Sharpness of some properties of Wiener amalgam and modulation spaces

We prove sharp estimates for the dilation operator $f(x)\longmapsto f(λx)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

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