Which space is $\mathcal{D}^\prime_t(I; \mathscr{S}(\mathbb R^n))$?

Question

In the paper "Remarks on the Solvability of Linear Equations of Evolution" (L. Nirenberg, F. Trèves, Symposia Math. 7, 1970 or 1971, 325–338) the authors make the following (paraphrased) comment (pg. 22) about a notion of solvability of an equation:

$Lu=f$ is solvable in the interval $I$ if, to every $f\in C^\infty_0(I\times \mathbb R^n)$ there is $u\in\mathscr{D}^\prime_t(I; \mathscr{S}(\mathbb R^n))$ such that $Lu=f.$

Above $L$ is a given pseudo-differential operator.

Which space is $\mathcal{D}^\prime_t(I; \mathscr{S}(\mathbb R^n))$? The authors don't define it.

Thanks

P.s.: References are welcome.

Answer 1

This is a fairly typical notation for function space-valued things. The idea is that for each fixed time $t$, we have a function of a particular type with respect to $x$. So here the notation $\mathcal{D}^\prime_t(I;\mathcal{S}(\mathbb{R}^n))$ would mean distributions of the kind $u(t,x)$ such $u$ is a distribution with respect to $t$, but for each fixed $t$, $u$ is an element of the function space $\mathcal{S}(\mathbb{R}^n)$ with respect to $x$. This allows for (seemingly) strange constructs such as $u(t,x) = \delta(t)\phi(x)$ where $\phi$ is some Schwartz function, which would represent an "impulse" or initial condition. For each $t$, $u$ is a Schwartz function, but with respect to $t$ it's a distribution.

In general, the notation $X_t(I,Y)$ should be taken to mean "objects of type $X$ with respect to $t$, which for fixed $t$ are objects of type $Y$". See for instance Evans PDE, section 5.9.2, where he discusses exactly this type of construct.

Answer 2

Definitions for $\mathcal{D}$ (test functions) and $\mathcal{D}'$ (distributions) can be found in Laurent Schwartz's book, Théorie des distributions, chapter 3 (Espaces topologiques de distributions, structure des distributions), p. 63, Hermann, 1966. I do not possess a version in English. They are mentioned as well in F. Trèves, Basic Linear Partial Differential Equations, Academic Press, 1975 and Introduction to Pseudodifferential and Fourier Integral Operators: Pseudodifferential Operators, Springer, 1980, with few details.