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...

In this paper, we study the reduit, the thinness and the non-tangential limit associated to a harmonic structure given by coupled partial differential equations. In particular, we obtain such results for biharmonic equation (i.e. ${▵}^2\varphi =0$) and equations of ${▵}^2\varphi =\varphi$ type.

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.