Inversion Formula For The Windowed Linear Canonical Transform And Boundedness Of Vector-Valued Calderón-Zygmund Operators

The linear canonical transform is a generalization of the classical Fourier trans-form.It can be used to study many practical transforms in optics and engineering.The fractional Fourier transform,the affine Fourier transform,the generalized Fresnel transform and the Lorentz transform are all its special cases.With more degrees of freedom compared to the Fourier transform and the fractional Fourier transform,the linear canonical transform is more flexible.The windowed linear canonical transform was introduced to study the local properties of the linear canonical transform.Since the windowed linear canonical transform can reflect the local properties,it has high reso-lution and eliminates the cross terms,moreover,the calculation is simple,it is widely used in practice.Reconstructing a function from its windowed linear canonical trans-form is important in both theory and applications.Inspired by this,we study the inverse formula of the windowed linear canonical transform.Since the Fourier transform is a special case of linear canonical transform,this result is also a generalization of the inverse formula of the windowed Fourier transform.Function spaces and singular integral operator theory play an important role in both harmonic analysis and partial differential equations.The metric space satisfy-ing the measure doubling condition is called the homogeneous type space.The dou-bling condition on measure is one of the most important assumptions in the classi-cal harmonic analysis.Later,it was found that many results are still valid in non-doubling measure spaces which we replace the measure satisfying the doubling con-dition with another measure satisfying the polynomial growth condition.However,the non-doubling measure spaces is not a generalization of the homogeneous type space and does not have more generality.To this end,Hyt¨onen introduce the non-homogeneous metric measure spaces in 2010,which include both homogeneous type spaces and non-doubling measure spaces.On the other hand,Calderón-Zygmund oper-ator,as a generalization of Hilbert transform and many other transforms,is very impor-tant in many studies.In this paper,we prove that the vector-valued Calderón-Zygmund operator is bounded on the non-homogeneous metric measure spaces X satisfying non-atomic condition,which further generalize the boundedness of Calderón-Zygmund op-erator.In Chapter one,we mainly introduce the background of this research,the basic content of this paper,some definitions and symbols.In Chapter two,we first prove that every signal can be recovered from its win-dowed linear canonical transform with a univariate integral.For 1<p<∞,we prove that the integral involved is convergent almost everywhere as well as in L^p（R）.For functions in L¹,we obtain the corresponding results by using the method of Ces`aro summability.Furthermore,we prove that every signal can be recovered from its win-dowed linear canonical transform with a discrete series.We introduce two kinds of discrete series.For the first,we prove that the discrete series involved is convergent almost everywhere as well as in L<sup>p</sup>（R）for 1<p<∞.Especially for p=2,we give a necessary and sufficient condition for the convergence of the series involved.For functions in L<sup>1</sup>,we obtain the corresponding results by using the method of Ces`aro summability.For the second,we prove that the discrete series involved is convergent in L^p（R）for 1<p<∞.In Chapter three,we first introduce the non-homogeneous metric measure space which satisfies non-atomic condition,vector-valued Calderón-Zygmund kernel,vector-valued Calderón-Zygmund operator and some related basic knowledge.We prove that the boundedness of the vector-valued Calderón-Zygmund operator from L²（X,B₁）to L²（X,B2）is equivalent to the boundedness from L¹（X,B₁）to L^1,∞（X,B2）,and from L^p（X,B₁）to L^p（X,B2）for p∈（1,∞）.As an applications,we obtain two corollaries.