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We establish optimal local and global Besov–Lipschitz and Triebel–Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov–Lipschitz spaces and certain Banach and quasi-Banach scales of Triebel–Lizorkin spaces .
Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap , then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an -space with or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap .
This criterion applies to random groups in the triangular density model for densities . In this way, we are able to generalize recent results of Druţu and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on -spaces, our results are quantitatively stronger, even in the case . This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model.
Additionally, we obtain results on the eigenvalues of the -Laplacian on graphs, and on the spectrum and degree distribution of Erdős–Rényi graphs.
We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the torsion of the corresponding long exact sequence of cohomology equipped with -metrics. As an application, we give a purely analytic proof of the gluing formula for analytic torsion.
We study the local propagation of conormal singularities for solutions of semilinear wave equations , where is a polynomial of degree in with coefficients. We know from the work of Melrose and Ritter and Bony that if is conormal to three waves which intersect transversally at point , then after the triple interaction is a conormal distribution with respect to the three waves and the characteristic cone with vertex at . We compute the principal symbol of at the cone and away from the hypersurfaces. We show that if , then is an elliptic conormal distribution.
We establish interior Schauder estimates for kinetic equations with integrodifferential diffusion. We study equations of the form , where is an integrodifferential diffusion operator of order acting in the -variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of , we obtain an a priori estimate for in a properly scaled Hölder space.
Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes’ laws, and it is expected to be singular if the slope has the same inclination as the group velocity. We prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.
We construct Green’s functions for elliptic operators of the form in domains , under the assumption or . We show that, in the setting of Lorentz spaces, the assumption is both necessary and optimal to obtain pointwise bounds for Green’s functions. We also show weak-type bounds for the Green’s function and its gradients. Our estimates are scale-invariant and hold for general domains . Moreover, there is no smallness assumption on the norms of the lower-order coefficients. As applications we obtain scale-invariant global and local boundedness estimates for subsolutions to in the case .
We complete the microlocal study of the geodesic x-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou (J. Differential Geom.105:2 (2017), 177–208) and pursued by Guillarmou and Lefeuvre in (Ann. of Math.190:1 (2019), 321–344). We prove new stability estimates and clarify some properties of the operator — the generalized x-ray transform. These estimates rely on a refined version of the Livšic theorem for Anosov flows, especially on a new quantitative finite-time Livšic theorem.
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