The treatise deals with translation-invariant operators on various function spaces (including Besov, Lebesgue-Bôchner, and Hardy), where the range space of the functions is a possibly infinite-dimensional Banach space X. The operators are treated both in the convolution form T f = k ∗ f and in the multiplier form in the frequency representation, T^f = m f̂, where the kernel k and the multiplier m are allowed to take values in ℒ(X) (bounded linear operators on X). Several applications, most notably the theory of evolution equations, give rise to non-trivial instances of such operators.Verifying the boundedness of operators of this kind has been a long-standing problem whose intimate connection with certain randomized inequalities (the notion of "R-boundedness" which generalizes classical square-function estimates) has been discovered only recently. The related techniques, which are exploited and developed further in the present work, have proved to be very useful in generalizing various theorems, so far only known in a Hilbert space setting, to the more general framework of UMD Banach spaces.The main results here provide various sufficient conditions (with partial converse statements) for verifying the boundedness of operators T as described above. The treatment of these operators on the Hardy spaces of vector-valued functions is new as such, while on the Besov and Bôchner spaces the convolution point-of-view taken here complements the multiplier approach followed by various other authors. Although general enough to deal with the vector-valued situation, the methods also improve on some classical theorems even in the scalar-valued case: In particular, it is shown that the derivative condition|ξ||α| |Dαm(ξ)| ≤ C ∀α ∈ {|α|∞ ≤ 1} ∩ {|α|1 ≤ ⌊n/2⌋ + 1}is sufficient for m to be a Fourier multiplier on Lp(ℝn), p ∈ ]1,∞[ – the set of required derivatives constitutes the intersection of the ones in the classical theorems of S. G. Mihlin and L. Hörmander.
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机译：该论文涉及各种函数空间（包括Besov，Lebesgue-Bôchner和Hardy）上的平移不变算符，其中这些函数的范围空间可能是无限维的Banach空间X。这些算符都以卷积形式处理T f = k ∗ f且在频率表示中为乘数形式，T ^ f = m f̂，其中内核k和乘数m可以取ℒ（X）的值（X上的有界线性算子）。几种应用，最著名的是演化方程的理论，引起了这类算子的非平凡实例。验证这类算子的有界性是一个长期存在的问题，它与某些随机不等式密切相关（“ R”概念广义经典平方函数估计的“有界””是最近才发现的。在当前工作中进一步开发和利用的相关技术已被证明在将迄今为止仅在希尔伯特空间设置中已知的各种定理推广到UMD Banach空间的更通用框架方面非常有用。提供如上所述的各种充分条件（带有部分相反的语句）来验证算子T的有界性。在矢量值函数的Hardy空间上对这些算符的处理是新的，而在Besov和Bôchner空间上，此处采取的卷积视点补充了乘数法，随后其他许多作者也纷纷采用。尽管足以处理向量值情况，但这些方法甚至在标量值情况下也对一些经典定理进行了改进：特别是，证明了导数条件|ξ||α| |Dαm（ξ）| ≤C∈∈{|α|∞≤1}∩{|α| 1≤⌊n/2⌋+ 1}足以使m成为Lp（ℝn），p∈[1]，∞[ –所需导数集构成SG Mihlin和L.Hörmander经典定理中那些导数的交集。
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