Reverse Carleson Inequalities For Weighted Bergman Spaces With Békollé Weights

In this paper,we mainly use the characteristics of the Békollé weight function to prove the special form of reverse Carleson inequality and the general form of reverse Carleson inequality in weighted Bergman space.First of all,this paper uses the characterization of the reverse Carleson inequality of the special case in the unweighted Bergman space,combines the characteristics of the Békollé weight function,and promotes the three key lemmas of Luecking to give the dominating set on the Békollé type weighted Bergman space.Characterization of the dominating set by generalizing these three key lemmas,and obtaining the sufficiency and necessity proof of the dominating set,and then using this result to discuss the reversible Toeplitz operator on weighted Bergman space with Békollé weights.Secondly,this paper gives a sufficient condition that the positive Borel measure ? is the reverse Carleson measure on the weighted Bergman space,and the general reverse Carleson inequality is obtained.That is,by giving the relation between the condition of ? and the satisfaction of the set G,using the proof and conclusion of the dominating set during the proof process,we get that any function on weighted Bergman space with Békollé weights satisfies the form of the reverse Carleson inequality,and the general situation of the reverse Carleson inequality is obtained.Finally,this paper obtain the Carleson inequality of the derivative of the function on weighted Bergman space with Békollé weights.By changing the variables of function and finding the n order derivative of function,we can obtain a key proposition of the proof.According to the characterization of the Carleson inequality of the derivative of the function in the unweighted Bergman space,the key proposition is applied to the proof of the Carleson inequality of the derivative of the function,and it is finally proved.Then we propose the idea of studying the reverse Carleson inequality of the derivative of the function.Applying the reverse Carleson inequality of the general case and the Carleson inequality of the derivative of the function to the proof of the reverse Carleson inequality of the derivative of the function,we will get its characterization of the reverse Carleson inequality.