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

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

Thus, the last quantity is bounded by⎛∑C ⎜⎝Q b ∈Gg(Q) q′ µ(T (Q b )) q′l(Q b ) q′ ω(Q b ) q′ /2⎛∫⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ⎞⎟⎠1/q ′ ,and this is supposed to be less than or equal to⎛⎞⎝ ∑ Q b ∈Gg(Q) q′ µ(T (Q b ))⎠1/q ′ .Comparing the sums termwise, we see that our inequality will follow if⎛∫µ(T (Q b )) q′⎜l(Q b ) q′ ω(Q b ) q′ /2⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ≤ cµ(T (Q b ))ong>forong> all Q b ∈G; i.e., that⎛∫µ(T (Q b )) q′ −1 ⎜⎝∂Ω⎡⎤∞∑⎣2 −j(2α−τ)ω(2 j Q b ) χ R j (Q b )(x) ⎦j=0p ′ /2⎞⎟σdω⎠q ′ /p ′ ≤ cl(Q b ) q′ ω(Q b ) q′ /2 ,or, taking q ′ roots and simplifying,⎛∫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 the sufficient condition we stated in Theorem 3.1.Let’s now consider the easier case, 2

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