Inversion and Representation Theorems for the Laplace Transformation

A study is made of the Laplace transformation on Banach-valued functions of a real variable, with particular reference to inversion and representation theories. First a new type of integral for Banach-valued functions of a real variable, the "Improper Bochner" integral is defined. The relations between the Bochner, Improper Bochner, Riemann-Graves, and Riemann-Stieltjes integrals are studied. Next, inversion theorems are proved for a new "real" inversion operator when the integral in the Laplace transformation is each of the above mentioned types. Lastly, representation of Banach-valued functions by Laplace integrals of functions in Bp([0,∞);¥), 1 ≤ p < ∞, is studied, and theorems are very like those proved, for numerically-valued functions, by D. V. Widder in his book "The Laplace Transform" (Princeton, 1941) page 312. The classes Hp(α ; ¥), 1 ≤ p < ∞, are also studied in this section as is the representation of numerically-valued functions by Laplace-Stieltjes integrals

Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator (A) with certain off-diagonal bounds, such that (A) always has a bounded (H^{\infty})-functional calculus on these spaces. This provides a new way of proving functional calculus of (A) on the Bochner spaces (L^p(\R^n;X)) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMD-valued context. Even when (X=\C), our approach gives refined (p)-dependent versions of known results.Comment: 28 pages; submitted for publicatio

Order-type Henstock and McShane integrals in Banach lattice setting

We study Henstock-type integrals for functions defined in a compact metric space $T$ endowed with a regular $\sigma$-additive measure $\mu$, and taking values in a Banach lattice $X$. In particular, the space $[0,1]$ with the usual Lebesgue measure is considered.Comment: 5 page

Banach-valued multilinear singular integrals

We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued H\"ormander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed $L^p$-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rules of Muscalu et. al.. Our approach works just as well for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform, extending results of Silva. We also prove several operator-valued $T (1)$-type theorems both in one parameter, and of multi-para\-meter, mixed-norm type. A distinguishing feature of our $T(1)$ theorems is that the usual explicit assumptions on the distributional kernel of $T$ are replaced with testing-type conditions. Our proofs rely on a newly developed Banach-valued version of the outer $L^p$ space theory of Do and Thiele.Comment: 44 pages. Final version, to appear in Indiana Univ. Math.

Multifunctions determined by integrable functions

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee i