Weighted Sobolev spaces on metric measure spaces

Abstract

We investigate weighted Sobolev spaces on metric measure spaces (X, d, m). Denoting by ρ the weight function, we compare the space W 1,p (X, d, ρm) (which always concides with the closure H 1,p (X, d, ρm) of Lipschitz functions) with the weighted Sobolev spaces W 1,p ρ (X, d, m) and H 1,p ρ (X, d, m) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W 1,p (X, d, ρm) = H 1,p ρ (X, d, m). We also adapt the results in [22] and in the recent paper [26] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality W 1,p ρ (X, d, m) = H 1,p ρ (X, d, m).

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

“…Invoking again Fatou's Lemma we can say that both ∇u and u belong to L 2 (B 1 , |y| a dz). This is enough to conclude when a ∈ (−1, 0), thanks to the W = H theorem (see [2] and references therein). Finally, when a ≤ −1 we perform the change of variable v ε = √ ρ a ε u ε (see [23]) and easily obtain weak convergence of the corresponding sequence in…”

“…It is now established that also when ε = 0 and a ∈ (−1, 1), then property (2.3) still holds (see e.g. [2,15]). When ε = 0, in the super singular case a ≤ −1, by (2.2) and the isometry T a ε we will have a useful tool to work without property (2.3).…”

Conformal Transformation on Metric Measure Spaces

“…Remark 3.4. General weighted Sobolev space is studied in [7]. Thus we can also characterize the Sobolev space under conformal transformation for unbounded v. In case w is unbounded, the problem is more complicated.…”

“…Thus we can also characterize the Sobolev space under conformal transformation for unbounded v. In case w is unbounded, the problem is more complicated. Here we introduce a possible approach, the idea comes from [7].…”

Sobolev and bounded variation functions on metric measure spaces

“…One can say that this notion is in some sense an extrinsic one, because one appeals to the standard Euclidean structure to define the weak derivative and then exploits the weight w only for the the integrability of the function and its gradient. In the recent work [18] a precise comparison is made between this point of view and the totally intrinsic point of view of Sobolev spaces in metric measure spaces. , where (M, g) is a Riemannian manifold, m = wvol, where vol denotes the Riemannian volume on M and w(x) = e −V (x) , for some potential V .…”