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We study mapping properties of finite field k-plane transforms. Usinggeometric combinatorics, we do an elaborate analysis to recover the criticalendpoint estimate. As a consequence, we obtain optimal L^p-L^r estimates forall k-plane transforms in the finite field setting. In addition, applyingHolder's inequality to our results, we obtain an estimate for multilineark-plane transforms.

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