The sharp Riesz potential estimates in metric spaces. (English) Zbl 1038.46027

The classical Sobolev inequality for \(C^{\infty}_0(\Omega)\)-functions \(u\) has a slightly sharper version
\[ \int_0^{\infty} \alpha^{p-1} \operatorname{mes} (| u| > \alpha)^{1-p/n}\,d\alpha \leq C\tag \(*\) \] with \(\| \nabla u\|_p \leq 1\) and \(1 < p < n\) in terms of the distribution function of \(| u| \). There is a corresponding exponential result, due to N. S. Trudinger [J. Math. Mech. 17, 473–483 (1967; Zbl 0163.36402)], in the boarderline case \(p = n\). The authors extend these results to metric measure spaces \((X,d,\mu)\) where \(\mu\) is a doubling measure and satisfies a lower estimate \(\mu (B(x,r)) \geq cr^n\); it is enough to assume that \(d\) is a quasimetric, see J. Heinonen [Lectures on Analysis on Metric Spaces, Springer-Verlag (2001; Zbl 0985.46008)]. Inequality \((*)\) is replaced by \(\int_0^{\infty} \alpha^{p-1}\mu (G_{\alpha})^{1-p/n}d\alpha \leq C\) where \(\| g\|_p \leq 1\) , \(G_{\alpha}=\{x \in X: (Ig)(x) > \alpha\}\) and \(Ig\) refers to a Riesz potential of the form \(\int_0^r \int_{B(x,t)} g\,dy\,dt\), \(r = \text{diam}(X)/2\). The proof makes use of the Hardy-Littlewood maximal function; the main difficulty is that there is no upper bound for \(\mu (B(x,r))\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
47G10 Integral operators
31C15 Potentials and capacities on other spaces
Full Text: DOI Link