Endpoint Aubin–Lions compactness under uniform integrability

Establish an endpoint variant of the Stieltjes Aubin–Lions compactness theorem for the cases involving p_0=1 and/or p_1=1, using uniform integrability of the state and derivative families to replace reflexive weak compactness and the Hölder temporal modulus, and verify stability of the Stieltjes evolution graph under the resulting weak convergence together with validity of the final compactness argument.

Background

The main compactness theorem assumes 1<p_0,p_1<\infty and reflexivity of the outer Banach spaces. The paper explains that p_1\>1 supplies a uniform temporal modulus through Hölder’s inequality, while p_0>1 supplies weak compactness of the state component in its Bochner space. At the endpoints, boundedness in L^1 permits concentration and does not by itself provide either property.

The paper proposes uniform integrability as a possible replacement: uniform integrability of the derivative family would yield a vanishing temporal modulus and, in the reflexive setting, relative weak compactness in L_g^1(B_1); uniform integrability of the state family would similarly provide the weak compactness needed at p_0=1. However, the author explicitly states that a complete endpoint result must still establish graph stability under the modified weak convergence and complete the compactness argument.