Non-Euclidean unification of isoperimetric profiles and grand Lebesgue-Sobolev scales
Let $(X,d,\mu)$ be a complete separable metric measure space satisfying a doubling condition and a $(1,1)$-Poincar\'e inequality. We develop a rigorous framework unifying two lines of analysis: the isoperimetric-profile approach of Coulhon-Grigor'yan-Levin \cite{CGL2003} and the grand/small Lebesgue-Sobolev scale introduced by Fiorenza-Formica-Gogatishvili \cite{FFG2018}. An explicit profile-to-scale transform $\PhiX$, defined via an inverse integral of $\IX$, converts geometric data into grand Lebesgue parameters. Sharp, up to universal constants, embeddings $W^{1,1}(X) \hookrightarrow \mathcal{G}_X$ with explicit constants (Theorem \ref{thmmain}). A converse: controlled grand embeddings imply explicit lower bounds on $\IX$ (Theorem \ref{thmconverse}). Concrete examples in genuinely non-Euclidean settings: the Heisenberg group $\mathbb{H}^1$, a model manifold with logarithmic volume growth, and Gaussian measure on $\R^n$ treated as a locally doubling space. All arguments are carried out on general metric measure spaces without reference to charts or a smooth structure; the gradient is the upper gradient in the sense of Heinonen-Koskela, and perimeter is the outer Minkowski content.