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Is there some analogon of the Riesz Representation theorem for nonlinear functionals on $L^p$ spaces?

I would expect something like: A smooth (e.g. in the Fréchet sense) nonlinear functional $\Psi: L^p \rightarrow \mathbb{R}$ can be written as $\Psi[f] = \sum_i \int dx_1 ... dx_i ~ A_i(x_1, ... x_i)~f(x_1)...f(x_i)$ for $A_i(x_1, ...,x_i)$ in some $L^{q_i}$ with $q_i$ being related to $p_i$ by some Hoelder stuff.

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