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On singular integrals with rough kernels in Triebel-Lizorkin weighted spaces. (English) Zbl 1278.42016

Summary: Let \(\Omega \in L^1(\mathbb S^{n-1})\) have mean value zero and satisfy the condition \[ \sup_{\zeta' \in \mathbb S^{n-1}} \int_{\mathbb S^{n-1}} |\Omega(y')| (\ln |\zeta' \cdot y'|^{1})^{(\ln(e + \ln |\zeta' \cdot y'|^{1}))^{\beta}} \, d\sigma(y') <\infty \] for some \(\beta > 0\). Under certain conditions on the measurable function \(h\), we show that the singular integral \[ Tf(x) = \text{p. v.} \int_{\mathbb{R}^n} \frac{h(|y|) \Omega (y')}{|y|^n} f(x y)\,dy \] is bounded on the Triebel-Lizorkin weighted spaces \(\dot{F}^{\alpha,w}_{p,q}\mathbb{R}^n\). We also study the Marcinkiewicz integral (with the same kernel \(\Omega\) as above) in the \(L^p\) weighted spaces.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
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References:

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