| The properties of the commutators and composite operators have been a hot topic of research in harmonic analysis,and there have been quite a lot of results so far.This paper extends the previous works and focuses on the Lorentz boundedness and compactness of the commutator of the ω-type Calderón-Zygmund operator on space of homogeneous type,and the quantitative weighted bounds for the composition of CalderónZygmund operator and fractional integral operator.The main contents are as follows:First research work,we study the Lorentz boundedness and compactness of the commutator of the ω-type Calderón-Zygmund operator on space of homogeneous type.First,we prove the Lorentz boundedness of the ω-type Calderón-Zygmund operator on space of homogeneous type,and establish the pointwise estimation of the sharp maximal function of the commutator,then we use these results to prove the Lorentz boundedness of the commutator; Next,about the compactness of the commutator,we get the pointwise estimation of the related maximal truncated operator,then by using the compactness criterion to prove that the commutator is a compact operator.Second research work,we provide the quantitative weighted bounds for the composition of Calderón-Zygmund operator and fractional integral.First,we establish a bilinear sparse domination of the composite operators,and estimate the constant in the two-weight norm inequality,then we use these results to prove the quantitative weighted bounds for the composition of Calderón-Zygmund operator and fractional integral; Next,we prove the related inequality of the sharp fractional maximal operator,then by using the existing results we get the weighted weak type endpoint estimates of the composite operators. |
