AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$
d
ω
(
y
)
d
x
with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$
ω
(
0
,
2
t
)
≤
C
ω
(
0
,
t
)
. The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.
AbstractWe determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators on weighted Bergman spaces over bounded symmetric domains $$D=G/K,$$
D
=
G
/
K
,
for the irreducible K-types indexed by all partitions of length $$r={\mathrm {rank}}(D)$$
r
=
rank
(
D
)
.
AbstractIn this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.
Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure