English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We use a straightforward variation on a recent argument of Hezari andRivi\`ere~\cite{HR} to obtain localized $L^p$-estimates for all exponentslarger than or equal to the critical exponent $p_c=\tfrac{2(n+1)}{n-1}$. We areable to this directly by just using the $L^{p}$-bounds for spectral projectionoperators from our much earlier work \cite{Seig}. The localized bounds weobtain here imply, for instance, that, for a density one sequence ofeigenvalues on a manifold whose geodesic flow is ergodic, all of the $L^p$,$2<p\le \infty$, bounds of the corresponding eigenfunctions are relativelysmall compared to the general ones in \cite{Seig}, which are saturated on roundspheres. The connection with quantum ergodicity was established for exponents$2<p<p_c$ in the recent results of the author \cite{SK} and Blair and theauthor \cite{BS2}; however, the article of Hezari and Rivi\`ere~\cite{HR} wasthe first one to make this connection (in the case of negatively curvedmanifolds) for the critical exponent, $p_c$. As is well known, and we indicatehere, bounds for the critical exponent, $p_c$, imply ones for all of the otherexponents $2<p\le \infty$. The localized estimates involve $L^2$-norms oversmall geodesic balls $B_r$ of radius $r$, and we shall go over what happens forthese in certain model cases on the sphere and on manifolds of nonpositivecurvature. We shall also state a problem as to when one can improve on thetrivial $O(r^{\frac12})$ estimates for these $L^2(B_r)$ bounds. If$r=\lambda^{-1}$, one can improve on the trivial estimates if one has improved$L^{p_c}(M)$ bounds just by using H\"older's inequality; however, obtainingimproved bounds for $r\gg \lambda^{-1}$ seems to be subtle.

Localized $L^p$-estimates of eigenfunctions: A note on an article of Hezari and Rivi\`ere