Weighted modulation space Ms p,q(w) with w ∈ Aloc

In the present paper, we present a natural extension of the modulation spaces Ms p,q with 0 < p < ∞, 0 < q ∞ and s ∈ R. The weight we take up here belongs to the class Aloc ∞ , which is even wider and contains weights of exponential growth. After defining the function spaces, we investigate their atomic and molecular decomposition as well as some elementary properties