Some Aspects of Vector-Valued Singular Integrals

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.