Self-improving properties of weighted Gehring classes with applications to partial differential equations

Abstract

Abstract In this paper, we prove that the self-improving property of the weighted Gehring class G λ p $G_{\lambda }^{p}$ with a weight λ holds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt class A q $A^{q}$ . By using the rearrangement (nonincreasing rearrangement) of the functions and applying the Jensen inequality, we show that the results cover the cases of non-monotonic functions. For applications, we prove a higher integrability theorem and report that the solutions of partial differential equations can be solved in an extended space by using the self-improving property. Our approach in this paper is different from the ones used before and is based on proving some new inequalities of Hardy type designed for this purpose.

Keywords

• Reverse Hölder’s inequality
• Muckenhoupt type inequality
• Reverse Hölder’s inequality
• Higher integrability
• Gehring type inequalities