In this paper, we introduce vector-valued multiresolution analysis and vector-valued wavelets for vector-valued signal spaces. We construct vector-valued wavelets by using paraunitary vector lter bank theory. In particular, we construct vector-valued Meyer wavelets that are band-limited. We classify and construct vector-valued wavelets with sampling property. As an application of vector-valued ...

Boundedness results for bilinear square functions and vector-valued operators on products of Lebesgue, Sobolev, and other spaces of smooth functions are presented. Bilinear vector-valued Calderón-Zygmund operators are introduced and used to obtain bounds for the optimal range of estimates in target Lebesgue spaces including exponents smaller than one.

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vectorvalued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBS. The theory is then applied to multi-task machine learning. Especially, the...

We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided L-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-I...

The aim of the paper is to characterize the trace space of vector-valued Sobolev spaces W p (R , E) , where E is an arbitrary Banach space. In particular, we do not assume that the underlying Banach space E has the UMD property. Vector-valued Sobolev and Besov spaces are widely used in abstract evolution equations, cf. e.g. Amann [1, 2, 4], Veraar and Weis [57] or Denk, Hieber, Prüss, Saal, and...

The aim of this paper is to establish the vector-valued inequalities for Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g∗μ-functions, and their commutators on the Herz-Morrey spaces with variable exponentMK̇ p,q(·)(R n). By applying the properties of Lp(·)(Rn) spaces and the vector-valued inequalities for Littlewood-Paley operators and their...

We obtain several Banach-Stone type theorems for vector-valued functions in this paper. Let X,Y be realcompact or metric spaces, E,F locally convex spaces, and φ a bijective linear map from C(X,E) onto C(Y, F ). If φ preserves zero set containments, i.e., z(f) ⊆ z(g)⇐⇒ z(φ(f)) ⊆ z(φ(g)), ∀ f, g ∈ C(X,E), then X is homeomorphic to Y , and φ is a weighted composition operator. The above conclusio...