We study a capacity theory based on a definition of Hajłasz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are γ-medians, for which we also prove a new version of a Poincaré type inequality.

Dans cette partie de la théorie des potentiels besseliens on considère les restrictions de potentiels de la classe $P^a(R^n)$ aux domaines ouverts $D\subset R^n$. On cherche à caractériser de manière intrinsèque la classe $P^a\left(D\right)$ ainsi obtenue.On attaque ce problème en définissant de manière directe (§ 2) une classe ${\check{P}}^a\left(D\right)\subset P^a\left(D\right)$ qui, pour des domaines assez réguliers, est égale à $P^a\left(D\right)$.L’égalité $P^a\left(D\right)=P^a\left(D\right)$ est équivalente à l’existence d’un opérateur-extension $E:{\check{P}}^a\left(D\right)\to P^a(R^n)$, linéaire et continu, tel que $Eu$ soit une extension de $u$. Si un tel opérateur $E$ transforme...

In the previous parts of the series on Bessel potentials the present part was announced as dealing with manifolds with singularities. The last notion is best defined in the more general framework of subcartesian spaces. In a subcartesian space $X$ we define the local potentials of reduced order$\alpha :u\in P_{\mathrm{loc}}^{⟨\alpha ⟩}\left(X\right)$, if for any chart $(U,\phi ,{\mathbf{R}}^n)$ of the structure of $X,u\circ {\gamma }^{-1}$ can be extended from $\phi \left(U\right)$ to the whole of ${\mathbf{R}}^n$ as potential in $P_{\mathrm{loc}}^{\alpha +(n/2)}({\mathbf{R}}^n)$. This definition is not intrinsic. We obtain an intrinsic characterization of $P_{\mathrm{loc}}^{⟨\alpha ⟩}\left(X\right)$ when $X$ is with singularities...

Let $L_{\alpha }^q(R^D),\alpha \>0,1\<q\<\infty$, denote the space of Bessel potentials $f=G_{\alpha }*g$, $g\in L^q$, with norm $\parallel f{\parallel }_{\alpha ,q}=\parallel g{\parallel }_q$. For $\alpha$ integer $L_{\alpha }^q$ can be identified with the Sobolev space $H^{\alpha ,q}$.One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space $H^{1;2}$, and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the positive cone in the space...

Necessary and sufficient conditions governing two-weight $L^p$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.