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Yuanqi Wang : A critical elliptic theory and its applications in higher-dimensional gauge theory.

The celebrated result of Lockhart-Mcowen says that on a non-compact complete manifold, an elliptic operator (with proper asymptotic conditions) is Fredholm between weighted Sobolev spaces if and only if the weight is not an indicial root. We show that a proper weighted Sobolev-theory exists even when the weight is an indicial root. We also discuss some applications to singular $G_{2}-$instantons which converges to their tangent cones in polynomial rates.

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