We will study metric measure spaces \((X,\mathsf{d},{\mathfrak {m}})\) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds \(\mathsf{BE}_1(\kappa ,\infty )\)

• for which we prove the equivalence with sharp gradient estimates,

• the class of which will be preserved under time changes with arbitrary \(\psi \in \mathrm {Lip}_b(X)\), and

• which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets \(Y\subset X\).

In the latter case, the distribution-valued Ricci bound will be given by the signed measure \(\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}\) where k denotes a variable synthetic lower bound for the Ricci curvature of X and \(\ell \) denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

我们将研究具有合成的较低Ricci边界的空间范围之外的度量度量空间\（（X，\ mathsf {d}，{\ mathfrak {m}}）\）。特别是，我们引入了分布值下Ricci界\（\ mathsf {BE} _1（\ kappa，\ infty）\）

• 为此，我们用陡峭的梯度估计证明了它的等效性，

• 随时间变化的类将被任意\（\ psi \ in \ mathrm {Lip} _b（X）\）保留，并且

• 它们对于任意半凸子集\（Y \ subset X \）上的Neumann Laplacian都满足。

在后一种情况下，分布值的Ricci界将由有符号的测度\（\ kappa = k \，{\ mathfrak {m}} _ Y + \ ell \，\ sigma _ {\ partial Y} \）给出k表示X的Ricci曲率的可变合成下限，而\（\ ell \）表示Y的“边界曲率”的下限，以纯度量术语定义。我们还提出了一个新的本地化参数，该参数允许我们将RCD属性传递给RCD空间的任意开放子集。并且，我们引入了有关边界曲率，第二基本形式以及RCD空间子集的边界度量的新综合概念。