The Cauchy-Riemann equations in the unit ball of \(l^2\)
The author investigates the solvability of the equation \({\bar \partial }F=\omega \) on \(B\), the closed unit ball of \(\ell^2\), where \(\omega \) is a closed \((0,1)\)-form of class \(C^\infty \) and of the type \(\omega (z)= \sum_i z_i\omega^i(z)d{\bar z_i}\) for \(z=(z_i) \in \ell^2\) with the following assumptions.\newline (i) Let \(I_1,I_2,\ldots\) be a partition of \({\mathbb N}\) such that all \(I_k\) are finite and take \(z^k = (z_i)_{i\in I_k}\). Assume that \(\omega ^i\) is smooth on \(B\) and \(\omega ^i(z)=\sum_k \omega_k^i(z^k)\) such that the series is absolutely convergent, where \(\omega_k^i\) is defined on the closed unit ball of \({\mathbb C}^{|I_k|}\).\newline (ii) \(\omega_k^i=\sum_{\alpha \in {\mathbb N}^{|I_k|}}(z^k)^\alpha \omega^i_{k,\alpha}(|z^k|^2)\).\newline Under these conditions it is shown that, if there is a positive integer \(q\) such that the derivatives \(D^h\omega^i\) are uniformly bounded in \(i\) on \(B\) for \(h \leq q\), and \(\omega^i_{k,\alpha}=0\) for \(|\alpha |>q\) and all \(i\in I_k\) \((|I_k|>1)\), then \({\bar \partial }F=\omega \) has a solution on a smaller ball \(rB\) and \[ |F(z)|\leq C\|z\|^2 \sup\big\{\|D^h\omega^i\|_\infty : i, h\leq q\big\},\quad\|z\|<r, \] where \(C\) is a constant.
\({\bar \partial }\)-exactness on Hilbert spaces of infinite dimension
scientific article; zbMATH DE number 6351274