Support and kernel theorem for Fourier hyperfunctions (Q5949472)

The existence and the characterization of the support of a Fourier hyperfunction \(u\) is not trivial; the support can contain or consists of infinite points. By definition, if \(u\) is a Fourier hyperfunction \((u\in Q' (\mathbb{D}^n)\), \(\mathbb{D}^n= \mathbb{R}^n\cup S^{n-1})\) and if it can be identified by an element of \(Q'(K)\), where \(K\) is a compact set in \(\mathbb{D}^n\), then \(K\) is a carrier of \(u\). The minimun carrier of \(u\) defines the support of \(u\). The authors prove two theorems which give sufficient condition that a closed set \(K_0\) in \(\mathbb{D}^n\) be the support of \(u\in Q'(\mathbb{D}^n)\). With regard to the kernel theorem they also prove:NEWLINENEWLINENEWLINELet \(L_1\) and \(L_2\) be compact sets in \(\mathbb{D}^n\) and \(\mathbb{D}^m\) respectively, and \(B\) a separately continuous bilinear form on \(0(L_1)\times 0(L_2)\). Then there exists \(F\in Q'\), \(\text{supp} F \subset L_1\times L_2\) satisfying \(B(\beta_1,\beta_2)= F(\beta\otimes \beta_2)\), \(\beta_1 \in 0(L_1)\), \(\beta_2 \in 0(L_2)\).