Product Rule and Chain Rule Estimates for Fractional Derivatives on Spaces that Satisfy the Doubling Condition
Product Rule and Chain Rule Estimates for Fractional Derivatives on Spaces that Satisfy the Doubling Condition
复制标题
满足倍增条件的空间上的分数阶导数的乘积法则和链式法则估计
DOI：
10.1006/jfan.2001.3836
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发表时间：
2002
影响因子：
1.7
通讯作者：
A.Eduardo Gatto
中科院分区：
文献类型：
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作者：
A.Eduardo Gatto
本文的目的是证明定义在齐型Coifman-Weiss空间上的函数的分数导数的一些经典估计，特别是(T.Kato和G.Ponce，1988，Comm.)中的乘积规则和链规则估计。纯苹果。数学41,891-907)和(F.M.基督和M.I.温斯坦，1991，J.Funct.分析.100，87-109)。M.Riesz的分数阶微积分在(A.E.Gatto，C.Segovia和S.Vagi，1996，Rev.Mat.)中被推广到这些空间。伊比利亚-美洲。我们的主要工具是分数差商和R.Strichartz在(1967，J.Math.Mech.16、9)扩展到这一上下文。对于Rn的特殊情况，我们的方法统一了这些估计的证明，并澄清了Calderon公式对这些结果的作用。由于平方分数导数易于离散化，我们还证明了齐型空间上分数阶Sobolev空间的离散和连续的Triebel-Lizorkin范数是等价的。
The purpose of this paper is to prove some classical estimates for fractional derivatives of functions defined on a Coifman–Weiss space of homogeneous type, in particular the product rule and chain rule estimates in (T. Kato and G. Ponce, 1988, Comm. Pure Appl. Math.41, 891–907) and (F. M. Christ and M. I. Weinstein, 1991, J. Funct. Anal.100, 87–109). The fractional calculus of M. Riesz was extended to these spaces in (A. E. Gatto, C. Segovia, and S. Vagi, 1996, Rev. Mat. Iberoamericana12). Our main tools are fractional difference quotients and the square fractional derivative of R. Strichartz in (1967, J. Math. Mech.16, 9) extended to this context. For the particular case of Rn, our approach unifies the proofs of these estimates and clarifies the role of Calderon's formula for these results. Since the square fractional derivative can be easily discretized, we also show that the discrete and continuous Triebel–Lizorkin norms for fractional Sobolev spaces on spaces of homogeneous type are equivalent.