Papers

Let $\lambda>-1/2$, $\lambda\neq0$, and let $\Delta_\lambda=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx}$ be the Bessel operator on $\mathbb R_+=(0,\infty)$ studied by Muckenhoupt and Stein (TAMS 1965). Andersen and Kerman (Studia Math. 1981) proved that for $1<p<\infty$, the Bessel Riesz transform $R_\lambda=\frac{d}{dx}\Delta_\lambda^{-1/2}$ is bounded on $L^p(\mathbb R_+,w(x)\,dx)$ if and only if $w$ is in the intrinsic class $A_{p,\lambda}$. However, the sharp quantitative weighted bound via $[w]_{A_{p,\lambda}}$ was not addressed before. In this paper, we give a positive answer to this question by proving the sharp quantitative estimate $$ \|R_\lambda f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,\lambda} [w]_{A_{p,\lambda}}^{\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ Moreover, for a real-valued function $b$ in the Bessel BMO space ${\rm BMO}_\lambda$, the sharp weighted bound for the Riesz commutator is $$ \|[b,R_\lambda]f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,\lambda}\|b\|_{{\rm BMO}_\lambda} [w]_{A_{p,\lambda}}^{2\cdot\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ The argument uses the exact conjugation $U(x)=x^{p-2\lambda-1}w(x), \ d\nu_\lambda=x^{2\lambda+1}\,dx, \ [U]_{A_p(d\nu_\lambda)}=[w]_{A_{p,\lambda}},$ which reduces the Andersen–Kerman estimate to an $A_p$ weighted estimate for the auxiliary operator $$ \mathcal R_\lambda F(x)=\frac{1}{x}R_\lambda(yF(y))(x) $$ on the space of homogeneous type $(\mathbb R_+,|x-y|,d\nu_\lambda)$, whose kernel is a standard Calderón–Zygmund kernel with respect to $\nu_\lambda$.

Fix $I = (0,\rho)$, where $0<\rho\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserves positive semidefiniteness for all $A \in \mathbb{P}_n(I)$. We characterize these functions exactly: if $f\in C^{2n-2}(I)$ and $\mathcal{E} = x\frac{d}{dx}$, then this holds if and only if $$ f^{(k)}(x)\geq 0 \quad (0\leq k\leq n-1) \qquad\text{and}\qquad \bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j = 0}^{n-1}\succeq 0 $$ for every $x\in I$. Regularization then removes all a priori smoothness: for $n\geq2$, every preserver belongs to $C^{2n-4}(I)$ and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald–Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton–Guillot–Khare–Putinar and Khare–Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside $I$ can detect positivity preservation on $\mathbb{P}_n(I)$.

We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$ \|(\log|a_f|^{2})^{1/2}\|_{L^{p'}(\mathbb{R})} \le \|f\|_{L^{p}(\mathbb{R})},\quad 1\le p<2, $$ for all $f\in L^{p}(\mathbb{R})$, where $a_f$ denotes the transmission coefficient of the nonlinear Fourier transform. In particular, this resolves the Muscalu-Tao-Thiele uniformity problem. The proof uncovers a hidden Hilbert-space structure that reduces the nonlinear inequality to classical interpolation.

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn–Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn–Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.

Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\varphi_d(z)=\Phi(z)\Phi(-z)$, and $\Phi$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute $\mathcal C_3$ to high accuracy. Even dimensions remain open within this approach.

In this project, I shall establish a large number of new Fejér-type inequalities which reduce all results of the following Subsections 1.1-1.3. A number of applications for the extended incomplete Beta function and the extended special means are given.

This paper addresses the Turán extremal problem on the symmetric three-interval set $\Omega_\lambda = (-1,1) \cup (\lambda-1,\lambda+1) \cup (-\lambda-1,-\lambda+1)$, $\lambda \geq 0$. We establish a discrete approximation principle relating the continuous Turán problem on $\Omega_\lambda$ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters $\lambda$. We further analyze the dependence of the Turán constant on $\lambda$ and derive upper bounds for all $\lambda > 5$.

It is well known the local integral inequality which is called Poincaré–Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigated similar weighted generalization for this inequality with the local weights which are depending on the domain cube without assuming the Muckenhoupt condition. Moreover, we considered the weighted generalization for the homogeneous and inhomogeneous-type Sobolev embedding theorems as some applications.

For every positive integer $d$ divisible by $4$, we prove the following new upper bound for the Bourgain-Clozel-Kahane sign uncertainty constant: \[ \mathrm{A}_+(d) \le \sqrt{2 \left\lfloor \frac{d}{16} \right\rfloor + 2}. \] It recovers the optimal bound $\mathrm{A}_+(12) \le \sqrt{2}$ in dimension $12$ and improves the previously best known bound $\sqrt{(d+2)/(2\pi)}$ for all $d \ge 52$ divisible by $4$. The proof uses Fourier eigenfunctions and associated quasimodular forms constructed by Feigenbaum, Grabner, and Hardin.

Uniform asymptotic expansions are derived as $\kappa \to \infty$ for the Whittaker functions $W_{\kappa,\mu}(z)$, $M_{\kappa,\mu}(z)$, as well as related functions including generalized Laguerre polynomials. The results are uniformly valid for $0\leq\mu\leq(1-\delta_0)\kappa<\kappa$, where $\delta_0\in(0,1)$ is arbitrary and fixed. The analysis is based on the associated differential equation, which has a double pole and two turning points. At one of the turning points, which may coalesce with the double pole, a recently developed asymptotic theory is applied to obtain expansions involving Bessel functions. At the second turning point, expansions involving Airy functions are obtained. In both cases, the coefficients are readily computable, in contrast to those occurring in earlier results. The expansions, when taken together, uniformly cover the entire complex $z$-plane on the principal branch. Standard analytic continuation and connection formulas extend the results to all branches of $z$ and to negative $\mu$.

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are$\delta$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_\nu f\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_\nu} \leq Ch^\beta\|f\|_{L^2_\nu}. \] The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.

We address the question of which common differences can arise in arithmetic progressions contained in fractal sets. For a compact set $C\subset\mathbb R$, we investigate not only whether arithmetic progressions occur in $C$, but the full collection of their common differences. More generally, for a finite pattern $P$, we study the set of scales at which affine copies of $P$ appear in $C$. For affine self-similar sets satisfying strong separation, we obtain explicit restrictions on admissible common differences. Specializing to middle-$\varepsilon$ Cantor sets, we prove that the longest arithmetic progression has length four whenever $3-2\sqrt2<\varepsilon\le 1/3$, showing that the maximal progression length drops immediately from six at the critical parameter $\varepsilon=3-2\sqrt2$. We further develop recursive bounds for the sets of admissible common differences and derive explicit blackout intervals, namely ranges of scales for which arithmetic progressions cannot occur. On the positive side, sufficiently thick Cantor sets exhibit the opposite behavior. Combining a refinement of the Hunt-Kan-Yorke construction with the Newhouse Gap Lemma, we prove that every sufficiently small common difference occurs in a three-term arithmetic progression. In particular, if the largest bounded gap of a Cantor set is at most $0.067 diam(C)$ and its thickness is at least $6.96268\ldots$, then every common difference in $(0,0.435 diam(C)]$ occurs in a three-term arithmetic progression contained in $C$. Analogous interval results are obtained for four-term arithmetic progressions and asymmetric three-point patterns.

We extend the classical Taibleson-Weiss molecular theory to arbitrary radii, reaching the critical decay |x|^{-n/p}, with the resulting embedding shown sharp. This molecular theory governs the largest class of functions certifying membership in h^p(R^n), 0 < p <= 1, through Goldberg's convolution splitting: a critical Hölder modulus and an \ell^p-Dini decay majorant, sharp on the regularity, decay, and cancellation axes.

Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,\epsilon}(\# X)^{2+\epsilon}. \] In particular, this answers a question of Bourgain and Demeter concerning finite subsets of the unit sphere.

We study purely atomic measures whose range is a central Cantor set and characterize those central Cantor sets that are the range of exactly one such measure. Next, we extend the characterization to the case of symmetric Cantor sets using a different method of proof. Finally, we make a few initial observations and remarks on recovering a measure whose range is a Cantorval.

Let $f$ and $g$ be nonzero functions in $L^2([0,1])$. The woven weighted exponential system (associated with $f$ and $g$) is defined by $$\Wc(f,g)=\bigset{\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}\,|\,J\subset\Z}.$$ We say that $\Wc(f,g)$ is wovenly complete, (resp. wovenly minimal, a woven frame) if the weaving $\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}$ is complete, (resp. minimal, a frame) for all $J\subseteq \Z.$ In this paper, we study conditions that imply certain approximation properties of $\Wc(f,g)$, such as completeness, minimality and the frame property. We first provide a complete characterization of the woven weighted exponential systems that are wovenly complete. We also show that $\Wc(f,g)$ is a woven frame if $f/g$ is strictly positive or strictly negative over $[0,1].$ Additionally, several counterexamples are provided to show that certain seemingly correct conditions do not imply the desired approximation properties of $\Wc(f,g).$ All results presented in this paper apply equivalently to systems of regular translates and Gabor systems at critical density in $L^2(\R)$.

We study the zeros of functions in the Dirichlet space. Using extremal functions, we produce a necessary and sufficient condition for a sequence of points in the unit disk to be a zero set of the classical Dirichlet space. This Shapiro-Shields type condition involves kernels of the harmonic Dirichlet space associated with measures depending on the zero sequence.

Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $\lambda>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geq\lambda \right\} \right| \lesssim_n q^{2n-1}|E|\lambda^{-2n}. \] The second aim is to determine, for every $1\leq u,v\leq\infty$, the sharp exponent of $q$ in the corresponding $\ell^u\to\ell^v$ estimate. More precisely, we prove that \[ A_n^{\mathrm{rd}}(u,v) = \max\left\{ \frac{2n-1}{v},\ 1-\frac1u,\ \frac{2n}{v}-\frac1u,\ 1+\frac{2n}{v}-\frac{2n+1}{u} \right\}. \] The proof combines the polynomial method with multiplicities and a probabilistic covering argument based on the action of the affine symplectic group.

In 1936, Zalcwasser proved the almost everywhere convergence of the arithmetic means of the square subsequence of trigonometric Fourier partial sums and asked whether this result extends to higher powers and to Cesàro means of fractional order. We give affirmative answers to both questions in a stronger pointwise form. Let $0<\alpha\leq 1$, and let $P$ be an integer-valued polynomial of degree with positive leading coefficient. We prove that the $(C,\alpha)$ means of the Fourier partial sums along any sequence whose $k$-th term equals $P(k)$ for all sufficiently large $k$ converges to $f(x)$ at every Lebesgue point $x$ of every $f\in L^{1}(\T)$.

Let $n\ge3$, and let $u$ be a bounded hyperbolic harmonic mapping on the real hyperbolic upper half-space $\mathbb H^n$. First, for $0<\alpha<1$, we establish quantitative transversal and modulus criteria for the global $\alpha$-Hölder continuity of $u$. Then, we construct lacunary examples to show that these criteria are invalid when $\alpha=1$. Finally, under additional integral conditions, we obtain the corresponding global Lipschitz continuity and give Dini–Zygmund conditions on the boundary trace that ensure the Lipschitz continuity.

In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $\Omega \subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(\Omega) \le \frac{\pi^{s-\frac{1}{2}}\Gamma\left(\frac{3-s}{2}\right)} {s(1-s)\Gamma\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partial\Omega)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $\Gamma$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.

Let $N\in\mathbb N$ and $\gamma\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $\gamma\in[-1,0)$, $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy–Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy–Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$, and neither $\dot{W}^{1,1}(\gamma)/\mathbb R$ nor $\dot{BV}(\gamma)/\mathbb R$ is normable, which answers the above two questions.

Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg–Landau equation $$ -\Delta u = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\,dx < \infty, $$ then it has the asymptotic property that $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as Open Problem 2.5 in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the $L^4$-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the $L^2$-integrability of the function $1-|u|$ on an exterior domain, without any a priori integrability assumption.

For every $0\leq\beta\leq1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_\beta=f_\beta$ and $f_\beta(\sqrt{n}/[\log(e+n)]^{\beta})=0$ for all $n\geq 0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula. The construction uses a scale of reproducing kernel Hilbert spaces generated by the Fourier-invariant Hermite functions. Applying the Mehler formula, we identify the reproducing kernels of these spaces. By suitable estimates of these kernels, we show that $(\sqrt{n}/[\log(e+n)]^{\beta})$, with one auxiliary point added to it, is a universal interpolating sequence for at least one of the Hilbert spaces under consideration. However, this result fails when $\beta>1/2$.

We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.

We study two-weight weak-type estimates for the operator $S_v f = \mathcal{T}(fv)/v$, where $\mathcal{T}$ is the Hardy-Littlewood maximal operator or a Calderón-Zygmund operator (CZO) and $v$ is a weight. Concretely, under certain conditions on the weights involved, we prove that $S_v$ is bounded from $L^{1}(wv)$ to $L^{1,\infty} (uv)$. We also consider the corresponding inequalities when $\mathcal{T}$ is a higher-order commutator of a CZO. These types of results are inspired by the article of Sawyer in [21], (see also [17]).

For any given time lag $\gamma\in(0,1)$, we prove that the one-sided parabolic BMO space $\mathrm{BMO}^+(\gamma)$ coincides with the parabolic BMO space $\mathrm{PBMO}^-(\gamma)$ with equivalent norms, the parabolic Muckenhoupt class $A_{\infty}^+(\gamma)$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^+(\gamma)$ with $r\in[1,\infty)$, and the parabolic reverse Hölder classes $\bigcup_{q\in(1,\infty]}RH_q^+$ coincide with the parabolic Muckenhoupt classes $\bigcup_{r\in[1,\infty)}A_r^+(\gamma)$, and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016)]. To show them, we establish the uniform parabolic space-time shifting property for parabolic reverse Hölder weights, and develop the one-sided stopping time argument which yields a new parabolic John–Nirenberg inequality for $\mathrm{BMO}^+(\gamma)$. As applications, we obtain John–Nirenberg and exponential integrability characterizations of $\mathrm{BMO}^+(\gamma)$, prove that $\mathrm{BMO}^+(\gamma)$ is independent of the positive time lag, and identify its null space.

We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.

We prove that the conformal dimension of every locally finite Borel measure is either zero or infinite. The main ingredient is a quasisymmetric dimension-reduction theorem: every full-support probability measure of finite Hausdorff dimension on a separable metric space admits quasisymmetrically equivalent metrics in which its Hausdorff dimension is arbitrarily small. In particular, every locally finite Borel measure on a doubling metric space has conformal dimension zero. We also prove that for every $n\geq 1$ and $p>0$ there are a metric space $X$, a quasisymmetric homeomorphism $f:[0,1]^n\to X$, and a Borel set $E\subset[0,1]^n$ such that $\dim_H f(E)\leq p$ and $\dim_H([0,1]^n\setminus E)\leq n-1+p$. The second bound is sharp up to $p$: if $\dim_H f(E)<1$, then $\dim_H([0,1]^n\setminus E)\geq n-1$.

We show that for every conformal map $\phi:\mathbb{D}\to\Omega\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $\Omega$, the length of $\phi^{-1}(\Omega\cap L)$ is at most $\pi^2$ for every line $L$.