Remove the restriction q < p + 1 in the parabolic setting

Determine whether the restriction q<p+1 can be removed for higher-integrability results for weak solutions of parabolic double-phase equations under the Hölder interpolative gap condition.

Background

The paper proves gradient higher integrability for weak solutions of parabolic double-phase equations under the assumptions that the solution is Hölder continuous, the coefficient is Hölder continuous, q≤p+α/(1−γ), and q<p+1. The restriction q<p+1 enters through the (p,q)-intrinsic parabolic Poincaré inequalities: a Hölder exponent θ must satisfy θ≥(q−1)/p, while the reverse Hölder argument requires θ<1.

The authors emphasize that this restriction is inherited from the underlying parabolic machinery rather than caused by the Hölder continuity assumption on the solution. They contrast this with the elliptic theory, where no analogous restriction appears, leaving open whether the parabolic restriction is merely technical or genuinely necessary.