  Full-history recursive multilevel Picard (MLP) approximation schemes have
been shown to overcome the curse of dimensionality in the numerical
approximation of high-dimensional semilinear partial differential equations
(PDEs) with general time horizons and Lipschitz continuous nonlinearities.
However, each of the error analyses for MLP approximation schemes in the
existing literature studies the $L^2$-root-mean-square distance between the
exact solution of the PDE under consideration and the considered MLP
approximation and none of the error analyses in the existing literature
provides an upper bound for the more general $L^p$-distance between the exact
solution of the PDE under consideration and the considered MLP approximation.
It is the key contribution of this article to extend the $L^2$-error analysis
for MLP approximation schemes in the literature to a more general $L^p$-error
analysis with $p\in (0,\infty)$. In particular, the main result of this article
proves that the proposed MLP approximation scheme indeed overcomes the curse of
dimensionality in the numerical approximation of high-dimensional semilinear
PDEs with the approximation error measured in the $L^p$-sense with $p \in
(0,\infty)$.
