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- W2807566078 abstract "Let $f in C^2(mathbb{T}^2)$ have mean value 0 and consider $$ sup_{gamma~{tiny mbox{closed geodesic}}}{~~~ frac{1}{|gamma|} left| int_{gamma}{ f ~~dmathcal{H}^1}right| },$$ where $gamma$ ranges over all closed geodesics $gamma:mathbb{S}^1 rightarrow mathbb{T}^2$ and $|gamma|$ denotes their length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $gamma$ attaining the supremum in terms of textit{smoothness} of the function: for all $s geq 2$, $$ |gamma|^{s} lesssim_s left( max_{|alpha| = s}{ | partial_{alpha} f |_{L^{1}(mathbb{T}^2)}} right) | nabla f |_{L^2}^{} |f|_{L^2}^{-2}.$$ We also prove a sharp bound for trigonometric polynomials. This seems like an interesting phenomenon. We do not know at which level of generality it holds or whether versions or variants of it could be established in other settings (hyperbolic surfaces, groups,...)." @default.
- W2807566078 created "2018-06-13" @default.
- W2807566078 creator A5054778975 @default.
- W2807566078 date "2018-05-07" @default.
- W2807566078 modified "2023-09-27" @default.
- W2807566078 title "A Compactness Principle for Maximizing Smooth Functions over Toroidal Geodesics" @default.
- W2807566078 hasPublicationYear "2018" @default.
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