Suppose $f$ and $g$ are bounded functions, having whatever niceness properties you want, on some space of finite measure. Assume they are normalized so that $\int f^2=\intg^2=1$. I am looking for sufficient conditions so that $$\intf+g\geq c(\intf+\intg)$$ for some positive $c$ bounded away from zero. The sort of thing I have in mind is, say, control over $\int f^pg^p$ for $p>1$, which I interpret as meaning $f$ and $g$ cannot be too correlated. Are there instances where this sort of thing can be done? If not, what do counterexamples look like?

This is asking when the signs of f and g are the same a lot  ie $\int_A f+g = \int_A f+ g$ where $A$ is the set on which $f$ and $g$ have the same sign. So you're asking for when $\int_A f$ and $\int_A g$ are large compared to $\int f$ and $\int g$ – James Kilbane Jan 10 at 15:04

Well, I suppose I am thinking of $f$ and $g$ as being complex valued. I am thinking that for them to interact destructively, they have to have similar arguments and order of magnitudes. The point of the $L^p$ estimate is to try to prohibit the order of magnitudes from lining up too much. – anon Jan 10 at 15:08

I would think that $f = g$ is a counter example unless you include positivity in you niceness conditions. – Jaap Eldering Aug 9 at 7:13
You definitely need certain "niceness" assumptions. For instance, you can look at $f$ and $g$ on $[0,1]$ with $$f(x)=\begin{cases} N & \text{ if }0\leq x\leq \frac{1}{2N^2}\\ \frac{1}{\sqrt{2}} &\text{ otherwise}\end{cases}$$
and $$g(x)=\begin{cases} N & \text{ if }1\frac{1}{2N^2}\leq x\leq 1\\ \frac{1}{\sqrt{2}} &\text{ otherwise}.\end{cases}$$
Both of these functions have $L^2$ norm about 1, $L^1$ norm about $1/\sqrt 2$, and $\intf+g$ is about $1/N$. Because their spikes are disjointly supported, $$\int f(x)^pg(x)^p\leq \frac{1}{2^p}+N^{p2}$$ which is pretty small. By comparison, if $f$ and $g$ had the same spikes, then this would be about$N^{2p2}$ in magnitude.
Let $(\Omega,\mu)$ be a measure space and let $f,g:\Omega\mapsto\mathbb{R}$ be two bounded real valued functions defined on the space $\Omega$ with $\mu(\Omega)<+\infty$. Since $f$ and $g$ are bounded then there exists $M,N\in\mathbb{R}$ such that $f\leqslant M$ and $g\leqslant N$ on $\Omega$. Therefore $$\int_{\Omega}f+g\,d\mu\leqslant \int_{\Omega}(M+N)\,d\mu=(M+N)\mu(\Omega)<+\infty$$ On the other hand from triangle inequality we have $$\int_{\Omega}f+g\,d\mu\leqslant\int_{\Omega}f\,d\mu+\int_{\Omega}g\,d\mu$$ Therefore a sufficient condition would be:
There exists some positive constant $c\leqslant 1$ such that $f+g\geqslant c(M+N)$.