= Weak Harnack inequality
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A <Weak Harnack inequality> controls a positive integral mean of a nonnegative <weak supersolution> by its <essential infimum>. For example, let $n\ge3$, let $A$ be a bounded measurable uniformly elliptic coefficient matrix, and let $u\in H^1_{\rm loc}$ be nonnegative with $\operatorname{div}(A\nabla u)\le0$ in the weak sense. If $B_{4R}\Subset\Omega$, then for $0<q<n/(n-2)$,
$$
\left(\frac1{|B_{2R}|}\int_{B_{2R}}u^q\,dx\right)^{1/q}\le C\operatorname*{ess\,inf}_{B_R}u.
$$
The constant depends on dimension, ellipticity/coefficient bounds and the chosen exponent. The normalization makes the estimate invariant under spatial rescaling. Its infimum is essential because a <function> in a <Sobolev space> is initially an almost-everywhere equivalence class; changing a single representative value cannot invalidate the estimate.
This is a one-sided statement for supersolutions, distinct from the sup-to-inf <Harnack inequality for uniformly elliptic divergence-form equations> for solutions. Combining it with the subsolution bound from <Moser iteration> yields that latter estimate. If the essential infimum is zero, the left side forces vanishing almost everywhere on the larger ball, giving the mechanism of <propagation of zeros by the weak Harnack inequality>. Lower-order terms or forcing require the corresponding modified assumptions and error terms.
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