Examples of functions with values in distributions

Question

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in \mathcal{M}\cap H^{-1}$ but the time dependence is assumed to be Lipschitz. I am wondering about the more general case, for instance how you can have a delta distribution in $x$ that is integrable in $t$ behaving like $\delta(x)f(t,x)$ for some $f\in L^1$, since multiplication with a distribution is only defined for $C^\infty$ functions. Thanks!

Answer 1

To answer your first question, suppose you have a map $$f:(0,T)\to D'(\Omega),\quad t\to f(t)$$such that $\forall \phi\in D(\Omega)$ the function $$g_\phi:(0,T)\to\Bbb R,\quad g_\phi(t) = \langle f(t),\phi\rangle,$$ belongs to $L^p(0,T)$. Then we can say that $$f\in L^p((0,T);D'(\Omega)).$$

Like this we can give sense to expessions like, for example, $f(t,x)=\sin t \cdot\delta_0(x)$.