Representations of the spaces \(C^\infty(\mathbb{R}^N)\cap H^{k,p}(\mathbb{R}^N)\) (Q2703081)

Let \(1\leq p\leq\infty\), \(k\in\mathbb{N}\). The space \(C^\infty(\mathbb{R}^N)\cap H^{k,p}(\mathbb{R}^N)\) with its natural intersection topology and ordinary Fréchet structure, is considered. Let \(I\) be the interval \([0,1]\) and \(Q_N:= I^N\). The first result obtained is the isomorphism of \(C^\infty(\mathbb{R}^N)\cap H^{k,p}(\mathbb{R}^N)\) with \((C^\infty(Q_N))^N\cap \ell^p(H^{k,p}(Q_N))\) and, as a consequence, it is shown that, for \(p<\infty\), \((C^\infty(\mathbb{R}^N)\cap K^{k,p}(\mathbb{R}^N)\) has a basis. For certain open subsets \(\Omega\) of \(\mathbb{R}^N\) the following isomorphism are given: \(C^\infty(\Omega)\cap H^{k,p}(\Omega)\simeq [C^\infty(I)]^N\cap \ell^p[L^p(I)]\), \(C^\infty(\Omega)\cap H^{k,p}(\Omega)\simeq C^\infty(\mathbb{R})\cap L^p(\mathbb{R})\), hence it follows that, for such open sets \(\Omega\), the isomorphy class depends only on \(p\).NEWLINENEWLINENEWLINELet \(P_N:= \{f\in C^\infty(\mathbb{R}^N):f\) is \(2\pi\)-periodic with respect to each variable\} with its usual Fréchet space structure. It is assumed that \(P_{N,k,p}:= [P_N]^{\mathbb{N}}\cap \ell^p[H^{k,p}(S_N)]\), where \(S_N:= \{x\in \mathbb{R}^n:- \pi\leq x_i\leq\pi\), \(i= 1,2,\dots,N\}\), is endowed with its usual Fréchet space structure. It is then shown that \(P_{N,k,p}\) is isomorphic to \(S^N\cap \ell^2(\ell^2)\), thus showing that the isomorphy class does not depend of the dimension \(N\).NEWLINENEWLINENEWLINEIn a final section the authors study some topological properties and some Montel subspaces of the spaces \(C^\infty(\mathbb{R}^N)\cap H^{k,p}(\mathbb{R}^N)\). We recall the following: Let \(k\in\mathbb{N}\) and \(1< p< \infty\). Then the spaces \(C^\infty(\mathbb{R}^N)\cap H^{k,p}(\mathbb{R}^N)\) are totally reflexive, hence distinguished.