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Rough oscillatory singular integrals on ℝⁿ

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Al-Qassem H., Cheng L., Pan Y.

Studia Mathematica

2014

tom 221

nr 3

249-267

We establish sharp bounds for oscillatory singular integrals with an arbitrary real polynomial phase P. The kernels are allowed to be rough both on the unit sphere and in the radial direction. We show that the bounds grow no faster than log deg(P), which is optimal and was first obtained by Papadimitrakis and Parissis (2010) for kernels without any radial roughness. Among key ingredients of our methods are an L¹ → L² estimate and extrapolation.

Estimates for oscillatory singular integrals on Hardy spaces

100%

Al-Qassem H., Cheng L., Pan Y.

Studia Mathematica

2014

tom 224

nr 3

277-289

For any n ∈ ℕ, we obtain a bound for oscillatory singular integral operators with polynomial phases on the Hardy space H¹(ℝⁿ). Our estimate, expressed in terms of the coefficients of the phase polynomial, establishes the H¹ 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 A₁.

Ograniczanie wyników

2 Studia Mathematica

2 Al-Qassem H.

2 Cheng L.

2 Pan Y.

2 2014

Wyniki wyszukiwania

Rough oscillatory singular integrals on ℝⁿ

Estimates for oscillatory singular integrals on Hardy spaces