arXiv subjects
Publications and source records attributed to Jayson Cunanan.
We establish a sharp bilinear estimate for the Klein--Gordon propagator in the spirit of recent work of Beltran--Vega. Our approach is inspired by work in the setting of the wave equation due to Bez, Jeavons and Ozawa. As a consequence of our main bilinear estimate, we deduce several sharp estimates of null-form type and recover some sharp Strichartz estimates found by Quilodr\'an and Jeavons.
Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.
We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterisation of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. Our argument is not particular to the geometry of the cone and we illustrate this by establishing analogous estimates for the paraboloid.
We discuss the asymptotic stability of stationary solutions to the incompressible Navier-Stokes equations on the whole space in Besov spaces with positive smoothness and low integrability. A critical estimate for the semigroup generated by the Laplacian with a perturbation is main ingredient of the argument.
The paper deals with trace operators of Wiener amalgam spaces using frequency-uniform decomposition operators and maximal inequalities, obtaining sharp results. Additionally, we provide the embeddings between standard and anisotropic Wiener amalgam spaces.
We extended the known result that symbols from modulation spaces $M^{\infty,1}(\mathbb{R}^{2n})$, also known as the Sjöstrand's class, produce bounded operators in $L^2(\mathbb{R}^n)$, to general $L^p$ boundedness at the cost of lost of derivatives. Indeed, we showed that pseudo-differential operators acting from $L^p$-Sobolev spaces $L^p_s(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ spaces with symbols from the modulation space $M^{\infty,1}(\mathbb{R}^{2n})$ are bounded, whenever $s\geq n|1/p-1/2|.$ This estimate is sharp for all $1\leq p\leq\infty$.