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Some Aspects of Vector-Valued Singular Integrals

  • Oscar BlascoEmail author
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Let A, B be Banach spaces and \(1 < p < \infty. \; T\) is said to be a (p, A, B)- CalderóLon–Zygmund type operator if it is of weak type (p, p), and there exist a Banach space E, a bounded bilinear map \(u: E \times A \rightarrow B,\) and a locally integrable function k from \(\mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, x): x \in \mathbb{R}^n\}\) into E such that
$$T\;f(x) = \int u(k(x, y), f(y))dy$$
for every A-valued simple function f and \(x \notin \; supp \; f.\)

The study of the boundedness of such operators between spaces of vector-valued functions under the classical assumption of the kernel is established.

Given a bounded operator T from \(L_A^p (\mathbb{R}^n) \; {\rm to} \; L^p_B(\mathbb{R}^n), \; {\rm and} \; \mathcal{L}(A) \; {\rm and} \; \mathcal{L}(B)\)-valued functions b 1 and b 2, we define the commutator
$$T_{b_1,b_2}(f) = b_2T(f) - T(b_1 f)$$
for any A-valued simple function f. The boundedness of commutators of (p, A, B)- CalderöLon–Zygmund type operators and operator-valued functions in a space of bounded mean oscillation (BMO), under certain commuting properties on the couple (b 1, b 2), are also analyzed.

Keywords

Banach Space Hardy Space Singular Integral Weak Type Commute Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad de ValenciaValenciaSpain

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