The extended Carleson measure and the derivatives of \(L^ p\) functions
The author presents a proof of the fact that, for \(p\geq 2\) and \(n\in \mathbb{Z}_ +\), a positive measure \(\mu\) on \(\mathbb{R}^{n+1}_ +\) is \((n+pr)\)-Carleson if and only if \[ \int_{\mathbb{R}^{n+1}_ +} \bigl(|\partial^ r u/\partial y^ r|^ p+ \sum_{1\leq i\leq n} |\partial^ r u/\partial x^ r_ i|^ p\bigr) d\mu\leq C^ p\| f\|^ p_ p, \] \(u\) being the Poisson integral of \(f\).
scientific article; zbMATH DE number 60795