Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations

Abstract

We introduce modulation type spaces associated with the generators of polynomially bounded groups. Besides strongly continuous groups we study in detail the case of bi-continuous groups, e.g. weak$^*$-continuous groups in dual spaces. It turns out that this gives new insight in situations where generators are not densely defined. Classical modulation spaces are covered as a special case but the abstract formulation gives more flexibility. We illustrate this with an application to a nonlinear Schrödinger equation.

Appendix A. Two auxiliary results

Here we provide proofs for a well-known result on the short time Fourier transform and for the contruction of $\tau _X$-Riemann type integrals with respect to measures $\mu \in {\mathscr {M}}({\mathbb {R}})$.

Fix $g\in {\mathscr {S}}({\mathbb {R}}){\setminus }\{0\}$. The short-time Fourier transform with window g of $f\in {\mathscr {S}}'({\mathbb {R}})$ is the function $V_gf:{\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {C}}$ defined by

$$\begin{aligned} V_gf(x,\xi ):=\int _{\mathbb {R}}e^{-iy\xi } f(y)\overline{g(y-x)}\,dy=\langle {f}{e^{i\xi (\cdot )}g(\cdot -x)}\rangle , \end{aligned}$$

(19)

where the duality bracket extends the usual scalar product in $L^2({\mathbb {R}})$. We shall only need that $V_g$ maps ${\mathscr {S}}({\mathbb {R}})$ to rapidly decreasing functions.

Lemma A.1

For $g\in {\mathscr {S}}({\mathbb {R}}){\setminus }\{0\}$ and $f\in {\mathscr {S}}({\mathbb {R}})$ the function

$$\begin{aligned} (x,\xi )\mapsto \langle x \rangle ^k\langle \xi \rangle ^l V_gf(x,\xi ) \end{aligned}$$

is bounded on ${\mathbb {R}}\times {\mathbb {R}}$ for all $k,l>0$.

This is well-known but we reprove it here for convenience.

Proof

We have

$$\begin{aligned} |V_gf(x,\xi )|\le \int _{\mathbb {R}}|f(y)||g(y-x)|\,dy =|f|*|\sigma g|(x), \end{aligned}$$

where $\sigma g(y)=g(-y)$. By $f,g\in {\mathscr {S}}({\mathbb {R}})$ we infer that $\langle x \rangle ^k V_gf(x,\xi )$ is bounded for any $k>0$. For any $l\in {\mathbb {N}}$ we use integration by parts to obtain

$$\begin{aligned} (-i\xi )^l V_gf(x,\xi )= & {} (-1)^l\int _{\mathbb {R}}e^{-iy\xi }\frac{d^l}{dy^l}\big (f(y)\overline{g(y-x)}\big )\,dy\\= & {} (-1)^l \sum _{j=0}^l {l\atopwithdelims ()j} V_{g^{(l-j)}}\big (f^{(j)}\big )(x,\xi ). \end{aligned}$$

Since ${\mathscr {S}}({\mathbb {R}})$ is invariant under taking derivatives we can combine both arguments to obtain the assertion.$\square$

For the following result on $\tau _X$-Riemann type integrals we suppose that Assumption 4.1 holds.

Proposition A.2

Let $f:[a,b]\rightarrow X$ be a $\tau _X$-continuous and norm-bounded function and let $\mu \in {\mathscr {M}}({\mathbb {R}})$ be a complex Borel measure. For any partition $a=t_0<t_1<\ldots <t_n=b$ and any vector $(\xi _1,\ldots ,\xi _n)$ with $\xi _j\in [t_{j-1},t_j)$ we define the Riemann type sum

$$\begin{aligned} S(f, t_j,\xi _j):=\sum _{j=1}^n f(\xi _j)\mu \big ([t_{j-1},t_j)\big ) + f(b)\mu \big (\{b\}\big ). \end{aligned}$$

Then the $\tau _X$-limit of $S(f,t_j,\xi _j)$ exists in X as $\max _j|t_j-t_{j-1}|$ tends to 0. This limit is denoted $\int _{[a,b]} f(t)\,d\mu (t)$. We have the estimate

$$\begin{aligned} \left\| \int _{[a,b]} f(t)\,d\mu (t)\right\| _{X}\le \sup _{t\in [a,b]}\Vert f(t)\Vert _{X} \,|\mu |([a,b]) \end{aligned}$$

and for any $\tau _X$-continuous seminorm p we have

$$\begin{aligned} p \left( \int _{[a,b]} f(t)\,d\mu (t)\right) \le \int _{[a,b]}p\big (f(t)\big )\,d|\mu |(t) \le \sup _{t\in [a,b]}p\big (f(t)\big ) \,|\mu |([a,b]). \end{aligned}$$

Proof

If $\Vert f(t)\Vert _{X}\le C$ we easily get

$$\begin{aligned} \Vert S(f,t_j,\xi _j)\Vert _{X}\le C|\mu |\big ([a,b]\big ). \end{aligned}$$

By Assumption 4.1 (i) it thus suffices to show the Cauchy property. So we let $\delta >0$ and take two partitions $(t_j)$ and $(s_k)$ such that $\max _j|t_j-t_{j-1}|$ and $\max _k|s_k-s_{k-1}|$ are $\le \delta$ and we take two corresponding vectors $(\xi _j)$ and $(\eta _k)$. Then we rewrite

$$\begin{aligned} S(f,t_j,\xi _j)-S(f,s_k,\eta _k)=\sum _{l} \big (f(\widetilde{\xi }_l)-f(\widetilde{\eta }_l)\big )\mu \big ([u_l,u_{l-1})\big ), \end{aligned}$$

where $(u_l)$ is a partition obtained by the union of the $t_j$ and the $s_k$ and we have $\{\widetilde{\xi }_l\}=\{\xi _j\}$, $\{\widetilde{\eta }_l\}=\{\eta _k\}$ as sets, but we have to repeat $\xi _j$ according to the splitting of the interval $[t_{j-1},t_j)$. Then not necessarily $\widetilde{\xi }_l\in [u_{l-1},u_l)$ but we have at least $|\widetilde{\xi }_l-u_l|\le \delta$ and $|\widetilde{\xi }_l-u_{l-1}|\le \delta$. The same holds for the $\widetilde{\eta }_l$ so that $|\widetilde{\xi }_l-\widetilde{\eta }_l|\le 2\delta$. Taking a $\tau _X$-continuous seminorm p we thus have

$$\begin{aligned} p\big (S(f,t_j,\xi _j)-S(f,s_k,\eta _k)\big )\le & {} \sum _l p\big (f(\widetilde{\xi }_l)-f(\widetilde{\eta }_l)\big )|\mu |\big ([u_{l-1},u_l)\big )\\\le & {} \sup _{|\xi -\eta |\le 2\delta } p\big (f(\xi )-f(\eta )\big ) |\mu |\big ([a,b)\big ). \end{aligned}$$

The assertion thus follows from uniform $\tau _X$-continuity of f on [a, b]. The p-estimates are immediate, for the norm estimate we use Assumption 4.1 (iii) and the p-estimates for $p:=|\left\langle \cdot ,\psi \right\rangle |$ where $\psi \in \Phi (\tau _X)$. $\square$