Sharp 𝐿^{𝑝}→𝐿^{𝑟} estimates for 𝑘-plane transforms in finite fields

Koh, Doowon; Kwak, Dongyoon

doi:10.1090/proc/14072

Sharp $L^p\to L^r$ estimates for $k$-plane transforms in finite fields
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by Doowon Koh and Dongyoon Kwak PDF

Proc. Amer. Math. Soc. 146 (2018), 3381-3391 Request permission

Abstract:

We study mapping properties of finite field $k$-plane transforms. Using geometric combinatorics, we do an elaborate analysis to recover the critical endpoint estimate. As a consequence, we obtain optimal $L^p\to L^r$ estimates for all $k$-plane transforms in the finite field setting. In addition, applying Hölder’s inequality to our results, we obtain an estimate for multilinear $k$-plane transforms.

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