Some Aspects of Vector-Valued Singular Integrals

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


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.


Banach Space Hardy Space Singular Integral Weak Type Commute Property 


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© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad de ValenciaValenciaSpain

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