Higher integrability theorems on time scales from reverse Hölder's inequalities

Abstract

In this paper, we establish some new reverse dynamic inequalities and use them to prove some higher integrability theorems for decreasing functions on time scales. In order to derive our main results, we first prove a new dynamic inequality for convex functions related to the inequality of Hardy, Littlewood and Pólya, known from the literature. Then, we prove a refinement of the famous Hardy inequality on time scales for a class of decreasing functions. As an application, our results are utilized to formulate the corresponding reverse integral and discrete inequalities, which are essentially new.

A Themed Issue on Mathematical Inequalities, Analytic Combinatorics and Related Topics in Honor of Professor Feng Qi

“…The paper [36] is a starting point of [38,39] and many other papers such as [40][41][42][43][44][45][46][47][48]] by other mathematicians.…”

“…Since 2002, his names "Feng Qi", "F. Qi", and "Qi" have appeared in titles of over 89 papers or preprints which were published or announced by hundreds of mathematicians in the globe. See, for example, the papers [40][41][42][43][44][45][46][47][48]88,[179][180][181][182][183][184][185].…”

Fundamental Properties of Muckenhoupt and Gehring Weights on Time Scales

“…In recent years, increasing interest has been paid to the study of properties of Muckenhoupt and Gehring weights on time scales. For example, the authors used the tools on time scales and proved the self-improving properties of the Muckenhoupt and Gehring weights in [13] and proved some higher integrability theorems on time scales in [17]. Motivated by this work, the natural question that arises now is: Is it possible to prove some new properties of Muckenhoupt and Gehring weights on time scales, which, as special cases, cover the properties of the continuous and discrete Muckenhoupt and Gehring weights?…”

Self-Improving Properties of Continuous and Discrete Muckenhoupt Weights: A Unified Approach

“…For more details we refer to the books [37,38] and the references they have cited. Very recently, the authors in [39][40][41][42][43] proved the time scale versions of the Muckenhoupt and Gehring inequalities and used them to prove some higher integrability results on time scales. This also motivated us to develop a new technique on time scales to prove some new results of inequalities with weights and use the new inequalities to formulate some conditions for the boundedness of the Hardy operator with negative powers on time scales and show the applications of the obtained results.…”