共查询到20条相似文献,搜索用时 15 毫秒
1.
The aim of this paper is to analyze the heat semigroup ${(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}$ generated by the usual Laplacian operator Δ on ${\mathbb{R}^{d}}$ equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup. 相似文献
2.
In this paper we prove the Poincaré-type weighted inequality for a locally Lipschitz function f with a weighted mean equal to zero over a convex bounded domain \(\Omega \); here the weights v, \(\omega \) are positive measurable functions which satisfy a certain compatibility condition. This result is a generalization of the well-known weighted Poincaré inequality to the case of more general weights in the sense that we do not use the traditional conditions of high summability \(v,\, \omega ^{-\frac{1}{p-1}}\in L^{r,loc}\) with \(r>1\) for \(q=p\) or the reverse doubling condition on the function v for \(q>p\) . In other words, a Sawyer type sufficient condition on weight functions is established.
相似文献
$$\begin{aligned} \Vert v^{1/q} f \Vert _{L^q(\Omega )} \le C \Vert \omega ^{1/p} \nabla f \Vert _{L^p(\Omega )}, \quad q\ge p>1, \end{aligned}$$
3.
The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p1, persists through to the limit of a sequence of converging pointed metric measure spaces — this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality.
Mathematics Subject Classification (2000):31C15, 46E35. 相似文献
4.
We give a sufficient condition on a closed subset R
n
for the weighted Poincaré inequality (1.5) below to be valid. As an application, we prove that, for any 2p<n and any such closed subset R
n
, if uC
1( , N) W
1,p
(, N) is a stationary p-harmonic map such that |Du|
p
(x) dx is sufficiently small, then uC
1(, N). This extends previously known removal singularity theorems for p-harmonic maps.
Mathematics Subject Classification (2000):58E20, 58J05, 35J60This revised version was published online in September 2003 with a corrected date of receipt of the article. 相似文献
5.
6.
We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths.
Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.
Both authors were partially supported by the University of Bologna, funds for selected research topics. 相似文献
7.
We give some integral representations of the form f(x) = P(f)+K(?f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups. 相似文献
8.
Camille Tardif 《Potential Analysis》2013,38(3):1001-1030
Ben Arous and Gradinaru (Potential Anal 8(3):217–258, 1998) described the singularity of the Green function of a general sub-elliptic diffusion. In this article we first adapt their proof to the more general context of a hypoelliptic diffusion. In a second time, we deduce a Wiener criterion and a Poincaré cone condition for a relativistic diffusion with values in the Poincaré group (i.e the group of affine direct isometries of the Minkowski space-time). 相似文献
9.
Maurício CorrêaJr. Márcio G. Soares 《Bulletin of the Brazilian Mathematical Society》2011,42(3):485-503
We study one-dimensional holomorphic foliations on products of complex projective spaces and present results giving the number
of singularities, counting multiplicities, of a generic foliation, a criterion for a foliation to be Riccati and a Poincaré
type inequality, relating degrees of foliations to degrees of hypersurfaces which are invariant by them. 相似文献
10.
Ulrich Menne 《Calculus of Variations and Partial Differential Equations》2010,38(3-4):369-408
In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp. 相似文献
11.
12.
Feng-Yu Wang 《中国科学 数学(英文版)》2016,59(2):261-280
As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator L :=1/2 sum from i=1 to m X_i~2 on R~(m+d):= R~m× R~d is investigated, where X_i(x, y) = sum (σki?xk) from k=1 to m+sum (((A_lx)_i?_(yl)) from t=1 to d,(x, y) ∈ R~(m+d), 1 ≤ i ≤ m for σ an invertible m × m-matrix and {A_l}_1 ≤ l ≤d some m × m-matrices such that the Hrmander condition holds.We first establish Bismut-type and Driver-type derivative formulas with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group. 相似文献
13.
We study the problem of finding the best constant in the generalized Poincaré inequality
lpqr = min\frac|| y¢ ||Lp[0,1]|| y ||Lp[0,1], ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0, 相似文献
14.
Feng-Yu Wang 《Potential Analysis》2005,22(1):1-15
A generalized Beckner-type inequality interpolating the Poincaré and the log-Sobolev inequalities is studied. This inequality possesses the additivity property and characterizes certain exponential convergence of the corresponding Markov semi-group. A correspondence between this inequality and the so-called F-Sobolev inequality is presented, with the known criteria of the latter applying also to the former. In particular, an important result of Lataa and Oleszkiewicz is generalized. 相似文献
15.
16.
《Chaos, solitons, and fractals》2001,12(11):2023-2038
In the case of fully chaotic systems, the distribution of the Poincaré recurrence times is an exponential whose decay rate is the Kolmogorov–Sinai (KS) entropy. We address the discussion of the same problem, the connection between dynamics and thermodynamics, in the case of sporadic randomness, using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincaré recurrence times to “thermodynamics”, in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Today 52 (8) (1999) 39] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensitivity to initial conditions and the distribution of the Poincaré recurrence times are exponential. In the case of sporadic randomness, resulting at long times in the Lévy diffusion processes, the sensitivity to initial conditions is initially a power-law, but it becomes exponential again in the long-time scale, whereas the distribution of Poincaré recurrence times keeps, or gets, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show that a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict with the dynamic approach to Lévy statistics. The adoption of heuristic arguments indicates that this effect is possible, as a form of genuine equilibrium, after completion of the process of memory erasure. 相似文献
17.
Jacques Féjoz 《Regular and Chaotic Dynamics》2013,18(6):703-718
This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants “∮γ p dq”. The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did. 相似文献
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19.
S. G. Basalaev 《Siberian Mathematical Journal》2014,55(2):215-229
We obtain the Poincaré inequality for the equiregular Carnot-Carathéodory spaces spanned by vector fields with Hölder class derivatives. 相似文献
20.
B. Fleury 《Probability Theory and Related Fields》2012,152(1-2):141-178
Let K N = [±G 1, . . . , ±G N ] be the absolute convex hull of N independent standard Gaussian random points in ${\mathbb R^n}$ with N ≥ n. We prove that, for any 1-Lipschitz function ${f:\mathbb R^n\rightarrow\mathbb R}$ , the polytope K N satisfies the following Poincaré inequality in mean value: $$\mathbb {E}_{\omega} \int\limits_{K_N(\omega)} \left( f(x) - \frac{1}{\textup{vol}_n\left(K_N(\omega)\right)} \int\limits_{K_n(\omega)}f(y)dy \right)^2 dx \leq \frac{C}{n} \mathbb E_{\omega} \int\limits_{K_N(\omega)}|x|^2dx$$ where C?>?0 is an absolute constant. This Poincaré inequality is the one suggested by a conjecture of Kannan, Lovász and Simonovits for general convex bodies. Moreover, we prove in mean value that the volume of the polytope K N is concentrated in a subexponential way within a thin Euclidean shell with the optimal dependence of the dimension n. An important tool of the proofs is a representation of the law of (G 1, . . . , G n ) conditioned by the event that “the convex hull of G 1, . . . , G n is a (n ? 1)-face of K N ”. As an application, we also get an estimate of the number of (n ? 1)-faces of the polytope K N , valid for every N ≥ n. 相似文献
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