Papers

We show that the parabolic uniform rectifiability of a set can be characterized by an interior PDE property, namely Carleson measure estimates for bounded solutions to the heat equation. In particular, under very mild background hypotheses on an open set $\Omega$ and its boundary $\partial \Omega$, we show that the following are equivalent: (i) The quantity $|\nabla u(\cdot)|^2 \text{dist}(\cdot, \partial\Omega)$ is the density of a Carleson measure on $\Omega$ for all bounded solutions to the heat equation in $\Omega$. (ii) The caloric measure for $\Omega$ admits a corona decomposition. (iii) $\partial \Omega$ is parabolic uniformly rectifiable. The implication (iii) implies (i) is the main result of the cited paper [Bortz, Hoffman, Hofmann, Luna-García, Nyström, Anal. PDE 2023]. The remaining implications are new, and are direct parabolic analogues of results of Garnett, Mourgoglou and Tolsa [Duke Math. J. 2018] in the elliptic setting for the Laplacian; however, our proofs require several new ideas. In particular, we must adapt arguments to account both for time-lag, and for the fact that some characterizations of (elliptic) uniform rectifiability have either not been developed or have been shown to be untrue in the parabolic setting. Our background assumptions, as noted above, are very mild: we assume that the domain satisfies the time-symmetric capacity density condition and has interior corkscrews, and that the boundary is (non time-directed) Ahlfors-David regular. These are weaker than the background assumptions of Bortz, Hofmann, Martell and Nyström in arXiv:2510.22047, and are essentially optimal for the boundary continuity properties of caloric functions on which our arguments rely.

We present an example that shows that the double Hilbert transform $\H$ can have the characteristic function of an open set of finite measure in $\mathbb{R} \times \mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of $\H$, the double dyadic shift. We further construct non-diagonal sets whose indicators are fixed by $\H$ – strips bent along the Boole curve $x=y-1/y$ and, more generally, along level curves of real Herglotz functions with purely singular measures – and study the class of invariant sets which they generate. At positive density we construct an example built on three families of parallel lines, related to the carry cocycle $\{u\}+\{v\}-\{u+v\}$, and the fans – alternate sectors of an odd number of concurrent lines – which are the simplest members of the invariant cones, a class that we describe completely. We then study the classification of the invariant sets whose boundary lies on countably many lines. Its combinatorial half is settled completely: a locally finite arrangement of lines in which no point lies on exactly two lines is a family of parallel lines, a pencil of concurrent lines, or an affine image of the triangular lattice. The classification is further connected, via projective duality, with configurations of Sylvester–Gallai type, and, via a change of variables, with two-valued superpositions of traveling waves.

The Komlós conjecture is a classic problem in discrepancy theory; it asks whether an absolute constant $K$ exists such that given any $n$ vectors $a_1,\ldots,a_n$ inside the $m$-dimensional Euclidean ball, regardless of how large $m,n$ are, there is always a selection of signs $\varepsilon_1,\ldots,\varepsilon_n$ guaranteeing $$\|\varepsilon_1a_1+\ldots+\varepsilon_na_n\|_\infty \leq K.$$ We show that if the $\varepsilon_i$'s are allowed to take not just the values of $\pm 1$ but any unit modulus complex number, which we refer to as complex discrepancy, then the above inequality holds for a finite, explicit constant $K_{\mathbb{C}}$. Here, the $\ell^\infty$ norm of the resulting vector in $\mathbb{C}^m$ is the largest modulus of its entries, and thus the complex discrepancy of real vectors is equivalent to their rank-$2$ vector discrepancy. This quantity provides an upper bound (modulo uniform constant prefactor) on Gaussian discrepancy – a discrepancy measure introduced by Chewi, Gerber, Rigollet and Turner. Thus, we also resolve the Komlós conjecture for Gaussian discrepancy. Our paper builds upon the recent work of Bansal and Jiang on the Beck-Fiala and Komlós conjectures, which we approach from the formalism of Burkholder and the Bellman function method from probability and harmonic analysis. Our work was in part motivated by the realization that the complex discrepancy of the columns of any unitary matrix is equal to 1, a fact that follows from a straightforward calculation based on Idel and Wolf's generalization of the Sinkhorn normal form for unitary matrices.

A hypergraph Laplacian was introduced to investigate the structure of networks written as hypergraphs. It was shown that the behavior of solutions to an evolution equation associated with the hypergraph Laplacian quite resembles that of solutions to the classical heat equation. Hence by replacing the Laplacian in PDEs with the hypergraph Laplacian, we might be able to introduce various diffusion models on hypergraphs (discrete domains), whose behavior of solution is similar to PDEs. In this paper, we consider a system of equations obtained by replacing the Laplacian in the original Cahn–Hilliard equation with the hypergraph Laplacian. Due to the nonlinearity and multivaluedness of the hypergraph Laplacian, it is difficult to apply methods for the original Cahn–Hilliard equation to our problem. To cope with these difficulty, we shall introduce a new proof by using properties of the hypergraph Laplacian in this paper.

The polynomial Hardy–Littlewood inequalities bound coefficient norms of homogeneous polynomials on $\ell_p$ balls by their supremum norms, with constants independent of the dimension. Their $p=\infty$ endpoint is the Bohnenblust–Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving $D_{m,p_m}\le(1+o(1))D_m$ whenever $p_m/m^2\to\infty$, where $D_{m,p}$ and $D_m$ are the optimal complex polynomial Hardy–Littlewood and Bohnenblust–Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, $D_{m,m}=m$. We then prove the full first-order asymptotic $D_{m,p_m}\sim m$ whenever $p_m\ge m$ and $p_m-m\to0$, and the exact polynomial growth $D_{m,p_m}=m^{1+o(1)}$ on the wider scale $p_m-m=o((m/\log m)^{1/3})$.

We study the monic polynomials $\Sob{n}$ orthogonal with respect to a Sobolev-type modification of the discrete Krawtchouk inner product in which a single point mass at an exterior point $\alpha<0$, outside the support $\{0,1,\dots,N\}$ of the binomial measure, acts through the $j$-th forward difference $\dif^{j}$ (here $\Nzero=\{0,1,2,\dots\}$, $\Z_{+}=\{1,2,\dots\}$, $\R_{+}=(0,\infty)$). In the joint scaling $n,N\to\infty$, $n/N\to r$ with $0<p<r<1$ (the reciprocal-Gamma regime of Dominici's classification), we prove the Mehler–Heine formula. The exterior mass survives the large-degree limit exactly through the linear factor $z-\alpha$, with the explicit universal constant $C_{\ast}$, and the limit is independent of both $\lambda>0$ and $j\ge1$; the same conclusion holds in the Uvarov case $j=0$ (Corollary~\ref{cor:uvarov}). The derivation is pointwise and self-contained: it uses only a rank-one connection formula (Theorem~\ref{thm:connection}), the reproducing-kernel recursion, Dominici's pointwise limit and the classical three-term recurrence; the normalised Sobolev correction obeys an exact affine recursion at fixed $N$, controlled by a triangular contraction argument over two backward windows with an explicit boundary estimate, and the contraction ratio $\varrho<1$ is exactly the condition $r>p$. We also discuss the scale of the theorem: in this normalisation the classical term is asymptotically negligible, and the independence of $\lambda$ concerns each fixed $\lambda>0$, not the continuity at $\lambda=0$. Multiprecision computations illustrate the convergence and the exact value $C_{\ast}=2.8$ for $p=0.3$, $r=3/5$, $\alpha=-2.5$, $j=2$, $\lambda=5$.

We determine the optimal exponential scale in Erdős's robust polynomial interpolation problem and identify the leading exponential coefficient for Chebyshev–Lobatto grids. Given $C>0$ and $0\le\rho<1$, put $H=C/(1-\rho)$. We prove that every configuration of $n\ge n_0$ nodes in $[-1,1]$ admits sign data for which every polynomial of degree at most $(1+\epsilon)n$ and uniform norm at most $C$ has error greater than $\rho$ at more than $\epsilon n$ nodes. One may choose $\epsilon\ge e^{-A(1+H)}$ and $n_0\le e^{A(1+H)}$, where $A$ is an absolute constant, and a matching upper bound shows that the exponential order is optimal. For the same obstruction restricted to Chebyshev–Lobatto grids, the optimal parameter $\epsilon_{\mathrm{Ch}}(C,\rho)$ satisfies $\log(1/\epsilon_{\mathrm{Ch}}(C,\rho))=\frac{\pi}{2}H+O(\log(H+1))$ as $H\to\infty$. The proofs combine finite interpolation estimates in Bernstein spaces with periodic sign constructions and averaging. We also establish obstructions for random signs and stability under angular perturbations of the nodes.

This paper provides a solution to the matrix $A_p$ conjecture. For every \(1<p<\infty\), we construct a family of \(2\times2\) matrix \(A_p\) weights $W$ with arbitrarily large characteristics for which \[\|H\|_{L^p(W)\to L^p(W)}\ge c_p[W]_{A_p}^{1+1/[p(p-1)]},\] where \(H\) is the Hilbert transform and \(c_p>0\) depends only on \(p\).

The functions $\varphi_\ell(z)=(e^z-s_\ell(z))/z^\ell$, with $s_\ell$ the Taylor polynomial of $e^z$ of degree $\ell-1$, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis $\mathbb{R}_-$. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9'-problem: $E_n(\varphi_\ell)^{1/n}\to H$, Halphen's constant, for every $\ell$. We prove it. The theorem behind it gives the same rate for $f=u_0+u_1\exp$ with $u_0$ rational and $u_1\not\equiv0$ meromorphic with finitely many poles and $\log|u_1(z)|=o(|z|)$, the lower bound required only away from $\mathbb{R}_-$, whenever $f$ has no pole on $\mathbb{R}_-$ (Theorem~4.4). For rational $u_1$ this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, $u_1(z)=\sinh(\pi\sqrt z)/(\pi\sqrt z)$, whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight $\Phi_n$, and a contour identity of Schmelzer and Trefethen presents $u_1\exp$, once its principal parts are removed, in exactly that form, with $\Phi_n(t)=u_1(-nt)e^{-nt}$. The factor $u_1(-nt)$ grows subexponentially in $n$, and their hypothesis on $\Phi_n$ divides by $2n$, so it drops out of the external field: field, extremal arc, $S$-property and constant are all theirs, unchanged. For $\varphi_\ell$ the factor is $t^{-\ell}$, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.

We present an explicit construction of examples showing that the estimate $ \|S(f)\|_{L^2(w)\to L^{2,\infty}(w)}\lesssim \big([w]_{A_2} \log(1 + [w]_{A_2})\big)^{1/2}$ for the dyadic square function is sharp in terms of the characteristic $[w]_{A_2}$.

We prove a Hörmander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^\sigma(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(d\omega)$ boundedness for $1<p<\infty$ and weak type $(1,1)$. No radiality or reflection-group invariance is assumed, and the dyadic pieces may meet the reflecting hyperplanes. The main new estimate controls the Dunkl kernel $E_\kappa(i\xi,x)$ and its first Euclidean derivatives in $x$, uniformly near arbitrary intersections of reflecting hyperplanes. Here $\xi$ lies in a fixed compact annulus. Writing $w_\kappa$ for the Dunkl weight and $\Omega_\kappa$ for its inhomogeneous counterpart, it gives the bound $C\Omega_\kappa(x)^{-1/2}w_\kappa(\xi)^{-1/2}$. The proof decomposes each root contribution into Hermitian two-dimensional blocks, applies oscillatory changes of variables near the rootwise transition points, and uses a commutator identity to cancel the non-integrable phase derivative. Chamber lifting also yields endpoint bounds for the pullbacks of the componentwise Coifman–Weiss atomic $H^1$ space and the corresponding BMO space on a fixed chamber.

A minimal surface in $\mathbb{R}^3$ is $\textit{separable}$ if it is the zero set of $f(x)+g(y)+h(z)$, and $\textit{isotropic}$ if moreover $f=g=h$. We announce that there are exactly two non-planar isotropic separable minimal surfaces up to homothety and rigid motion, that they are conjugate, and that they are Schwarz's diamond surface $\mathrm{D}$ and Schwarz's primitive surface $\mathrm{P}$. In lattice-normalized coordinates their implicit equations are \[\mathrm{D}:\; \mathcal{D}(x)\mathcal{D}(y)\mathcal{D}(z)=1,\] \[\mathrm{P}:\; \mathcal{C}(x)+\mathcal{C}(y)+\mathcal{C}(z) =3\,\mathcal{C}(x)\mathcal{C}(y)\mathcal{C}(z),\] where $\mathcal{D}(t)=\frac{\operatorname{sn}\operatorname{dn}}{\operatorname{cn}}(K[\frac{1}{4}]t,\frac{1}{4})$ and $\mathcal{C}(t)=\operatorname{cn}(2K[\frac{3}{4}]t,\frac{3}{4})$. The first equation is classical – it is due to Schwarz, later Cayley, and appears in Nitsche's $\textit{Lectures}$ – we identify Nitsche's transcendental function exactly as $\operatorname{sn}\operatorname{dn}/\operatorname{cn}$ at parameter $\frac{1}{4}$. The second appears to be new; it is the symmetric member of the two-parameter family of Kim and Ogata (2024), and it proves their assertion that the family contains $\mathrm{P}$. Both surfaces are governed by one differential equation with two signs, \[\varphi'^{\,2}=1+2\cosh 2\varphi\ \ (\mathrm{D}),\] \[\varphi'^{\,2}=1+2\cos 2\varphi\ \ (\mathrm{P}),\] the two signs of the separation constant of the classical reduction; and in that common normalization the two lattice periods are $2K[\frac{1}{4}]$ and $2K[\frac{3}{4}]$, whose ratio is Schwarz's 1866 constant $K'[\frac{1}{4}]/K[\frac{1}{4}]=1.2792615...$, the necessary scale ratio for a conjugate pair. Proofs will appear in [D, P].

In this paper, we show that for every p>2, there is an orthonormal basis of spherical harmonics on Sn with n>=2 which achieves the maximal Lp norm growth. More precisely, one basis achieves this growth for all p<=2(n+1)/(n-1), and another for all p>=2(n+1)/(n-1). This extends the results in Han (arXiv:1404.5016) to all dimensions and all p>2, and improves the positive density subsequence there to a whole basis.

We show that the strong maximal operator on ${\mathbb R}^2$ does not satisfy a linear weighted $L^2$ estimate in terms of the rectangular $A_2$ characteristic. More precisely, we construct weights with arbitrarily large characteristic for which the operator norm is bounded below by $[w]_{A_2^{\mathrm{str}}}\sqrt{\log [w]_{A_2^{\mathrm{str}}}}$. The proof is elementary. It uses an explicit two-parameter mass recurrence on a product of geometric interval partitions, together with a direct comparison of arbitrary rectangular averages with anchored averages.

We prove a quantitative polynomial Roth theorem for corners in \(\mathbb F_p^2\) for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials \(\phi_1,\phi_2\in \mathbb \mathbb{F}_p [y]\) are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every \(A\subset\mathbb F_p^2\) with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\qquad (x_1+\phi_1(y),x_2),\qquad (x_1,x_2+\phi_2(y)) $$ for some \(y\in\mathbb F_p^\times\). This improves the estimate $p^{2-1/16}$ of Han–Lacey–Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han–Lacey–Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map \(\widetilde F:W\to\mathbb A^3\) from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin–Schreier sheaf and Katz–Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An \(\ell^2\) matrix estimate adapted to such sets completes the resonant case.

We prove the sharp $\ell^p$ bound $p^\star-1$ for certain second-order discrete Riesz transforms on products of integers, in all dimensions. The family includes the imaginary part of the discrete Beurling-Ahlfors operator in dimension two as a special case. While these operators admit a martingale representation via compound Poisson jump processes and heat extensions, the martingale attached to a function and the martingale attached to its transform do not enjoy strong differential subordination, so the operator-norm estimate is not accessible through prior work.

We prove three results as part of the program of diameter-free estimates initiated in [CDW26]. The first two are reverse square function estimates for the light cone in $\mathbb{R}^3$. We first establish an abstract $L^4$ inequality of independent interest, under an ordered superorthogonality hypothesis: for every four distinct indices, only the two nonalternating pairings are required to vanish. The loss is $C(1+\log N)^2$, where $N$ is the number of functions. An alternating-determinant argument verifies this hypothesis for separated cone sectors. This gives a diameter-free estimate for arbitrary disjoint angular intervals, at the thickness determined by their smallest width, with no upper restriction on the radial parameter. For the canonical equal-width partition on a fixed radial annulus, we obtain the loss $C(1+\log N)^{1/4}$, which is sharp up to constants. This refines the estimate by Guth-Wang-Zhang, via a different approach. The improvement uses additional orthogonality between diagonal and off-diagonal differences, together with bounded overlap of dyadic difference shells. Our third result is the diameter-free $\ell^2L^6$ decoupling for arbitrary partitions of the parabola, with an $N^\varepsilon$ loss independent of the interval widths. The argument is an adaptation of the method from \cite{Cushman-Demeter-Wu} and implies their three-fold additive-energy estimate for the parabola. The three proofs use variants of interlacing and special orthogonality in place of wave packet analysis, multilinearity and parabolic/Lorentz rescaling. Together, these results provide further evidence for the scope of the paradigm introduced in [CDW26].

For the Jacobi family with parameters $(0,\beta)$, $\beta\geq-1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi–Radau functions. When two consecutive functions are parametrised by the same Prüfer phase, the function of higher degree attains that phase closer to $\theta=0$ and has strictly larger amplitude. In particular, the moduli of all corresponding relative extrema increase strictly with the degree. The lower bound $-1/3$ is sharp for this continuous statement: when $-1<\beta<-1/3$, the amplitude inequality is reversed at sufficiently small positive phases. This local reversal does not determine the optimal range for the discrete extremal inequalities. For $-1/3\leq\beta\leq0$, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so $(\Lambda_n^{(0,\beta)})_{n\geq0}$ is strictly increasing. At $\beta=0$ the weighted Radau functions reduce to $P_m^{(0,-1)}$. We thereby recover the theorem of Wong and Zhang and obtain, through an exact total-variation formula, a short proof of the Qu–Wong theorem on Legendre Lebesgue constants. The representation and the termwise comparison together realise the alternative-expression approach proposed by Qu and Wong, without asymptotic expansions, error bounds, or finite numerical verification. The proof is based on the equal-phase Prüfer architecture developed in the first author's earlier preprint arXiv:2608.01404; it applies that architecture to the problem posed by Qu and Wong.

In this work, we establish the sharp $L^2 \to L^4$ extension inequality for a large class of quadratic surfaces in $\mathbb{F}^3$, where $\mathbb{F}$ is a finite field of odd characteristic. These surfaces include the cones without the origin and the spheres with nonzero radius $j$ such that $-j$ is not a square in $\mathbb{F}$. We also show that the maximizers of these inequalities must have constant modulus.

We quantify the failure of interpolation between Hardy spaces $H^p(\mathbb T^\infty)$ on the infinite-dimensional torus. For any set $A\subseteq[1,\infty]$ such that $A\setminus\{\infty\}$ is closed in $[1,\infty)$, we show that there exists an operator that is densely defined on $H^p(\mathbb T^\infty)$ for all $1\leq p<\infty$ and weak-$\ast$ densely defined on $H^\infty(\mathbb T^\infty)$ that extends to a bounded linear operator on $H^p(\mathbb T^\infty)$ if and only if $p\in A$.

Let $k\geq 2$ be an integer, $T$ an infinite rooted $k$-ary tree, and $M$ the Hardy–Littlewood maximal operator on $T$. For any $p\in(0,\infty)$, we characterize the weight $w$ such that $M$ is bounded on $L^p(w)$. To this end, we introduce a two-layer Muckenhoupt weight class $\mathscr A_p$ and prove that, for any $p\in(\frac{1}{2},\infty)$, the boundedness of $M$ on $L^p(w)$, $w\in\mathscr A_p$, and the exponential decay boundedness of spherical averaging operators on $L^p(w)$ are mutually equivalent, and that, when $p\in(0,\frac{1}{2}]$, there exists no weight $w$ such that $M$ is bounded on $L^p(w)$. Moreover, for any $p\in(\frac{1}{2},\infty)$, we establish the quantitative estimate, with the optimal exponent $\frac{1}{p}$ of the weight constant, for the boundedness of $M$ on $L^p(w)$. For any $p\in(1,\infty)$, we also obtain two further equivalent characterizations of the boundedness of $M$ on $L^p(w)$, respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any $p\in(\frac{1}{2},\infty)$, under the assumption that $M$ is bounded on $L^p(w)$, we establish the boundedness of exponentially decaying kernel operators on $L^p(w)$ and weighted Fefferman–Stein vector-valued inequalities.

Let $0<\alpha<2$ and $p=2/(2-\alpha)$. We study critical nonlinear convolution inequalities for kernels $K:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^m$ and functions $\Phi:\mathbb{R}^m\to\mathbb{R}$ positively homogeneous of degrees $\alpha-2$ and $p$. We assume angular Lipschitz regularity for $\Phi$ and angular $C^{0,\gamma}$ regularity for $K$, where $0<\gamma\le1$ and $\gamma=1$ if $p<2$. On bounded finitely cornered planar domains with a common Dini tangent modulus on the boundary branches, we characterize $|\int_\Omega\Phi(K*f) dx|\le C\|f\|_{L^1}^p$ for real densities $f\in L^1(\mathbb{R}^2)\cap L^s_{\mathrm{loc}}(\mathbb{R}^2)$, $1<s\le\infty$, by signed angular cancellation on the plane, tangent half-planes, and full vertex tangent cones. An analogous criterion holds on infinite sectors for compactly supported mean-zero densities. For the Newton kernel and each nonzero angularly Lipschitz, positively two-homogeneous $\Phi:\mathbb{R}^2\to\mathbb{R}$ with vanishing half-circle moments, we construct a bounded convex $C^1$ domain $\Omega_\Phi$, independent of $\varepsilon$, and smooth tests $u_\varepsilon$ satisfying $\|\Delta u_\varepsilon\|_{L^1(\Omega_\Phi)}+\|\partial_nu_\varepsilon\|_{L^1(\partial\Omega_\Phi)}\le2$, while, despite tangent cancellation, their nonlinear integrals grow in magnitude like $\log\log(1/\varepsilon)$. On the Dini class, a constant-preserving linear extension with compact support modulo the domain mean and controlled measure-valued Laplacian yields the same necessary-and-sufficient criterion for the Newton-kernel PDE inequality, without exact corner charts. We also construct infinite-dimensional cancellation spaces containing nonquadratic functions and classify the quadratic locus in polygon moduli.

We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley $g$-functions in the spirit of Stein's work for compact Lie groups and prove their $L^{p}$-boundedness for $1<p\leq 2$ and for some of them also for $1<p<\infty$. As an application, we define associated Riesz transforms and imaginary powers and prove their $L^{p}$-continuity for $1<p<\infty$. Passing to the average with respect to the action of the associated reflection group, we obtain $L^{p}$-boundedness of $g$-functions for the symmetric Poisson semigroup for all $1<p<\infty$. In particular, our framework covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.

This paper is a direct continuation of papers arxiv:2401.00053 and arxiv:2507.09332. By this reason, neither the introductory part of the paper nor the bibliography are duplicated here. However, for the reader's convenience, the formulas from the previous papers that are referenced are given in a special addendum at the end of the paper under their original numbers. In this paper, we investigate new local foliations: major and medium pockets and several bilinearity regions. The appearance of such local foliations is illustrated by further analysis of examples with third-degree polynomials. At the end of the paper, a complete overview of possible foliations is given when the boundary values are third-degree polynomials.

The asymptotic behavior of the coefficients of a sequence of polynomials is given under the conditions that all the zeros are real and positive and the zeros have an asymptotic zero distribution on $(0,\infty)$. We show how the limit for the coefficients is related to the Stieltjes transform of the asymptotic zero distribution and make a connection with the $S$-transform in free probability. We illustrate with some examples involving orthogonal and multiple orthogonal polynomials.

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

We establish a weighted restriction estimate in $\mathbb{R}^2$. Our proof combines new two-ends Furstenberg estimates with refined decoupling.As an application, our result yields decay estimates for the circular $L^p$-means of the Fourier transforms of fractal measures in $\mathbb{R}^2$.

We give a self-contained proof of Bourgain's finite $\Lambda(p)$ selection theorem with an explicit probability bound. For every $p>2$ and $N\ge2$, given $N$ pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality $\lceil N^{\frac{2}{p}}\rceil$ satisfies the $\Lambda(p)$ inequality with probability at least $1-\frac{1}{N}$, with a constant depending only on $p$. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed $A>0$, the success probability is at least $1-\frac{1}{N^A}$ with a constant depending only on $p$ and $A$, and independent of $N$. The proof uses a greedy approximation in $L^r$, with $r>2$, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.

We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy–Littlewood maximal operator for every $1<p<\infty$, proving that it equals $p/(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly larger than the constant associated with the critical power $|x|^{-1/p}$, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.

In this present paper, we study the weighted weak type $(1,\,1)$ bounds for rough maximal singular integral operators as well as the the corresponding variation and jump operators. Our first result is the weighted weak type $(1,\,1)$ bound for the maximal singular integral $$T_{\Omega}^{*}f(x)=\sup\limits_{\varepsilon>0}|T_{\varepsilon,\Omega}f(x)|=\sup\limits_{\varepsilon>0}\bigg|\int_{|x-y|>\varepsilon} \frac{\Omega(x-y)}{|x-y|^{d}}f(y)dy\bigg|,$$ where $\Omega\in L^\infty(\mathbb{S}^{d-1})$, homogeneous of degree zero, and satisfies the cancellation condition. We show that $$\|T_{\Omega}^{*}\|_{L^1(w)\rightarrow L^{1,\infty}(w)}\lesssim[w]_{A_1}[w]_{A_\infty}\log([w]_{A_\infty}+1),$$ where $w$ belongs to the Muckenhoupt class $A_1(\mathbb{R}^d)$. This represents an essential improvement of a result (Honzík, Inter.Math. Res. Not. 2020) and a result (Bhojak and Mohanty, J. Funct. Anal.2023). Our second one is the weighted weak type $(1,\,1)$ bounds for variation and jump operators corresponding to $\{T_{\varepsilon,\Omega}\}_{\varepsilon\in2^{\mathbb{Z}}}$ and $\{T_{\varepsilon, \Omega}^{\phi}\}_{\varepsilon\in\mathbb{R}^{+}}$ under the condition $\Omega\in L^\infty(\mathbb{S}^{d-1})$, where $T_{\epsilon,\Omega}^{\phi}$ represents a smooth truncation of rough singular integral operator.These results of this part are the first weighted weak type $(1,\,1)$ variation inequalities and jump inequalities for rough singular integrals.