One can view a 2-parameter Brownian sheet {W (s, t); s, t ≥ 0} as a stream of interacting Brownian motions {W (s, •); s ≥ 0}. Given this viewpoint, we aim to continue the analysis of Walsh (1978) on the local times of the stream W (s, •), near time s = 0. Our main result is a kind of maximal inequality that, in particular, verifies the following conjecture of Khoshnevisan (1995a): as s → 0, the...

Abstract: One can view a 2-parameter Brownian sheet {W (s, t); s, t ≥ 0} as a stream of interacting Brownian motions {W (s, •); s ≥ 0}. Given this viewpoint, we aim to continue the analysis of [20] on the local times of the stream W (s, •) near time s = 0. Our main result is a kind of maximal inequality that, in particular, verifies the following conjecture of [10]: As s → 0, the local times of...

In this paper, we will prove some new dynamic inequalities on a time scale T. These inequalities, as special cases, when T= R contain some integral inequalities and when T= N contain the discrete inequalities due to Leindler. The main results will be proved by using the Hölder inequality and a simple consequence of Keller’s chain rule on time scales. From our results, as applications, we will d...

It is known that the classic Korn inequality is not valid for Hölder α domains. In this paper we prove a family of weaker inequalities for this kind of domains, replacing the standard L-norms by weighted norms where the weights are powers of the distance to the boundary. In order to obtain these results we prove first some weighted Poincaré inequalities and then, generalizing an argument of Kon...

Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to Heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential s...

We consider an elliptic system of equations on the torus [ −L2 , L 2 )d with random coefficients A, that are assumed to be coercive and stationary. Using two different approaches we obtain moment bounds on the gradient of the corrector, independent of the domain size L. In the first approach we use Green function representation. For that we require A to be locally Hölder continuous and distribu...

Estimating divergences in a consistent way is of great importance in many machine learning tasks. Although this is a fundamental problem in nonparametric statistics, to the best of our knowledge there has been no finite sample exponential inequality convergence bound derived for any divergence estimators. The main contribution of our work is to provide such a bound for an estimator of Rényi-α d...

where the potential V belongs to Bq1 for q1 ≥ n/2. We are interested in the L boundedness of the operator∇4H−1, where the potential V satisfies weaker condition than that in 5, Theorem 1, 2 . The estimates of some other operators related to Schrödinger-type operators can be found in 2, 5 . Note that a nonnegative locally L integrable function V on R is said to belong to Bq 1 < q < ∞ if there ex...