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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 \(\mu (B(x,r))\).

MSC:

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
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