arXiv subjects

Prabath Silva

Publications and source records attributed to Prabath Silva.

3 recordsLinked to original sources

arXiv2013

On Gevrey Regularity of the Supercritical SQG equation in Critical Besov Spaces

In this paper we show that the solution of the supercrti- cal surface quasi-geostrophic (SQG) equation, starting from initial data in homogeneous critical Besov spaces belong to a subanalytic Gevrey class. In particular, we improve upon the result of Dong and Li in [26], where they showed that the solutions of Chen-Miao-Zhang (cf. [11]) are classical solutions. We extend the approach of Biswas (cf. [7]) to critical, L^p -based Besov spaces, and adapt the point of view of Lemarie- Rieusset (cf. [36]), who treated the operator arising from applying the analytic Gevrey operator to a product of analytic functions as a bilinear multiplier operator. In order to obtain L^p bounds, we prove that our bilinear multiplier operator is of Marcinkiewicz type, and show that due to additional localizations inherited from working in Besov spaces, this condition implies boundedness.

math.AP↗

arXiv2013

Some new light on a few classical results

The purpose of this paper is to describe a unified approach to proving vector-valued inequalities without relying on the full strength of weighted theory. Our applications include the Fefferman-Stein and Cordoba-Fefferman inequalities, as well as the vector-valued Carleson operator. Using this approach we also produce a proof of the boundedness of the classical bi-parameter multiplier operators, that does not rely on product theory. Our arguments are inspired by the vector valued restricted type interpolation used in [1].

math.CA↗

arXiv2012

Vector valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems

Muscalu, Pipher, Tao and Thiele \cite{MPTT} showed that the tensor product between two one dimensional paraproducts (also known as bi-parameter paraproduct) satisfies all the expected $L^p$ bounds. In the same paper they showed that the tensor product between two bilinear Hilbert transforms is unbounded in any range. They also raised the question about $L^p$ boundedness of the bilinear Hilbert transform tensor product with a paraproduct. We answer their question by obtaining a wide range of estimates for this hybrid bilinear operator. Our method relies on new vector valued estimates for a family of bilinear Hilbert transforms.

math.CA↗