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Register a new userSzego Limit Theorem on the Heisenberg Group
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Let $mathcal{H}_0=V, mathcal{H}_1=B+V$ and $mathcal{H}_2=mathcal{L}+V$ be the operators on the Heisenberg group $mathbb{H}^n$, where $V$ is the operator of multiplication growing like $|g|^kappa, 0<kappa<1$, $B$ is a bounded linear operator and $mathcal{L}$ is the sublaplacian on $mathbb{H}^n$. In this paper we prove Szego limit theorem for the operators $mathcal{H}_0, mathcal{H}_1$ and $mathcal{H}_2$ on $L^2(mathbb{H}^n).$
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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 characterize even measures $mu=wdx+mu_s$ on the real line with finite entropy integral $int_{R} frac{log w(t)}{1+t^2}dt>-infty$ in terms of $2times 2$ Hamiltonian generated by $mu$ in the sense of inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.
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.
We establish the necessary and sufficient conditions for those symbols $b$ on the Heisenberg group $mathbb H^{n}$ for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson--Wolff and Rochberg--Semmes, which address the same question in the Euclidean setting. Moreover, the approach that we develop bypasses the use of Fourier analysis, and can be applied to characterise that the commutator is of the Schatten class in other settings beyond Euclidean.
Versions of well known function theoretic operator theory results of Szego and Widom are established for the Neil algebra. The Neil algebra is the subalgebra of the algebra of bounded analytic functions on the unit disc consisting of those functions whose derivative vanishes at the origin.
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