For each $$\alpha \in \mathbb {T}$$ α ∈ T consider the discrete quadratic phase Hilbert transform acting on finitely supported functions $$f : \mathbb {Z} \rightarrow \mathbb {C}$$ f : Z → C according to $$\begin{aligned} H^{\alpha }f(n):= \sum _{m \ne 0} \frac{e^{i\alpha m^2} f(n - m)}{m}. \end{aligned}$$ H α f ( n ) : = ∑ m ≠ 0 e i α m 2 f ( n - m ) m . We prove that, uniformly in $$\alpha \in \mathbb {T}$$ α ∈ T , there is a sparse bound for the bilinear form $$\left\langle H^{\alpha } f , g \right\rangle $$ H α f , g for every pair of finitely supported functions $$f,g : \mathbb {Z}\rightarrow \mathbb {C}$$ f , g : Z → C . The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes.
Analysis and Mathematical Physics – Springer Journals
Published: Sep 25, 2017
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