. Consider a bounded, strongly pseudoconvex domain D ⊂ C n with minimal smoothness (namely, the class C 2 ) and let b be a locally integrable function on D . We characterize boundedness (resp., compactness) in L p ( D ) , p > 1, of the commutator [ b, P ] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b . We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

Some characterizations of Bloch functions on strongly pseudoconvex domains

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

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Equivalent characterizations of Bloch functions

This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field.Originally published in 1972.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Boundary Behavior of Holomorphic Functions of Several Complex Variables.

Bloch Functions in Several Complex Variables, I

Soit D le disque unite ouvert dans le plan complexe C. Soit H ∞ (D) l'ensemble des fonctions analytiques bornees sur D. On caracterise les fonctions feH ∞ (D) telles que T f *T f -TfT f * est un operateur compact, ou T f est l'operateur multiplication T f :L a 2 →L a 2 defini par T f (g)=fg

The Bergman space, the Bloch space, and commutators of multiplication operators

A necessary and sufficient condition is given for a subset E C C to satisfy Sup{(jf(z)j (1 Z12)Iz E f (E)) C satisfying Sup{If'(z)I (1 IZ12)IZ E D) < oo. See [1] for other characterizations and some interesting properties of Bloch functions. The following theorem is the main result of this note. THEOREM 1. Suppose E C C. Then the condition SUp{If'(Z)I (1 Iz12)Iz Ef-'(E)} K ? (*) is a sufficient condition for an analytic function f: D -) C to be a Bloch function if and only if the radii of the discs contained in C E are bounded above. For example E = Z2 = the set of Gaussian integers works, but no finite set E will work. This theorem was motivated by a theorem of P. Lappan [3] about normal functions. A meromorphic function f on D is said to be normal if Sup{If'(Z)I (1 Iz12)/ (1 + If(z)12)Iz E D } < oo (see [1]). THEOREM [LAPPAN 5-VALUE THEOREM]. If f is a meromorphic function defined on the unit disc D and Sup{If'(z)I(l Iz12)/(I + If(Z)12)IZ Ef-l(E)} = M < X for any subset E of the Riemann sphere containing at least 5 points (3 finite points if f is analytic) then f is a normal function. Since Bloch functions are closely related to normal functions it is natural to Received by the editors January 23, 1978 and, in revised form, March 3, 1978. AMS (MOS) subject classifications (1970). Primary 30A74, 30A78.

/pdf/a-necessary-and-sufficient-condition-for-bloch-functions-517qw6wkli.pdf

Some characterizations of Bloch functions on strongly pseudoconvex domains

Citations

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

Equivalent characterizations of Bloch functions

References

Boundary Behavior of Holomorphic Functions of Several Complex Variables.

Bloch Functions in Several Complex Variables, I

The Bergman space, the Bloch space, and commutators of multiplication operators

A necessary and sufficient condition for Bloch functions

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