Honors Project - Open Access
We examine the $k=1$ case of a conjecture by Baernstein and Loss pertaining to the operator norm of the $k$-plane transform from $L^p(\R^d)$ space to $L^q(\M)$ space. Previous work on this problem by Carlen and Loss, as well as by Drouot, has used an iterative technique known as the ``competing symmetries argument’’ to prove this conjecture in the $q=2$ and $q=d+1$ cases. We summarize the conjecture and this proof technique, then perform work that strongly suggest that no sufficiently ``nice” transformation exists that can be used to apply the competing symmetries argument to other cases of the conjecture.
DressenWall, Arthur D., "Sharp Inequalities of the X-Ray Transform and the Competing Symmetries Argument" (2022). Mathematics, Statistics, and Computer Science Honors Projects. 67.
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