arXiv subjects

Lixin Yan

Publications and source records attributed to Lixin Yan.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the generalized Hausdorff dimension of Besicovitch sets

Keich (1999) showed that the sharp gauge function for the generalized Hausdorff dimension of Besicovitch sets in $\mathbb R^2$ is between $r^2\log 1/r$ and $r^2(\log 1/r) (\log\log 1/r)^{2+\varepsilon}$ by refining an argument of Bourgain (1991). It is not known whether the iterated logarithms in Keich's bound are necessary. In this paper we construct a family of Besicovitch line sets whose sharp gauge function is smaller than $r^2(\log 1/r) (\log\log 1/r)^{\varepsilon}$. Moreover, these Besicovitch sets are minimal in the sense that there is essentially only one line in the set pointing in each direction.

math.CA↗

arXiv2023

Local smoothing and Hardy spaces for Fourier integral operators on manifolds

We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic curvature condition, and for wave equations on compact manifolds. The estimates are essentially sharp, for all $2<p<\infty$ and on each compact manifold. We also apply our local smoothing estimates to nonlinear wave equations with initial data outside of $L^{2}$-based Sobolev spaces.

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↗

arXiv2023

The Garnett-Jones Theorem on BMO spaces associated with operators and applications

Let $X$ 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. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with the operator $L$ and we define its distance from the subspace $L^{\infty}(X)$ under the $ {\rm BMO}_L(X)$ norm as follows: $$ {\rm dist} (f, L^{\infty}):= \inf_{g\in L^{\infty}} \|f -g\|_{{\rm BMO}_L(X)}. $$ In this paper we prove that ${\rm dist} (f, L^{\infty})$ is equivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberg inequality for the space ${\rm BMO}_L(X)$: $$ \sup_B { μ\big(\{ x\in B: |f(x)-e^{-{r_B^2}L}f(x)|>λ\}\big) \over μ(B)} \leq e^{-λ/\varepsilon}\ \ \ \ {\rm for\ large\ } λ. $$ This extends the well-known result of Garnett and Jones \cite{GJ1} for the classical ${\rm BMO}$ space (introduced by John and Nirenberg). As an application, we show that a ${\rm BMO}_L(X)$ function with compact support can be decomposed as the summation of an $L^\infty$-function and the integral of the heat kernel (associated with $L$) against a finite Carleson measure on $X\times[0,\infty)$. The key new technique is a geometric construction involving the semigroup $e^{-tL}$. We also resort to several fundamental tools including the stopping time argument and the random dyadic lattice.

math.CA↗

arXiv2023

On smoothing estimates for Schrödinger equations on product spaces $\mathbb{T}^m\times \mathbb{R}^n$

Let $Δ_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{itΔ_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{α,p}(\mathbb{T}^m\times\mathbb{R}^n)} $$ holds if $p\geq 2(m+n+2)/(m+n)$ and $α> (m+2n)(1/2-1/p)-2/p$. Furthermore, we apply the $\ell^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schrödinger operator $e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}$ in modulation spaces $M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)$: $$ \|e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}f\|_{L^p(\mathbb{T}^m\times\mathbb{R}^n\times [0,1])}\leq C \|f\|_{M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)} $$ for some range of $α$ and $p, q$. The smoothing estimates in $L^p$-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of $p$ and $q$.

math.CA↗

arXiv2022

Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

Let $G\cong\mathbb{R}^{d} \ltimes \mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot(y,v)\rightarrow(x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\leq {\rm rank}\, J 0}\big|\int_Σ f(x-ty, u- t x^{T}Jy) dμ(y)\big|, $$ where $Σ$ is a smooth convex hypersurface and $dμ$ is a compactly supported smooth density on $Σ$ such that the Gaussian curvature of $Σ$ is nonvanishing on supp $dμ$. In this paper we prove that when $d\geq 4$, the maximal operator ${\frak M}$ is bounded on $L^{p}(G)$ for the range $(d-1)/(d-2)<p\leq\infty$.

math.CA↗

arXiv2022

The commutators of Bochner-Riesz Operators for Hermite operator

In this paper, we study the $L^p$-boundedness of the commutator $[b, S_R^δ(H)](f) = bS_R^δ(H) f - S_R^δ(H)(bf)$ of a BMO function $b$ and the Bochner-Riesz means $S_R^δ(H)$ for Hermite operator $H=-Δ+|x|^2$ on $\mathbb{R}^n$, $n\geq2$. We show that if $δ>δ(q)=n(1/q -1/2)- 1/2$, the commutator $[b,S_R^δ(H)]$ is bounded on $L^p(\mathbb{R}^n)$ whenever $q<p<q'$ and $1\leq q\leq 2n/(n+2)$.

math.AP↗

arXiv2022

Towards a compact all optical terahertz-driven electron source at Tsinghua University

We propose a physical design of a compact all optical terahertz (THz)-driven electron source. The 300 mm accelerator beamline, powered by Joule level laser system, is easily to be integrated to tabletop scale. A dual-feed THz-driven electron gun with an exponential impedance, a tapered dielectric loaded cylindrical waveguide, THz-driven bunch compressors and permanent magnet solenoids (PMS) have been designed and optimized. Dynamics simulations show that the electron source can deliver a 19 fC, 3 MeV electron beams with a normalized transverse emittance of 0.079 π.mm.mrad. A minimum relative energy spread of 0.04% or a minimum root-mean-square bunch length of 6.1 fs can be achieved by adjusting the beam shaping line. Sensitivity analysis shows that the THz-driven electron source can effectively work under a 1.5% energy jitter of the THz power system. Simulated diffraction pattern up to the fourth order of an aluminum sample based on the beamline can be clearly distinguished. A prototype THz gun has beam fabricated and is now under testing, more results will be reported in future works.

physics.acc-ph↗

arXiv2022

The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left\| e^{itL} (I+L)^{-{σn}} f\right\|_{p} \leq C(1+|t|)^{σn} \|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ σ\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.

math.AP↗

arXiv2021

Hardy spaces meet harmonic weights

We investigate the Hardy space $H^1_L$ associated with a self-adjoint operator $L$ defined in a general setting in [S. Hofmann, et. al., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78.]. We assume that there exists an $L$-harmonic non-negative function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space $H^1_L$ in terms of a simple atomic decomposition associated with the $L$-harmonic function $h$. Our approach also yields a natural characterisation of the $BMO$-type space corresponding to the operator $L$ and dual to $H^1_L$ in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in $\mathbb{R}^d$, Schrödinger operators with certain potentials, and Bessel operators.

math.FA↗

arXiv2021

Almost everywhere convergence of spectral sums for self-adjoint operators

Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} λdE_{ L}(λ)$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R dE_{ L}(λ) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a sufficient condition on $L$ such that $$ \lim_{R\rightarrow \infty} S_R({ L})f(x) =f(x),\ \ {\rm a.e.} $$ for any $f$ such that ${\rm log } (2+L) f\in L^2(X)$. These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.

math.CA↗

arXiv2021

Sharp endpoint estimates for Schrödinger groups on Hardy spaces

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies the Davies-Gaffney estimates of order $m\geq 2$. Let $H^1_L(X)$ be the Hardy space associated with $L.$ In this paper we show sharp endpoint estimate for the Schrödinger group $e^{itL}$ associated with $L$ such that \begin{eqnarray*} \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ L^1(X)} + \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ H^1_L(X)} \leq C(1+|t|)^{n/2}\|f\|_{H^1_L(X)}, \ \ \ t\in{\mathbb R} \end{eqnarray*} for some constant $C=C(n, m)>0$ independent of $t$. By a duality and interpolation argument, it gives a new proof of a recent result of \cite{CDLY} for { sharp} endpoint $L^p$-Sobolev bound for $e^{itL}$: $$ \left\| (I+L)^{-s }e^{itL} f\right\|_{ L^p(X)} \leq C (1+|t|)^{s} \|f\|_{ L^p(X)}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big| $$ for every $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound, which extends the classical results due to Miyachi ) for the Laplacian on the Euclidean space ${\mathbb R}^n$.

math.AP↗

arXiv2020

A sharp regularity estimate for the Schrödinger propagator on the sphere

Let $Δ_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it Δ_{\mathbb S^n}}f \|_{L^4([0, 2π) \times \mathbb S^n)} \leq C \| f\|_{W^{α, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $α> {(n-2)/4}.$ The range of $α$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schrödinger propagator $e^{it Δ_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schrödinger maximal operator $\sup_{0\leq t<2π} |e^{itΔ_{\mathbb S^n}} f|$ is bounded from $W^{α, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb S^n)$ whenever $α>{1/3}$.

math.AP↗

arXiv2020

Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $λ< λ(p)/2$ in the sense that there is an $f\in L^p(\mathbb R^n)$ for $2n/(n-1)\le p$ such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For $n\geq 2$ and $p\ge 2$ our result tells that the critical summability index for a.e. convergence for $S_R^λ(H)$ is as small as only the \emph{half} of the critical index for a.e. convergence of the classical Bochner-Riesz means. When $n = 1$, we show a.e. convergence holds for $f\in L^p({\mathbb R})$ with $ p\geq 2$ whenever $λ>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^λ(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.

math.CA↗

arXiv2020

Cascaded high-gradient terahertz-driven acceleration of relativistic electron beams

Terahertz (THz)-driven acceleration has recently emerged as a new route for delivering ultrashort bright electron beams efficiently, reliably, and in a compact setup. Many THz-driven acceleration related working schemes and key technologies have been successfully demonstrated and are continuously being improved to new limits. However, the achieved acceleration gradient and energy gain remain low, and the potential physics and technical challenges in the high field and high energy regime are still under-explored. Here we report a record energy gain of 170 keV in a single-stage configuration, and demonstrate the first cascaded acceleration of a relativistic beam with a 204 keV energy gain in a two-stages setup. Whole-bunch acceleration is accomplished with an average accelerating gradient of 85 MV/m and a peak THz electric field of 1.1 GV/m. This proof-of-principle result is a crucial advance in THz-driven acceleration with a major impact on future electron sources and related scientific discoveries.

physics.acc-ph↗

arXiv2020

Sharp endpoint $L^p$ estimates for Schrödinger groups

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. In this paper we prove {\it sharp} endpoint $L^p$-Sobolev bound for the Schrödinger group $e^{itL}$, that is for every $p\in (p_0, p'_0)$ there exists a constant $C=C(n,p)>0$ independent of $t$ such that \begin{eqnarray*} \left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*} As a consequence, the above estimate holds for all $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces ${\mathbb R}^n$. We also give an application to obtain an endpoint estimate for $L^p$-boundedness of the Riesz means of the solutions of the Schrödinger equations.

math.CA↗

arXiv2020

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_φ$. Moreover, we obtain the reducing subspaces of $S_φ$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$ on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

math.CV↗