arXiv subjects

Lixin Yan

Publications and source records attributed to Lixin Yan.

At least 19 recordsLinked to original sources

arXiv2026

Fefferman--Stein type inequalities via area and maximal functions for Schr\"odinger operators with applications

In this paper, we establish a Fefferman--Stein inequality in terms of area function and non-tangential maximal function associated with the Schr\"odinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ denotes the sub-Laplacian on $\mathcal G$ and $V$ is a nonnegative locally integrable function. As an application, we extend this inequality to the tensor product $\mathcal G_1 \times \mathcal G_2$ of two stratified Lie groups and develop atomic decompositions associated with the Schr\"odinger operator for functions in the Orlicz space $L\log^{+}L(\mathcal G_1 \times \mathcal G_2).$ Using these atomic decompositions, we further prove weak-type endpoint estimates for the area integral operator and the Riesz transforms associated with the Schr\"odinger operator on $L\log^{+}L(\mathcal G_1 \times \mathcal G_2),$ thereby extending the celebrated result of R.\,Fefferman and E.M.\,Stein \cite{FSt1982} to the setting of singular integrals with non-smooth kernels.

math.AP↗

arXiv2026

BMO Classification for Two-Dimensional Dunkl Newton and Green Kernels

Using Gaussian heat-kernel estimates, we classify planar Dunkl Newton kernels in weighted Euclidean BMO and obtain the corresponding local classification for unit-ball Green kernels. For atomic Newton potentials supported on a regular reflection orbit, orbit BMO detects exactly whether the coefficients are constant along the orbit. We also prove an intrinsic Dunkl--CLMS theorem in the heat-semigroup Hardy space and derive intrinsic and Euclidean-source Newton--Wente estimates, extending the BMO--Hardy-space method of Chanillo and Li to the Dunkl setting. Further consequences include sharp local atomic estimates, same-domain criteria, a localized $L^1$-to-BMO bound, and a Brezis--Merle-type estimate.

math.AP↗

arXiv2026

Herz versus Fefferman: Symmetric and asymmetric Bochner--Riesz theory

We study Bochner--Riesz summability for a one-dimensional model of noncompact manifolds with ends. Each end has an effective Euclidean dimension, and these dimensions may differ from one end to another. The central point is the contrast between the symmetric and asymmetric cases. When the end dimensions agree, the model follows Herz's radial theory for the Euclidean Laplacian. When they are unequal, a Fefferman-type obstruction appears, analogous to the ball multiplier obstruction in higher-dimensional Fourier analysis, even though the model itself is one-dimensional. We give a complete characterisation of the $L^p$-boundedness of the corresponding spectral projections and Bochner--Riesz means. In the asymmetric case, the boundedness range contains an additional restriction depending on the difference between the end dimensions. This restriction is absent from Herz's radial model and shows that Bochner--Riesz summability on spaces with unequal ends is governed not only by the maximal end dimension, but also by the interaction between the ends.

math.AP↗

arXiv2026

Longitudinal beam instability driven by coherent radiation in an SSMB laser modulator

Storage ring-based steady-state microbunching (SSMB) is a promising approach for generating high-average-power coherent radiation, while the instabilities driven by coherent undulator radiation in the laser modulator (LM) is important for the ring performance. In this paper we investigate the longitudinal single-bunch multi-turn LM instability using cavity mode decomposition techniques. The evolution of the wakefield in the longitudinal beam dynamics equations are derived, and the instability growth rates are analyzed. Numerical simulations show excellent agreement with the theoretical model, validating the mode decomposition approach. These findings provide critical insights into the design and operation of SSMB storage rings, suggesting effective mitigation strategies to suppress the instability and enhance the overall performance.

physics.acc-ph↗

arXiv2025

Growth Estimates for Solutions to the Wave Equation on Damek--Ricci Spaces

Let $\mathcal{L}$ be the positive definite left-invariant distinguished Laplacian, and let $\mathrm{d}\rho$ denote the right Haar measure on a Damek--Ricci space $S$. Let $u(t,x)$ denote the solution to the wave equation $\partial_t^2 u + \mathcal{L} u=0$ with initial data $(u,\partial_t u)|_{t=0}=(f,g)$. In this paper, we establish the sharp-in-regularity $L^p$-bounds \begin{align*} \|u(t,\cdot)\|_{L^p(S ,\mathrm{d}\rho)} \lesssim_p(1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_0}{2}}\!f\|_{L^p(S ,\mathrm{d}\rho)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_1}{2}}\!g\|_{L^p(S,\mathrm{d}\rho)} \end{align*} for all $t\in\mathbb{R}^*$ and $1<p<\infty$, where the exponents $\alpha_0 = (n-1)\left|1/p-1/2\right|$ and $\alpha_1 = (n-1)\left|1/p-1/2\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by M\"uller, Thiele, and Vallarino.

math.CA↗

arXiv2025

Terahertz-Driven Nano-tip Field-Emission Electron Gun and Cascaded Acceleration

This paper reports two versions of terahertz (THz)-driven nanotip field-emission electron guns: single-layer reflective guns (SLRGs) and double-layer reflective guns (DLRGs). Both guns use nanotip emitters and accelerate electrons through the electric field of the THz wave. SLRGs employ a reflective structure to superimpose the initial and subsequent half-cycles of the THz electric field, enhancing the field amplitude and acceleration efficiency. Experiments have demonstrated that SLRGs achieve higher acceleration efficiency than single-layer nonreflective guns (SLNRGs) for identical THz input energies. This constitutes direct experimental verification of the efficacy of the reflective structure. Theoretically, SLRGs operating in single-feed mode can match the acceleration efficiency of dual-feed SLNRGs while reducing operational complexity. DLRGs demonstrate THz-driven cascaded electron acceleration through precise scanning of the delay between two incident THz beams. This represents a direct experimental demonstration of cascaded acceleration in THz-driven electron sources. The experimental results of DLRGs align closely with the results of electron dynamics predicted by simulations. This establishes the foundation for developing multilayer high-acceleration-efficiency THz-driven high-energy electron guns. The ability to manipulate the THz for each layer individually holds promising potential for improving the beam quality of THz electron guns.

physics.acc-ph↗

arXiv2025

Observation and Interpretation of Field Emission Saturation Induced by an Ultra-fast Intense Terahertz Field

Field emission under ultra-fast intense terahertz fields provides a promising approach for generating electron bunches with ultrashort pulse duration and high charge densities. It is generally believed that the field emission current described by traditional field emission theory increases dramatically with the applied electric field. However, we conducted extensive field emission experiments using quasi-single-cycle strong-field terahertz radiation at various energy levels and different temperatures and observed an intriguing phenomenon where the emitted charge reached saturation. A novel model is proposed to interpret this phenomenon, which considers the contribution of surface valence electrons and the dynamic replenishment of free electrons from the bulk to the surface. The experimentally observed convex relationship between the emitted charge and terahertz energy is consistent with the model prediction, unlike the concave relationship derived from the traditional field emission formula. In addition, another observed counter-intuitive phenomenon, the inverse correlation between the cathode temperature and saturated emission charge, is also well interpreted by the model. This work offers comprehensive insights into field emission dynamics under ultra-fast intense fields, paving the way for generating electron bunches with unprecedented temporal resolution.

physics.acc-ph↗

arXiv2025

Sharp $L^p$-estimates for wave equation on $ax+b$ groups

Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} \rho$ which is of $ax+b$ type, and $L$ be the positive definite distinguished left invariant Laplacian on $G$. Let $u=u(t,\cdot)$ be the solution of $u_{tt}+Lu=0$ with initial conditions $u|_{t=0}=f$ and $u_t|_{t=0}=g$. In this article we show that for a fixed $t \in{\mathbb R}$ and every $1<p<\infty$, \begin{align*} \|u(t,\cdot)\|_{L^p(G)}\leq C_p\Big( (1+|t|)^{2|1/p-1/2|}\|f\|_{L^p_{\alpha_0}(G)}+(1+|t|)\,\|g\|_{L^p_{\alpha_1}(G)}\Big) \end{align*} if and only if \begin{align*} \alpha_0\geq n\left|{1\over p}- {1\over2}\right| \quad \mbox{and} \quad \alpha_1\geq n\left|{1\over p}- {1\over2}\right| -1. \end{align*} This gives an endpoint result for $\alpha_0=n|1/p-1/2|$ and $\alpha_1=n|1/p-1/2|-1$ with $1<p<\infty$ in Corollary 8.2, as pointed out in Remark 8.1 due to M\"{u}ller and Thiele [Studia Math. \textbf{179} (2007)].

math.CA↗

arXiv2025

Enabling Continuous THz Band Coverage via Precise Electron Beam Tailoring in Free-electron Lasers

High-power, continuously tunable narrowband terahertz (THz) sources are essential for advancing nonlinear optics, THz-driven material dynamics, and ultrafast spectroscopy. Conventional techniques typically impose a trade-off between pulse energy and frequency tunability. Here, we introduce a novel free-electron laser approach that overcomes these limitations by pre-modulating a relativistic electron beam with a frequency-beating laser pulse and leveraging bunch compression along with collective effects to enhance microbunching. Experimental results demonstrate that this technique generates narrowband THz emission with continuous frequency tunability from 7.8 to 30.8THz, achieving pulse energies up to 385{\mu}J while maintaining spectral bandwidths between 7.7% and 14.7%. Moreover, the method exhibits exceptional robustness and scalability, highlighting its unique ability to bridge the long-standing THz gap and offering a promising solution for diverse cutting-edge scientific applications.

physics.acc-ph↗

arXiv2025

$L^p\to L^q$ estimates for Stein's spherical maximal operators

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{\alpha -1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $\|{\mathfrak M}^{\alpha}_{[1,2]} f \|_{L^q({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})}$ holds for some $\alpha\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,\alpha\geq \sigma_n(p,q):=\max\left\{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$ Conversely, we show that ${\mathfrak M}^\alpha_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,\alpha>\sigma_2(p,q)$ for $n=2$; and ${\rm Re}\,\alpha>\max\left\{\sigma_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right\}$ for $n>2$. The range of $\alpha,p$ and $q$ is almost optimal in the case either $n=2$, or $\alpha=0$, or $(p,q)$ lies in some regions for $n>2$.

math.CA↗

arXiv2024

On maximal functions generated by H\"ormander-type spectral multipliers

Let $(X,d,\mu)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. We assume that there exists an $L$-harmonic function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. In this paper we apply the Doob transform and some techniques as in Grafakos-Honz\'ik-Seeger \cite{GHS2006} to obtain an optimal $\sqrt{\log(1+N)}$ bound in $L^p$ for the maximal function $\sup_{1\leq i\leq N}|m_i(L)f|$ for multipliers $m_i,1\leq i\leq N,$ with uniform estimates. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function $M_{m,L}f(x) = \sup_{t>0} |m(tL)f(x)|$ is bounded on $L^p(X)$. The applications include Schr\"odinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators.

math.CA↗

arXiv2024

The Spherical Maximal Operators on Hyperbolic Spaces

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We prove that when $n\geq 2$, for $\alpha\in\mathbb{R}$ and $1 1-n+n/p$ for $1 \max \{{(2-n)/p}-{1/(p p_n)},{(2-n)/p}- (p-2)/[p p_n(p_n-2)]\} $ for $2\leq p\leq \infty$, with $p_n=2(n+1)/(n-1)$ for $n\geq 3$ and $p_n=4$ for $n=2$.

math.FA↗

arXiv2024

Instantaneous and Retarded Interactions in Coherent Radiation

In coherent radiation of an ensemble of electrons, radiation field from electrons resonantly drives the other electrons inside to produce stimulated emission. The radiation reaction force on the electrons accounting for this stimulated radiation loss is classically described by the Lienard-Wiechert potential. Despite its being the foundation of beam physics for decades, we show that using the "acceleration field'' in Lienard-Wiechert potential to describe radiative interactions leads to divergences due to its implicit dependence on instantaneous interactions. Here, we propose an alternative theory for electromagnetic radiation by decomposing the interactions into instantaneous part and retarded part. It is shown that only the retarded part contributes to the irreversible radiation loss and the instantaneous part describes the space charge related effects. We further apply this theory to study the coherent synchrotron radiation wake, which hopefully will reshape our understanding of coherent radiation and collective interactions.

physics.acc-ph↗

arXiv2024

Bochner-Riesz means for critical magnetic Schr\"odinger operators in ${\mathbb R^2}$

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schr\"odinger operators $\mathcal{L}_{\bf A}$ in ${\mathbb{R}^2}$, which involve the physcial Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\neq 2$, the Bochner-Riesz operator $S_{\lambda}^\delta(\mathcal{L}_{\bf A})$ of order $\delta$ is bounded on $L^p(\mathbb{R}^2)$ if and only if $\delta>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient of the proof is to obtain the localized $L^4(\mathbb R^2)$ estimate of $S_{\lambda}^\delta(\mathcal{L}_{\bf A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_{\lambda}^\delta(\Delta)$ for the Laplacian $\Delta$ in $\mathbb{R}^2$.

math.AP↗

arXiv2024

On pointwise convergence of cone multipliers

For $p\ge 2$, and $\lambda>\max\{n|\tfrac 1p-\tfrac 12|-\tfrac12, 0\}$, we prove the pointwise convergence of cone multipliers, i.e. $$ \lim_{t\to\infty}T_t^\lambda(f)\to f \text{ a.e.},$$ where $f\in L^p(\mathbb R^n)$ satisfies $supp\ \widehat f\subset\{\xi\in\mathbb R^n:\ 1<|\xi_n|<2\}$. Our main tools are weighted estimates for maximal cone operators, which are consequences of trace inequalities for cones.

math.CA↗

arXiv2024

Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications

In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-\Delta u(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in \cite{OS, OS2}. In further developing our method originally formulated for the heat equation, we obtain analogous global ${\rm BMO}$-maximal regularity associated to the Schr\"odinger operator $\mathcal L=-\Delta+V$, where the nonnegative potential $V$ belongs to the reverse H\"older class ${\rm RH}_q$ for some $q> n/2$. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces. Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates.

math.AP↗

arXiv2023

Characterizations of product Hardy spaces on stratified groups by singular integrals and maximal functions

A large part of the theory of Hardy spaces on products of Euclidean spaces has been extended to the setting of products of stratified Lie groups. This includes characterisation of Hardy spaces by square functions and by atomic decompositions, proof of the duality of Hardy spaces with BMO, and description of many interpolation spaces. Until now, however, two aspects of the classical theory have been conspicuously absent: the characterisation of Hardy spaces by singular integrals (of Christ--Geller type) or by (vertical or nontangential) maximal functions. In this paper we fill in these gaps by developing new techniques on products of stratified groups, using the ideas of Chen, Cowling, Lee, Li and Ottazzi on the Heisenberg group with flag structure.

math.FA↗

arXiv2023

Lp bounds for Stein's spherical maximal operators

Let ${\frak M}^α$ be the spherical maximal operators of complex order $α$ on ${\mathbb R^n}$. In this article we show that when $n\geq 2$, suppose \begin{eqnarray*} \|{\frak M}^α f \|_{L^p({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})} \end{eqnarray*} holds for some $α$ and $p\geq 2$, then we must have ${\rm Re}\,α\geq \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$ When $n=2$, we prove that $\|{\frak M}^α f \|_{L^p({\mathbb R^2})} \leq C\|f \|_{L^p({\mathbb R^2})}$ if ${\rm Re}\ \ α>\max\{1/p-1/2,\ -1/p\}$, and hence the range of $α$ is sharp in the sense the estimate fails for ${\rm Re}\ α<\max\{1/p-1/2, -1/ p\}.$

math.AP↗