Consider the following operator $\mathcal{L}: L^\infty(\mathbb{R})\rightarrow L^\infty(\mathbb{R})$:
$$
\mathcal{L}v\equiv \int_{-\infty}^{\infty} K(x,y)v(y)dy.
$$
Is this a compact operator if $K\in L^1(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$? My particular $K$ is bounded and decays exponentially fast for large $x$ and $y$.
If not, what kind of $K$ would make it compact?
If the answer is yes, does that mean it is compact on $L^\infty(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$, which is really the space I am interested in right now.
Frankly, on the one hand, I am not sure how to tackle that. On the other hand, this must be a well-known result, which I cannot find. I found this question, which almost answers:
Integral operator on $L^p$ is compact
Best Answer
In $L^1$ and $L^\infty$, information about the magnitude of the kernel (the exponential decay stated in the question) isn't enough to establish compactness: some kind of uniformity is also needed. For the operator described:
Assume that there is a constant $M$ such that for almost all $x\in\mathbb{R}$, $K(x,\cdot)\in L^1(\mathbb{R})$ and $\|K(x,\cdot)\|_1\leq M$ (that's a sufficient condition for boundeness of the operator on $L^\infty(\mathbb{R})$). Then the operator is compact iff for any $\varepsilon>0$ there exist $\delta>0$ and $R>0$ such that for a.a. $x\in\mathbb{R}$ and all $h\in(-\delta,\delta)$ we have $$\int_{\mathbb{R}\setminus(-R,R)}|K(x,y)|\mathrm{d}y<\varepsilon\tag{1}$$ and $$\int_{\mathbb{R}}|K(x,y+h)-K(x,y)|\mathrm{d}y<\varepsilon\tag{2}$$ These two conditions on cross-sections through the kernel require (1) uniform decay at $\pm\infty$ and (2) uniform continuity of translation.
See my paper in Proc. Amer. Math. Soc. 123 (1995), 3709-3716 for details.
On $L^\infty\cap L^2$, presumably the intended norm is the sum, maximum or similar of the $L^2$ and $L^\infty$ norms? If so, a straightforward subsequence argument shows that any kernel that gives rise to a compact operator on $L^2$ and a compact operator on $L^\infty$ will give rise to a compact operator on the intersection. The Hilbert-Schmidt condition should be enough for $L^2$ compactness in this instance.