Boundedness of Operators over $$(\mathcal{X},\nu )$$

Abstract

AbstractIn this chapter, we consider the boundedness of Calderón–Zygmund operators over non-homogeneous spaces \((\mathcal{X},\nu )\). We first show that the Calderón–Zygmund operator T is bounded from \({H}^{1}(\mathcal{X},\nu )\) to \({L}^{1}(\mathcal{X},\,\nu )\). We then establish the molecular characterization of a version of the atomic Hardy space \(\tilde{H}_{\mathrm{atb}}^{1,\,p}(\mathcal{X},\nu )\), which is a subspace of \(H_{\mathrm{atb}}^{1,\,p}(\mathcal{X},\nu )\), and obtain the boundedness of T on \(\tilde{H}_{\mathrm{atb}}^{1,\,p}(\mathcal{X},\nu )\). We also prove that the boundedness of T on \({L}^{p}(\mathcal{X},\,\nu )\) with p ∈ (1, ∞) is equivalent to its various estimates and establish some weighted estimates involving the John–Strömberg maximal operators and the John–Strömberg sharp maximal operators, and some weighted norm inequalities for the multilinear Calderón–Zygmund operators. In addition, the boundedness of multilinear commutators of Calderón–Zygmund operators on Orlicz spaces is also presented.KeywordsSharp Maximal OperatorAtomic Hardy SpaceMultilinear CommutatorsWeighted Norm InequalitiesMarcinkiewicz IntegralThese 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.