A variation norm Carleson theorem
A variation norm Carleson theorem
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DOI：
10.4171/jems/307
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发表时间：
2009-10
影响因子：
2.6
通讯作者：
R. Oberlin;A. Seeger;T. Tao;C. Thiele;James Wright
中科院分区：
文献类型：
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作者：
R. Oberlin;A. Seeger;T. Tao;C. Thiele;James Wright
通过标准的近似论证，可以得出，S[f ]可以有意义地定义为几乎对任何x都是连续的函数，只要f ∈ L，并且定理的先验界对这样的函数继续成立。定理1.1与L[0，1]中函数的部分傅立叶和的几乎处处收敛密切相关。通过转移原理[12]，它确实等价于Carleson [2]的著名定理（p = 2）和Hunt [9]的Carleson定理的扩展（1 < p < ∞）;参见[7]，[15]和[8]。本文的主要目的是使定理1.1更加精确，以控制参数摄动中的变差范数。因此，我们考虑混合L和V r范数的类型：
By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by Hunt [9] for 1 < p < ∞; see also [7],[15], and [8]. The main purpose of this paper is to sharpen Theorem 1.1 towards control of the variation norm in the parameter ξ. Thus we consider mixed L and V r norms of the type: