On the structure of Bergman and poly-Bergman spaces
Let \(\Pi\) be the upper complex halfplane, \(\mathbb{R}\) the real axis, \(\mathbb{R}_+\) the positive real axis, and \(H^2_+(\mathbb{R})\) the Hardy space of those members of \(L_2(\mathbb{R})\) with analytic extensions of \(\Pi\). It is well known that the Fourier transform \(F\) is an isometric isomorphism of \(L_2(\mathbb{R})\) such that \(F(H^2_+(\mathbb{R}))= L_2(\mathbb{R}_+)\). The author's aim is to generalize this result to the space \(A^2_n(\Pi)\) of those \(\varphi= \varphi(u, v)\) in \(L_2(\Pi)\) such that \((\partial/\partial u+i\partial/\partial v)^n\varphi= 0\) for all \(u+iv\) in \(\Pi\). We call such a \(\varphi\) \(n\)-analytic and note that when \(n=1\), \(\varphi\) satisfies the Cauchy-Riemann equations, so \(\varphi\) is analytic in \(\Pi\). We say that \(\varphi\) is \(n\)-anti-analytic in case \((\partial/\partial u-i\partial/\partial v)^n\varphi= 0\) and write \(\overline A^2_n(\Pi)\) for the space of all such \(\varphi\). The author constructs isometric isomorphisms \(U\) of \(L_2(\Pi)\) such that \(U(A^2_n(\Pi))\), \(U(\overline A^2_n(\Pi))\) are direct products of \(L_2(\mathbb{R}_+)\), \(L_2(\mathbb{R}_-)\) with another factor. Other results extend the Szegö projection from \(L_2(\mathbb{R})\) onto \(H^2_+(\mathbb{R})\) to the situation of the present paper.
scientific article; zbMATH DE number 1310386
scientific article