Symmetric and Zygmund measures in several variables. (Q5956595)

In order to understand a measure \(\mu\) on \(\mathbb R^n\), we can consider its harmonic extension \(u(x,t)=\int _{\mathbb R^n} P(x-t,y) \,d\mu (t)\) on \(\mathbb R^{n+1}_+\). We are looking for properties of \(\mu\) in terms of the properties of \(u\). In the first part of the paper under review, the authors obtain this kind of results in the following situations. A function \(\omega\) is a regular gauge function, if it is a non-decreasing bounded function \(\omega \colon (0,\infty)\to (0,\infty)\) such that \(\omega(t)/t^{1-\epsilon}\) is decreasing for some \(\epsilon >0\). A (signed) Borel measure \(\mu\) on \({\mathbb R}^n\) is said to be a \(\omega\)-Zygmund measure if there exists \(C>0\) such that \(| \mu (Q_+)-\mu (Q_-)| \leq C \omega ( l(Q_+))| Q_+| \) for any pair of adjacent cubes \(Q_+, Q_-\) of the same volume. A positive Borel measure \(\mu \) on \(\mathbb R^n\) is said to be doubling if \(C^{-1} \leq \frac{\mu(Q_+)}{\mu (Q_-)}\leq C\), and \(\mu\) is called \(\omega \)-symmetric if \(| \frac{\mu (Q_+)}{\mu (Q_-)}-1| \leq C \omega ( l(Q_+))\) when \( l(Q_+)<1\).NEWLINENEWLINEThe main results of this paper are:NEWLINENEWLINE{(A)} \(\mu \) is \(\omega\)-Zygmund if and only if there exists a constant \(C>0\) such that NEWLINE\[NEWLINEy | \nabla u(x,y) | \leq C\omega (y)NEWLINE\]NEWLINE for all \((x,y)\in {\mathbb R}_+^{n+1}\).NEWLINENEWLINE{(B)} For a finite positive measure \(\mu\) on \(\mathbb R^n\), assuming that \(\omega (0^+)=0\), \(\mu\) is \(\omega\)-symmetric if and only if there exists a constant \(C>0\) such that NEWLINE\[NEWLINE\frac{y | \nabla u(x,y)| }{u(x,y)}\leq C\omega (y)NEWLINE\]NEWLINE for all \((x,y)\in {\mathbb R}_+^{n+1}\).NEWLINENEWLINE{(C)} For a positive measure \(\mu\) on \(\mathbb R^n\) such that \(\int _{\mathbb R^n} (1+| x| )^{-n-1}\,d \mu (x) < \infty\) and \(\mu\) is a doubling measure then it satisfies NEWLINE\[NEWLINEy| \partial_y u(x,y)| \leq Cu(x,y)NEWLINE\]NEWLINE for all \((x,y) \in {\mathbb R}_+^{n+1}\) and some constant \(0<C<n\). And conversely that if the above inequality holds for some constant \(C<1\) then \(\mu\) is doubling.NEWLINENEWLINEIn the second part of the paper, the authors also obtain properties of \(\omega\)-Zygmund measures and \(\omega\)-symmetric measures in terms of the properties of \(\omega\).NEWLINENEWLINE{(D)} Under the assumption that \(\int_0^\infty \frac {\omega^2(t)}t dt < \infty,\) any \(\omega\)-Zygmund [resp. \(\omega\)-symmetric] measure \(\mu\) is \(m\)-continuous and has a density \(d\mu =fdm\) such that \(\int \exp(A| f| ^2)<\infty\) [resp. \(\int \exp(A| \log f| ^2)<\infty\)]. And without the assumption above, there exist both singular \(\omega\)-Zygmund measures and singular \(\omega\)-symmetric measures.NEWLINENEWLINEThe paper also contains other results, most of them based on properties of harmonic functions. And the proof of the integrability condition for the density is done by using some martingale techniques.