Weighted Integral Inequality For The Result Of P-Harmonic Type Equation

Harmonic functions have wide applications in many fields, such as potentialtheory, quasiregular mapping, harmonic analysis and H~p-spaces. In recent years, thestudy of the integral properties that refer to the results of A harmonic equations andP-harmonic equations is popular, and Hardy-Littlewood integral inequality for theresult of conjugate harmonic functions has become a valid method to study differentialsystem, Schauder estimating in elliptic and parabolic forms, L~2 theorem about ellipticequations etc.In this paper, We consider weighted Hardy-Littlewood inequality for P-harmonictype tensors. P-harmonic type equations are important generalizations of A-harmonicequations and P-harmonic equations, so the weighted Hardy-Littlewood inequality forP-harmonic type tensors is more widespread, and when we set special values to theweights we can get some results of A-harmonic equations and P-harmonic equations.Firstly we use the generalized Ho¨lder inequality and the properties of the weightto prove a local A_r-weighed Hardy-Littlewood integral inequality for the result of P-harmonic type functions. Then by using the local weighted integral inequality and theWhitney decomposition, we prove a global weighted integral inequality for conjugateP-harmonic type tensors inδ-John domains.