-norms of periodizations over integer lattices

We show the boundedness of the mixed L p , p ′ ( [ 0 , 1 ] d × SO ( d ) ) norm of periodizations of a function f from L p ( R d ) ∩ L q ( R d ) where 1 ⩽ p ⩽ 2 and 1 ⩽ q < q ( p ) and d ⩾ 5 .