Let $0<σ<n/2$ and $H=(-Δ)^σ+V(x)$ be Schrödinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2σ}$ with a sharp coupling constant $a\ge -C_{σ,n}$ ($C_{σ,n}$ is the best constant of Hardy's inequality of order $σ$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{σ,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<σ<n/2$, which turn out to be equivalent to {\it Kato smoothing estimates} for the Cauchy problem. We then establish {\it Strichartz estimates} for $σ>1/2$ and {\it uniform Sobolev estimates} of Kenig--Ruiz--Sogge type for $σ\ge n/(n+1)$. These extend the same properties for the Schrödinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain {\it improved Strichartz estimates with a gain of regularities} for general initial data if $1<σ<n/2$ and for radially symmetric data if $n/(2n-1)<σ\le1$, which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e. $a=-C_{σ,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.
We prove an improved version of the trace-Hardy inequality, so-called Kato's inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by \cite{AFV} in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.
We study time decay estimates of the fourth-order Schrödinger operator $H=(-Δ)^{2}+V(x)$ in $\mathbb{R}^{d}$ for $d=3$ and $d\geq5$. We analyze the low energy and high energy behaviour of resolvent $R(H; z)$, and then derive the Jensen-Kato dispersion decay estimate and local decay estimate for $e^{-itH}P_{ac}$ under suitable spectrum assumptions of $H$. Based on Jensen-Kato decay estimate and local decay estimate, we obtain the $L^1\rightarrow L^{\infty}$ estimate of $e^{-itH}P_{ac}$ in $3$-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of $e^{-itH}…
Let $H$ be a selfadjoint operator and $A$ a closed operator on a Hilbert space $\mathcal{H}$. If $A$ is $H$-(super)smooth in the sense of Kato-Yajima, we prove that $AH^{-\frac14}$ is $\sqrt{H}$-(super)smooth. This allows to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schrödinger equations. We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag \cite{ErdoganGoldbergSchlag09-a}, we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on $\mathbb{R}^{n}$…
In this paper, we investigate a stochastic Hardy-Littlewood-Sobolev inequality. Due to the stochastic nature of the inequality, the relation between the exponents of intgrability is modified. This modification can be understood as a regularization by noise phenomenon. As a direct application, we derive Strichartz estimates for the white noise dispersion which enables us to address a conjecture from [3].
In this short note, we prove Strichartz estimates for Schrödinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than two. The main ingredient of the proof is a variant of Kato's smoothing estimate with a singular weight.
We look for the optimal range of Lebesque exponents for which inhomogeneous Strichartz estimates are valid. We show that it is larger than the one given by admissible exponents for homogeneous estimates. We prove inhomogeneous estimates adopting the abstract setting and interpolation techniques already used by Keel and Tao for the endpoint case of the homogenenous estimates. Applications to Schrodinger equations are given, which extend previous work by Kato.
We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$, and $2 < q, r \leq \infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > …
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequali…
In this paper, we prove that Kato smoothing effects for magnetic Schrödinger operators can yield the endpoint Strichartz estimates for linear wave equation with magnetic potential on two dimensional hyperbolic spaces. This result serves as a cornerstone for the author's work \cite{Lize} and collaborative work \cite{LMZ} in the study of asymptotic stability of harmonic maps for wave maps from $\Bbb R\times\Bbb H^2$ to $\Bbb H^2$.
In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space $\rnpiu =\R^{n-1}\times(0,\infty)$, we show that, if we replace the optimal constant $\frac{(n-2)^2}{4}$ with a smaller one $\frac{(β-2)^2}{4}$, $2\le β<n$, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when $β$ goes to $n$, while it is equal to the optimal constant in the Kat…
We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing $ L^{\infty}_{x}(\mathbb{R}^{n})$ by $BMO_{x}(\mathbb{R}^{n})$ and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an…
We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schrödinger-type equations we show its close relation with the Schrödinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary …
This paper applies Hermite function techniques to give elementary proofs of Kato type smoothing estimates for the Schrödinger equation with quadratic potential in R^n+1. This is equivalent to proving a uniform L^2(R^n) to L^2(R^n) boundedness for a family of singularized Hermite projection kernels. As an applicationas the above estimate, we also prove the R^9 collapsing variable type Strichartz estimate.
We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations $$ iu_t+u_{xx}=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}\times\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strichartz e…
In this paper we study the bilinear-control problem for the linear and non-linear Schr{ö}dinger equation with harmonic potential. By the means of different examples, we show how space-time smoothing effects (Strichartz estimates, Kato smoothing effect) enjoyed by the linear flow, can help to prove obstructions to controllability.
We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schrödinger equations with potential in three dimensions. Our main focus is on the small rough time-dependent potentials. Examples of such potentials are of the form $V(t,x)=T(t) V_0(x)$, where $T$ is quasiperiodic in time and $V_0$ is essentially an $L^{3/2}$ function of the spatial variables. We also prove the dispersive estimates for small time-independent potentials which belong to the interestion of the Rollnik and global Kato classes. Finally, we settle the question posed by Journe, Soffer, Sogge …
We consider the inhomogeneous biharmonic nonlinear Schrödinger equation $$ i u_t +Δ^2 u+λ|x|^{-b}|u|^αu = 0, $$ where $λ=\pm 1$ and $α$, $b>0$. In the subctritical case, we improve the global well-posedness result obtained in \cite{GUZPAS} for dimensions $N=5,6,7$ in the Sobolev space $H^2(\mathbb{R}^N)$. The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, $α=\frac{8-2b}{N-4}$ are also reported. More precisely, we show well-posedness and a st…
In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u), ~u(0)=u_{0} \in H^{s} (\mathbb R^{n} ),\] where $n\ge3$, $1\le s<\frac{n}{2} $, $0<b<2$ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^{σ} u$ with $λ\in \mathbb C$ and $σ=\frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $H^{s} (\mathbb R^{n} )$ with $1\le s<\frac{n}{2}$ for the critical INLS equation under some assumption on $b$. To this end, we first establi…
This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, Gérard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach, relying only on the heat semigroup in order to understand the analytic connexion between the heat semigroup and the unitary Schrödinger group (both related to a same self-adjoint operator). One of the novelty is to forget the endpoint $L^1-L^\infty$ dispersive estimates and to look for a weaker $H^1-BMO$ estimates (Hardy and BMO spaces both adapted to the hea…
Let L_U = -Delta+U be a Schrödinger operator on R^d, where U\in L^1_{loc}(R^d) is a non-negative potential and d\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\in L^1(R^d): \|f\|_{H^1(L_U)}:= \|sup_{t>0} |K_{t,U} f| \|_{L^1(R^d)} < \infty. Assume that U=V+W, where V\geq 0 satisfies the global Kato condition sup_{x\in R^d} \int_{R^d} V(y)|x-y|^{2-d} < \infty. We prove that, under certain assumptions on W\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is eith…
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u=λ|x|^{-b} |u|^{σ} u,\; u(0)=u_{0} \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $λ=\pm1$, $0<b<2$, $σ=\frac{4-2b}{d-2}$ and $c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^{1}$ for $c>-\frac{(d+2-2b)^{2}-4}{(d+2-2b)^{2}}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strich…
We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, $H_V=-Δ_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$, which extend the results of the third author~\cite{sogge88} corresponding to the case where $V\equiv 0$. We are able to handle critically singular potentials and consequently assume that $V\in L^{\tfrac{n}2}(M)$ and/or $V\in {\mathcal K}(M)$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $V\equiv 0$ that go back to the third author \cite{sogge88} as w…
We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate ou…
This paper is mainly devoted to study time decay estimates of the higher-order Schrödinger type operator $H=(-Δ)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen type estimates and local decay estimates for higher order Schrödinger propagator $e^{-itH}$ i…
Peral/Miyachi's celebrated theorem on fixed time $L^{p}$ estimates with loss of derivatives for the wave equation states that the operator $(I-Δ)^{- \fracα{2}}\exp(i \sqrt{-Δ})$ is bounded on $L^{p}(\mathbb{R}^{d})$ if and only if $α\geq s_{p}:=(d-1)|\frac{1}{p}-\frac{1}{2}|$. We extend this result to operators of the form $\mathcal{L} = -\sum \limits _{j=1} ^{d} a_{j+d}\partial_{j}a_{j}\partial_{j}$, such that, for $j=1,...,d$, the functions $a_{j}$ and $a_{j+d}$ only depend on $x_{j}$, are bounded above and below, but are merely Lipschitz continuous. This is below the $C^{1,1}$ regularity th…
Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in $L^2$ spaces, and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of $L^p$ spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator, to prov…
