The locally uniform space on $\mathbb{R}^N$ collects all functions $f\in L^p_{\textrm{loc}}(\mathbb{R}^N)$ with the condition $$\|f\|_{L_U^p(\mathbb{R}^N)}:=\sup_{x\in\mathbb{R}^N}\|f\|_{L^p(B(x,1))}<\infty.$$ I would like to find its dual space. But it seems improbable to write a linear functional $\ell\in L_U^p(\mathbb{R}^N)^*$ as an integral. Clearly $\ell(f)=\sup_{x\in\mathbb{R}^N}\int_{B(x,1)} f(y)g(y)\mathrm{d}y$ is not linear.
However, as in $L^p$ spaces, I have shown that $$\|f\|_{L_U^p(\mathbb{R}^N)}=\sup_{x\in\mathbb{R}^N,\|g\|_{L_U^q(\mathbb{R}^N)}\le 1}\left|\int_{B(x,1)} f(y)g(y)\mathrm{d}y\right|,$$where $p,q$ are dual exponents. So the expression above may be related to $\ell$.
Just an idea (written for $N=2$): Decompose the plane $\mathbb R^2$ into squares $Q_{a,b}=[a,a+1)\times [b,b+1)$ with $a,b\in\mathbb Z$, and consider the map $$\Phi:L^p_U(\mathbb R^N)\to \ell^\infty(L^p(Q_{a,b})_{a,b\in\mathbb Z}),\; f\mapsto (f|_{Q_{a,b}})_{a,b\in\mathbb Z})$$ where, for Banach spaces $X_i$, $\ell^\infty((X_i)_{i\in I})=\{ (x_i)_{i\in I}\in \prod_i X_i: \sup\{\|x_i\|_{X_i}:i\in I\}<\infty\}$.
I believe that $\Phi$ is an isomorphism, and on the range you can write down many functionals (with similar definitions, $\ell^1((X_i')_{i\in I})$ embeds into the dual of $\ell^\infty((X_i)_{i\in I})$). However, the dual of $\ell^\infty$ is already an intricate object so that it won't be easy to describe the dual of $\ell^\infty((X_i)_{i\in I})$ in a nice way.