Weighted inequalities for gradients on non-smooth domains ...

emba.uvm.edu

Weighted inequalities for gradients on non-smooth domains ...

Using our bound on λ Q , the last quantity is less than or equal toC ∑ Q b ∈G(|g(Q b )|µ(T (Q b ))l(Q b ) √ ω(Q b )) p′⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠ ;which, since p ′ ≥ q ′ , is bounded by⎛(∑C ⎜|g(Q b )|µ(T (Q b ))⎝l(Q b ) √ ω(Q b )Q b ∈G) q′⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ⎞⎟⎠p ′ /q ′ .In order ong>forong> our dual inequality to hold, the last quantity must be less than or equal to⎛⎞C ⎝ ∑ |g(Q b )| q′ µ(T (Q b ))Q b ∈G⎠p ′ /q ′ ,which will be true ong>forong> all g’s if(µ(T (Q b ))l(Q b ) √ ω(Q b )) q′⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q)(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ≤ Cµ(T (Q b ))ong>forong> all Q b ∈G; i.e., after some transposition,⎛∫µ(T (Q b )) 1/q ⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠1/p ′ ≤ Cl(Q b ) √ ω(Q b ),which is our condition from Theorem 3.1. QED.We shall now state the fairly straightong>forong>ward corollary of Theorem 3.1 to solutions of uniong>forong>mlyelliptic equations in divergence ong>forong>m.Recall that such an equation has the ong>forong>mLu ≡ div(A(x)∇u) =0, (3.7)where A(x) =[a i,j (x)] is a real, bounded, measurable, (d +1)× (d + 1) symmetric matrix satisfyingΛ −1 |ξ| 2 ≤ ∑ i,ja i,j ξ i ξ j ≤ Λ|ξ| 2ong>forong> all x ∈ Ω and all ξ ∈ R d+1 , where Λ is a fixed positive constant. A u that satisfies (3.7) iscalled elliptic.30

More magazines by this user
Similar magazines