最大单项振荡希尔伯特变换的稀疏界
Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms
摘要：
对于每个$ d \geq 2$，具有如下多项式振荡的希尔伯特变换满足$ (1, r )$稀疏界，对于所有$ r>1$ $$ H _{ \ast } f (x) = \sup _{ε} \Bigl\lvert \int_{|y| > ε} f (x-y) \frac { e ^{2 πi y ^d }} y\; dy \Bigr\rvert. $$这很快意味着最大截断的弱型不等式，其对于$A_1$权重成立，但即使在勒贝格测度的情况下也是新的。Chanillo和Christ（1987）给出了未加权的弱型估计\emph{without maximal truncations}，但使用了任意多项式。
英文摘要：
For each $ d \geq 2$, the Hilbert transform with a polynomial oscillation as below satisfies a $ (1, r )$ sparse bound, for all $ r>1$ $$ H _{ \ast } f (x) = \sup _{ε} \Bigl\lvert \int_{|y| > ε} f (x-y) \frac { e ^{2 πi y ^d }} y\; dy \Bigr\rvert. $$ This quickly implies weak-type inequalities for the maximal truncations, which hold for $A_1$ weights, but are new even in the case of Lebesgue measure. The unweighted weak-type estimate \emph{without maximal truncations} but with arbitrary polynomials, is due to Chanillo and Christ (1987).
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