Trace inequalities for weighted Hardy-Sobolev spaces in the non-diagonal case (Q2882513)

Let \(\mathbb{B}^n\) be the unit ball in \(\mathbb{C}^n\), and \(\mathbb{S}^n\) the unit sphere in \(\mathbb{C}^n\). Let \(w\) be a doubling weight on the \(\mathbb{S}^n\). For \(\alpha >1\) and \(\xi\in\mathbb{S}^n\), the admissible region \(D_{\alpha}(\xi)\) is given by NEWLINE\[NEWLINE D_\alpha(\xi):=\left\{z\in\mathbb{B}^n: \left| 1-z\overline{\xi}\right|<\frac{\alpha}{2}(1-\left| z\right|^2)\right\}. NEWLINE\]NEWLINE For a function \(f\) defined on \(\mathbb{B}^n\), the maximal function \(M_\alpha f\) is the function defined on \(\mathbb{S}^n\) by \(M_\alpha f(\xi)=\sup_{z\in D_\alpha(\xi)}\left| f(z)\right|\).NEWLINENEWLINEFor \(0\leq s\) and \(0<p<\infty\), the weighted Hardy-Sobolev space \(H^p_s(w)\), is the collection of holomorphic in \(\mathbb{B}^n\) such that NEWLINE\[NEWLINE \left\| f\right\|_{H^p_s(w)}:=\left\| M_\alpha[\mathcal{R}^s f]\right\|_{L^p(w)}<\infty. NEWLINE\]NEWLINE Here \(\mathcal{R}^sf(s)\) is defined by NEWLINE\[NEWLINE \mathcal{R}^s f(z):=(I+R)^sf(z), NEWLINE\]NEWLINE where \(R\) is the radial derivative.NEWLINENEWLINEA finite positive Borel measure \(\mu\) on \(\mathbb{S}^n\) is called a \(q\)-trace measure (sometimes a \(q\)-Carleson measure) for \(H^p_s(w)\) if and only if there exists a constant \(C>0\) such that for all \(f\in H^p_s(w)\) NEWLINE\[NEWLINE \left\| M_\alpha[\mathcal{R}^s f]\right\|_{L^q(\mu)}\leq C\left\| f\right\|_{H^p_s(w)}. NEWLINE\]NEWLINENEWLINENEWLINEThe authors obtain a characterization of the \(q\)-trace measures for \(H^p_s(w)\) for certain ranges of indices \(p,q,s\) and properties on the weight \(w\). Depending on the parameters \(p,q, s\) and the properties of the weight, the characterizations obtained are given in terms of capacitary information, in terms of geometric relationships between the \(\mu\) measure of a ball and the \(w\) measure of a ball, or in terms of a Wolff-type potential.