Abstract
We first introduce and study a new family of weights, the A(α, β, γ; E)-class which contains the well-known A r (E)-weight as a proper subset. Then, as applications of the A(α, β, γ ;E)-class, we prove the local and global Poincaré inequalities with the Radon measure for the solutions of the non-homogeneous A-harmonic equation which belongs to a kind of the nonlinear partial differential equations.2000 Mathematics Subject Classification: Primary 26D10; Secondary 35J60; 31B05; 58A10; 46E35.
Highlights
Let Ω be a domain in Rn, n ≥ 2, B be a ball and sB be the ball with the same center and diam(sB) = sdiam(B), s > 0
We say w is a weight if w ∈ L1loc(Rn) and w > 0 a.e
In 1972, Muckenhoupt [1] introduced the following Ar(E)-weight in order to study the properties of the
Summary
We say that a measurable function g(x) defined on a subset E ⊂ Rn satisfies the A(a, b, g; E)-condition for some positive constants a, b, g, write g(x) Î A (a, b, g; E) if g(x) > 0 a.e., and sup where the supremum is over all balls B ⊂ E. Which can be written as Similar to inequality (2.4), using (2.6) and the definitions of the Ar(E)-weight and the A(a, b,g; E)-class, we obtain that Ar(E) ⊂ A(a, 1/(r-1), a; E) for any a with 0
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