Papers

We prove the $L^p(\mathbb R^n\times\mathbb R^m)$ boundedness of the rough product singular integral $T_\Omega$, its maximal truncation $T_\Omega^*$, and the product Marcinkiewicz integral $\mu_\Omega$ for $1<p<\infty$, assuming that $\Omega\in H^1(S^{n-1}\times S^{m-1})$ satisfies separate cancellation. These results include kernels in $H^1(S^{n-1}\times S^{m-1})$ that lie outside the Orlicz classes required in earlier work. We also obtain equivalent characterizations of the product Hardy space in terms of restricted, conjugate, and spectral Riesz transforms, together with radial maximal functions, non-tangential maximal functions, and area integrals associated with the spectral and ball Poisson extensions. The operator bounds are proved using restricted Riesz transforms and the Calderón–Zygmund rotation method. For the Marcinkiewicz integral, we further establish integrable bounds for the radial profiles of the transformed kernels.

Dujella, Jak\v setić and Pe\v carić recently established in \cite{DJP}, a chain of power-sum inequalities, built from Hölder's and Cauchy's inequalities and their converse forms, and applied it to the Fibonacci sequence using the identities $\sum_{i=1}^nF_i^2=F_nF_{n+1}$ and $\sum_{i=1}^nF_iF_{i+1}=F_{n+1}^2-\frac{1+(-1)^n}{2}$. We show that this machinery applies uniformly across the one-parameter family of $k$-Fibonacci and $k$-Lucas numbers of Falcón and Plaza, via the generalized identities \begin{align*} \sum_{i=1}^nL_{k,i}^2=\frac{L_{k,n}L_{k,n+1}-2k}{k},\qquad \sum_{i=1}^nF_{k,i}^2=\frac{F_{k,n}F_{k,n+1}}{k}, \end{align*} and \begin{align*} \sum_{i=1}^nL_{k,i}L_{k,i+1}=\frac{L_{k,n+1}^2}{k}-k+\bigl((-1)^n-1\bigr)\left(\frac{2}{k}+\frac{k}{2}\right), \end{align*} which recover the classical Fibonacci and Lucas identities at $k=1$. We further exploit the cross-identity $F_{k,i}L_{k,i}=F_{k,2i}$, valid for every $k\ge1$, to obtain a Cauchy-Schwarz-type inequality linking the two families that has no counterpart in the Fibonacci-only setting. At $k=1$, this specializes to a fully explicit elementary inequality between ordinary Fibonacci and Lucas numbers, alongside the corresponding Hölder and Cauchy-conversion refinements we obtain for ordinary Lucas numbers.

Let $T$ be a convolution Calderón–Zygmund operator satisfying Hytönen's non-degeneracy condition. We prove that, for every $p\in(1,\infty)$ and $m\ge3$, there exists a positive constant $C$ such that, for any $t\in[1,\infty)$, \begin{equation*} \sup_{[W]_{\mathcal A_p}\le t} \|T\|_{L^p(W)\to L^p(W)} \ge C t^{1+\frac{1}{p(p-1)}}, \end{equation*} where the supremum is taken over all matrix weights $W:\ \mathbb R^n\to\mathbb C^{m\times m}$ with their weight constants $[W]_{\mathcal A_p}\le t$. If, in addition, the odd part of the kernel is non-degenerate (such as the Hilbert transform and the Riesz transforms), the same conclusion holds for every $m\ge2$.

The Favard length of a planar set is the average length of its orthogonal projections. We provide sharp estimates for the Favard length decay of a random variant of the non-homogeneous four corner Cantor sets introduced by Garnett in the 1970s. As a corollary, we obtain examples which show that Mattila's classical lower bound on the Favard length in terms of the Riesz capacity is sharp for all possible decay rates. We also obtain a new family of examples of sets with positive analytic capacity and zero Favard length.

We study the elementary continued fractions of Ramanujan. The entries considered are precursors of his hypergeometric and $q$-hypergeometric continued fractions. They appear in the first third of Chapter 12 of Ramanujan's second notebook.

It is shown that all solutions of the canonical equation y^a = 1 + a*x*y^b with arbitrary complex exponents a, b (case m = 1) and of its limiting cases - the binomial root y = (1 + a*x)^(1/a) at b = 0, the exponential y = e^x at a = 0, b = 0, and the Lambert equation y = e^(x*y^b) at a = 0 - as well as the logarithms of these solutions (case m = 0), are expressed by a single power master series M(m; a; b; x) = m + x + ((m - a + 2b)/2!) x^2 + ((m - a + 3b)(m - 2a + 3b)/3!) x^3 + ... For multivalued solutions beyond the radius of convergence, a geometric criterion for selecting the conjugate series (the same master series, but with a different set of parameters M(m; a'; b'; x')) is proposed, ensuring branch continuity on the complex plane. This representation of power series in terms of master numbers not only allows one to compute coefficients in degenerate cases where the gamma-function representation loses meaning, but also provides a transparent analytical algorithm for the merge operation on master series. The method is illustrated on problems in gas dynamics and plasma physics, where it is demonstrated that physically realizable states require the application of conjugate series beyond the radius of convergence of the basic expansion.

We study weighted $L^2$ Bernstein-type inequalities for algebraic polynomials on a trapezoidal domain. Using a parametrization by the unit square, we construct a natural symmetric second-order Sturm–Liouville operator and derive exact directional Bernstein inequalities with explicit constants. We also obtain strengthened estimates for the ordinary partial derivatives. Finally, we relate the corresponding boundary factors to Dini derivatives of the Siciak extremal function of the trapezoid.

Given a reproducing kernel Hilbert space $\mathcal H$ and a bounded operator $T$ on $\mathcal H$, we say that $T$ obeys the strong reproducing kernel thesis (SRKT) if $\|T\|=\sup_k\|Tk\|$, where $k$ ranges over the normalized reproducing kernels in $\mathcal H$. Two problems are posed. One of these is to determine which Hankel operators on the Hardy space $H^2$ obey the SRKT. The other concerns the geometry of the unit ball of $BMO$ with respect to the Garsia norm.

We study the recovery of a finite three-dimensional point cloud from its spherical Fourier signal subspace at a fixed wavenumber. Two ESPRIT constructions are analyzed. The first uses finite shifts between paired small circles; the second uses angular derivatives on the whole sphere. Both recover the same commuting coordinate matrices, either through their finite exponentials or directly from a coupled linear system. We prove that the continuous generator system is injective for every distinct point cloud and that one guard harmonic degree preserves the retained equations exactly. For $s$ targets, the cutoff $K\ge s$ gives generic finite identifiability, but not uniform conditioning. We also derive a finite-cutoff perturbation bound. The accuracy of the coordinate estimates is a separate issue. A one-target perturbation calculation shows that the original generator estimate can retain a nonzero bias as the wavenumber increases. To address this bias, we use the generator eigenbasis to separate approximate atoms and recover their locations from amplitude-normalized phases. Under pointwise atom dominance, the resulting error is $O(\kappa^{-1})$; for a dominant two-atom mixture at fixed separation, it is $O(\kappa^{-2})$. Experiments on structured point clouds with smooth perturbations compare the two constructions and their use as initializers for a common MUSIC refinement.

We study probability measures minimizing the energy with kernel $|x\cdot y|^p$ on the real unit sphere. For every positive rational $p=a/b$, each minimizer admits a finitely supported minimizing replacement. If $a/b$ is reduced and $b\ge d$, every minimizer on $\mathbb S^{d-1}$ is finitely supported. The latter assertion follows from a finite-rank algebraic lift, Nash curve selection, and a Wronskian multiplicity bound. For irrational exponents, we prove that a positive semidefinite restriction of finite rank cannot have infinite compact support. We also establish local stability and a quantitative support bound under positive-definite Hessians at every contact of every limiting minimizer. This hypothesis is stronger than finite support and does not prove unconditional openness of the discreteness regime. Finally, subtracting a single suitably scaled high even power produces a uniformly nearby kernel whose every minimizer has finite support. We identify the atomic-purity selection of these approximants and prove a target-centered recovery theorem, separating finite approximation from discreteness of the limiting problem.

We prove a sharp boundedness result on $L^4(\mathbb R^2)$ for mollifications of the disc multiplier.

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|\delta V\|_{L^p(\gamma_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(\gamma_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2\mathcal N^{-1}$, where $\mathcal N := -\Delta+x\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\geq2$, and $m_p := (\mathbb E|g|^p)^{1/p}$ for a standard Gaussian $g$, then \[ \|DP\|_{L^p(\gamma_d;\mathbb R^d)} \leq\frac{2\sqrt{2e\, \mathrm{deg}(P)}}{m_p}\,\|P\|_{L^p(\gamma_d)}\, . \] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\sqrt{\mathrm{deg}(P)/p}$ throughout the range $p\geq2$.

In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in\Omega: c\le f(x)<c+\delta\},\quad c\in\mathbb R,\;\delta>0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n-1}|\, r(\Omega)^{n-2}\,\delta,$$ where $r(\Omega)=\frac12\,{\mathrm{diam}\Omega}$. As an application, we derive $L^2$ estimates for oscillatory integrals with convex phases.

We prove that the strong spherical maximal operator in $\mathbb{R}^n$ is bounded on $L^p$ for $n\geq4$ and $p>(n+1)/(n-1)$. This establishes the conjecture of Hickman and Zahl in the remaining dimensions. The main ingredient is a local $L^2$ estimate for the pieces obtained by a dyadic decomposition in the normal variable. After squaring one input coordinate, we reduce this estimate to a $TT^*$ argument for diagonal quadratic phases.

This paper is concerned with the maximal regularity theory for non-autonomous stochastic evolution equations with a generalized fractional derivative in UMD spaces. The generalized time-fractional derivative provides a unified framework covering both the classical Riemann-Liouville and Caputo fractional derivatives, which accommodates a wider class of anomalous diffusion processes with intermediate memory effects. Based on the singularities of the initial term and the stochastic convolution kernel, a time-weighted space and a regular-singular decomposition are used to obtain the well-posedness, space-time regularity, and stochastic maximal $L^p$-regularity results. Our results are applied to non-autonomous stochastic diffusion equation and stochastic fractional reaction-diffusion SIR model.

Let \(0<\alpha\leq1\), and let \(g_\alpha\) be the Littlewood-Paley function associated with the fractional Poisson kernel, which reduces to the classical \(g\)-function when \(\alpha=1\). We establish dimension-free \(L^p\) bounds for \(g_\alpha\) for every \(1<p<\infty\) and identify its exact \(L^2\) norm as \(\sqrt{\alpha/2}\). We further determine the precise asymptotic behavior of the weak-type \((1,1)\) constant as \(n\to\infty\), yielding quantitative upper and lower bounds in terms of the dimension.

Let \begin{equation*} f_p(x)=\frac{\mathcal{K}(\sqrt{x})}{p-\ln\sqrt{1-x}}, \end{equation*} where $\mathcal{K}$ denotes the complete elliptic integral of the first kind, and set $g_p=1/f_p$. Tian and Yang conjectured that, at $p=\ln4$, both $-f_{\ln4}^{\prime\prime\prime}$ and $g_{\ln4}$ are absolutely monotonic on $(0,1)$. Writing \begin{equation*} \frac{2}{\pi}f_{\ln4}(x)=\sum_{n=0}^{\infty}a_n(\ln4)x^n,\quad \frac{\pi}{2}g_{\ln4}(x)=\sum_{n=0}^{\infty}b_n(\ln4)x^n, \end{equation*} we prove \begin{equation*} a_n(\ln4)<0\quad(n\geq3),\quad b_n(\ln4)>0\quad(n\geq0), \end{equation*} thereby settling both conjectures. The proof of the first inequality is based on the representation \begin{equation*} a_n(\ln4)=\frac{(-1)^nC}{15^{n+1}}-M_n, \end{equation*} where $C>0$ and $(M_n)$ is a positive moment sequence. Its strict log-convexity, together with estimates for the initial coefficients, determines the sign of every $a_n$. The second inequality follows from a factorization of the quadratic truncation of the normalized Taylor series and a reciprocal-series argument.

Operator learning of periodic solutions requires deciding how simulation data should be recorded and represented. A natural choice is to integrate long enough for transients to decay and record a window wide enough to contain at least one full period of all trajectories. We find that these conservative choices can make the resulting trajectories difficult to learn, even when the underlying periodic orbits vary regularly with system parameters. Unaligned trajectories generalize poorly even within the training distribution. Phase alignment substantially improves in-distribution generalization, but models trained on a fixed physical-time window still have large errors on trajectories with periods outside the training range. We explain both failures through a common mechanism: frequency differences accumulate over time, so the target phase varies rapidly with the parameters. Predictors that cannot track this variation incur a population MSE floor in both settings; for fixed window prediction, we also derive a per-sample lower bound. We then study one of the simplest representations that escape these floors: learning an aligned, normalized waveform and its period separately. We establish regularity of the decoupled targets under ODE assumptions and show experimentally that this approach avoids both failures in ODE systems and a PDE case study.

Let $H$ be an infinite-dimensional Hilbert space and let $\set{x_n}\inN$ be a frame for $H$. We say that a sequence $\set{y_n}\inN$ is an alternative dual of $\set{x_n}\inN$ if $$x=\sumli\ip{x}{y_n}x_n\quad \text{for all }x\in H,$$ with the convergence of the series in the norm of $H.$ We say that a sequence $\set{z_n}\inN$ is a pseudo-synthesis dual of $\set{x_n}\inN$ if $$x=\sumli\ip{x}{x_n}z_n \quad \text{for all }x\in H,$$ with the convergence of the series in the norm of $H.$ We prove that, at every level of smoothness of the generator, there exist Gabor frames for $L^2(\R^d)$ with infinitely many alternative duals (and pseudo-synthesis duals) such that none of them is a Gabor system

For a reduced crystallographic root system with Weyl group $W$, Weyl alternants are signed sums of exponentials indexed by $W$. They occur in the numerators of the Weyl character formula and the explicit formula for spherical functions on complex semisimple groups. We establish rootwise bounds that quantify cancellation in these oscillatory sums, with each positive root contributing a factor depending on both the spatial variable and the spectral parameter. More generally, we establish rootwise bounds for all mixed directional derivatives of normalized Weyl alternants. Our proof extends to arbitrary rank the descent strategy used by the second author for pointwise character bounds on $\mathrm{SU}(3)$. The key ingredients include the decomposition of the Weyl group into parabolic double cosets and the relative BGG–Demazure identity. For spherical functions on complex semisimple groups, these estimates recover the recent pointwise bound of Brumley, Marshall, Matz, and Peterson, and they refine the derivative bounds of Cowling and Nevo by controlling arbitrary mixed radial derivatives uniformly across spatial and spectral root hyperplanes. Furthermore, by a standard localization argument relative to the affine Weyl arrangement, we establish periodic alternant estimates for regular integral weights and corresponding rootwise bounds for all mixed radial derivatives of irreducible characters of compact connected semisimple groups.

In this paper, we focus on the positive reflectionless measures rigidity problem proposed by Xavier Tolsa in [16]. Let $1\leq n<d$, we consider positive reflectionless measures for the $n$-dimensional Riesz transform in $\mathbb R^d$. Under polynomial upper growth and a lower mass bound at large scales about one point, we obtain two flatness results for reflectionless measures. The first concerns globally defined Lipschitz graph measures with finite $\dot H^{1/2}$ energy and bounded, uniformly positive parameter densities whose deviations from a constant belong to $L^2$. The second case concerns a reflectionless measure that coincides with a flat measure outside a compact set. In both cases, the measure is a constant multiple of the $n$-dimensional Hausdorff measure on an affine $n$-plane. For integers $s>n/2$, we further obtain $H^s$ estimates controlling the graph gradient and density perturbation by the normalized Riesz field under smallness of the slope and density perturbation in $L^\infty$. The constants are independent of the supports.

We show that there exists a subadditive bijection $f:(0,\infty)\to(0,\infty)$ such that $\liminf_{x\to 0}f(x)=0$ and $\limsup_{x\to 0}f(x)=1$. This answers a question by Matkowski and Swiatkowski in [Proc. Amer. Math. Soc. 119 (1993), 187–197].

We prove sharp lower bounds for the Schrödinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.

We survey Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type, with particular emphasis on sharp constants and extremal functions. We discuss both classical results and recent developments, including limit relations between Nikolskii constants for polynomials and entire functions of exponential type, weighted and multidimensional versions, and inequalities on constrained classes of functions. Special attention is given to the difficult case $p=1$ and to methods based on reproducing kernels, duality, and extremal approximation. We also describe connections of sharp Bernstein–Nikolskii inequalities with approximation theory, the Riemann zeta function, spherical designs, Turán–Delsarte type problems, and sphere packing. Several open problems concerning sharp constants, extremal functions, and multidimensional extensions are formulated.

For $1<p<\infty$, we characterize the boundedness of the segment multiplier on $L^p(w)$ by a large-scale Muckenhoupt condition, obtained by restricting the classical $A_p$ condition to intervals of length at least one. The proof uses a discretization involving the discrete Hilbert transform on an associated weighted sequence space.

Motivated by the recent paper [Sharp endpoint extension inequalities for the moment curve on finite fields II: an extremal property of the uniform distribution, arXiv:2609.29882], which proves sharp extension inequalities in finite fields via a two-point symmetrization argument, we prove here a more general component-wise inequality for permutation matches that implies theirs.

We show that, on the Fourier side, the sharp second order Heisenberg Uncertainty Principle (HUP) and its sharp stability for functions are nothing but the first order $L^{2}$-Caffarelli–Kohn–Nirenberg inequality and its sharp stability, applied to their Fourier transforms. This gives a proof in a few lines of the sharp stability estimate that we established recently by a much longer argument. The same observation then produces two one-parameter families of sharp fractional Heisenberg Uncertainty Principles, in which the gradient is replaced by a fractional power of the Laplacian: for each of them we compute the sharp constants, characterize all the optimizers, and establish the sharp stability estimates. The stability constant of the first family is equal to $1$ for every nonnegative order and every dimension. Negative orders, where the fractional Laplacian becomes a Riesz potential, are treated as well. Finally, we show that the deficit carries much more information than the distance to the optimizers alone: it controls an explicit chain of successive remainder terms whose constants are the successive spectral gaps of an explicit operator. For the classical Heisenberg Uncertainty Principle this gives three remainder terms with optimal explicit constants, and for the second order HUP a chain of four remainder terms measured against explicit confluent hypergeometric profiles and coupled through one single set of parameters. As applications, we also establish the chain of stability of the HUP for curl-free vector fields.

We investigate orthogonality under the shift-invariant resolvent delta operator $\mathscr L=D(I-D)^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\mathscr L$-Appell if and only if it is a translated Laguerre sequence. We then move beyond the Appell setting and show that the diagonal family associated with $$\mathcal S_\beta=\frac{\beta}{\beta+1}\mathcal L^0+\frac{1}{\beta+1}\delta$$ satisfies the exact Hahn relation $$\mathscr L B_{n+1}(x;\beta)=(n+1)B_n(x;\beta+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=\theta\,\mathscr L P_{n+1}(c)$ already forces the orthogonality functional to be a translated Laguerre functional or a translated member of the diagonal family. Connection and structure relations, a semiclassical Pearson equation, a differential characterization, and a generating function complete the description. Thus the resolvent operator provides an explicit setting in which delta-operator lowering and semiclassical orthogonality are linked by exact structural identities.

We prove a prescribed projection theorem for any class of non-linear projections defined on open subsets of the plane which satisfies some mild geometric conditions. This generalizes Davies's classic result on prescribed projections for orthogonal projections, as well as a recent result of authors, Chang, McDonald, and Taylor, on a specific class of non-linear projection maps. We also give several geometric applications, including a proof that certain classes of plane curves can be used to cover arbitrary measurable sets with measure zero error.

Scott and Sokal asked whether every homogeneous polynomial with the half-plane property has a completely monotone negative power. We answer this question affirmatively and prove the Riesz-positivity conjecture of Michałek, Sturmfels, Uhler and Zwiernik, restated by Kozhasov, Michałek and Sturmfels: in $n$ variables, every exponent $\alpha\ge4096n^2$ is admissible, independently of the degree and coefficients. For complete hyperbolic polynomials the Riesz density is strictly log-concave, with relative Gaussian error at most $512n^2/\alpha$ and explicit curvature bounds. We characterize admissible exponents by a common spectral Dirichlet law, prove $n\le m+\alpha m(m-1)$ with its equality case, and obtain the sharp degree-dependent gap $0<\alpha<1/(2(m-1))$ whenever the degree-$m$ polynomial has a nonlinear irreducible factor. A nonnegative fourth-order defect of $-\log p$ vanishes at one point precisely for products of positive integer powers of Euclidean Jordan determinants. We classify the corresponding logarithmic Monge-Ampère equation and answer the question of Etingof, Kazhdan and Polishchuk about polynomial multiplicative Legendre transforms within irreducible complete hyperbolic polynomials; the general question has counterexamples, the Clifford quartics of Kogiso and Sato. The classification extends to reducible polynomials under a boundary-visibility hypothesis, and asymptotic common-power formulas for the Riesz densities force exact Jordan formulas.