Regularity of the $p$-Poisson equation in the plane

We study the regularity of the $p$-Poisson equation $Δ_p u = h, \quad h\in L^q$ in the plane. In the case $p>2$ and $2<q<\infty$ we obtain the sharp Hölder exponent for the gradient. In the other cases we come arbitrarily close to the sharp exponent.

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