Weak-Harnack exponent range and its dimension-dependent upper endpoint

The weak Harnack inequality of Weak Harnack inequality for nonnegative supersolutions is asserted only for the sourced range $0<p<n\mathrm{/}(n-2)$ when $n\ge 3$ (and for every finite $p$ when $n=2$). The endpoint $n\mathrm{/}(n-2)$ depends only on dimension; the seed exponent $p_0$ and constants depend on the coefficients. Starting from $p_0$, the higher exponents below that endpoint are obtained by the positive-integrability Sobolev transitions in the weak-Harnack proof, while Hölder interpolation supplies smaller exponents. No claim is made here that every positive exponent is admissible; in particular one must not restate the weak Harnack inequality with an arbitrary $p>0$, and the constant for the source $F\in L^q$ depends on $n,q,\theta ,M_a,p$ as recorded.

Krummel, DeGiorgi-Nash lecture notes, Theorem 2 (printed p. 1) states the weak Harnack inequality for $0<p<n\mathrm{/}(n-2)$ when $n\ge 3$, and its proof obtains higher exponents from the fixed seed by Sobolev transitions, with Hölder interpolation giving smaller exponents. Simon, Lectures on Partial Differential Equations, Lecture 17, Theorem 2 (printed pp. 199-210) states the same range. The remark records the exact range of the theorem of Weak Harnack inequality for nonnegative supersolutions and carries no proof obligation of its own.