Characterization of Boundedness on Weighted Modulation Spaces of τ-Wigner Distributions

Abstract

Abstract This paper is devoted to give several characterizations on a more general level for the boundedness of $\tau $-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of $\tau $-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained by Cordero and Nicola [10] and Cordero [6]. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operators are established. In particular, we give the sharp exponents for the boundedness of pseudodifferential operators with symbols in Sjöstrand’s class and the corresponding Wiener amalgam spaces.