Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity

We study a class of mixed local and nonlocal $p$-Laplace equations with prototype $$-{\mathrm{\Delta }}_{p}u+(-{\mathrm{\Delta }}_p{)}^{s}u=f\text{in }\mathrm{\Omega },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is bounded and open. We provide sufficient condition on $f$ to ensure weak Harnack inequality with a tail term for sign-changing supersolutions. Two different proofs are presented, avoiding the Krylov–Safonov covering lemma and expansion of positivity: one via the John–Nirenberg lemma, the other via the Bombieri–Giusti lemma. To our knowledge, these approaches are new, even for $p=2$ with $f\equiv 0$, and include a new proof of the reverse Hölder inequality for supersolutions. Further, we establish Harnack inequality for solutions by first deriving a local boundedness result, together with a tail estimate and an initial weak Harnack inequality.