Unimodular Fourier multipliers on Wiener amalgam spaces

We study the boundedness of unimodular Fourier multipliers on Wiener amalgam spaces. For a real-valued homogeneous function mu on R-n of degree alpha >= 2, we show the boundedness of the operator e(iu(D)) between the weighted Wiener amalgam space W-s(p,q) and W-P,W-q for all 1 <= p, q <= infinity and s > n(alpha - 2)vertical bar 1/p - 1/2 vertical bar + n vertical bar 1/p - 1/q vertical bar. This threshold is shown to be optimal for regions max(1/q,1/2) <= 1/p and min(1/q, 1/2) >= 1/p. Moreover, we give sufficient conditions for the boundedness of e(iu(D)) on W-p,W-q for alpha is an element of (0,2). (C) 2014 Elsevier Inc. All rights reserved.