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L^p bounds for eigenfunctions at the critical exponent 

Matthew Blair (U of New Mexico)
Wednesday, November 13, 2019 - 4:00pm to 5:00pm
PDL C-38

 Abstract: We consider upper bounds on the growth of L^p norms of
eigenfunctions of the Laplacian on a compact Riemannian manifold in the
high frequency limit. In particular, we seek to identify geometric or
dynamical conditions on the manifold which yield improvements on the
universal L^p bounds of C. Sogge.  The emphasis will be on bounds at the
"critical exponent", where a spectrum of scenarios for phase space
concentration must be considered.  We then discuss a recent work with C.
Sogge which shows that when the sectional curvatures are nonpositive,
there is a logarithmic type gain in the known L^p bounds at the critical
exponent.

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