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Register a new userAnalogues of theorems of Chernoff and Ingham on the Heisenberg group
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We prove an analogue of Chernoffs theorem for the Laplacian $ Delta_{mathbb{H}} $ on the Heisenberg group $ mathbb{H}^n.$ As an application, we prove Ingham type theorems for the group Fourier transform on $ mathbb{H}^n $ and also for the spectral projections associated to the sublaplacian.
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We prove an analogue of Chernoffs theorem for the sublaplacian on the Heisenberg group and use it prove a version of Inghams theorem for the Fourier transform on the same group.
We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $ mathbb{C}^n $ and Hermite operator on $ mathbb{R}^n.$
In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-tau_y A_rf$, where $tau_yf(x)=f(xy^{-1})$ is the right translation operator.
We provide a new geometric proof of Reimanns theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.
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