Weighted Bergman projections on the monogenic Bergman spaces over the real ball of ℝn have been already studied in a paper of Ren and Malonek. We extend and generalize their results by considering more general Bergman operators and stating a necessary and sufficient condition for them to be bounded in weighted Lebesgue spaces. Sharp estimates for the weighted monogenic Bergman kernel are also given.

Topics
Functional analysis, General topology
1.
K.L.
Avetisyan
,
Continuous inclusions and Bergman type operators in n-harmonic mixed norm spaces on the polydisc
,
J. Math. Anal. Appl
.
291
,
727
–
740
(
2004
).
https://doi.org/10.1016/j.jmaa.2003.11.039
Google Scholar
Crossref
Search ADS
 
2.
F.
Brackx
,
R.
Delanghe
 and
F.
Sommen
,
Clifford Analysis
(
Research Notes in Mathematics
, 76, Pitman,
Boston, MA
,
1982
).
Google Scholar
 
3.
F.
Brackx
,
F.
Sommen
 and
N.
Van Acker
,
Reproducing Bergman kernels in Clifford analysis
,
Complex Vari-ables Theory Appl
.
24
, no.
3-4
,
191
–
204
(
1994
).
Google Scholar
 
4.
B.R.
Choe
,
H.
Koo
 and
Y.J.
Lee
,
Positive Schatten class Toeplitz operators on the ball
,
Studia Math
.
189
,
65
–
90
(
2008
).
https://doi.org/10.4064/sm189-1-6
Google Scholar
Crossref
Search ADS
 
5.
B.R.
Choe
,
H.
Koo
 and
K.
Nam
,
Optimal norm estimate of operators related to the harmonic Bergman projection on the ball
,
Tohoku Math. J
.
62
,
357
–
374
(
2010
).
https://doi.org/10.2748/tmj/1287148616
Google Scholar
Crossref
Search ADS
 
6.
A.E.
Djrbashian
,
Integral representations for Riesz systems in the unit ball and some applications
,
Proc. Amer. Math. Soc
.
117
,
395
–
403
(
1993
).
https://doi.org/10.1090/S0002-9939-1993-1116256-2
Google Scholar
Crossref
Search ADS
 
7.
A.E.
Djrbashian
 and
F.A.
Shamoian
,
Topics in the Theory of Ap Spaces
(
Teubner-Texte zur Math.
, b.
105
,
Teubner, Leipzig
,
1988
).
Google Scholar
 
8.
K.
Gürlebeck
,
K.
Habetha
 and
W.
Sprössig
,
Holomorphic Functions in the Plane and n-dimensional Space
(
Birkhäuser Verlag
,
Basel
,
2008
).
Google Scholar
 
9.
K.
Gürlebeck
,
K.
Habetha
 and
W.
Sprössig
,
Applications of Holomorphic Functions in Two and Higher Dimensions
(
Birkhäuser, Springer
,
Switzerland
,
2016
).
Google Scholar
Crossref
Search ADS
 
10.
M.
Jevtić
 and
M.
Pavlović
,
Harmonic Bergman functions on the unit ball in
Rn, Acta Math. Hungar
.
85
,
81
–
96
(
1999
).
https://doi.org/10.1023/A:1006620929091
Google Scholar
Crossref
Search ADS
 
11.
G.
Ren
,
Harmonic Bergman spaces with small exponents in the unit ball
,
Collect. Math
.
53
,
83
–
98
(
2002
).
Google Scholar
 
12.
G.
Ren
 and
U.
Kähler
,
Weighted harmonic Bloch spaces and Gleason’s problem
,
Complex Variables Theory Appl
.
48
,
235
–
245
(
2003
).
Google Scholar
 
13.
G.
Ren
 and
H.R.
Malonek
, ”Bergman projection in Clifford analysis”, in:
Clifford algebras
(
Cookeville, TN
,
2002
),
Google Scholar
 
(
Prog. Math. Phys
.,
34
,
Birkhäuser
,
Boston, MA
,
2004
), pp.
125
–
139
.
14.
P.
Van Lancker
, ”
Clifford analysis on the sphere
”, in:
Clifford algebras and their application in mathematical physics, Proc. 4th conf., Aachen, Germany, May 28-31, 1996
, edited by
V.
Dietrich
 et al. (
Dordrecht, Kluwer
,
Fundam. Theor. Phys
.
94
,
1998
), pp.
201
–
215
.
Google Scholar
Crossref
Search ADS
 
15.
K.
Zhu
, Spaces of holomorphic functions in the unit ball (
Graduate Texts in Math
., vol.
226
,
Springer-Verlag
,
New York
,
2005
).
Google Scholar
 
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