Electron. J. Differential Equations,
Vol. 2018 (2018), No. 111, pp. 117.
Compactness of commutators of Toeplitz operators on qpseudoconvex domains
Sayed Saber
Abstract:
Let
be a bounded qpseudoconvex domain in
,
and let
.
If
is smooth, we find sufficient conditions for the
Neumann operator to be compact.
If
is nonsmooth and if
, we show
that compactness of the
Neumann operator,
,
on square integrable (0, p+1)forms is equivalent to compactness of the
commutators
,
,
on square integrable
closed (0, p)forms, where
is the
Bergman projection on (0, p)forms. Moreover, we prove that compactness of
the commutator of
with bounded functions percolates up in the
complex on
closed forms and square
integrable holomorphic forms. Furthermore, we find a characterization of
compactness of the canonical solution operator,
,
of the
equation restricted on (0, p+1)forms
with holomorphic
coefficients in terms of compactness of commutators
,
,
on (0, p)forms with holomorphic coefficients,
where
is the BergmanToeplitz operator acting on (0, p)forms
with symbol
.
This extends to domains which are not necessarily
pseudoconvex.
Submitted August 4, 2017. Published May 10, 2018.
Math Subject Classifications: 32F10, 32W05.
Key Words:
and
Neumann operator;
BergmanToeplitz operator; qconvex domains.
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Sayed Saber
Department of Mathematics
Faculty of Science and Arts in Baljurashi
Albaha University
Albaha, Saudi Arabia

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