Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new user$L_infty$-estimates for the torsion function and $L_infty$-growth of semigroups satisfying Gaussian bounds
65
0
0.0
(
0
)
Authors
Hendrik Vogt
Ask ChatGPT about the research
No Arabic abstract
We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_infty$ operator norm of the semigroup. As an application we prove a new $L_infty$-bound for the torsion function of a Euclidean domain that is close to optimal.
rate research
Read More
A result by Liskevich and Perelmuter from 1995 yields the optimal angle of analyticity for symmetric submarkovian semigroups on $L_p$, $1<p<infty$. C.~Kriegler showed in 2011 that the result remains true without the assumption of positivity of the semigroup. Here we give an elementary proof of Krieglers result.
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally $log$-Holder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(cdot)}(mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels ${e^{-t^2L}}_{tin (0,infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(cdot)}(mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(cdot)}(mathbb R^n)$ under an additional assumption that the heat kernels of $L$ have the Holder regularity.
The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between $L_infty$-morphisms that are twisted with gauge equivalent Maurer-Cartan elements.
In this short note we describe an alternative global version of the twisting procedure used by Dolgushev to prove formality theorems. This allows us to describe the maps of Fedosov resolutions, which are key factors of the formality morphisms, in terms of a twist of the fiberwise quasi-isomorphisms induced by the local formality theorems proved by Kontsevich and Shoikhet. The key point consists in considering $L_infty$-resolutions of the Fedosov resolutions obtained by Dolgushev and an adapted notion of Maurer-Cartan element. This allows us to perform the twisting of the quasi-isomorphism intertwining them in a global manner.
It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an $L_infty$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_infty$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_infty$ algebra is $L_infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
