Smooth pointwise multipliers of modulation spaces

Let 1 < p, q < ∞ and s, r ∈ ℝ. It is proved that any function in the amalgam space W(H^r_p(ℝ^d), ℓ_∞), where p' is the conjugate exponent to p and H^r_p′ (ℝ^d) is the Bessel potential space, defines a bounded pointwise multiplication operator in the modulation space M^s_p,q(ℝ^d), whenever r > |s| + d