Extension Theorem For Functions Research Articles

ON EXTENSION THEOREMS FOR ANISOTROPIC BESOV–MORREY TYPE SPACES

We will establish some extension theorems for functions belonging to anisotropic Besov–Morrey type spaces and defined on bounded domains in ℝn which satisfying certain geometric properties.

Hartogs Extension Theorem for Functions with Values in Complex Clifford Algebras

A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic function, i.e. a null solution of the firstorder, vector-valued Dirac operator in \({\mathbb{R}}^m\).

An Extension Theorem for Functions on Semigroups

For $S$ a semitopological semigroup, a continuous function on $S$ is said to be in $LMC(S)$ if its set of right translates is relatively compact in $C(S)$ for the topology of pointwise convergence on $S$. It is proved here that, if $S$ is a dense subsemigroup of a topological group $G$, then every function in $LMC(S)$ extends to a function continuous on $G$. This result generalizes earlier results that were arrived at independently by A. T. Lau and the present author. Some corollaries of this result are also presented.

An extension theorem for functions on semigroups

For S S a semitopological semigroup, a continuous function on S S is said to be in L M C ( S ) LMC(S) if its set of right translates is relatively compact in C ( S ) C(S) for the topology of pointwise convergence on S S . It is proved here that, if S S is a dense subsemigroup of a topological group G G , then every function in L M C ( S ) LMC(S) extends to a function continuous on G G . This result generalizes earlier results that were arrived at independently by A. T. Lau and the present author. Some corollaries of this result are also presented.

Extending minimal varieties

Introduction The purpose of this paper is to study the general problem of extending minimal varieties across closed subsets in riemannian manifolds. The analogous theory in the complex analytic case falls into two categories: extension theorems for functions (Hartog's phenomenon) and extension theorems for varieties (the RemmertStein-Shiffman Theorem). The discussion here splits along similar lines. In w 1 we analyze the Bers, deGiorgi-Stampacchia results on removing singularities of solutions to the non-parametric minimal surface equation in codimension one. This theorem does not carry over directly to higher codimension. However, certain parts of their argument can be generalized and lead to a strong reflection principle for minimal submanifolds and a uniqueness result for minimal cones. Also a theorem for surfaces in general codimension is proved. The remainder of the paper is devoted to proving extension theorems of the following type for varieties. Let M be a riemannian manifold and A ~ M a compact subset of sufficient smoothness and appropriate Hausdorff dimension. Then any k-dimensional minimal variety which is stationary (or area minimizing) in M A extends to a minimal variety which is stationary (or, resp., area minimizing) in M. The proof falls into two parts. The first is to show that the mass of the variety is finite across A so that an extension exists (w 3). The second is to prove the properties of minimality for the extension. The main conclusions are stated precisely in w 5.

Sobolev inequalities and extension theorems for functions with certain $L^{p}$ derivatives

Sobolev inequalities and extension theorems for functions with certain $L^{p}$ derivatives