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Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures
Journal of Inequalities and Applications volume 2008, Article number: 141379 (2008)
Abstract
Let be a nonnegative Radon measure on
which only satisfies the following growth condition that there exists a positive constant
such that
for all
and some fixed
. In this paper, the authors prove that for suitable indexes
and
, the parametrized
function
is bounded on
for
with the assumption that the kernel of the operator
satisfies some Hörmander-type condition, and is bounded from
into weak
with the assumption that the kernel satisfies certain slightly stronger Hörmander-type condition. As a corollary,
with the kernel satisfying the above stronger Hörmander-type condition is bounded on
for
. Moreover, the authors prove that for suitable indexes
and
is bounded from
into
(the space of regular bounded lower oscillation functions) if the kernel satisfies the Hörmander-type condition, and from the Hardy space
into
if the kernel satisfies the above stronger Hörmander-type condition. The corresponding properties for the parametrized area integral
are also established in this paper.
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Lin, H., Meng, Y. Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures. J Inequal Appl 2008, 141379 (2008). https://doi.org/10.1155/2008/141379
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DOI: https://doi.org/10.1155/2008/141379