2,016 research outputs found
Fourier multiplier theorems involving type and cotype
In this paper we develop the theory of Fourier multiplier operators
, for Banach spaces
and , and an operator-valued symbol. The case has been studied
extensively since the 1980's, but far less is known for . In the scalar
setting one can deduce results for from the case . However, in the
vector-valued setting this leads to restrictions both on the smoothness of the
multiplier and on the class of Banach spaces. For example, one often needs that
and are UMD spaces and that satisfies a smoothness condition. We
show that for other geometric conditions on and , such as the
notions of type and cotype, can be used to study Fourier multipliers. Moreover,
we obtain boundedness results for without any smoothness properties of
. Under smoothness conditions the boundedness results can be extrapolated to
other values of and as long as remains
constant.Comment: Revised version, to appear in Journal of Fourier Analysis and
Applications. 31 pages. The results on Besov spaces and the proof of the
extrapolation result have been moved to arXiv:1606.0327
Variable-Sweep Transition Flight Experiment (VSTFE): Stability code development and clean-up glove data analysis
The primary objective of the Variable Sweep Transition Flight Experiment (VSTFE) was to establish an improved swept wing transition criterion. The development of the Unified Stability System gave a way of quickly examining disturbance growth for a wide variety of laminar boundary layers. The disturbance growth traces shown are too scattered to define a transition criteria to replace the F-111 data band, which has been used successfully to design NLF gloves. Still, a careful review of the clean-up glove data may yield cases for which the transition location is known more accurately. Liquid crystal photographs of the clean-up glove show much spanwise variation in the transition front for some conditions, and this further complicates the analyses. Several high quality cases are needed in which the transition front is well defined and at a relatively constant chordwise station
Disintegration of positive isometric group representations on -spaces
Let be a Polish locally compact group acting on a Polish space with a
-invariant probability measure . We factorize the integral with respect
to in terms of the integrals with respect to the ergodic measures on ,
and show that () is -equivariantly
isometrically lattice isomorphic to an -direct integral of the
spaces , where ranges over the ergodic
measures on . This yields a disintegration of the canonical representation
of as isometric lattice automorphisms of as an
-direct integral of order indecomposable representations.
If is a probability space, and, for some , acts in a strongly continuous manner on
as isometric lattice automorphisms that
leave the constants fixed, then acts on
in a similar fashion for all . Moreover, there exists an alternative model in which these
representations originate from a continuous action of on a compact
Hausdorff space. If is separable, the representation of
on can then be disintegrated into order
indecomposable representations.
The notions of -direct integrals of Banach spaces and
representations that are developed extend those in the literature.Comment: Section on future perspectives added. 35 pages. To appear in
Positivit
Operator Lipschitz functions on Banach spaces
Let , be Banach spaces and let be the space of
bounded linear operators from to . We develop the theory of double
operator integrals on and apply this theory to obtain
commutator estimates of the form for a large class of functions
, where , are scalar type
operators and . In particular, we establish this
estimate for and for diagonalizable operators on and
, for and , and for . We also
obtain results for . We also study the estimate above in the setting
of Banach ideals in . The commutator estimates we derive hold
for diagonalizable matrices with a constant independent of the size of the
matrix.Comment: Final version published in Studia Mathematica, with some minor
change
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