Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHardy-Rellich and second order Poincare identities on the hyperbolic space via Bessel pairs
85
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove a family of Hardy-Rellich and Poincare identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincare inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second ord
rate research
Read More
We prove infinite-dimensional second order Poincare inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Steins method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.
We establish sharp Hardy-Adams inequalities on hyperbolic space $mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $alpha>0$ there exists a constant $C_{alpha}>0$ such that [ int_{mathbb{B}^{4}}(e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2})dV=16int_{mathbb{B}^{4}}frac{e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2}}{(1-|x|^{2})^{4}}dxleq C_{alpha}. ] for any $uin C^{infty}_{0}(mathbb{B}^{4})$ with [ int_{mathbb{B}^{4}}left(-Delta_{mathbb{H}}-frac{9}{4}right)(-Delta_{mathbb{H}}+alpha)ucdot udVleq1. ] As applications, we obtain a sharpened Adams inequality on hyperbolic space $mathbb{B}^{4}$ and an inequality which improves the classical Adams inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.
We study the second order hyperbolic equations with initial conditions, a nonhomogeneous Dirichlet boundary condition and a source term. We prove the solution possesses $H^1$ regularity on any piecewise $C^1$-smooth non-timelike hypersurfaces. We generalize the notion of energy to these hypersurfaces, and establish an estimate of the difference between the energies on the hypersurface and on the initial plane where the time $t = 0$. The energy is shown to be conserved when the source term and the boundary datum are both zero. We also obtain an $L^2$ estimate for the normal derivative of the solution. In the proofs we first show these results for $C^2$-smooth solutions by using the multiplier methods, and then we go back to the original results by approximation.
We propose and prove an identity relating the Poincare polynomials of stabilizer subgroups of the affine Weyl group and of the corresponding stabilizer subgroups of the Weyl group.
We prove Rellich and improved Rellich inequalities that involve the distance function from a hypersurface of codimension $k$, under a certain geometric assumption. In case the distance is taken from the boundary, that assumption is the convexity of the domain. We also discuss the best constant of these inequalities.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
