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A nonnegative weak solution of a uniformly elliptic divergence-form equation satisfieswhenever a fixed larger concentric ball lies in the domain. Combine the subsolution bound from Moser iteration with the Weak Harnack inequality for supersolutions.

For a weak solution of a uniformly elliptic divergence-form equation, local boundedness and the Weak Harnack inequality implyfor a universal $0<\theta <1$. Iteration gives local Hölder continuity.

The Weak Harnack inequality states that there are $p>0$ and $C=C(n,\lambda ,\Lambda )$ such that every nonnegative weak supersolution satisfiesThe radii may be replaced by any fixed nested pair of balls, with the constant adjusted accordingly.

On a ball $B_{2r}(x_0)⋐B$, bothare nonnegative solutions and hence supersolutions. Combining the Weak Harnack inequality with the local boundedness estimate for subsolutions gives an oscillation decay estimatewhere $\theta$ depends only on $n,\lambda ,\Lambda$. Iteration yields Hölder continuity with some exponent $\mu \in (0,1)$. The local $L^2$-to-$L^{\infty }$ estimate controls the initial oscillation and gives

Use the Weak Harnack inequality: for some $q>1$ and every nonnegative weak supersolution,where $C$ and $q$ depend only on $n,\lambda ,\Lambda$. A weak solution is both a subsolution and a supersolution. Applying part (i), after rescaling from $B_1$ to $B_{1/2}$, and then the weak Harnack inequality givesThis is the Harnack inequality for uniformly elliptic divergence-form equations.