Two-weight Caccioppoli inequalities for solutions of nonhomogeneous 𝐴-harmonic equations on Riemannian manifolds

Abstract

In this paper, we first prove the local two-weight Caccioppoli inequalities for solutions to the nonhomogeneous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -harmonic equation of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Superscript star Baseline upper A left-parenthesis x comma d omega right-parenthesis equals upper B left-parenthesis x comma d omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋆ </mml:mo> </mml:mrow> </mml:msup> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">d^{\star } A(x, d \omega ) =B(x,d\omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Then, as applications of the local results, we prove the global two-weight Caccioppoli-type inequalities for these solutions on Riemannian manifolds.