Papers

Let $0\leq\alpha<n$, and let $\sigma$ and $\omega$ be locally finite positive Borel measures on $\R^n$, possibly with common point masses. Suppose that one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve. We prove that the $\alpha$-fractional Riesz vector maps $L^2(\sigma)$ to $L^2(\omega)$ if and only if the two $\cA_2^\alpha$ conditions with holes, the two punctured $A_2^\alpha$ conditions, and the forward and dual testing conditions over rotated cubes hold. The theorem includes curves whose tangent is not Hölder continuous of any positive order. Straightening such a curve produces a kernel whose first derivatives need not satisfy the Hölder estimates required by the energy-side-condition $T1$ theorem. We prove both strong energy conditions in the straightened coordinates using only a sufficiently small graph slope, and show that these conditions are preserved under biLipschitz changes of variables. The Dini condition makes the errors in the kernel and testing comparisons summable. We then transfer the energy conditions back and apply the $T1$ theorem to the original smooth Riesz kernel. We also obtain a two-weight criterion for Green gradients with curve-supported sources. For arclength measure on the curve and observation measure $\dist(x,\Gamma)^\beta\,dx$, the range is $\beta>n-3$ in a bounded tube and $n-3<\beta<n-2$ in $\R^n$. The endpoints fail logarithmically whenever the measure is locally finite. In dimension three, the corresponding whole-space range for the Biot–Savart field with arbitrary vector densities is $0<\beta<1$.

Let \(A,B\subset\R^2\) be nonempty Borel sets. We prove that \[ \dimH(gA-B)\ge\min\{\dimH A+\dimH B,2\} \] for almost all \(g\in\mathrm O(2)\). This bound is sharp in general, and the proof gives a corresponding statement for the lower Hausdorff dimension of convolutions of Frostman measures. We also prove that \(\LL^2(gA-B)>0\) for almost all \(g\) if both dimensions exceed \((\sqrt{13}-1)/2\). The second result follows from an angular decomposition, refined decoupling and a two-ends Furstenberg estimate. Some applications on intersection patterns are also addressed.

We develop a unified multi-scale framework for sparse domination of multilinear operators beyond the multilinear Calderón–Zygmund theory. More precisely, under mild hypotheses on a support condition, weak type and restricted strong type uniform bounds for finite partial sums, and single-scale improving and regularity estimates, we establish $(r_1, \ldots, r_m, s')$-sparse form bound for a family of localized $m$-linear operators uniformly over all finite partial sums. This sparse bound yields weighted norm inequalities for the corresponding multilinear Muckenhoupt classes $A_{\vec{p}, (\vec{r}, s)}$, including the quasi-Banach range. As applications, we determine the optimal open sparse range for multilinear rough singular integrals, and characterize sparse domination for bilinear pseudo-differential operators. Further applications include sparse bounds for bilinear oscillatory Fourier multipliers, bilinear spherical maximal operators, and bilinear Bochner–Riesz means.

We prove that round circles $C_L$ are the unique minimisers of integral Menger curvature among embedded closed $C^1$ curves $\gamma$ of prescribed length $L$ for every $p\geq 2$, i.e. $M_p(C_L) \leq M_p(\gamma)$, with equality if and only if $\gamma$ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For $p>3$, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals $L$. In particular, $M_p$ detects the unknot and is basic for $p>3$, thereby answering a question posed repeatedly in the literature.

This paper proves the weak type (1,1) boundedness for maximal truncated singular integral operators with rough non-convolution kernels of the form $\Omega(x-y) K(x,y)$, where $\Omega$ is homogeneous of degree zero with $\Omega \in L\log L(\sS^{d-1})$ and $K$ is a standard Calderón-Zygmund kernel. This extends the recent resolution of the classical convolution case by Lai \cite{Lai2025weak} to a wider class of operators.

We introduce upper and lower discrete intermediate dimensions for unbounded subsets of Euclidean space and extend them to metric spaces. The definitions restrict the covering radii within a scale-dependent range of dyadic shells and take a supremum over an auxiliary parameter. The resulting dimensions are monotone in the parameter and recover the Barlow–Taylor discrete Hausdorff dimension at zero. We prove continuity at every positive parameter, with a sharp quantitative bound, and give an example with a jump at zero. At the upper endpoint, the upper dimension equals upper mass dimension for uniformly discrete Euclidean sets; the analogous lower mass identity fails. We establish a mass distribution principle and prove quasi-isometric invariance under a uniform coarse packing condition. For spaces with uniform polynomial volume growth, we construct clustered subsets attaining the continuity bound. Independent thinning truncates these spectra at the resulting mass dimension, almost surely for all parameters simultaneously. Applications include the discrete Heisenberg group with its word metric and the ranges of simple symmetric random walks.

Let $\lambda>-1/2,\neq0$. Consider the Bessel operators \begin{align*} \Delta_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx} \end{align*} on $\mathbb{R}_+$. In this paper, we first establish several characterizations for $\widetilde{A}_{\infty,\lambda}$ weights introduced by Li et al. (Math. Z. 309(1) (2025), Paper No. 13, 29pp), which are different from the previous $A_p$ class with respect to the measure $x^{2\lambda}dx$. Among these characterizations, we obtain the sharp reverse Hölder inequality (RHI) for $\widetilde{A}_{\infty,\lambda}$ weights. As applications of RHI, we derive a precise open property of $\widetilde{A}_{p,\lambda}$ weights, and obtain a sharp weighted bounds for the Hardy-Littlewood maximal operator $\mathcal{M}_\lambda$ associated to $\Delta_\lambda$, which was missing in the previous papers. Next, we also establish several extrapolation theorems related to $\widetilde{A}_{p,\lambda}$ weights. Furthermore, we characterize the restricted weighted weak type estimates for $\mathcal{M}_\lambda$ via a new class of weights $\widetilde{A}_{p,\lambda}^{\mathcal{R}}$, which is strictly larger than $\widetilde{A}_{p,\lambda}$. Meanwhile, we obtain quantitative restricted weighted weak type $(p,p)$ bounds for sparse operators. As applications, we derive various kinds of weighted inequalities for Riesz transforms, maximal operators, square functions, variation operators and their commutators associated to $\Delta_\lambda$.

In 2012, D. Karp studied a family of polynomials constructed from Polya frequency sequences and conjectured that all their zeros are real and negative. We prove this conjecture for positive sequences and establish further results concerning the location of these zeros.

We prove a compact two-weight $T1$ theorem for the $\alpha$-fractional Riesz vector, $0\leq\alpha<n$, when one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve $\Gamma$. The measures may have common point masses. Under the global boundedness assumptions, we give a two-sided estimate for the essential norm in terms of Muckenhoupt and testing constants computed after restricting both measures to small cubes. Compactness is equivalent to the uniform vanishing of these local bounds as the cube side length tends to zero. The smooth truncations have a unique weak operator limit and converge in operator norm in the compact case. For arclength measure and distance power weights, we determine the compactness ranges for Green gradients on bounded tubes and for Newtonian gradients and Biot–Savart fields on the whole space. The whole-space operators in these ranges are compact but are not Hilbert–Schmidt. When $\alpha=n-1$ and both measures have positive continuous arclength densities $a,b$ on the same closed curve, we obtain the exact essential norm $\pi c_{n-1,n}\max_\Gamma\sqrt{ab}$.

We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant real-valued $\varphi\in A(\mathbb{T})$, $\frac{1}{T}\int_a^{a+T}\|e^{it\varphi}\|_{A(\mathbb{T})}dt \ge c_\varphi\log T \quad(a\in\mathbb{R},\ T\ge T_\varphi)$, where $c_\varphi>0$ and $T_\varphi>1$ are independent of $a$. The proof uses exact sampling on finite subgroups of the circle, uniformly bounded localized modulations, and the weighted non-harmonic Littlewood inequality of Jaming–Kellay–Saba. A suitable cutoff and a small linear perturbation of the phase ensure that a fixed positive proportion of the coefficients at the beginning of the frequency ordering have modulus one. Passing to integer powers and removing the degree of the circle map yield Kahane's conjecture without any additional smoothness assumptions. We also give equivalent phase, convolution, and operator formulations, and derive consequences for homomorphisms of weighted Wiener algebras and for higher-dimensional tori.

In this paper, new existence and multiplicity results for positive periodic solutions of singular second-order problems under sublinear and superlinear growth assumptions at infinity are established. Two aspects of our results should be highlighted in comparison with the existing literature. On the one hand, they allow for the consideration of systems exhibiting mixed behavior: sublinear in one component and superlinear in another. On the other hand, in the superlinear case, the existence of up to four positive solutions is established, whereas only two were obtained in previous works. Our approach is based on a recently established vectorial version of the Krasnoselskii-Guo fixed point theorem in product spaces.

We study how many bits are needed to write down an elementary analytic expression that uniformly approximates a Lipschitz function. Expressions are finite trees over the basis $\{+,-,\times,\div,\exp\}$ with the variables and rational constants, and their size is $\dc(f)$, the length of the shortest description of $f$ in a fixed binary description language. For functions of one variable we prove that every $L$-Lipschitz $F:[0,1]\to[0,1]$ has an everywhere-defined approximant of accuracy $\Delta$ with $\dc(f)\le C\,(L/\Delta+\lg(1/\Delta+2))$ for all $L,\Delta>0$, by a linear spline smoothed with logistic ramps in a description language with one repetition construct. We prove a matching lower bound valid for every description language: for $\Delta\le 1/36$ and every $L>0$ some $L$-Lipschitz function needs at least $\tfrac1{16}(L/\Delta+\lg(1/\Delta+2))$ bits. Hence the worst-case description complexity is $\Theta(L/\Delta+\lg(1/\Delta+2))$. For functions on $[0,1]^d$ the same method gives $\Theta_d\bigl((L/\Delta)^d+\lg(1/\Delta+2)\bigr)$, using a tensor-product spline whose coefficients are mixed second differences and cost $O(d)$ bits each. Without the repetition construct the upper bounds carry an additional logarithmic factor, whose necessity for plain expression trees is left open. A reference encoder, decoder and evaluator confirm the constructions numerically, and figures and tables report the checks.

We proved recently that there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega)$, $1 \le p < 2$, for any set $\Omega \subset \mathbb{R}^d$ of finite measure. For $p > 2$, we proved that the same holds provided that $\Omega$ has nonempty interior, but the general case remained open. In this note we close this gap, i.e., we establish that exponentials cannot form an unconditional Schauder frame in $L^p(\Omega)$, $p \ne 2$, without any additional assumptions on the set $\Omega$ of finite measure other than measurability.

We prove decay estimates for oscillatory integral operators along $(y_1, y_2, y_1^2+y_2^2)$ in the finite field and the continuous settings. In the continuous setting, our result allows a bi-parameter maximal modulation, which extends the one-parameter version previously established by Pierce and Yung. Our approach is different from theirs. Then we discuss three applications of our estimates: a power saving bound for certain pattern finding problem in finite fields, a short approach to $L^p$ boundedness of an one-parameter Pierce–Yung operator, and $L^p$ boundedness of a bi-parameter Pierce–Yung operator.

We determine the best constants in estimates of the $L^\infty$ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension $d\geq2$, these constants are expressed through weighted $L^1$ extremal polynomials. On $S^2$ the constant is an explicit positive series, and the best coefficient of $n^{-2}$ is $7\zeta(3)/4$. On $S^3$ the best coefficient of $n^{-2}$ is a positive integral involving the Hörmander–Bernhardsson extremal function. In higher dimensions the rescaled extremal polynomials converge to weighted entire extremals, which give asymptotic formulas for the Bohr constants on $S^4$ and $S^6$. The most unexpected and, in our view, interesting observation is that, for every fixed $n\geq2$ on $S^2$ and $S^3$, the constants in best approximation estimates are strictly smaller than the corresponding Bohr–Favard constants.

We develop a finite-tree formalism for sharp inequalities on real-valued, mean-centered $\operatorname{L}^1$-BMO over homogeneous $q$-ary filtrations. A reduction theorem identifies local and global sharp constants with the supremum of finite-depth constants. Finite anchoring gives compact optimization problems and rational extremizers under rational piecewise-linear hypotheses, while stable centering shows that imposing zero root average preserves the supremum over all depths. The principal application is the absolute-value map: we construct centered examples in which taking absolute values increases the BMO seminorm and obtain bounds strictly below the universal factor $2$. Jensen-defect Carleson functionals and martingale-difference envelopes provide further applications, illustrating distinct anchoring mechanisms and the differences between finite-depth rationality, stabilization, and global attainment.

We study minimal positive cubature formulas on the two-dimensional sphere of degrees four and six, with ten and eighteen nodes, respectively. We prove the regularity of Popov's symmetric formulas and establish that their local families of deformations are two-dimensional, up to rotations. We obtain algebraic relations between the weights and pairwise inner products of the nodes. For degree four, we construct a one-parameter family with half-turn symmetry connecting the formulas of Popov and Heo–Xu. We also give a self-contained proof that eighteen nodes are minimal for degree six.

The second Jones problem (2011) consists in determining whether there exists a convergent positive series for which the ratio of consecutive terms has a limit and the set of subsums is a Cantorval; in the survey by Gł{\k a}b and Prus-Wiśniowski (2026) this problem is included in the list of open problems. In this paper, a family of such series is constructed explicitly as combinations of blocks of the form $\frac{2^k-1}{2^t} + \frac{2^{k-1}}{2^t} + \frac{2^{k-2}}{2^t} + \cdots + \frac{2}{2^t}$, where $t$ is the index of the last term of the block in the series and $k\ge2$ is the length of the block. For each series of the family it is proved that its set of subsums contains an interval and is not a finite union of closed intervals, and hence is a Cantorval. The proof relies on mixed-radix numerical systems with a redundant alphabet and uses an idea of Guthrie and Nymann (1988). If the lengths of the blocks tend to infinity, then the limit of the ratio of consecutive terms of such series is $\tfrac{1}{2}$, which, by a result of Jones, is the only possible value of the limit for which the set of subsums of a series can be a Cantorval. Thus, an affirmative answer to the second Jones problem is obtained.

For $1\leq r\leq n-1$ we study shrinking self-similar ends of the $r$-mean curvature flow, $S_r=c\langle X,N\rangle$ with $c<0$, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect $xu'-u$ is strictly negative, which yields $r$-admissibility, and the profile has a complete asymptotic expansion for $u(x)/x$ in powers of $x^{-(r+1)}$, valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly $r$-admissible cones whose links have constant principal curvatures.

For the bilinear spherical averages in dimension $d\geq2$, we show that the lacunary maximal operator is not bounded from $L^1(\mathbb R^d)\times L^q(\mathbb R^d)$ to $L^r(\mathbb R^d)$ whenever $1\leq q\leq\infty$ and $1/r=1+1/q$. The obstruction is realized by fixed nonnegative inputs; a second construction proves quantitative failure on smooth compactly supported functions and finitely many dyadic scales. The same conclusions hold for the full maximal operator. We give a sharp Lorentz-index criterion in the range $q>d/(d-1)$ and extend the negative result to all multilinear spherical maximal operators with an $L^1$ input. We indicate the precise strong-type boundary statements to which these conclusions apply, without drawing conclusions about separate weak-type or interior estimates.

We resolve the long-standing problem of Rudin concerning uniformly bounded orthonormal bases of homogeneous polynomials on complex unit spheres. Specifically, we show that for every $K \geq 2$ and $N \geq 0$ there exists a basis of degree $N$ homogeneous polynomials in $\mathbb{C}^K$ that is orthonormal with respect to the surface measure of the unit sphere of $\mathbb{C}^K$ and uniformly bounded by a constant depending only on $K$ on the unit ball of $\mathbb{C}^K$. Bourgain resolved the case $K = 2$ in 1985 and the case $K = 3$ in 2016. No cases $K \geq 4$ were previously known.

In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a $C^{1,\beta}$-manifold. This extends the analysis in \cite{Ca24} to dimensions $\geq 3$ and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones $\beta$-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.

We prove a dimension-free Markov–Bernstein-type inequality for polynomial mappings $(f_1,f_2)\colon\mathbb R^n\to\mathbb R^2$ of degree at most $d$. For product probability measures whose one-dimensional marginals have densities bounded by $1$, we obtain a lower bound for the integral of the Gram determinant of $\nabla f_1$ and $\nabla f_2$ in terms of the Gram determinant of the $d$-th derivative tensors $D^df_1$ and $D^df_2$, with a constant depending only on $d$. For pairs in a fixed Wiener chaos, the resulting Gaussian inequality answers a question of Nourdin, Nualart, and Poly concerning a lower bound for the expectation of the Malliavin determinant in terms of the determinant of the covariance matrix.

In this paper, by using signs rule for a class of special power series and for the Laplace integrals, we find the best power mean bounds for the hypergeometric function (sharp at the right endpoint). Our results contain many known conclusions.

We prove optimal dual Furstenberg estimates for Frostman and Katz–Tao configurations for which the shadings are not locally concentrated near pencils. The starting point is the sticky estimate of Orponen and Shmerkin in a companion paper. We deduce restricted projection theorems in the plane and in higher dimensions. In particular, Ahlfors-regular measures on the circle satisfy a Marstrand-type projection theorem. Applications include restricted projections to surfaces in the $\mathbb{G}(4,2)$ Grassmannian, pinned distances, discretized sum-product, and a new proof of the dimension formula for planar self-affine sets.

We introduce spread-out measures, a notion of local non-concentration near pencils for measures on Grassmannians. We establish an optimal $\delta$-discretized dual Furstenberg estimate for “sticky spread-out” shadings. A simple-to-state corollary is the following. Assume that $\nu$ is an Ahlfors $s$-regular measure on $\mathbb{G}(2,1)$ with $s > 0$. Let $E \subset \mathbb{R}^{2}$ be a Borel set with equal Hausdorff and packing dimension. Then, $\dim_{\mathrm{H}} \Pi_{V}(E) = \min\{\dim_{\mathrm{H}} E,1\} \quad \text{for $\nu$-a.e. } V \in S^1$. The general $\delta$-discretized estimate for “sticky spread out” configurations here is the starting point for a more general theorem in the companion paper of Cohen, Shmerkin, Wang and Zakharov, where the “stickiness” hypothesis is removed. For instance, the assumption of equal Hausdorff and packing dimension in the result stated above will be dispensed with in the companion paper.

A Hsiang cubic is a cubic form on $\mathbb{R}^{n}$ whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on $S^{n-1}$: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial $T_3$. For the spherical analogue of the problem of least deviation, the minimization of $\|p\|_\infty^2/\|p\|_2^2$ over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.

For $0\leq\alpha<n$, let $A_{*}^{\alpha}$ denote the fractional spherical maximal operator. In this paper we characterise the range of $\beta$ for which the $L^p-L^q$ power-weight estimate \[\left\|A_{*}^{\alpha} f\right\|_{L^q(|x|^{q\beta})} \leq C \left\|f\right\|_{L^p(|x|^{p\beta})}\] holds. In the diagonal case $p=q$, our work gives an affirmative answer to a conjecture of Lee. We prove analogous results for the lacunary fractional spherical maximal operator.

For every fixed real quintic polynomial $F$, we prove that the slow–fast Liénard system $\dot{x}=y-F(x),\quad \dot{y}=-\varepsilon x$ admits at most two limit cycles whenever $0<\varepsilon<\varepsilon_0(F)$ for a suitable $\varepsilon_0(F)>0$. Using scalar comparison near the equilibrium and a logarithmic fold-passage correction, we exclude \rev{canard singular limits}. Every sequence of limit cycles with $\varepsilon\to0^+$ has a subsequence converging in the Hausdorff metric to a common slow–fast path, whose regular slow arcs are either all normally attracting or all normally repelling. We also establish uniform Floquet separation for fixed $F$. The global counting argument uses shared regular arcs, nesting, and stability alternation. These results give a global small-parameter bound and describe the singular-limit structure, complementing the local cyclicity analysis of Li and Lu(J. Differential Equations, 2014).

Given an integer $k \geq 2$ and a prime $p$ large enough depending on $k$, we show that, if $A \subseteq \mathbb F^n_p$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \\ y_2^2 \\ \vdots \\ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \\ y_2^{k-1} \\ \vdots \\ y_n^{k-1} \end{pmatrix}, \end{align*} where $x \in \mathbb F^n_p$ and $y \in \{0,2, 3, \ldots, M_k\}^n \backslash \{\vec{0} \}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{(\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\geq 4$, the best previously known bound came from the density Hales–Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.