Calderón-Zygmund theory for operator-valued kernels. (English) Zbl 0627.42008

Rubio de Francia, José L.; Ruiz, Francisco J.; Torrea, José L

Adv. Math. 62, 7-48 (1986).

The authors present a new angle of the classical paper of Benedek- Calderón-Panzone on convolution operators on Banach space valued functions; with BMO, UMD, atoms, weights, Littlewood-Paley operators, operators with variable kernels, in the vector-valued setting. From the new results: For every \(\epsilon >0\), there exists \(C_{\epsilon}>0:\) \((T^*f)^{\#}\leq C_{\epsilon}M_{1+\epsilon}f\), \(f\in L_ c^{\infty}\); then \(T^*\) is bounded from \(L_ c^{\infty}\) to BMO. And \[ \int_{R^ n}T^*f(x)^ pw(x)dx\leq C_{p,w}\int_{R^ n}| f(x)|^ pw(x)dx, \] for all \(1<p<\infty\), \(w\in A_ p\).

Reviewer: Bui Doan Khanh

MSC:

42B20

Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords:

convolution operators on Banach space valued functions

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