We introduce a notion of modular with a corresponding modular function space in order to build a modular capacity theory. We give two different definitions of capacity, one of them of variational type, the other one through either the modular of the test functions, or the modular of their gradients. We study, in both cases, the removability of sets of zero capacity in fairly general abstract Sobolev spaces with zero boundary values. As a key tool, we establish a modular Poincaré inequality. With the notion of modular function space in hands, we find a way to introduce a Banach function space, which allows to compare the zero capacity sets with respect to both notions. Thanks to this comparison, we characterize the compact sets of zero variational type capacity as removable sets. The paper is enriched with several examples, extending and unifying many results already known in literature in the settings of Musielak–Orlicz–Sobolev spaces, Lorentz–Sobolev spaces, variable exponent Sobolev spaces.

为了建立模块化的容量理论，我们引入了带有相应模块化功能空间的模块化概念。我们给出了两种不同的容量定义，一种是变化类型的，另一种是通过测试功能的模块或它们的梯度的模块定义的。在这两种情况下，我们研究具有零边界值的相当通用的抽象Sobolev空间中零容量集合的可移动性。作为关键工具，我们建立了模块化的庞加莱不等式。有了模块化函数空间的概念，我们找到了一种引入Banach函数空间的方法，该方法可以比较两种概念的零容量集。由于进行了此比较，我们将零变型容量的紧凑集表征为可移动集。本文包含了许多示例，