Inclusion relations between LpLp-Sobolev and Wiener amalgam spaces

We determined optimal inclusion relations between LpLp-Sobolev and Wiener amalgam spaces. For applications, we discuss mapping properties of unimodular Fourier multipliers ei|D|αei|D|α between LpLp-Sobolev and Wiener amalgam spaces and derive some Littlewood–Paley type inequalities.