Unified Framework for Continuous and Discrete Relations of Gehring and Muckenhoupt Weights on Time Scales

Abstract

This article contains some relations, which include some embedding and transition properties, between the Muckenhoupt classes Mγ;γ>1 and the Gehring classes Gδ;δ>1 of bi-Sobolev weights on a time scale T. In addition, we establish the relations between Muckenhoupt and Gehring classes, where we define a new time scale T˜=v(T), to indicate that if the Δ˜ derivative of the inverse of a bi-Sobolev weight belongs to the Gehring class, then the Δ derivative of a bi-Sobolev weight on a time scale T belongs to the Muckenhoupt class. Furthermore, our results, which will be established by a newly developed technique, show that the study of the properties in the continuous and discrete classes of weights can be unified. As special cases of our results, when T=R, one can obtain classical continuous results, and when T=N, one can obtain discrete results which are new and interesting for the reader.

Keywords

• Muckenhoupt class
• Gehring class
• inclusion and transition properties
• weighted functions
• time scales
• mathematical operators