Hyperbolic Green Function Estimates

Ryznar, Michał; Serafin, Grzegorz; Żak, Tomasz

doi:10.1007/s11118-020-09837-5

Cited by 2 publications

(2 citation statements)

References 23 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the hyperbolic space, the Green function is given by G B n (x, y) := ∞ r (sinh s) −(n−1) ds and r := d(x, y) denotes the hyperbolic distance between the points x, y ∈ B n . Hence the following sharp estimates of the Green function G B n holds for n ≥ 3 (see also [61,Eq 17]) : 2) , for all (x, y) ∈ B n : 0 < d(x, y) ≤ 1, e −(n−1)d(x,y) , for all (x, y) ∈ B n : d(x, y) > 1. (…”

Section: Smoothing Effect For the Relative Error And Rate Of Convergencementioning

confidence: 99%

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Bhakta¹,

Ganguly²,

Karmakar³

et al. 2022

Preprint

View full text Add to dashboard Cite

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n ≥ 3 and 1 < p ≤ n+2 n−2 , there exists a best constant S n,p,λ (B n ) > 0 such that, where (n−1) 2 4is the bottom of the L 2 -spectrum of −∆ B n . It is known from the results of Mancini and Sandeep [56] that under appropriate assumptions on n, p and λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant S n,p,λ (B n ). In this article we investigate the quantitative gradient stability of the above inequality and the associated Euler-Lagrange equation locally around a bubble. In our first result, we prove a sharp quantitative stability of the above Poincaré-Sobolev inequality: if u ∈ H 1 (B n ) almost optimizes the above inequality then u is close to the manifold of optimizers in a quantitative way. Secondly, we prove the quantitative stability of its critical points: if u ∈ H 1 (B n ) almost solves the Euler-Lagrange equation corresponding to the above Poincaré-Sobolev inequality and the energy of u is close to the energy of an extremizer, then we derive the following quantitative bound dist (u, Z) ≤ C(n, p, λ), where Z denotes the manifold of non-negative finite energy solutions of −∆ B n w−λw = |w| p−1 w. Our result generalizes the sharp quantitative stability of Sobolev inequality in R n of Bianchi-Egnell [13] and to the Poincaré-Sobolev inequality on the hyperbolic space.Furthermore, combining our stability results and implementing a refined smoothing estimates, we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.

show abstract

Section: Smoothing Effect For the Relative Error And Rate Of Convergencementioning

confidence: 99%

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Bhakta¹,

Ganguly²,

Karmakar³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…More recent developments further addressed low-energy scattering theory [54], embedded eigenvalues and Neumann-Wigner type potentials [50], decay rates when magnetic potentials and spin are included [33], a relativistic Kato-inequality [34], Carleman estimates and unique continuation [25,55], or nonlinear relativistic Schrödinger equations [1,19,59]. Given its relationship with random processes with jumps, the V = 0 case has received much attention also in potential theory [14,31,56].…”

Section: Introductionmentioning

confidence: 99%

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

Ascione,

Lőrinczi

2023

Ann. Henri Poincaré

View full text Add to dashboard Cite

We propose a probabilistic representation of the ground states of massive and massless Schrödinger operators with a potential well in which the behaviour inside the well is described in terms of the moment-generating function of the first exit time from the well and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part, we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schrödinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.

show abstract

Hyperbolic Green Function Estimates

Abstract: For a hyperbolic Brownian motion in the hyperbolic space $\mathbb {H}^{n}, n\ge 3$ ℍ n , n ≥ 3 , we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brown… Show more

Cited by 2 publications

References 23 publications

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

Contact Info

Product

Resources

About