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

• Published in: Axioms 2024-11, Vol.13 (11), p.754
• Main Authors: Saker, Samir H., Mohammed, Naglaa, Rezk, Haytham M., Saied, Ahmed I., Aldwoah, Khaled, Alahmade, Ayman
• Format: Article
• Language: English
• Subjects: Functional equations; Functions; Gehring class; inclusion and transition properties; Inequalities (Mathematics); Mathematical functions; mathematical operators; Muckenhoupt class; Partial differential equations; Time; time scales; weighted functions
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13110754

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.