On the Vector valued Fourier Transform And Compatibility of Operators
In Sook Park
备注：
Submitted at June, 2002. Changes in this revision: i) I correct a mistake in the proof of Lemma 13, but the statement is not changed. (ii) I rewrite the abstract more simple and clearly understood. But the meaning is not changed
Let $\mathbb{G}$ be a locally compact abelian group and let $1<p\leq 2$. $\mathbb{G}^{'}$ is the dual group of $\mathbb{G}$, and $p^{'}$ the conjugate exponent of $p$. An operator $T$ between Banach spaces $X$ and $Y$ is said to be compatible with the Fourier transform $F^{\mathbb{G}}$ if $F^{\mathbb{G}}\otimes T: L_p(\mathbb{G})\otimes X\to L_{p^{'}}(\mathbb{G}^{'})\otimes Y $ admits a continuous extension $[F^{\mathbb{G}},T]:[L_p(\mathbb{G}),X]\to [L_{p^{'}}(\mathbb{G}^{'}),Y]$. $\mathcal{FT}_p^{\mathbb{G}}$ denotes the set of such $T$'s. We show that $\mathcal{FT}_p^{\mathbb{R}\times\mathbb{G}}=\mathcal{FT}_p^{\mathbb{Z}\times\m athbb{G}} =\mathcal{FT}_p^{\mathbb{Z}^n \times\mathbb{G}}$ for any $\mathbb{G}$ and positive integer $n$. And if the factor group of $\mathbb{G}$ with respect to its component of the identity element is a direct sum of a torsion free group and a finite group with discrete topology then $\mathcal{FT}_p^{\mathbb{G}}=\mathcal{FT}_p^{\mathbb{Z}}$ .
We define a scalar valued Fourier transform for functions on the Heisenberg group and establish some of its basic properties like inversion formula, Plancherel theorem and Riemann-Lebesgue lemma. We also restate certain well known theorems for the group Fourier transform in terms of the new transform which we would like to call Strichartz Fourier transform.
We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, Bäumer, Lumer and Neubrander and Langenbruch.
We give an overview of some recent results on operator-valued $(L^{p},L^{q})$ Fourier multipliers and stability theory for evolution equations. The aim is to provide a relatively nontechnical introduction to the underlying ideas, emphasizing the connection between the two areas. We also indicate how operator-valued $(L^{p},L^{q})$ Fourier multipliers can be applied to functional calculus theory.
We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.
In this paper, we investigate the nuclear trace of vector-valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo-differential operators. First, we characterise the nuclearity of pseudo-differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector-valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector-valued Fou…
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This paper is devoted to the study of operator-valued Triebel-Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel-Lizorkin spaces on $\mathbb{R}^d$. As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood-Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for th…
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Let $G$ be a locally compact unimodular group, and let $ϕ$ be some function of $n$ variables on $G$. To such a $ϕ$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an $n$-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear Sch…
We explore the Bohr inequality involving the Fourier transforms of complex valued integrable and square integrable functions defined on a second countable compact topological group. We also investigate the connection of the Bohr phenomenon with a modulus of convexity of the space of bounded linear operators defined on a complex Hilbert space.
On a Class of Fourier Integral Operators on Manifolds with Boundary
本文研究了紧流形上一类具有边界的傅里叶积分算子，它与一类保边界的自然辛同胚有关。可以为这样的傅里叶积分算子定义一个Boutet de Monvel类型的微积分，并建立适当的连续性性质。该演算的一个关键特征是，这些算子的局部表示是由作用于施瓦茨函数或温带分布的算子值符号给出的。在这里，我们专注于相应的本地相位函数的属性，这使得证明这一结果在一个相当简单的方式。
The main result is a Paley's theory for lacunary Fourier series using semigroup-BMO and $H^1$ spaces. This interpretation allows an extension of Paley's theory to general discrete groups, complementing the work of Rudin for abelian groups with a total order, and Lust-Piquard and Pisier's work for lacunary Fourier series with operator-valued coefficients.
We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maxi…
We consider the $1$- and $2$-d bicomplex analogs of the classical Fourier--Wigner transform. Their basic properties, including Moyal's identity and characterization of their ranges giving rise to new bicomplex--polyanalytic functional spaces are discussed. Particular case of special window is also considered. An orthogonal basis for the space of bicomplex--valued square integrable functions on the bicomplex numbers is constructed by means of the polyanalytic complex Hermite functions.
The quadratic phase Fourier transform QPFT is a neoteric addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic phase Fourier transform to quaternion valued signals, known as the quaternion QPFT QQPFT. We initiate our investigation by studying the QPFT of 2D quaternionic signals, then we introduce the QQPFT of 2D quaternionic signals. Using the fundamental relationship between the QQPFT and quaternion Fourier transf…
Fourier multipliers in Banach function spaces with UMD concavifications
证明了Banach函数空间上算子值乘子的Coifman-Rubio de Francia-Semmes乘子定理的各种推广。我们的结果涉及到一个新的有界性条件的运营商，我们称之为$\ell^{r}(\ell^{s})$ -有界性，这意味着$\mathcal{R}$ -有界性在许多情况下。证明是基于作者最近在Banach函数空间中得到的新的Littlewood-Paley-Rubio de Francia型估计。
In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.
