removes the coefficient bounds from the zero theorem, whose proof is independent of ratio asymptotics. Uniform bounds give the version with compact support. Power-law limits and their explicit transform formula follow as a special case.

In this paper, we consider the endpoint Sobolev regularity of Radon transforms associated with real-analytic functions in the plane. We establish all of the failure of endpoint regularity for real-analytic phases. On the one hand, the critical endpoint regularity $L^p(\mathbb{R}^2)\rightarrow L^p_s(\mathbb{R}^2)$, corresponding to extreme points of the reduced Newton polyhedron of phase function, is true if only and only if $p=2$. For generic extreme points, this failure of endpoint regularity generalizes a result of Christ \cite{christ}. On the other hand, we find that the endpoint Sobolev regularity also fails on a bounded segment of the reduced Newton polyhedron when the bisectrix $y=x$ intersects one of unbounded faces at a relative interior point. In particular, our results imply that the Radon transform is not bounded from $L^p$ into $L^p_{1/n}$ with $p=n/(n-1)$ or $p=n$, where the phase functions are real-valued homogeneous polynomials $S(x,y)=a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}$ with $n\geq 3$ and $a_1a_{n-1}\neq0$. This answers a problem of Phong and Stein in 1994. Moreover, the main results of this paper also give examples which show failure of critical Sobolev regularity for degenerate Fourier integral operators satisfying finite type conditions formulated in \cite{Seeger93,seeger2}. Based on earlier results and our proof of several remaining cases, we conclude that all of other critical endpoint Sobolev estimates are true for Radon transforms associated with real-analytic phases in the plane. Therefore, all failures of endpoint regularity estimates are clearly determined.

We study the sharp Fourier extension inequality from $L^2(\mathbb S^{d-1})$ to the radial–angular mixed space $L^q_{\mathrm{rad}}L^2_{\mathrm{ang}}$. Our work is motivated by and builds upon the foundational analysis of Carneiro, Oliveira e Silva and Sousa \cite{COS}, who initiated the sharp mixed-norm spherical restriction problem from the viewpoint of spherical-harmonic decomposition. Their reduction revealed that the extremal problem is governed by a competition among the individual harmonic modes, and their results established degree-zero extremizers in several important ranges of exponents, including even integer exponents and the endpoint case. The present paper addresses this question by establishing a complete modal ordering result. Building on their spherical-harmonic reduction, the sharp-constant problem is reduced to the weighted Bessel moments \[ I_{\nu,k}(q) := \int_0^\infty |J_{\nu+k}(r)|^q r^{2\nu+1-\nu q}\,dr, \qquad \nu:=\frac d2-1. \] For every integer $d\ge2$ and every finite admissible exponent $q>2d/(d-1)$, we prove \[ I_{\nu,k}(q)<I_{\nu,0}(q), \qquad k\ge1, \] and hence constants are the unique nonzero extremizers of the sharp mixed-norm extension inequality. In dimensions $d\ge3$ we prove the stronger strict ordering \[ I_{\nu,0}(q)>I_{\nu,1}(q)>I_{\nu,2}(q)>\cdots. \] We resolve this difficulty by combining exact Hankel asymptotics with a finite remainder comparison. The modal ordering also yields the optimal quadratic global stability inequality. For every finite admissible $q$, \[ C_{d,q}^2\|f\|_2^2 -\|\widehat{f\sigma}\|_{L^q_{\mathrm{rad}}L^2_{\mathrm{ang}}}^2 \ge \Gamma_{d,q}\|f-P_0f\|_2^2, \] with the best possible coefficient. When $d\ge3$, this coefficient is explicitly \[ \Gamma_{d,q} =(2\pi)^d \left[ I_{\nu,0}(q)^{2/q}-I_{\nu,1}(q)^{2/q} \right]. \]

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\LL = ({\bf d}+ iA)^*({\bf d}+iA) + V$ where $A$ is a real $1$-differential form in $L^2_{loc}$ and $V \in L^1_{loc}$ is a nonnegative electric potential. We prove that the magnetic Riesz transform $({\bf d} + iA) \LL^{-\einhalb}$ is of weak type $(1,1)$ with bound at most $2$. It extends to a bounded operator on $L^p$ for $p \in (1, 2]$ with norm bounded by a constant $c_p$ depending only on $p$. It is then independent of the dimension of the manifold and independent of $A$ and $V$. We also prove weak type $(1,1)$ bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of $\RR^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.

In this paper, we establish sparse bounds for singular integral operators with rough non-translation-invariant Calderón-Zygmund kernels. Our approach verifies the localized hypothesis in the sparse domination framework introduced by Conde-Alonso, Di Plinio et al. The proof refines Seeger's microlocal decomposition using a Littlewood-Paley analysis adapted to the non-translation-invariant setting. As applications, we obtain quantitative weighted estimates for Calderón-type commutators and other singular integrals with rough kernels.

We prove an endpoint Pucci maximum principle for domains in $\mathbb R^2$, formulated using a generalization of elementary symmetric polynomials to noninteger orders. As a result, we sharpen an earlier result of Astala, Iwaniec and Martin, and deduce a new Sobolev inequality for $q$-Hessians where $1 < q < 2$. The fractional elementary symmetric functions we use are derived through fractional differentiation. Their signed singular value extension coincides with the Burkholder functional introduced by Burkholder and gives us access to a local Burkholder inequality due to Astala, Iwaniec, Prause and Saksman, leading to a different proof of the Pucci maximum principle from the one in that used quasiconformal maps. We include an elementary real-variable proof of this local Burkholder inequality, obtained by combining a few known techniques from existing literature.

We study positive limit cycles of the Abel equation $$\dot{x}=A(t)x^{3}+B(t)x^{2},$$ where $A$ and $B$ are $2\pi$-periodic functions. By developing a half-period convexity method, we establish a new criterion ensuring that the equation has at most one positive limit cycle. Moreover, if such a limit cycle exists, it is hyperbolic. As applications, we consider two classes of planar polynomial systems that can be reduced to Abel equations. For a subfamily of cubic rigid systems, we prove that there is at most one limit cycle, thereby giving a partial answer to the first problem in Gasull's list of 33 open problems. For quadratic systems, we derive a new uniqueness criterion for limit cycles surrounding the origin and provide explicit parameter regions under which the criterion is satisfied.

For $1\le p < \infty$, $\varphi_p$ is the function of minimal $L^p$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. We start from the ansatz that the positive zeros $\tau_n$ of $\varphi_p$ are close to \(n+1/p-1/2\) and use a weighted convex correction to first place them in an \(\ell^2\)-neighborhood of this sequence. A bootstrap argument then improves this to a {Poincaré asymptotic} expansion of $\tau_n-(n+\frac1p-\frac12)$ in terms of odd powers of $(n+\frac1p-\frac12)^{-1}$. We also get precise information about the Fourier transform $\widehat{\varphi}_p$, a result that in particular implies that for all integers $n\ge 2$, $\widehat{\varphi}_{2n}$ is an integrable function which, when convolved with itself $2n-2$ times, is constant on the interval $[-\pi,\pi]$. In particular, we identify the extremizer in an autoconvolution problem posed by B. Green in 2001.

For integers $\alpha,d \geqslant1$, we determine the best constants in the inequalities $||(a_j)_{j=1}^{d}||_{\ell^1} \leqslant C_{\alpha,d} ||((\Delta^\alpha a)_j)_{j=1}^{d+\alpha}||_{\ell^2},$ where $\Delta$ is the difference operator. We find that these constants correspond to the $\ell^2$ norm of the discrete Chebyshev polynomials, and show that the extremal sequences may be obtained by appropriate sampling of the discrete Landau polynomials. From beginning to end, this project was carried out without use of AI.

We introduce a spectrum of dimensions of measures, the packing intermediate dimensions $\dim_{P,\theta}\mu$, $\theta\in(0,1]$, of a Borel probability measure $\mu$ on $\mathbb{R}^n$ and the quasi-Hausdorff dimension $\dim_{qH}\mu$. We show that this spectrum encodes the information on $\mu$ that governs the packing dimension of its projections and Hölder images. First, $\dim_{qH}\mu$ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real $0<s\leq n$, $\dim_P^s\mu=s$ if and only if $s\leq\dim_{qH}\mu$. Second, we prove a lower bound for the profiles in terms of $\dim_H\mu$, $\dim_{P,\theta}\mu$ and the Assouad dimension of $\operatorname{supp}\mu$, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever $\dim_H\mu<\dim_P\mu$ and $\dim_A(\operatorname{supp}\mu)<n$, and when $\dim_A(\operatorname{supp}\mu)=n$ it occurs exactly when $\dim_{P,\theta}\mu$ exceeds an explicit universal lower bound for some $\theta$ in an explicit range. We present examples of measures with $\dim_H\mu<\dim_P\mu<\dim_A(\operatorname{supp}\mu)=n$ for which our bound gives the exact value $\dim_P\mu_V=\min\{\dim_P\mu,m\}$ for almost every $m$-dimensional subspace $V$, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random $\alpha$-Hölder maps $f_\omega:\mathbb{R}^n\to\mathbb{R}^m$, we prove that, for compactly supported $\mu$, $\dim_P\mu_{f_\omega}=m$ almost surely if and only if $\alpha m\leq\dim_{qH}\mu$, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if $\dim_{qH}\mu=\dim_P\mu$. Analogous results are obtained for analytic sets.

The Klein-Gordon equation is a second order differential equation that appears in a variety of applications in mathematics, physics and engineering. This equation is often solved using numerical methods, since a general analytical solution is unattainable. However, one can find very useful analytical solutions for some special cases. In cosmology, the Klein-Gordon equation of a field can explain the physics of inflation and has important consequences on understanding the origin and evolution of the universe. An analytical solution to the Klein-Gordon equation in cosmology has been obtained in the small-field excursion approximation, and it turns out that the solution can be written in terms of the confluent hypergeometric limit function. In this work, we derive this result again while showing the details of the calculations and then generalize to the case of two coupled Klein-Gordon equations. We show that the solution of this coupled system can be written in terms of Kummer's hypergeometric limit function while considering minimal assumptions. We carry out a full numerical calculation of the coupled system and compare it to the results from the analytical expressions.

In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type $\mathrm{M}_{\varphi}$, associated to non-negative and radial decreasing kernels $\varphi$ of integral $1$. In dimension $d=1$ we prove that $\mathrm{var}(\mathrm{M}_{\varphi} f) \leq \mathrm{var}(f)$ when $f$ is a characteristic function. In higher dimensions we prove that $\mathrm{var}(\mathrm{M}_{\varphi} f) \leq C_d \mathrm{var}(f)$ when $f$ is a radial characteristic function, with a constant $C_d$ depending only on the dimension $d$ but not on the kernel $\varphi$. We further show that $C_d \sim \sqrt{d}\log d $ is the sharp dependence on $d$ for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in $\mathbb R^d$, a result of independent interest.

We study coding multiplicity for self-similar sets on the real line. When the similarity dimension exceeds one, almost every point with respect to the natural self-similar measure has infinitely many codings. We focus on the case when the similarity dimension is at most one. Our analysis proceeds through the recurrence properties and the size of the critical set, defined as the union of the pairwise intersections of the first-level pieces. Our main structural result shows that when the critical set is finite, unboundedness of coding multiplicity is equivalent to periodic recurrence to the critical set. Moreover, every commensurable system with a finite critical set has uniformly bounded coding multiplicity. This conclusion is sharp in the incommensurable setting. We construct a one-parameter family of four-map systems such that for Lebesgue almost every parameter, the critical set is a singleton and the open set condition fails, while the attractor contains a point with infinitely many codings. More generally, for transversal families, the set of parameters admitting an infinite coding fibre is dense among those for which the strong separation condition fails. We also obtain a quantitative lower bound for the Hausdorff dimension of this parameter set. Such parameters can form a meagre set. We continue the study of the geometric size of branching, obtaining dimension formulae for typical coding fibres and almost-sure upper bounds on the dimension of the critical set in transversal families. In contrast, we construct an exponentially separated system with no dimension drop whose critical set has the same Hausdorff dimension as the attractor and whose coding map has a fibre of continuum cardinality. Although we work in the one-dimensional setting, several results and arguments extend to higher dimensions.

We prove sharp $L^p(|x|^\alpha)$ estimate for spherical maximal functions. We resolve the remaining open problem for the full maximal operator for $\alpha=1-d$ and $p<2$ and prove a more general result involving a local version of Assouad dimension of the underlying dilation set. We also make a case study with endpoint bounds for a model case involving convex sequences in $[1,2]$.

We continue our study of multilinear rough homogeneous singular integrals by establishing weighted estimates for their maximal truncations. Let $m\ge2$ and let $\Omega\in L^q(\mathbb{S}^{mn-1})$, $1<q\le\infty$, have mean zero. For $1<p_1,\dots,p_m<\infty$ with $p_j\ge q'$ and $1/p=\sum_{j=1}^m1/p_j$, we prove $$\big\Vert \mathcal{L}_\Omega^*(f_1,\dots,f_m)\big\Vert_{L^p(v_{\vec{\boldsymbol{w}}})} \lesssim \Vert\Omega\Vert_{L^q(\mathbb{S}^{mn-1})} \prod_{j=1}^m\Vert f_j\Vert_{L^{p_j}(w_j)}$$ whenever $\vec{\boldsymbol{w}}=(w_1,\dots,w_m)\in A_{(p_1/q',\dots,p_m/q')}$, where $q'$ is the Hölder conjugate of $q$ and $v_{\vec{\boldsymbol{w}}}=\prod_{j=1}^m w_j^{p/p_j}$. The estimate holds under the same multiple weight condition as the corresponding principal value estimate.

We investigate a direct two-ends approach to restriction for the truncated cone in $\R^n$, which gives the diagonal restriction range \[ p>2+\frac{4(7n+15)}{11n^2-n-24},\qquad n\ge10. \] This improves the range obtained by applying the result of Wang–Wu and then slicing.

We study linear Stieltjes control systems in which actuation is concentrated at finitely many temporal events, and we ask what happens when one of those events progressively loses its atomic weight in the associated Stieltjes measure. For constant coefficients, ordinary finite-atomic reachability is governed by a truncated Krylov space and is therefore insensitive to the positive values of the event weights. A different picture emerges once trajectory regularity is imposed. In the supercritical super-$g$-Hölder regime, we minimize the largest event-normalized state jump and characterize exactly the terminal directions for which this cost remains uniformly bounded as a prescribed event weight tends to zero. The characterization combines what can be reached before the vulnerable event, a drift-cancellability condition at that event, and the directions that remain available afterwards. Ordinary reachability of the deleted-event system can therefore be strictly larger than bounded-cost reachability in the singular limit. We also separate two mechanisms of divergence: loss of ordinary reachability after deletion and failure of the intermediate compatibility condition. For compatible targets, the minimum cost converges to a constrained deleted-event problem. A dimensionless position–velocity benchmark motivated by pulsed proximity maneuvers shows the distinction explicitly.

We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.

The Dotsenko-Fateev equation is a third-order linear ordinary differential equation studied in two-dimensional conformal field theory. We study the confluent Dotsenko-Fateev system, a rank-three system of linear ordinary differential equations obtained by confluence from a first-order system equivalent to the Dotsenko-Fateev equation. It has an irregular singular point at infinity, and its solutions admit double integral representations of confluent Selberg type. Using these representations, we construct sectorial fundamental solution matrices around infinity, establish their asymptotic expansions, and determine the associated Stokes matrices explicitly. We further study three aspects of the confluent Dotsenko-Fateev system: its equivalence to the degree-two polynomial reduction of the Schrödinger equation for the quantum Painlevé equation of type V, its realization as a restriction of a confluent version of the linear Pfaffian system associated with Appell's hypergeometric function $F_4$, and its reduction to the trivial rank-one equation via integral transforms and a quadratic change of the independent variable.

Under the stated regularity assumptions, the full orthogonality functionals of the five finite para-families considered here are classical for the respective affine, quadratic, $q$-quadratic, $q$-linear and alternating calculi. For each of the para-Krawtchouk, para-Racah, $q$-para-Racah and $q$-para-Krawtchouk families, two finite solutions of one global structural equation have $N+1$ nodes altogether and common moments through degree $N-1$. Their relative mass changes only the central recurrence coefficients. We prove regularity for every normalised combination with both component masses non-zero and recover the full published functionals and recurrences. The alternating construction identifies para-Bannai-Ito within the same structural framework. For the four non-alternating families, equal positive combinations give persymmetric Jacobi matrices; the known spectral gap criterion then decides perfect state transfer. Explicit parameter correspondences locate the classical components and resonant limits of the full systems. A positive seven-point example shows how further named families can be manufactured without a new governing principle. Counterexamples to mathematical assertions in three of four recent preprints expose the consequences of insufficient hypotheses for finite representations and physical models. In an era of AI-assisted symbolic production, we call for a standard that rewards structural explanation and verification rather than the proliferation of examples.

We study the membership of commutators $[b,T]$ of Calderón–Zygmund operators in the Schatten–Lorentz classes in the matrix-weighted setting. Working on a space of homogeneous type $(X,\rho,\mu)$ with a matrix $A_2$ weight $W$, we characterise, for $p\in(1,\infty)$ and $q\in[1,\infty)$, the Schatten–Lorentz norm $\|[b,T]\|_{S^{p,q}(L^2(W))}$ in terms of an oscillation (Besov-type) norm $\|b\|_{\mathrm{Osc}^{p,q}}$ of the symbol: for all $p\in(1,\infty)$, finiteness of $\|b\|_{\mathrm{Osc}^{p,q}}$ is necessary; and for $p>p(\eta)$, where $p(\eta)$ depends on the regularity of the kernel of $T$ and a dimensional parameter of the space, the two-sided estimate $\|[b,T]\|_{S^{p,q}(L^2(W))}\approx\|b\|_{\mathrm{Osc}^{p,q}}$ holds. Extending a recent unified framework for unweighted and scalar-weighted versions of such results by one of us to the matrix-weighted setting, our approach combines the dyadic representation of singular integral operators with the nearly weakly orthogonal (NWO) sequences of Rochberg and Semmes.

Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of \(\mathbb R^n\) with zero \(n\)-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-\(\lambda\) inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class $\mathcal R\mathcal H_{q,\delta}$ associated with Hausdorff content, and then prove the self-improving property of $\mathcal R\mathcal H_{q,\delta}$ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman–Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-\(\lambda\) principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.

Two basic problems in the theory of integral self-affine tiles are to characterize the digit sets that generate translational tiles and to understand whether such tiles admit periodic translational tilings. We give a complete solution to the first problem: for an expanding integer matrix $\mathbf{A}$ and a digit set $\mathcal{D}\subset\mathbb{Z}^n$ with $\#\mathcal{D}=|\det\mathbf{A}|$, the self-affine set $T(\mathbf{A},\mathcal{D})$ is a tile if and only if, for some $N\geq1$, the iterated digit set $\mathcal{D}_{{\mathbf{A}},N}$ is of skew-product form with respect to $\mathbf{A}^N$. As a consequence, every integral self-affine tile admits a fully periodic self-replicating $q$-fold tiling. Moreover, $q$ divides $|T(\mathbf{A},\mathcal{D})|$. In dimension two we prove more: every integral self-affine tile admits a fully periodic tiling. In particular, the periodic tiling conjecture holds for integral self-affine tiles in $\mathbb{R}^2$.

We study Fourier extension estimates for the truncated cone in $\R^{n+1}$ with weights satisfying an $m$-dimensional ball condition only at radii between $1$ and $R^{1/2}$, where $0<m\le n-1$. Within the discrete induction framework of Du–Li–Wang–Zhang, we obtain a family of bounds by combining multilinear refined Strichartz inequalities, narrow decoupling, and Lorentz rescaling. We also give lightplank, lattice, and hybrid lattice–Knapp examples.

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $\lambda$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(\lambda^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\log\lambda)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.