Representation of functionals on the spaces \(L_ p^ m(E_ n)\) (Q1360836)

A representation formula \[ (l,f)=\sum _{{}\alpha{}=m }{m!\over\alpha_1!\dots\alpha_n!} \int_{E_n} u^{(\alpha )} (x)f^{(\alpha )}(x) dx \] is established for functionals \(l\) on the space \(L_p^m(E_n)\) of functions with weak derivatives of order \(m\) summable to the power \(p\). The derivatives \(u^{(\alpha )}\) of \(u\) in the above representation are expressed in terms of the fundamental solution to the polyharmonic equation \(\Delta^m=0\). The result strengthens the author's previous results [Embedding theorems and their applications to problems of mathematical physics, Collect. Sci. Works, Novosibirsk, 137-139 (1989; Zbl 0778.46020)]\ devoted to the case of compactly-supported functionals.