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1. CMB Online first
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators |
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
second-order Riesz transforms $VA^{-1}$ and
$(\nabla-i\vec{a})^2A^{-1}$ are bounded from the
Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 |
2. CMB 2013 (vol 56 pp. 827)
Erratum to ``Quantum Limits of Eisenstein Series and Scattering States'' This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 8G25, 35P25 |
3. CMB 2012 (vol 56 pp. 814)
Quantum Limits of Eisenstein Series and Scattering States We identify the quantum limits of scattering states
for the modular surface. This is obtained through the study of quantum
measures of non-holomorphic Eisenstein series away from the critical
line. We provide a range of stability for the quantum unique
ergodicity theorem of Luo and Sarnak.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 58G25, 35P25 |
4. CMB 2011 (vol 56 pp. 378)
Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases In this paper, we consider the Gross-Pitaevskii equation for the
trapped dipolar quantum gases. We obtain the sharp criterion for the
global existence and finite time blow up in the unstable regime by
constructing a variational problem and the so-called invariant
manifold of the evolution flow.
Keywords:Gross-Pitaevskii equation, sharp threshold, global existence, blow up Categories:35Q55, 35A05, 81Q99 |
5. CMB 2011 (vol 56 pp. 659)
Asymptotics and Uniqueness of Travelling Waves for Non-Monotone Delayed Systems on 2D Lattices We establish asymptotics and uniqueness (up
to translation) of travelling waves for delayed 2D lattice equations
with non-monotone birth functions. First, with the help of
Ikehara's Theorem, the a priori asymptotic behavior of
travelling wave is exactly derived. Then, based on the obtained
asymptotic behavior, the uniqueness of the traveling waves is
proved. These results complement earlier results in the literature.
Keywords:2D lattice systems, traveling waves, asymptotic behavior, uniqueness, nonmonotone nonlinearity Category:35K57 |
6. CMB 2011 (vol 56 pp. 3)
Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimension-one simplices satisfying a certain restriction on an
$n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.
Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits Categories:58G25, 81Q50, 35P20, 42B05 |
7. CMB 2011 (vol 55 pp. 555)
Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the
scope of CalderÃ³n-Zygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
Hardy-Littlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.
Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates Categories:42B20, 42B25, 35S05, 47G30 |
8. CMB 2011 (vol 55 pp. 663)
An Onofri-type Inequality on the Sphere with Two Conical Singularities In this paper, we give a new proof of the Onofri-type inequality
\begin{equation*}
\int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{
\frac{1}{4\pi(\beta+1)} \int_S |\nabla u|^2 \,ds^2 +
\frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\}
\end{equation*}
on the sphere $S$ with Gaussian curvature $1$ and with conical
singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for
$\beta\in (-1,0)$; here $p_1$ and $p_2$ are antipodal.
Categories:53C21, 35J61, 53A30 |
9. CMB 2011 (vol 55 pp. 736)
Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous
neutral functional differential equations. An application to partial neutral differential equations is considered.
Keywords:neutral equations, mild solutions, classical solutions Categories:35R10, 34K40, 34K30 |
10. CMB 2011 (vol 55 pp. 537)
Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents In this paper, we investigate
a semilinear elliptic equation that involves multiple
Hardy-type terms and critical Hardy-Sobolev exponents. By the
Moser iteration method and analytic techniques, the asymptotic
properties of its nontrivial solutions at the singular points are
investigated.
Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iteration Categories:35B33, 35B40, 35J60 |
11. CMB 2011 (vol 55 pp. 623)
The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations In this paper, we consider the Cauchy problem
$$
\begin{cases}
u_{t}=\Delta(u^{m}), &x\in{}\mathbb{R}^{N}, t>0, N\geq3,
\\
% ^^----- here
u(x,0)=u_{0}(x), &x\in{}\mathbb{R}^{N}.
\end{cases}
$$
We will prove that:
(i) for
$m_{c} Keywords:fast diffusion equations, Cauchy problem, continuous dependence on nonlinearity Categories:35K05, 35K10, 35K15 |
12. CMB 2011 (vol 55 pp. 249)
Description of Entire Solutions of Eiconal Type Equations The paper describes entire solutions to the eiconal type non-linear partial differential
equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases,
where
$X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$,
$X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$
are linearly independent operators with $p_j$ being arbitrary
polynomials in $\mathbf{C}^2$.
Keywords:entire solution, eiconal equation, polynomial, transcendental function Categories:32A15, 35F20 |
13. CMB 2011 (vol 55 pp. 3)
On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations We improve several recent results in the
asymptotic integration theory of nonlinear ordinary differential
equations via a variant of the method devised by J. K. Hale and
N. Onuchic The results
are used for investigating the existence of positive solutions to
certain reaction-diffusion equations.
Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equation Categories:34E10, 34C10, 35Q35 |
14. CMB 2011 (vol 55 pp. 88)
Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 |
15. CMB 2011 (vol 54 pp. 249)
A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies This paper deals with the analytic solvability of a special class of
complex vector fields defined on the real plane, where they are
tangent to
a closed real curve, while off the real curve, they are elliptic.
Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$) Categories:35A01, 58Jxx |
16. CMB 2010 (vol 54 pp. 28)
Generalized Solution of the Photon Transport Problem
The purpose of this paper is to show the existence of a
generalized solution of the photon transport problem. By means of the theory of
equicontinuous $C_{0}$-semigroup on a sequentially complete locally convex
topological vector space we show that the perturbed abstract Cauchy problem
has a unique solution when the perturbation operator and the forcing term
function satisfy certain conditions. A consequence of the abstract result is
that it can be directly applied to obtain a generalized solution of the photon
transport problem.
Keywords:photon transport, $C_{0}$-semigroup Categories:35K30, 47D03 |
17. CMB 2010 (vol 54 pp. 126)
Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups
We prove that the fundamental solutions
of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$
on the anisotropic Heisenberg groups are tempered distributions and have
meromorphic continuation in $\alpha$ with simple poles. We compute the
residues and find the partial fundamental solutions
at the poles. We also find formulas for the
fundamental solutions for some matrix-valued
Kohn type sub-Laplacians
on H-type groups.
Categories:22E30, 35R03, 43A80 |
18. CMB 2010 (vol 53 pp. 674)
Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary |
Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 |
19. CMB 2010 (vol 53 pp. 737)
On the Negative Index Theorem for the Linearized Non-Linear SchrÃ¶dinger Problem
A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky,
and Vougalter based on the variational principle for the
quadratic form of a self-adjoint operator.
It is the negative index theorem for a linearized NLS operator in
three dimensions.
Categories:35Q55, 81Q10 |
20. CMB 2009 (vol 53 pp. 153)
Several Hardy Type Inequalities with Weights Related to Generalized Greiner Operator In this paper, we establish several weighted $L^p (1\lt p \lt \infty)$
Hardy type inequalities related to the generalized Greiner operator
by improving the method of Kombe. Then the best
constants in inequalities are discussed by introducing new polar
coordinates.
Keywords:generalized Greiner operator, polar coordinates, Hardy inequality Categories:35B05, 35H99 |
21. CMB 2009 (vol 53 pp. 163)
Variants of Arnold's Stability Results for 2D Euler Equations We establish variants of stability estimates in norms
somewhat stronger than the $H^1$-norm under Arnold's stability hypotheses on
steady solutions to the Euler equations for fluid flow on planar
domains.
Category:35Q35 |
22. CMB 2009 (vol 53 pp. 295)
The Global Attractor of a Damped, Forced Hirota Equation in $H^1$ The existence of the global attractor of a damped
forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The
main idea is to establish the so-called asymptotic compactness
property of the solution operator by energy equation approach.
Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness Categories:35Q53, 35B40, 35B41, 37L30 |
23. CMB 2009 (vol 52 pp. 555)
Boundary Behavior of Solutions of the Helmholtz Equation This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewood-type theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 |
24. CMB 2008 (vol 51 pp. 249)
On the Inner Radius of a Nodal Domain Let $M$ be a closed Riemannian manifold.
We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$.
We give upper and lower bounds on the inner radius of the type
$C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on
a local behavior of eigenfunctions discovered by Donnelly and
Fefferman and a Poincar\'{e} type inequality proved by Maz'ya.
Sharp lower bounds are known
only in dimension two. We give an account of this case too.
Categories:58J50, 35P15, 35P20 |
25. CMB 2008 (vol 51 pp. 140)
First Variations of the Best Sobolev Trace Constant with Respect to the Domain In this paper we study the best constant of the Sobolev trace
embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$
is a bounded smooth domain in $\RR^N$. We find a formula for the
first variation of the best constant with respect to the domain.
As a consequence, we prove that the ball is a critical domain when
we consider deformations that preserve volume.
Keywords:nonlinear boundary conditions, Sobolev trace embedding Categories:35J65, 35B33 |