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Lixin Yan

Publications and source records attributed to Lixin Yan.

At least 55 records · Page 3Linked to original sources
arXiv2019

BMO spaces associated to operators with generalised Poisson bounds on non-doubling manifolds with ends

Consider a non-doubling manifold with ends $M = \mathfrak{R}^{n}\sharp\, {\mathbb R}^{m}$ where $\mathfrak{R}^n=\mathbb{R}^n\times \mathbb{S}^{m-n}$ for $m> n \ge 3$. We say that an operator $L$ has a generalised Poisson kernel if $\sqrt{ L}$ generates a semigroup $e^{-t\sqrt{L}}$ whose kernel $p_t(x,y)$ has an upper bound similar to the kernel of $e^{-t\sqrtΔ}$ where $Δ$ is the Laplace-Beltrami operator on $M$. An example for operators with generalised Gaussian bounds is the Schrödinger operator $L = Δ+ V$ where $V$ is an arbitrary non-negative locally integrable potential. In this paper, our aim is to introduce the BMO space ${\rm BMO}_L(M)$ associated to operators with generalised Poisson bounds which serves as an appropriate setting for certain singular integrals with rough kernels to be bounded from $L^{\infty}(M)$ into this new ${\rm BMO}_L(M)$. On our ${\rm BMO}_L(M)$ spaces, we show that the John--Nirenberg inequality holds and we show an interpolation theorem for a holomorphic family of operators which interpolates between $L^q(M)$ and ${\rm BMO}_L(M)$. As an application, we show that the holomorphic functional calculus $m(\sqrt{L})$ is bounded from $L^{\infty}(M)$ into ${\rm BMO}_L(M)$, and bounded on $L^p(M)$ for $1 < p < \infty$.

math.AP↗
arXiv2019

Weak type $(1,1)$ bounds 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 kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s> n|{1/ 2}-{1/p}| $ and $ p\in (1, \infty)$ (see for example, \cite{CCO, H, Sj}). The index $s= n|{1/ 2}-{1/p}|$ was only obtained recently in \cite{CDLY, CDLY2}, and this range of $s$ is sharp since it is precisely the range known in the case when $L$ is the Laplace operator $Δ$ on $X=\mathbb R^n$ (\cite{Mi1}). In this paper we establish that for $p=1,$ the operator $(1+L)^{-n/2}e^{itL}$ is of weak type $(1, 1)$, that is, there is a constant $C$, independent of $t$ and $f$ so that \begin{eqnarray*} μ\Big(\Big\{x: \big|(I+L)^{-n/2 }e^{itL} f(x)\big|>λ\Big\}\Big) \leq Cλ^{-1}(1+|t|)^{n/2} {\|f\|_{L^1(X)} }, \ \ \ t\in{\mathbb R} \end{eqnarray*} (for $λ> 0$ when $μ(X) = \infty$ and $λ>μ(X)^{-1}\|f\|_{L^1(X)}$ when $μ(X) < \infty$). Moreover, we also show the index $n/2$ is sharp when $L$ is the Laplacian on ${\mathbb R^n}$ by providing an example. Our results are applicable to Schrödinger group for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with non-negative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces.

math.AP↗
arXiv2019

Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1<p<\infty$. We then show that the corresponding $H_{L}^{1}(\mathbb{R}^n)$-norm (resp. $L^p(\mathbb{R}^n)$-norm) of a function $f$ in terms of the frame coefficients is equivalent to the $H_{L}^{1}(\mathbb{R}^n)$-norm (resp. $L^p(\mathbb{R}^n)$-norm) of $f$. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of $L$.

math.AP↗
arXiv2019

Sharp spectral multipliers without semigroup framework and application to random walks

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates à la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results. Finally, we consider the random walk on the integer lattice $\mathbf{Z}^n$ and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on $\mathbb{R}^n$.

math.AP↗
arXiv2018

Terahertz streaking of few-femtosecond relativistic electron beams

Streaking of photoelectrons with optical lasers has been widely used for temporal characterization of attosecond extreme ultraviolet pulses. Recently, this technique has been adapted to characterize femtosecond x-ray pulses in free-electron lasers with the streaking imprinted by farinfrared and Terahertz (THz) pulses. Here, we report successful implementation of THz streaking for time-stamping of an ultrashort relativistic electron beam of which the energy is several orders of magnitude higher than photoelectrons. Such ability is especially important for MeV ultrafast electron diffraction (UED) applications where electron beams with a few femtosecond pulse width may be obtained with longitudinal compression while the arrival time may fluctuate at a much larger time scale. Using this laser-driven THz streaking technique, the arrival time of an ultrashort electron beam with 6 fs (rms) pulse width has been determined with 1.5 fs (rms) accuracy. Furthermore, we have proposed and demonstrated a non-invasive method for correction of the timing jitter with femtosecond accuracy through measurement of the compressed beam energy, which may allow one to advance UED towards sub-10 fs frontier far beyond the ~100 fs (rms) jitter.

physics.acc-ph↗
arXiv2018

On the boundary Strichartz estimates for wave and Schrödinger equations

We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable Lévy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.

math.AP↗
arXiv2018

Bounds on the maximal Bochner-Riesz means for elliptic operators

We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we obtain the sharp $L^p$ maximal bounds for the Schrödinger operators on asymptotically conic manifolds, the harmonic oscillator and its perturbations or elliptic operators on compact manifolds.

math.AP↗
arXiv2017

Experimental Polarization Control of Thomson Scattering X/γ-ray Source

Thomson scattering of intense laser pulses from relativistic electrons allows us to generate high-brightness and tunable-polarization X/γ-ray pulses. This paper demonstrates the polarization control of the Thomson scattering source experimentally. The polarization of X/γ-ray is related to incident polarized laser beams, which is controlled by rotating a quarter-wave plate. In the experiment, the polarization of X-ray is determined by recording the spatial distribution of scattered photons, produced by X-ray irradiating on a target. According to modulation curves analysed from experiment results, the conclusion is that the polarization of Thomson scattering source is tunable and controllable. Meanwhile, stokes parameters of X/γ-ray whose energy varies between tens of keV and MeV are simulated. Considering the paraxial photons, it is showed that the polarization is nearly the same value in a wide range of X-ray energy according to simulation results.

physics.acc-ph↗
arXiv2017

On characterization of Poisson integrals of Schrödinger operators with Morrey traces

Let $L$ be a Schrödinger operator of the form $L=-Δ+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\geq n.$ In this article we will show that a function $f\in L^{2, λ}({\mathbb{R}^n}), 0<λ<n$ is the trace of the solution of ${\mathbb L}u=-u_{tt}+L u=0, u(x,0)= f(x),$ where $u$ satisfies a Carleson type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-λ}\int_0^{r_B}\int_{B(x_B, r_B)} t|\nabla u(x,t)|^2 {dx dt} \leq C <\infty. \end{eqnarray*} Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces $\mathscr{L}_L^{2,λ}({\mathbb{R}^n})$ associated to the operator $L$, i.e. $$\mathscr{L}_L^{2,λ}(\mathbb{R}^n)= {L}^{2,λ}(\mathbb{R}^n). $$ Conversely, this Carleson type condition characterizes all the ${\mathbb L}$-harmonic functions whose traces belong to the space $L^{2, λ}({\mathbb{R}^n})$ for all $ 0<λ<n$. This extends the previous results of [FJN, DYZ, JXY].

math.AP↗
arXiv2017

Experiments on bright field and dark field high energy electron imaging with thick target material

Using a high energy electron beam for the imaging of high density matter with both high spatial-temporal and areal density resolution under extreme states of temperature and pressure is one of the critical challenges in high energy density physics . When a charged particle beam passes through an opaque target, the beam will be scattered with a distribution that depends on the thickness of the material. By collecting the scattered beam either near or off axis, so-called bright field or dark field images can be obtained. Here we report on an electron radiography experiment using 45 MeV electrons from an S-band photo-injector, where scattered electrons, after interacting with a sample, are collected and imaged by a quadrupole imaging system. We achieved a few micrometers (about 4 micrometers) spatial resolution and about 10 micrometers thickness resolution for a silicon target of 300-600 micron thickness. With addition of dark field images that are captured by selecting electrons with large scattering angle, we show that more useful information in determining external details such as outlines, boundaries and defects can be obtained.

physics.acc-ph↗
arXiv2017

Carleson measures, BMO spaces and balayages associated to Schrodinger operators

Let $Ł$ be a Schrödinger operator of the form $Ł=-Δ+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{\mathcal{L}}(\RR)$ denote the BMO space associated to the Schrödinger operator $Ł$ on $\RR$. In this article we show that for every $f\in {\rm BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in L^{\infty}(\RR)$ and a finite Carleson measure $μ$ such that $$ f(x)=g(x) + S_{μ, {\mathcal P}}(x) $$ with $\|g\|_{\infty} +\||μ\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ where $$ S_{μ, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) dμ(y, t), $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup $\{e^{-t\sqrtŁ}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Conversely, if $μ$ is a Carleson measure, then $S_{μ, {\mathcal P}}$ belongs to the space ${\rm BMO}_{\mathcal{L}}(\RR)$. This extends the result for the classical John--Nirenberg BMO space by Carleson \cite{C} (see also \cite{U,GJ,W}) to the BMO setting associated to Schrödinger operators.

math.AP↗
arXiv2017

Generation of High-Power, Tunable Terahertz Radiation from Laser Interaction with a Relativistic Electron Beam

We propose a method based on the slice energy spread modulation to generate strong subpicoseond density bunching in high-intensity relativistic electron beams. A laser pulse with periodic intensity envelope is used to modulate the slice energy spread of the electron beam, which can then be converted into density modulation after a dispersive section. It is found that the double-horn slice energy distribution of the electron beam induced by the laser modulation is very effective to increase the density bunching. Since the modulation is performed on a relativistic electron beam, the process does not suffer from strong space charge force or coupling between phase spaces, so that it is straightforward to preserve the beam quality for further applications, such as terahertz (THz) radiation and resonant excitation of plasma wakefield. We show in both theory and simulations that the tunable radiation from the beam can cover the frequency range of 1-10 THz with high power and narrow-band spectra.

physics.acc-ph↗
arXiv2016

Multilinear Multiplier Theorems and Applications

We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calderón and Calderón-Coifman-Journé.

math.AP↗
arXiv2016

Spectral multipliers via resolvent type estimates on non-homogeneous metric measure spaces

We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator. We show that, in this setting, spectral multipliers follow from resolvent type estimates. The most notable point of the paper is that our approach is very flexible and can be applied even if the corresponding ambient space does not satisfy the doubling condition or if the semigroup generated by an operator is not uniformly bounded. As a corollary we obtain the $L^p$ spectrum independence for several second order differential operators and recover some known results. Our examples include the Laplace-Belltrami operator on manifolds with ends and Schrödinger operators with strongly subcritical potentials.

math.AP↗
arXiv2016

Exponential-square integrability, weighted inequalities for the square functions associated to operators and applications

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder's continuity in $x$ but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (i) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (ii) weighted norm inequalities for the square functions; and (iii) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

math.AP↗
arXiv2016

Maximal function characterizations for Hardy spaces associated to nonnegative self-adjoint operators on spaces of homogeneous type

Let $X$ be a metric measure space with a doubling measure and $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. In this article we continue a study in \cite{SY} to give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(X)$ in terms of the nontangential maximal function associated with the heat semigroup of $L$, and hence we establish characterizations of Hardy spaces associated to an operator $L$, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of $ H^p_{L, max}(X)$ in terms of the radial maximal function.

math.AP↗
arXiv2015

Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type

Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space $X_1\times X_2$. We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner--Riesz means.

math.CA↗