863 research outputs found

    Sharp quantitative stability of Struwe's decomposition of the Poincar\'e-Sobolev inequalities on the hyperbolic space: Part I

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    A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincar\'e-Sobolev equation on the hyperbolic space −ΔBnu−λu=∣u∣p−1u,u∈H1(Bn), -\Delta_{\mathbb{B}^n} u-\lambda u = |u|^{p-1}u, \quad u\in H^1(\mathbb{B}^n), are unique up to hyperbolic isometries where n≄3,n \geq 3, 1<p≀n+2n−21 < p \leq \frac{n+2}{n-2} and λ≀(n−1)24.\lambda \leq \frac{(n-1)^2}{4}. We prove under certain bounds on ∄∇u∄L2(Bn)\|\nabla u \|_{L^2(\mathbb{B}^n)} the inequality ÎŽ(u)â‰Č∄ΔBnu+λu+up∄H−1, \delta(u) \lesssim \|\Delta_{\mathbb{B}^n} u+ \lambda u + u^{p}\|_{H^{-1}}, holds whenever p>2p >2 and hence forcing the dimensional restriction 3≀n≀5,3 \leq n \leq 5, where ÎŽ(u)\delta(u) denotes the H1H^1 distance of uu from the manifold of sums of hyperbolic bubbles. Moreover, it fails for any n≄3n \geq 3 and p∈(1,2].p \in (1,2]. This strengthens the phenomenon observed in the Euclidean case that the (linear) quantitative stability estimate depends only on whether the exponent pp is >2>2 or ≀2\leq 2. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [Arch. Ration. Mech. Anal, 237 (2020)] but we notice a striking dependence on the exponent pp in the subcritical regime as well which is not present in the flat case. Our technique is an amalgamation of Figalli and Glaudo's method and builds upon a series of new and novel estimates on the interaction of hyperbolic bubbles and their derivatives and improved eigenfunction integrability estimates. Since the conformal group coincides with the isometry group of the hyperbolic space, we perceive a remarkable distinction in arguments and techniques to achieve our main results compared to that of the Euclidean case.Comment: 70 pages, 4 figures. This is the updated version of our previous submission arXiv:2211.14618. New results have been added e.g., Section 9 and Section 10. The main new result is contained in Theorem~9.

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

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    Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every n≄3n \geq 3 and 1<p≀n+2n−2,1 < p \leq \frac{n+2}{n-2}, there exists a best constant Sn,p,λ(Bn)>0S_{n,p, \lambda}(\mathbb{B}^{n})>0 such that Sn,p,λ(Bn)( ∫Bn∣u∣p+1 dvBn)2p+1≀∫Bn(∣∇Bnu∣2−λu2) dvBn,S_{n, p, \lambda}(\mathbb{B}^{n})\left(~\int \limits_{\mathbb{B}^{n}}|u|^{{p+1}} \, {\rm d}v_{\mathbb{B}^n} \right)^{\frac{2}{p+1}} \leq\int \limits_{\mathbb{B}^{n}}\left(|\nabla_{\mathbb{B}^{n}}u|^{2}-\lambda u^{2}\right) \, {\rm d}v_{\mathbb{B}^n}, holds for all u∈Cc∞(Bn),u\in C_c^{\infty}(\mathbb{B}^n), and λ≀(n−1)24,\lambda \leq \frac{(n-1)^2}{4}, where (n−1)24\frac{(n-1)^2}{4} is the bottom of the L2L^2-spectrum of −ΔBn.-\Delta_{\mathbb{B}^n}. It is known from the results of Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] that under appropriate assumptions on n,pn,p and λ\lambda there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant Sn,p,λ(Bn).S_{n,p,\lambda}(\mathbb{B}^n). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in Rn\mathbb{R}^n of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] and Ciraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to the Poincar\'{e}-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

    Measurement of the top quark forward-backward production asymmetry and the anomalous chromoelectric and chromomagnetic moments in pp collisions at √s = 13 TeV

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    Abstract The parton-level top quark (t) forward-backward asymmetry and the anomalous chromoelectric (d̂ t) and chromomagnetic (Ό̂ t) moments have been measured using LHC pp collisions at a center-of-mass energy of 13 TeV, collected in the CMS detector in a data sample corresponding to an integrated luminosity of 35.9 fb−1. The linearized variable AFB(1) is used to approximate the asymmetry. Candidate t t ÂŻ events decaying to a muon or electron and jets in final states with low and high Lorentz boosts are selected and reconstructed using a fit of the kinematic distributions of the decay products to those expected for t t ÂŻ final states. The values found for the parameters are AFB(1)=0.048−0.087+0.095(stat)−0.029+0.020(syst),Ό̂t=−0.024−0.009+0.013(stat)−0.011+0.016(syst), and a limit is placed on the magnitude of | d̂ t| &lt; 0.03 at 95% confidence level. [Figure not available: see fulltext.

    Measurement of t(t)over-bar normalised multi-differential cross sections in pp collisions at root s=13 TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions