×

A note on functions of vanishing mean oscillation on the bidisk. (English) Zbl 0608.32003

For a function \(f\in L^{\infty}(T^ 2)\), T the unit circle, S.-Y. A. Chang has proved [Ann. Math., II. Ser. 109, 613-620 (1979; Zbl 0401.28004)] that the Poisson integral \(\Lambda\) is a bounded operator from \(L^ 2(T^ 2)\) to \(L^ 2(d\mu_ f)\), where \[ d\mu_ f= | \nabla_ 1\nabla_ 2 \Lambda f(z_ 1,z_ 2)|^ 2 \log (1/| z_ 1|) \log (1/| z_ 2|) dV(z_ 1) dV(z_ 2) \] and where dV is area measure on the disk. The present author proves that if \(f\in C(T^ 2)\), then the above operator \(\Lambda\) is compact. An immediate consequence is an extension of a one variable result of S. C. Power [Bull. Lond. Math. Soc. 12, 207-210 (1980; Zbl 0438.47033)].
Reviewer: D.Clark

MSC:

32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A30 Other generalizations of function theory of one complex variable
30D55 \(H^p\)-classes (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI