Gaussian Bound Research Articles

Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let X , d , μ be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ . Let L be a non-negative self-adjoint operator on L 2 X . Assume that the (heat) kernel associated to the semigroup e − t L satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space H φ , L X associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space H L , G , φ X associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.

Gaussian bounds for noise correlation of resilient functions

Gaussian bounds on noise correlation of functions play an important role in hardness of approximation, in quantitative social choice theory and in testing. The author (2008) obtained sharp Gaussian bounds for the expected correlation of l low influence functions f(1), …, f(l) : Ωn → [0, 1], where the inputs to the functions are correlated via the n-fold tensor of distribution $${\cal P}$$ on Ωl in the following way: For each 1 ≤ i ≤ n, the vector consisting of the i’-th inputs to the l functions is sampled according to $${\cal P}$$. It is natural to ask if the condition of low influences can be relaxed to the condition that the function has vanishing Fourier coefficients. Here and g we further show that if f, g have a noisy inner product that exceeds the Gaussian bound, then the Fourier supports of their large coefficients intersect.

LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

Let$(X,d,\unicode[STIX]{x1D707})$be a metric measure space endowed with a distance$d$and a nonnegative, Borel, doubling measure$\unicode[STIX]{x1D707}$. Let$L$be a nonnegative self-adjoint operator on$L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup$e^{-tL}$satisfies a Gaussian upper bound. In this paper, we prove that for any$p\in (0,\infty )$and$w\in A_{\infty }$, the weighted Hardy space$H_{L,S,w}^{p}(X)$associated with$L$in terms of the Lusin (area) function and the weighted Hardy space$H_{L,G,w}^{p}(X)$associated with$L$in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duonget al.[‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’,J. Geom. Anal.26(2016), 1617–1646], who proved that$H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$for$p\in (0,1]$and$w\in A_{\infty }$by imposing an extra assumption of a Moser-type boundedness condition on$L$. Our result is new even in the unweighted setting, that is, when$w\equiv 1$.

Wiener-type tests from a two-sided Gaussian bound

In this paper, we are concerned with hypoelliptic diffusion operators \(\mathcal {H}\). Our main aim is to show, with an axiomatic approach, that a Wiener-type test of \(\mathcal {H}\)-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of \(\mathcal {H}\) with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Holder estimates of the solution under a suitable exterior cone condition.

Heat kernel for the elliptic system of linear elasticity with boundary conditions

We consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second Korn inequality holds. We construct heat kernel of the system subject to Dirichlet, Neumann, or mixed boundary condition under the assumption that weak solutions of the elliptic system are Hölder continuous in the interior. Moreover, we show that if weak solutions of the mixed problem are Hölder continuous up to the boundary, then the corresponding heat kernel has a Gaussian bound. In particular, if the domain is a two dimensional Lipschitz domain satisfying a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary condition, then we show that the heat kernel has a Gaussian bound. As an application, we construct Green's function for elliptic mixed problem in such a domain.

Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions

Dans ce papier nous étudions des bornes supérieures pour la densité d’une solution déquation différentielle conduite par un mouvement brownien fractionnaire d’indice de Hurst $H>1/3$. Nous montrons, que sous certaines conditions géomètriques, dans le cas régulier $H>1/2$, la densité de la solution satisfait l’inégalité de log-Sobolev, l’inégalité de concentration gaussienne et admet une borne supérieure gaullienne. Dans le cas $H>1/3$ et sous la même condition géomètrique, nous montrons que la densité est infiniment différentiable et admet une borne supérieure sous-gaussienne.

Impact of carrier frequency offset on the capacity of MIMO-OFDM systems

Carrier frequency offset (CFO) is a factor that corrupts subchannel orthogonality resulting to performance degradation in multiple-input multiple-output orthogonal frequency division multiplexing systems. We investigate the effect of CFO on the channel estimation and the effect of both CFO and channel estimation on the system capacity. The main result of the paper is a lower bound on the capacity for a given CFO and channel estimation error covariance. The lower bound is based on the Gaussian bound of the multiplicative noise because of channel estimation errors. Copyright © 2012 John Wiley & Sons, Ltd.

Concentration inequalities for Markov processes via coupling

We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order $1+a$ $(a > 0)$ of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order $1+ a$ is finite, uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.

Generation of the integrated semigroups by superelliptic differential operators

Let A be a superelliptic differential operator of order 2 m introduced by E.B. Davies [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141–169]. In the case of 2 m > N , he obtained the upper Gaussian bound of the integral kernel representing ( e − z A ) z ∈ C + and the estimates of the L p -operator norm of the semigroup for all p ∈ [ 1 , ∞ ) . The purpose of the present paper is to show that − i ( A + k ) (for some constant k > 0 ) generates an integrated semigroup on L α , p (weighted L p space) and l p ( L α , q ) . To prove this we need norm estimates of ( e − z A ) z ∈ C + on each of these spaces. Also we get another norm estimate of ( e − z A ) z ∈ C + on L p when 2 m > N without using the integral kernel. This norm estimate is better than that in [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141–169] and gives a better “times of the integration” of the integrated semigroup.

Littlewood-Paley theorem for Schrödinger operators

Let H be a Schrodinger operator on ℝ n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.

A family of diffusion processes on Sierpinski carpets

We construct a family of diffusions Pα = {Px} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d H < α < ∞, where d H = log(3 d − 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions Pα are reversible with invariant measures μ = μ[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3α·μ(A) for all A∈ℬ(F^). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet.

Heat kernels and maximal lp—lqestimates for parabolic evolution equations

Let A be the generator of an analytic semigroup Ton L2(Ω), where Ω is a homogeneous space with doubling property. We prove maximal Lp-Lp a—priori estimates for the solution of the parabolic evolution equation u'(t)=Au(t)+f(t), u(0)=0 provided Tmay be represented by a heat—kernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification:35K22, 58D25, 47D06

A bound on the rate-distortion function and application to images

An upper bound on the rate-distortion function for discrete ergodic sources with memory is developed by partitioning the source sample space into a finite number of disjoint subsets and bounding the rates for each subset. The bound depends only on the mean vectors and covariance matrices for the subsets and is easy to compute. It is tighter than the Gaussian bound for sources that exhibit clustering of either the values or covariances of successive source outputs. The bound is evaluated for a certain class of pictorial data using both one-dimensional and two-dimensional blocks of picture elements. Two-dimensional blocks yield a tighter bound than one-dimensional blocks; both result in a significantly tighter bound than the Gaussian bound.