Resolvent estimates and smoothing for homogeneous partial differential operators on graded Lie groups

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators H on graded groups, including Rockland operators, sublaplacians, and many others. Left or right invariance is not required. Typically the globally smooth operator has the form T = V|H|^1∕2, where V only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group ?^N some new classes of partial differential operators are treated.     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

A multiplier theorem for the sublaplacian on the Heisenberg group

A multiplier theorem for the sublaplacian on the Heisenberg group is proved using Littlewood-Paley-Stein theory ofg-functions.     

Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

Fractional time-dependent Schrödinger equation on the Heisenberg group

Let Δ be the Kohn sublaplacian on the Heisenberg group , . In this paper we estimate the L ²-norm of the local maximal function of the unitary group of operators generated by L, by the Sobolev W _γ,ε -norm for some γ > 0 and for all ε > 0. Research supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. The first author was also supported by the MNiSW research grant N201 012 31/1020.     

广义Sublaplace算子的谱乘子定理

本文通过建立与广义Ｓｕｂｌａｐｌａｃｅ算子Ｌ相关的Ｌｉｔｌｅｗｏｏｄ－Ｐａｌｅｙ理论，得到Ｌ的谱乘子定理，作为该结果的应用，并给出一类Ｌａｇｕｅｒｅ函数展开的乘子定理