Estimates for oscillatory singular integrals on Hardy spaces
Volume 224 / 2014
Studia Mathematica 224 (2014), 277-289
MSC: Primary 42B20; Secondary 42B30.
DOI: 10.4064/sm224-3-5
Abstract
For any $n \in \mathbb {N}$, we obtain a bound for oscillatory singular integral operators with polynomial phases on the Hardy space $H^1(\mathbb {R}^n)$. Our estimate, expressed in terms of the coefficients of the phase polynomial, establishes the $H^1$ boundedness of such operators in all dimensions when the degree of the phase polynomial is greater than one. It also subsumes a uniform boundedness result of Hu and Pan (1992) for phase polynomials which do not contain any linear terms. Furthermore, the bound is shown to be valid on weighted Hardy spaces as well if the weights belong to the Muckenhoupt class $A_1$.
