TY - JOUR

T1 - Factorization theorems for functions in the Bergman spaces

AU - Horowitz, Charles

PY - 1977/3

Y1 - 1977/3

N2 - 1. For 0 < p < o and 0 _< a o, we consider the Bergman spaces A of
functions analytic in the unit disc U of C and satisfying
U
where dA denotes the normalized Lebesgue measure (1/) dxdy. The spaces
A' are denoted simply A.
In the present paper, we discuss factorization properties of functions in these
spaces. Our first main result is the following:
THEOnEM 1. Let p and a be fixed, and let n 2 be an integer. Then there
is a constant C depending only on p, a, and n, such that i/] A'" there exist
functions ], 1, A with
and
i=l
Theorem 1 has a somewhat interesting history. The analogous theorem for
H spaces over U in place of A is elassicM. Rudin [9] showed that this result
does not generalize to H spaces over a polydisc, at least in dimensions greater
than 3. Somewhat later, Miles [7] and Rosay [8] extended Rudin's counterexample
to dimensions 2 and 3. Interest in the present problem arose partly
because of connections between A spaces and H spaces in several variables.
Indeed, in view of the theorem in [5], a negative result in place of Theorem 1
would have reproved the result of Rudin, Miles and Ilosay. In the other
direction, recent work of Coifman, Rochberg, and Weiss [2] on H spaces in
several variables has yielded as a corollary a weaker version of Theorem 1.
Our second main theorem deals with division by Blaschke products, which
products we define here mainly for purposes of fixing notation. For a U,
we define first the M6bius transformations
amz (1.1) C(z) 1 az

AB - 1. For 0 < p < o and 0 _< a o, we consider the Bergman spaces A of
functions analytic in the unit disc U of C and satisfying
U
where dA denotes the normalized Lebesgue measure (1/) dxdy. The spaces
A' are denoted simply A.
In the present paper, we discuss factorization properties of functions in these
spaces. Our first main result is the following:
THEOnEM 1. Let p and a be fixed, and let n 2 be an integer. Then there
is a constant C depending only on p, a, and n, such that i/] A'" there exist
functions ], 1, A with
and
i=l
Theorem 1 has a somewhat interesting history. The analogous theorem for
H spaces over U in place of A is elassicM. Rudin [9] showed that this result
does not generalize to H spaces over a polydisc, at least in dimensions greater
than 3. Somewhat later, Miles [7] and Rosay [8] extended Rudin's counterexample
to dimensions 2 and 3. Interest in the present problem arose partly
because of connections between A spaces and H spaces in several variables.
Indeed, in view of the theorem in [5], a negative result in place of Theorem 1
would have reproved the result of Rudin, Miles and Ilosay. In the other
direction, recent work of Coifman, Rochberg, and Weiss [2] on H spaces in
several variables has yielded as a corollary a weaker version of Theorem 1.
Our second main theorem deals with division by Blaschke products, which
products we define here mainly for purposes of fixing notation. For a U,
we define first the M6bius transformations
amz (1.1) C(z) 1 az

UR - http://www.scopus.com/inward/record.url?scp=84972573151&partnerID=8YFLogxK

U2 - 10.1215/S0012-7094-77-04409-X

DO - 10.1215/S0012-7094-77-04409-X

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84972573151

SN - 0012-7094

VL - 44

SP - 201

EP - 213

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 1

ER -