Some conjectures aboutL p norms of k-plane transforms

For 1≤k≤n-1, the k-plane transformT _k,n carries functionsf defined onR ⁿ to functionsT _k,n f defined on the set of affine k-planes inR ⁿ . It is known thatT _k,n mapsL ^p intoL ^q for certain values ofp andq. In this article we formulate conjectures for the exact values of the norm of theT _k,n , and state also a conjecture asserting that theL ^q norm ofT _k,n f changes monotonically whenf is replaced by its symmetric decreasing rearrangement. Conferenza tenuta il 6 marzo 1997 The first author was supported in part by NSF grants DMS-9206319 and DMS-9501293. The second author was supported in part by NSF grant DMS-9500840