| Popis: |
We have by now defined both the weighted and unweighted versions of the modulation spaces and indicated in Chaps. 4 and 5 their place in the time-frequency analysis of pseudodifferential operators and their good properties regarding certain nonlinear operations stemming from PDEs. Hidden in the seemingly simple definition of modulation spaces lies the practical problem of computing the STFT of a function or distribution and then further calculating its appropriate mixed Lebesgue norm. Therefore, a natural question arising in this context is whether any embeddings between such modulation spaces and other classical function spaces of analysis exist, something that we have alluded to already in Sects. 2.3 and 2.4. Let us recall also immediately that Sjostrand’s class, \(\mathcal {M}^{\infty , 1}\), contains the Hormander class \(S^0_{0,0}\), a fact that played a relevant role throughout Chap. 4; see again the comments after Corollary 4.22 and Sjostrand (Math Res Lett 1(2):185–192, 1994). |
