Paper's Title:

A Gradient Estimate for Riemannian Manifolds

Author(s):

Rene Erlin Castillo, Hector Camilo Chaparro

Department of Mathematics,
Universidad Nacional de Colombia,
Bogota,
Colombia.
recastillo@unal.edu.co

Program of Mathematics,
Universidad de Cartagena,
Cartagena de Indias,
Colombia.
hchaparrog@unicartagena.edu.co

Abstract:

In this paper we introduce the Kato class and the non-linear Kato class, on a Riemannian manifold of dimension n. We also obtain a gradient estimate for the non-linear Kato class.

Paper's Title:

On the Boundedness of the Discrete Hilbert Transform: An Elementary Proof

Author(s):

R. E. Castillo and H. C. Chaparro

Department of Mathematics,
Universidad Nacional de Colombia,
Bogota,
Colombia..
E-mail: recastillo@unal.edu.co

Program of Mathematics,
Universidad de Cartagena,
Cartagena de Indias,
Colombia.
E-mail: hchaparrog@unicartagena.edu.co

Abstract:

In this short note, we present an elementary proof of the boundedness of the discrete Hilbert transform on l_p(Z)$ spaces for 1 < p < ∞. Our approach relies solely on Hölder's inequality, avoiding more sophisticated tools from harmonic analysis. This offers a simplified and accessible pathway to understanding a classical result in operator theory.

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