Abstract This paper is devoted to the study of weak Harnack inequalities for minimizers of nonlocal double phase functionals, whose prototype is given by ₑⁿ {Rⁿ} (|u (x) -u (y) |ᵖ|x-y|^{n+sp}+a (x, y) |u (x) -u (y) |q|x-y|^{n+tq}) \, dx\, dy, ∬ R n × R n | u (x) - u (y) | p | x - y | n + s p + a (x, y) | u (x) - u (y) | q | x - y | n + t q d x d y, with a 0 a ≥ 0 and 0 0 s, t 1 p ≤ q ∞. The core of our approach is based on expansion of positivity and several measure theoretic estimates stemming from a nonlocal Caccioppoli-type inequality. The main challenge lies in controlling the subtle interaction between the pointwise behaviour of the modulating coefficient a (, ) a (·, ·) and the structural exponents. In addition, we discuss a quantitative boundedness result for minimizers of such functionals.
Fang et al. (Tue,) studied this question.