Subspace of Tempered Distributions Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all $\phi$ compactly supported smooth functions.
For $\phi \in C_{c}^{\infty}(\mathbb{R}^{n})$, we define $\phi(\lambda D)u$ by the Fourier transform $F[\phi(\lambda D)u] = \phi(\lambda t)F[u](t)$.
What do distributions in this subspace behave like? Also, how can I show that any function in $L^{p} + L^{q}$ belongs to this subspace for $p$ and $q$ finite? What happens in the case of $L^{\infty}$?
 A: $$\phi(\lambda D)u = \tfrac1{\lambda^n} \check \phi\left(\tfrac\cdot\lambda\right) * u := x \mapsto \int \tfrac1{\lambda^n} \check \phi\left(\tfrac y\lambda\right) u(x-y) \, dy .$$
Since $\phi$ is Schwartz, so is $\check\phi$, in particular $\check \phi \in L^p$ for all $1 \le p \le \infty$, and
$$\left\|\tfrac1{\lambda^n} \check \phi\left(\tfrac\cdot\lambda\right)\right\|_p = \lambda^{-n+n/p}\left\|\check \phi\right\|_p .$$
Hence by Young's convolution inequality
$$ \|\phi(\lambda D)u\|_\infty \le \lambda^{-n+n/p'}\|\check\phi\|_{p'} \|u\|_p ,$$ 
where $p'$ is the conjugate index to $p$.  (Note $n-n/p' = n/p$.)  And this converges to zero if $p < \infty$.
And of course, if $L^p$ and $L^q$ are subsets, then so is $L^p + L^q$.
Also $\phi(\lambda D)1 = \phi(0)$, where $1$ denotes the constant function.  So taking any $\phi$ such that $\phi(0) \ne 0$, we see that $L^\infty \not \subset {S_{h}}'(\mathbb{R}^{n})$.
Roughly speaking, being in ${S_{h}}'(\mathbb{R}^{n})$ is saying something like the distribution tends to $0$ in the average at infinity.
