Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces

Blasco, Óscar

doi:10.1016/0022-1236(88)90123-1

Cited by 71 publications

(23 citation statements)

References 9 publications

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“…The reason for the extra care is the fact that, contrary to the scalar case, the functions in H p (X) need not in general have radial boundary values a.e.. In fact, elements in H p (X) can be identified with X-valued L p -functions on the boundary with vanishing negative Fourier coefficients if and only if X has the so called analytic Radon-Nikodým property (see for instance [2]). The boundedness of C φ on the Bloch spaces reduces to the scalar case treated in [10] by noting that From here on we may and shall always assume in the proofs of our theorems that φ(0) = 0, since neither the results nor the conditions on φ will change if φ is composed with a conformal self-map of D.…”

Section: Resultsmentioning

confidence: 99%

“…Given a Banach space E, recall that an operator T ∈ L(E) is weakly conditionally compact (abbreviated T is w.c.c) if any bounded sequence (x n ) in E contains a subsequence (x n k ) such that the sequence (T x n k ) is weakly Cauchy. Rosenthal's 1 -criterion (see [8,2.e.5]) implies that T is not w.c.c if and only if T fixes a copy of 1 in E. Note that if 1 does not embed into E, then every T ∈ L(E) is w.c.c. Hence the set of w.c.c.…”

Section: Weakly Compact If and Only If X Is Reflexive Andmentioning

confidence: 99%

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Small composition operators on analytic vector-valued function spaces

Лю¹,

Saksman²,

Tylli³

1998

Pacific J. Math.

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Section: Resultsmentioning

confidence: 99%

Section: Weakly Compact If and Only If X Is Reflexive Andmentioning

confidence: 99%

Small composition operators on analytic vector-valued function spaces

Лю¹,

Saksman²,

Tylli³

1998

Pacific J. Math.

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“…By contrast, every f ∈ h 1 (X) admits a radial limit function lim r→∞ f (rξ) a.e. on T if and only if X has the Radon-Nikodým property (RNP), see [B3,Thm. 2.2], [H1,Satz 1.12] or [H3,Thm.…”

Section: Composition Operators On Harmonic Hardy Spacesmentioning

confidence: 99%

Composition operators on vector-valued harmonic functions and Cauchy transforms

Laitila¹,

Tylli²

2006

Indiana Univ. Math. J.

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Abstract. Let ϕ be an analytic self-map of the unit disk. The weak compactness of the composition operators Cϕ : f → f •ϕ is characterized on the vector-valued harmonic Hardy spaces h 1 (X), and on the spaces CT (X) of vector-valued Cauchy transforms, for reflexive Banach spaces X. This provides a vector-valued analogue of results for composition operators which are due to Sarason, Shapiro and Sundberg, as well as Cima and Matheson. We also consider the operators Cϕ on certain spaces wh 1 (X) and wCT (X) of weak type by extending an alternative approach due to Bonet, Domański and Lindström. Concrete examples based on minimal prerequisites highlight the differences between h p (X) (respectively, CT (X)) and the corresponding weak spaces.

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“…If 1<p< , the M. Riesz theorem states that the Hilbert transform is bounded from L p (T) into itself, which is equivalent to the fact that the Riesz projection is bounded from L p (T) onto H p (T). The problem of extending classical theorems in Harmonic Analysis to the setting of Banach space valued functions or measures has been considered by several authors [1], [2], [3], [7], [8], ... . It is known that for every F in the Bochner space L p (T, X ), 1 p< , the convolution…”

Section: Introductionmentioning

confidence: 99%