The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the...

We extend the results in [5] to non-compactly supported perturbations for a class of symmetric first order systems.

Let $ℒ$ be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure $\gamma$ on ${ℝ}^d.$ We prove a sharp estimate of the operator norm of the imaginary powers of $ℒ$ on $L^p\left(\gamma \right),$$1\<p\<\infty ...$

Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem [2] on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and finally of the empirical...

In the context of the spaces of homogeneous type, given a family of operators that look like approximations of the identity, new sharp maximal functions are considered. We prove a good-λ inequality for Muckenhoupt weights, which leads to an analog of the Fefferman-Stein estimate for the classical sharp maximal function. As a consequence, we establish weighted norm estimates for certain singular integrals, defined on irregular domains, with Hörmander conditions replaced by some estimates which do...

We describe the asymptotic distribution of eigenvalues of self-adjoint elliptic differential operators, assuming that the first-order derivatives of the coefficients are Lipschitz continuous. We consider the asymptotic formula of Hörmander's type for the spectral function of pseudodifferential operators obtained via a regularization procedure of non-smooth coefficients.

We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let $H_L^p\left(X\right)$ (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on $H_L^p\left(X\right)$ follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on $H_L^p\left(X\right)$.

We consider some descriptive properties of supports of shift invariant measures on ${ℂ}^{\mathbb{Z}}$ under the assumption that the closed linear span (in $L^2$) of the co-ordinate functions on ${ℂ}^{\mathbb{Z}}$ is all of $L^2$.

The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal $\left(H\right):=T\in \left(H\right):su{p}_{1\le m<\infty }1/(logm+1){\sum }_{n=1}^{m}aₙ\left(T\right)<\infty$ can be reduced to the theory of shift-invariant functionals on the Banach sequence space $\left(\mathbb{N}₀\right):=c=\left({\gamma }_l\right):su{p}_{0\le k<\infty }1/(k+1){\sum }_{l=0}^k\left|{\gamma }_l\right|<\infty$. The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier traces...

We prove a characterisation of sets with finite perimeter and $BV$ functions in terms of the short time behaviour of the heat semigroup in ${\mathbf{R}}^N$. For sets with smooth boundary a more precise result is shown.