[Show abstract][Hide abstract] ABSTRACT: We prove bounds for twisted ergodic averages for horocycle flows of
hyperbolic surfaces, both in the compact and in the non-compact finite area
case. From these bounds we derive effective equidistribution results for
horocycle maps. As an application of our main theorems in the compact case we
further improve on a result of A. Venkatesh, recently already improved by J.
Tanis and P. Vishe, on a sparse equidistribution problem for classical
horocycle flows proposed by N. Shah and G. Margulis, and in the general
non-compact, finite area case we prove bounds on Fourier coefficients of cups
forms which are off the best known bounds of A. Good only by a logarithmic
term. Our approach is based on Sobolev estimates for solutions of the
cohomological equation and on scaling of invariant distributions for twisted
[Show abstract][Hide abstract] ABSTRACT: We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the existence of some integral identities. We study the identities that lead to symmetry results, the functionals that can be considered and the function spaces that can be used. Then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of Laplacian, for the generalized Choquard functional and for the standing waves of the Davey–Stewartson equation.
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