Journal of Functional Analysis最新文献

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.

We prove the local Gan–Gross–Prasad conjecture for Fourier–Jacobi models of real unitary groups.

Let A and B be $C^{⁎}$-algebras with A separable, let I be an ideal in B, and let $\psi :A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family ${\Theta }_t:A\to B$, for $t\in [1,\infty )$, of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then ${\Theta }_t$ can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family ${\Theta }_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that ${\Theta }_t$ is linear and completely positive for all $t\in [1,\infty )$. In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

For any $p\in (0,1]$ and $s\in R$, the authors prove two types of characterizations of the pointwise multiplier space $M\left(B_{p,p}^s\right(R^n\left)\right)$ of the Besov space $B_{p,p}^s\left(R^n\right)$. One type is based on wavelet analysis and is an extension of a well-known argument of Yves Meyer. The other type works with Fourier analytic terms. As an application of the above two types of characterizations, the authors further obtain a characterization of bounded functions in the uniform space $B_{p,p,\mathrm{unif}}^{s,\tau }\left(R^n\right)$ via Haar wavelets in the critical index $\tau =\frac{1}{p}-\frac{s}{n}$.

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [16], which provides more elaborate and natural weighted estimates in the multilinear setting.