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Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group
In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities
on the Heisenberg group. On one hand, we apply the concentration compactness
principle to prove the existence of the maximizers. While the approach here
gives a different proof under the special cases discussed in a recent work of
Frank and Lieb, we generalize the result to all admissible cases. On the other
hand, we provide the upper bounds of sharp constants for these inequalities.Comment: To be published in Indiana University Mathematics Journa
Small scale equidistribution of random eigenbases
We investigate small scale equidistribution of random orthonormal bases of
eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group
of isometries acts transitively on M and the multiplicity of eigenfrequency
tends to infinity at least logarithmically. We prove that, with respect to the
natural probability measure on the space of eigenbases, almost surely a random
eigenbasis is equidistributed at small scales; furthermore, the scales depend
on the growth rate of multiplicity. In particular, this implies that almost
surely random eigenbases on the n-dimensional sphere (n>=2) and the
n-dimensional tori (n>=5) are equidistributed at polynomial scales.Comment: 13 page
Small scale quantum ergodicity in negatively curved manifolds
The main theorem has been slightly generalized to include a larger class of
symbols.Comment: 22 page
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