2 THEOREMS ON LATTICE EXPANSIONS

It is shown that there is a trade-off between the smoothness and decay properties of the dual functions, occurring in the lattice expansion problem. More precisely, it is shown that if g and g are dual, then 1) at least one of H1/2 g and H1/2 g is not in L2(R), 2) at least one of Hg and g is not in L2(R). Here, H is the operator -1(4pi2)d2/(dt2) + t2. The first result is a generalization of a theorem first stated by Balian and independently by Low, which was recently rigorously proved by Coifman and Semmes; a new, much shorter proof was very recently given by Battle. Battle suggests a theorem of type (i), but our result is stronger in the sense that certain implicit assumptions made by Battle are removed. Result 2) is new and relies heavily on the fact that, when G is-an-element-of W2,2(S) with S = [-1/2, 1/2] x [-1/2, 1/2] and G(0) = 0, then 1/G is-not-an-element-of L2(S). The latter result was not known to us and may be of independent interest.