Carleson measures and the fractional derivatives of holomorphic functions in the unit ball of \(\mathbb{C}^n\) (Q1266283)

Let \(B\subset\mathbb{C}^{n}\) be the unit Euclidean ball and let \(\mu\) be a finite positive Borel measure on \(B\). We say that \(\mu\) is a Carleson measure of order \(s\) if there exists \(C>0\) such that \(\mu(S(\zeta,t))\leq Ct^{s}\) for any \(\zeta\in\partial B\) and \(t\in(0,1)\), where \(S(\zeta,t):=\{z\in B\: \| z\| >1-t, | 1-\langle z/\| z\| ,\zeta\rangle| <t\}\). Let \(0<p\leq q<+\infty\), \(\alpha>-1\), \(\beta\geq 0\), \(m_\alpha:=(1-\| z\|)^\alpha\cdot\)(the normalized Lebesgue measure on \(B\)). The main results of the paper are the following two theorems. (1) \(\mu\) is a Carleson measure of order \((n+1+\alpha)q/p+\beta q\) iff there exists a constant \(C>0\) such that \(\| f^{[\beta]}\| _{L^{q}(B,\mu)}\leq C\| f\| _{L^{p}(B,m_\alpha)}\) for any \(f\in\mathcal O(B)\), where \(f^{[\beta]}\) denotes the \(\beta\)-th fractional derivative of \(f\). In the case \(\beta=0\) the result remains true for arbitrary bounded symmetric domain in \(\mathbb{C}^{n}\). (2) If \(2\leq p\) or \(p<q\), then \(\mu\) is a Carleson measure of order \(nq/p+\beta q\) iff there exists a constant \(C>0\) such that \(\| f^{[\beta]}\| _{L^{q}(B,\mu)}\leq C\| f\| _{H^{p}(B)}\) for any \(f\in\mathcal O(B)\).