For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.

中文翻译：

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分布的高阶的点状可微性

对于分布，我们建立了高阶逐点可微性的理论，对于零阶，包括Łojasiewicz点值的概念。结果包括微分的Borel正则性，相关喷射器的高阶可整流性，Rademacher-Stepanov型可微性定理和Lusin型近似。这种发展的很大一部分对于零阶也是新的。此外，我们建立了涉及可逆性的负阶自然准则的庞加莱不等式。作为推论，我们用分布偏导数的点值来表征逐点可微性。