The dual of a Bergman space on simply connected domains.
Let \(\Omega\) be a domain (open connected subset) in the complex plane. For \(1\leq p < \infty\), the Bergman \(p\)-space, denoted \(A^p(\Omega)\), consists of all holomorphic functions on \(\Omega\) such that \[ \| f\| _{A^p(\Omega)}= (\int_\Omega | f(z)| ^p\, dA(z))^{1/p}<\infty, \] where \(dA(z)\) is normalized area measure in the plane. The question studied in this interesting paper is whether one can represent the dual space of \(A^p(\Omega)\) as \(A^{p'}(\Omega)\) with respect to the dual action induced by \(A^2(\Omega)\): \[ \langle f,g\rangle= \int_{\Omega}f(z) \overline g(z) \,dA(z), \quad f\in A^p(\Omega),\;g\in A^{p'}(\Omega), \] where \(p'\) is the dual exponent to \(p\) defined by \(1/p + 1/p'=1.\) Unless \(p=2\), some kind of geometric requirement on \(\Omega\) is needed to have duality. The above question can be reformulated in the following form: For which \(\Omega\) and \(p\) does the Bergman projection \(\Pi: L^2(\Omega)\to A^2(\Omega)\) extend continuously to an operator \(L^p(\Omega) \to A^p(\Omega)\)? The author obtains the following main result: Theorem. Let \(\Omega\) be a simply connected domain in the complex plane other than the plane itself. There exists a universal constant \(p_0\), \(4/3 \leq p_0 < 2\), independent of \(\Omega\), such that the Bergman projection \(\Pi: L^2(\Omega)\to A^2(\Omega)\) extends to a bounded projection \(L^p(\Omega) \to A^p(\Omega)\) for all \(p\) in the interval \(p_0 < p < p'_0\). Moreover, the boundedness of the Bergman projection fails in general outside the closed interval \([p_0,p'_0]\). From this theorem follows that \(A^p(\Omega)^*= A^{p'}(\Omega)\) holds for all \(p\) with \(p_0 < p < p_0'\) and all simply connected domains. Moreover, this identification of the dual space fails generally outside the interval \([p_0, p_0']\). The exact value of the constant \(p_0\) is not known, but the estimate \(p_0 \leq p_1 \approx 1,413\) is obtained. Further, it is proved that the statement \(p_0 = 4/3\) is equivalent to a famous conjecture, due to James Brennan, that \(1/\varphi'\) belongs to \(A^p(\Omega)\) for all \(p\) with \(0 < p < 2\) and for all univalent functions \(\varphi\) on the disk. Finally, for bounded domains \(\Omega\), it is shown that the above main result can be formulated in terms of the solvability of the Dirichlet problem.
scientific article; zbMATH DE number 1998323
scientific article