Fourier multiplier theorems involving type and cotype

In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. 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 $X$ and $Y$ are UMD spaces and that $m$ satisfies a smoothness condition. We show that for $p<q$ other geometric conditions on $X$ and $Y$, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for $T_m$ without any smoothness properties of $m$. Under smoothness conditions the boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ 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 Lp\mathrm{L}^p-spaces

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $\mu$. We factorize the integral with respect to $\mu$ in terms of the integrals with respect to the ergodic measures on $X$, and show that $\mathrm{L}^p(X,\mu)$ ($1\leq p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an $\mathrm{L}^p$-direct integral of the spaces $\mathrm{L}^{p}(X,\lambda)$, where $\lambda$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of $\mathrm{L}^p(X,\mu)$ as an $\mathrm{L}^p$-direct integral of order indecomposable representations. If $(X^\prime,\mu^\prime)$ is a probability space, and, for some $1\leq q<\infty$, $G$ acts in a strongly continuous manner on $\mathrm{L}^q(X^\prime,\mu^\prime)$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on $\mathrm{L}^{p}(X^{\prime},\mu^{\prime})$ in a similar fashion for all $1\leq p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^\prime,\mu^\prime)$ is separable, the representation of $G$ on $\mathrm{L}^p(X^\prime,\mu^\prime)$ can then be disintegrated into order indecomposable representations. The notions of $\mathrm{L}^p$-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 $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. We also study the estimate above in the setting of Banach ideals in $\mathcal{L}(X,Y)$. 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