Carleson measures on product domains and 2-parameter Brownian martingales
In this paper we study Carleson measures on a certain product domain. Recently, \textit{S.-Y. A. Chang} [Ann. Math., II. Ser. 109, 613-620 (1979; Zbl 0401.28004)] proved the following Theorem C: Let \(\Delta^ 2\) be the bi-disc. For every \(f\in L^{\infty}(\partial \Delta \times \partial \Delta)\) the measure \[ d\mu_ f=| \nabla_ 1\nabla_ 2PP[f](z_ 1,z_ 2)|^ 2 \log (1/| z_ 1|)\log (1/| z_ 2|)dz_ 1d\bar z_ 1dz_ 2d\bar z_ 2 \] is a Carleson measure on \(\Delta^ 2\), where PP[f] is the double Poisson integral of f. Hence PP[.] is a bounded operator from \(L^ p(\partial \Delta \times \partial \Delta)\) to \(L^ p(\Delta^ 2,\mu_ f)\), \(1<p<\infty\). In our paper we extend the result as follows: Main Theorem. Let \(D=D_ 1\times D_ 2\) be the product of two bounded Dirichlet domains \(D_ 1\subset R^{n(1)}\) and \(D_ 2\subset R^{n(2)}\) (n(1),n(2)\(\geq 2)\). Let \(d\omega\) be the biharmonic measure relative to D and a point \(\xi =(\xi_ 1,\xi_ 2)\in D\). Then for every \(f\in L^{\infty}(d\omega)\), \[ d\mu_ f=| \nabla_ 1\nabla_ 2PP[f](z_ 1,z_ 2)|^ 2 G_ 1(\xi_ 1,z_ 1)G_ 2(\xi_ 2,z_ 2)dV_ 1(z_ 1)dV_ 2(z_ 2) \] satisfies that PP[.] is a bounded operator from \(L^ p(d\omega)\) to \(L^ p(D,\mu_ f)\), \(1<p<\infty\), where PP[.] is the double Poisson integral operator, \(G_ j\) is the Green function for \(D_ j\) and \(V_ j\) is the n(j) dimensional Lebesgue measure, \(j=1,2.\) In order to prove the theorem we use 2-parameter Brownian martingales. In our paper we also discuss Carleson measures on products of two NTA domains (NTA domains were defined in \textit{D. S. Jerison} and \textit{C. E. Kenig} [Adv. Math. 46, 80-147 (1982; Zbl 0514.31003)].)
Carleson measures on products of two NTA domains
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