Journal of Functional Analysis最新文献

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential$u_{tt}+({\Delta }^2+V)u=0,u(0,x)=f\left(x\right),u_t(0,x)=g\left(x\right)$ in dimension three, where V is a real-valued and decaying potential. Assume that zero is a regular point of $H={\Delta }^2+V$, we first prove the following optimal time decay estimates of the solution operators${\Vert \cos \left(t\sqrt{H}\right)P_{ac}\left(H\right)\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-\frac{3}{2}}\text{and}{\Vert \frac{\sin \left(t\sqrt{H}\right)}{\sqrt{H}}{P}_{ac}\left(H\right)\Vert }_{L^1\to L^{\infty }}\lesssim |t{|}^{-\frac{1}{2}}.$ Moreover, if zero is a resonance of H, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators $\cos \left(t\sqrt{H}\right)$ and $\frac{\sin \left(t\sqrt{H}\right)}{\sqrt{H}}$, but their decay will be significantly changed for the second and third-kind resonances.

We prove that any normalized block sequence in a Schreier space $X_{\xi }$, of arbitrary order $\xi <{\omega }_1$, admits a subsequence equivalent to a subsequence of the canonical basis of some Schreier space. The analogous result is proved for dual spaces to Schreier spaces. Using these results, we examine the structure of strictly singular operators on Schreier spaces and show that there are $2^c$ many closed operator ideals on a Schreier space of any order, its dual and bidual space.

We extend the notion of a spectral triple to that of a higher-order relative spectral triple, which accommodates several types of hypoelliptic differential operators on manifolds with boundary. The bounded transform of a higher-order relative spectral triple gives rise to a relative K-homology cycle. In the case of an elliptic differential operator on a compact smooth manifold with boundary, we calculate the K-homology boundary map of the constructed relative K-homology cycle to obtain a generalization of the Baum-Douglas-Taylor index theorem.

Let $\Omega \subset R^{n+1}$, $n\ge 2$, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space $M^{1,1}(\partial \Omega )$ and the weak-$A_{\infty }$ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in $M^{1,1}(\partial \Omega )$ is equivalent to the solvability of the regularity problem in $M^{1,p}(\partial \Omega )$ for some $p>1$. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak $(1,1)$-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space $M^{1,1}(\partial \Omega )$ is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions $3\le d\le 5$, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Let $p\in (1,\infty )$. We show that there is an isomorphism from any separable unital subalgebra of $B\left({\ell }^2\right)/K\left({\ell }^2\right)$ onto a subalgebra of $B\left({\ell }^p\right)/K\left({\ell }^p\right)$ that preserves the Fredholm index. As a consequence, every separable $C^{⁎}$-algebra is isomorphic to a subalgebra of $B\left({\ell }^p\right)/K\left({\ell }^p\right)$. Another consequence is the existence of operators on ${\ell }^p$ that behave like the essentially normal operators with arbitrary Fredholm indices in the Brown-Douglas-Fillmore theory.

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis exclusively in physical-space. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis [13] through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin [63] in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.