Extension from linear subvarieties for the Bergman scale of spaces on convex domains (Q2875404)

Assume that \(D\) is a bounded convex domain of finite type defined by a function \(r\) which is smooth on some neighborhood of \(\bar{D}\) and \(dr \neq 0\) on \(bD\). The Bergman space \(H^p(D)\) is the space of all holomorphic functions \(F\) in \(D\) such that \(\int_D |F|^p dV < \infty\). The symbol \(dV\) stands for the volume measure in \(\mathbb{C}^n\). Let \(e \in \mathbb{C}^n\) be a unit vector and \(\omega \in D\). Let NEWLINE\[NEWLINE A=A(\omega,e):= \big\{ z \in \mathbb{C}^n : h_{\omega,e} := \langle z-\omega,e\rangle=0 \big\}.NEWLINE\]NEWLINE The operator \(R_{D \cap A}\) of restriction to the subvariety \(A\) is defined for all holomorphic functions in \(D\) in the following way: NEWLINE\[NEWLINER_{D \cap A} : H(D) \ni f \mapsto f|D \cap A \in H(D \cap A).NEWLINE\]NEWLINE In the paper, the authors examine for which functions \(f\) holomorphic in \(D \cap A\) there is a function \(F\) in the Bergman space \(H^p(D)\), \(1 \leq p < \infty\), such that \(f = R_{D \cap A}F\). Let \(\nu\) be a positive Borel measure on \(D \cap A(\omega,e)\). Then, for \(1 \leq p < \infty\),NEWLINENEWLINE\[NEWLINEH^p(D \cap A(\omega,e), d\nu) := \bigg\{ f \in H(D \cap A(\omega,e)) : \int_{D \cap A(\omega,e)} |f|^p d\nu < \infty \bigg\}NEWLINE\]NEWLINE is the space of all functions holomorphic in \(D \cap A(\omega,e)\) and \(p\)-integrable with respect to the measure \(\nu\). In Theorem 1.1, the authors prove that there exists a bounded operator NEWLINE\[NEWLINEE_{A(\omega,e)} : H^1(D \cap A(\omega,e), |\partial h_{\omega,e}|_{\mathcal{N}}^2 dV_{A(\omega,e)}) \rightarrow H^1(D)NEWLINE\]NEWLINE such that NEWLINE\[NEWLINER_{D \cap A} \circ E_{A(\omega,e)} = \text{id}_{H^1(D \cap A(\omega,e), |\partial h_{\omega,e}|_{\mathcal{N}} ^2 dV_{A(\omega,e)})}.NEWLINE\]NEWLINENEWLINE Let \(e \in \mathbb{C}^n\) with \(e \neq 0\). In Theorem 1.2, they prove that a function \(f \in H(D \cap A(\omega,e))\) admits an extension which belongs to the Bergman space \(H^1(D)\) if and only if NEWLINE\[NEWLINE\int_{D \cap A(\omega,e)} |f| |\partial h_{\omega,e}|_{\mathcal{N}}^2 dV_{D \cap A} < \infty .NEWLINE\]NEWLINE NEWLINEIn other words, NEWLINE\[NEWLINER_{D \cap A} [H1(D)] = H^1\big(D \cap A(\omega,e), |\partial h_{\omega,e}|_{\mathcal{N}}^2 dV_{D \cap A}\big).NEWLINE\]NEWLINE For other Bergman spaces they obtain an \(\epsilon\)-optimal result. In Theorem 1.4, they prove that for any \(\epsilon >0\) and \(1<p< \infty\) there exists a bounded operator NEWLINE\[NEWLINEE_{A(\omega,e)} : H^p\big(D \cap A(\omega,e), |\partial h_{\omega,e}|_{\mathcal{N}} ^{2-\epsilon} dV_{A(\omega,e)}\big) \rightarrow H^p(D)NEWLINE\]NEWLINE such that NEWLINE\[NEWLINER_{D \cap A} \circ E_{A(\omega,e)} = \text{id}_{H^p(D \cap A(\omega,e), |\partial h_{\omega,e}|_{\mathcal{N}} ^{2-\epsilon} dV_{A(\omega,e)})}.NEWLINE\]