Sharp estimates for oscillatory integral operators via polynomial partitioning

Sharp Estimates for One-dimensional Oscillatory Integral Operators with C∞ Phase

It turns out that the answer to this question depends on the behavior of the second derivative ∂ 2S ∂x∂y (x, y). If ∂2S ∂x∂y (0, 0) 6= 0, then by a straightforward TT ∗ argument, one has that the best in (1.2) is exactly 12 . On the other hand, if each multiindex (α, β) with α, β ≥ 1 satisfies ∂ α+βS ∂xα∂yβ S(0, 0) = 0, then it is not too hard to show that there is no > 0 for which (1.2) holds....

Damping Estimates for Oscillatory Integral Operators with Finite Type Singularities

We derive damping estimates and asymptotics of L operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into L almost orthogonality technique of Cotlar-Stein.

Damping Estimates for Oscillatory Integral Operators with Finite Type Singularities

Oscillatory Integral Operators with Homogeneous Polynomial Phases in Several Variables

We obtain L decay estimates in λ for oscillatory integral operators Tλ whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of (2+2)–dimensions for any m, while in higher dimensions the result is sharp for m sufficiently large. The proof for large m follows from essentially algebraic considerat...

Some sharp inequalities for multilinear integral operators

In this paper, some sharp inequalities for certain multilinear operators related to the Littlewood-Paley operator and the Marcinkiewicz operator are obtained. As an application, we obtain the (L p , L q)-norm inequalities and Morrey spaces boundedness for the multilinear operators.