Abstract
We establish the Poincare-type inequalities for the composition of the Laplace-Beltrami operator and the Green's operator applied to the solutions of the non-homogeneous A-harmonic equation in the John domain. We also obtain some estimates for the integrals of the composite operator with a singular density.
Highlights
The purpose of the article is to develop the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator Δ = dd* + d*d and Green’s operator G over the δ-John domain. Both operators play an important role in many fields, including partial differential equations, harmonic analysis, quasiconformal mappings and physics [1,2,3,4,5,6]
We consider the composite operator with a singular factor
Lemma 2.1 [17]Let j be a strictly increasing convex function on [0, ∞) with j(0) = 0, and D be a domain in Rn
Summary
The purpose of the article is to develop the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator Δ = dd* + d*d and Green’s operator G over the δ-John domain. Both operators play an important role in many fields, including partial differential equations, harmonic analysis, quasiconformal mappings and physics [1,2,3,4,5,6].
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