A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces
© Chen and H. Li. 2009
Received: 27 April 2009
Accepted: 2 July 2009
Published: 4 August 2009
We investigate the functions spaces on for which the generalized partial derivatives exist and belong to different Lorentz spaces , where and is nonincreasing and satisfies some special conditions. For the functions in these weighted Sobolev-Lorentz spaces, the estimates of the Besov type norms are found. The methods used in the paper are based on some estimates of nonincreasing rearrangements and the application of , weights.
It is easy to see that inequality (1.5) fails to hold for , but, it was proved in  that (1.5) is true for and .
Now we state the main theorem in .
In many cases, the Lorentz space should be substituted by more general space, the weighted Lorentz space. In this paper, we will generalize the above result when the weighted Lorentz spaces take place of , where is a weight on which satisfies some special conditions.
2. Auxiliary Proposition
Lemma 2.1 (Generalized Hardy's inequalities).
It is easy to obtain this result applying Hardy's inequality .
Finally, using Lemma 2.1 and (2.19), we get (iii). The Lemma 2.2 is proved.
We will use the notations (2.23).
The proof is similar to [15, Lemma ]. All the argument holds true when we substitute the weight in this lemma for .
So we immediately get (2.34).
Inequality (2.53) now follows from (2.44) and (2.55).
This statement can be easily got from Lemma 2.6. Inequality (2.58) gives a generalization of Remark of  when because satisfies the preceding conditions.
Beyond constant weights, there are many weights satisfying conditions of Corollary 2.7. For example,
3. The Main Theorem
(3.10) then follows.
we get the inequality (3.6). The theorem is proved.
Then we have the following.
We can easily obtain the similar result to Lemma in  by substituting for there. Now the corollary is obvious using the Hardy's inequality and Theorem 3.1.
This work is supported by NSFC (no. 10571156, 10871173), Natural Science Foundation of Zhejiang Province (no. Y606117), Foundation of Zhejiang Province Education Department (no. Y200803879) and 2008 Excellent Youth Foundation of College of Zhejiang Province (no. 01132047).
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