Research Interest
My research interests center around Harmonic Analysis with topics such as singular integral operators, multilinear estimates, Littlewood-Paley theory, weighted norm inequalities, almost everywhere convergence, and broader function space estimates.
Mathematical Articles
Submitted
FRACTIONAL CHAIN RULE FOR TRIEBEL-LIZORKIN SPACES
In this paper we establish a fractional chain rule in the setting of weighted Triebel-Lizorkin spaces under various smoothness conditions. Notably, this provides a fractional chain rule for weighted Lebesgue spaces, L^p(w), for p\leq 1. Additionally, an explicit relationship between the smoothness index, integrablity index, and the choice of weights is determined. Download
Submitted Revision
REMARKS ON ALMOST EVERYWHERE CONVERGENCE
We prove almost everywhere convergence for convolutions of locally integrable functions with shrinking L^1 dilations of a fixed integrable kernel with an integrable radially decreasing majorant. The set on which the convergence holds is an explicit subset of the Lebesgue set of the locally integrable function of full measure. This result can be viewed as an extension of the Lebesgue differentiation theorem in which the characteristic function of the unit ball is replaced by a more general kernel. We obtain a similar result for multilinear convolutions. Download
Submitted Revision
KATO-PONCE INEQUALITY WITH MULTIPLE WEIGHTS
We prove the Kato-Ponce inequality (normed Leibniz rule) for multiple factors in the setting of multiple weights (A_{\vec P} weights). This improves existing results by extending the class of known weights for which the inequality holds. Download
Submitted Revision
WEIGHTED MULTIFACTOR KATO-PONCE INEQUALITIES
In this paper, we establish weighted Kato-Ponce inequalities for multiple factors. Our work is carried out in the framework of Muckenhoupt weights with particular focus on the weighted L^1 endpoint case. In this work we also extend the validity of the Kato-Ponce inequality from the class of Schwartz functions to the broader class of weighted Sobolev spaces. Download
Submitted
NORM ESTIMATES FOR THE FRACTIONAL DERIVATIVE OF MULTIPLE FACTORS
We extend the Kato-Ponce inequality to a product of m functions, proving an estimate currently missing from the literature. This study is motivated by the fact that the 3-factor Kato-Ponce does not follow directly from the 2-factor version in the full range of permissible indices. Our methodology is based upon that by Grafakos and Oh but our extension entails a novel decomposition that elegantly and effectively handles the technical difficulties that arise from the combinatorial complexity of the possibly large number of factors. Download
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