In this paper we study the vector-valued analogues of several inequalities for Fourier transform. particular, consider Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt include Hausdorff–Young Hardy–Littlewood state that transform is bounded from $$L^p({\mathbb {R}}^d,|\cdot |^{\beta p})$$ into $$L^q({\mathbb |^{-\gamma q})$$ under certain condition on $$p,q,\beta ...

1. Topological Vector Spaces 1 1.1. The Krein-Milman theorem 7 2. Banach Algebras 11 2.1. Commutative Banach algebras 14 2.2. ∗–Algebras (over complexes) 17 2.3. Problems on Banach algebras 20 3. The Spectral Theorem 21 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 26 3.2. Integration with respect to a Projection Valued Measure 27 3.3. The Functional Calculus 34 4. Unboun...

Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is....

The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are K-quasiconvex with respect to a closed convex cone K in a Banach space. Our main result extends a wellknown characterization of K-quasiconvexity by means of extreme directions of the polar cone of K, obtained by Dinh The Luc in the particular case when K is a polyhedral cone generate...

Let 2 ≤ p < ∞ and let X be a complex Banach space. It is shown that X is p-uniformly PL-convex if and only if there exists λ > 0 such that ‖f‖Hp(X) ≥ ( ‖f(0)‖p + λ ∫ D (1− |z|2)p−1‖f ′(z)‖pdA(z) )1/p , for all f ∈ Hp(X). Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided. AMS Subject Class. 46B20,46L52

Suppose that B is a separable Banach space and (S, 6 a, P) a probability space. H is a measurable symmetric kernel function from S" into B. In this paper we shall further study some limit theorems for B-valued U-statistics U",,H based on P and H. Special attention is paid upon the Marcinkiewicz type law of large numbers and the law of the iterated logarithm. Our results can be regarded as exten...