Duality Conditions for Morrey Measures via Parabolic Capacity

Abstract

We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$, we prove that \[ |\mu|\bigl(Q_{r,r^p}(z)\cap(\Omega\times (0,T))\bigr) \leq Mr^{n+p-\vartheta}, \qquad \vartheta<p, \] implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to $r^q$, $q>1$, the sufficient threshold is $\vartheta<\min\{p,q\}$. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.