MAXIMAL AND SINGULAR INTEGRAL-OPERATORS VIA FOURIER-TRANSFORM ESTIMATES
MAXIMAL AND SINGULAR INTEGRAL-OPERATORS VIA FOURIER-TRANSFORM ESTIMATES
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DOI：
10.1007/bf01388746
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发表时间：
1986-01-01
影响因子：
3.1
通讯作者：
DEFRANCIA, JLR
中科院分区：
文献类型：
--
作者：
DUOANDIKOETXEA, J;DEFRANCIA, JLR
在奇异积分算子Tf= K，f的经典理论中，起始点是傅立叶变换估计：l ~ eL，由此，不等式I] Tf} l 2< Cllf] l~紧接着得出。这一事实加上关于核的弱正则性假设，得到一个弱（1，1）型不等式（或称L~~不等式），其基本成分是Calderdn-Zygmund分解（见[16]）。利用插值和对偶的方法，证明了T对所有1< p<~ 1是L有界的.对于更奇异的算子尝试了相同的方案，如dt Tf（x）= pv f（x-7（t））·= K* f（x）-wd（沿IR”中曲线7沿着的希尔伯特变换）。在关于7的曲率的某些假设下，/s~的事实仍然可以被证明，但核K现在是曲线中支持的分布，/（一组零测量），这对于Calder 6 n-Zygmund机器来说太奇异了，无法适用。然而，f ~（~）= m（~）的衰减使得可以恶化乘子m，如果0> Re（c 0>-a，则获得一定的m~ _L~，而对于Re（c~）为正，m提高，并且实际上具有m，= f ~，具有局部可积核K~，其足够规则以落入经典理论的范围内。然后通过算子族T，（T 'f）^= fm~的解析插值得到了关于T的Lp不等式.类似的思想导致了相关极大算子的Lp不等式
In the classical theory of singular integral operators Tf= K, f, the starting point was a Fourier transform estimate:/~ eL, from which, the inequality I] Tf} l 2< Cllf] l~ follows immediately. This fact together with mild regularity assumptions on the kernel yield a weak type (1, 1)(or L~~ inequality by a nowadays standard procedure whose basic ingredient is the so-called Calderdn-Zygmund decomposition (see [16]). By interpolation and duality, the L rboundedness of T is proved for all 1< p<~. The same scheme was tried for more singular operators, like dt Tf (x)= pv f (x-7 (t))•= K* f (x)--wd (Hilbert transform along the curve 7 in IR"). Under some hypothesis on the curvature of 7, the fact that/s~ can still be proved, but the kernel K is now a distribution supported in the curve,/(a set of measure zero), which is too singular for the Calder6n-Zygmund machinery to be applicable. However, the decay of/~(~)= m (~) is such that one can worsen the multiplier m, obtaining certain m~ _L~ if 0> Re (c0>-a, while, for Re (c~) positive, m, improves, and one actually has m,=/~, with locally integrable kernels K~ which are regular enough to fall under the scope of the classical theory. The L p inequalities for T are then obtained by analytic interpolation of the family of operators T,(T'f)^= fm~. Similar ideas lead to L p inequalities for the associated maximal operators