Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators –(a)+b ₁+b ₂+V with real measurable (singular) coefficients, on an open set R ^N . The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L ^p () for p only from an interval in [1,), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L ^p (). We show however that the semigroup is ultracontractive in the scale of weighted spaces L ^p (,²dx) with a suitable weight and derive an upper and lower bound on its integral kernel.     

Riesz and Poisson-Jensen Representation Formulas for a Class of Ultraparabolic Operators on Lie Groups

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators on a homogeneous Lie group . Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for -Green functions and for -Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for . Investigation supported by University of Bologna. Funds for selected research topics.     

Boundeness of a Class of Maximal Operators on H~p Spaces on Compact Lie Groups

In this paper we discuss the weak type(H^p,L^p) boundedness of a class of maximal operators T*^ψ and themaximal strong mean boundedness of a family of the operators ｛T^ψ｝ on the atomic H^p spaces on compact Lic groups.Also,we obtain the correspoding convergent results.     

Continuous and Discrete Wavelets for Acceleration Kinematics: A Motion Estimation Based on Lie Groups and Variational Principles

A group-theoretic framework is presented for acceleration transformations. The main purpose is to show the existence of families of spatio-temporal continuous wavelets, frames, and discrete wavelets related to these transformations. The main application of interest is the analysis of motion in space–time signals. The construction of this framework starts with the enumeration of Lie algebras as building blocks that provide all the observable kinematics that comply with the properties of the space–time under analysis. These classes of accelerated kinematics generalize the kinematics defined in the Galilei group. Exponentiation from Lie algebras defines locally compact exponential groups. Unitary, irreducible, and square-integrable group representations are thereafter derived in the function spaces and the signals to be analyzed, leading to the existence of continuous and discrete wavelets, frames all indexed with higher orders of temporal derivatives of the translational motion. Group representations and wavelets are tools that perform the local optimum estimation of pieces of trajectory. The adjunction of a variational principle of optimality is further necessary for building a global trajectory and for performing tracking. The Euler–Lagrange equation provides the motion equation of the moving system and the Noether's theorem derives the related constants of motion. Dynamic programming implements the algorithms for tracking and constructing the global trajectory. Finally, tight frames and bases enable signal decompositions along the trajectory of interest.