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 this paper Bessel potentials on $C^{\infty }$-Riemannian manifolds (open or bordered) are studied. Let $\mathbf{M}$ be an $n$-dimensional manifold, and $\mathbf{N}$ a submanifold of $\mathbf{M}$ of dimension $k$. Sufficient conditions are given for: 1) the restriction to $\mathbf{N}$ of any potential of order $\alpha$ on $\mathbf{M}$ to be a potential of order $\alpha -\frac{n-k}{2}$ on $\mathbf{N}$ ; 2) any potential of order $\alpha -\frac{n-k}{2}$ on $\mathbf{N}$ to be extendable to a potential of order $\alpha$ on $\mathbf{M}$. It is also proved that for a bordered manifold $\mathbf{M}$ the restriction to its interior ${\mathbf{M}}^i$ is an isometric isomorphism between the...

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 outline some recent results concerning the existence of steady solutions to the Euler equation in ${ℝ}^3$ with a prescribed set of (possibly knotted and linked) thin vortex tubes.

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.

Let $U$ be a domain of type $H$ in a Brelot potential theory. A compact $K$ in $U$ is a $G_{\delta }$ in $U$ iff $U-K$ has at most countably many components. If $F$ is a relatively closed locally polar subset of $U$, any $G_{\delta }$ in $F$ is a $G_{\delta }$ in $U$. If $V$ is a domain in $U$, all Borel subsets of $\partial V\cap U$ are Baire even if $\partial V\cap U$ is not metrizable. The known results concerning equivalences between weak thinness, thinness, and strong thinness of a set $A$ at a point $x\notin A$ are extended from the case where $\left\{x\right\}$ is a $G_{\delta }$ to the cases in which $A$ meets only countably...

Nous commençons par définir la notion d’espaces ${\mathbf{L}}^1\left(\gamma \right)$ où $\gamma$ est une capacité, ce qui permet d’introduire la notion de mesure d’énergie finie par rapport à $\gamma$, et de parler d’espaces de Dirichlet basés sur $\gamma$.Soit d’autre part un espace de Dirichlet en ce sens avec potentiels s.c.i. : on étudie les espaces de Dirichlet sur les ouverts fins correspondants à l’aide d’une compactification. On retrouve plus facilement et on généralise les résultats de D. Feyel et A. de La Pradelle, (Lecture Notes).

On définit sur un espace vectoriel $E$ une classe de topologies qui rendent la multiplication continue, mais ne sont pas vectorielles en général. Sur un espace complexe $E$ elles permettent d’obtenir encore les principales propriétés des fonctions plurisousharmoniques. De telles topologies séparées sont localement pseudo-convexes (mais non localement convexes en général) : cette notion intervient dans les extensions données récemment par l’auteur du théorème de Banach-Steinhaus aux familles de polynômes...

rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p),\theta }(X,\mu )$ to $L^{q),q\theta /p}(X,\nu )$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace inequalities...