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Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori
This paper deals with the operator-valued Hardy spaces introduced and studied
by Tao Mei. Our principal result shows that the Poisson kernel in Mei's
definition of these spaces can be replaced by any reasonable test function. As
an application, we get a general characterization of Hardy spaces on quantum
tori. The latter characterization plays a key role in our recent study of
Triebel-Lizorkin spaces on quantum tori
Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori
This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin
spaces on a noncommutative -torus (with a
skew symmetric real -matrix). These spaces share many properties
with their classical counterparts. We prove, among other basic properties, the
lifting theorem for all these spaces and a Poincar\'e type inequality for
Sobolev spaces. We also show that the Sobolev space
coincides with the Lipschitz space of order
, already studied by Weaver in the case . We establish the embedding
inequalities of all these spaces, including the Besov and Sobolev embedding
theorems. We obtain Littlewood-Paley type characterizations for Besov and
Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms
of the Poisson, heat semigroups and differences. Some of them are new even in
the commutative case, for instance, our Poisson semigroup characterizations
improve the classical ones. As a consequence of the characterization of the
Besov spaces by differences, we extend to the quantum setting the recent
results of Bourgain-Br\'ezis -Mironescu and Maz'ya-Shaposhnikova on the limits
of Besov norms. The same characterization implies that the Besov space
for is the quantum
analogue of the usual Zygmund class of order . We investigate the
interpolation of all these spaces, in particular, determine explicitly the
K-functional of the couple , which is the quantum analogue of a classical
result due to Johnen and Scherer. Finally, we show that the completely bounded
Fourier multipliers on all these spaces do not depend on the matrix ,
so coincide with those on the corresponding spaces on the usual -torus
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