A Converse Extrapolation Theorem for Translation-invariant Operators
نویسنده
چکیده
Theorem 1.1. Let G, X, p0, and r be as above. Suppose T is translation invariant, maps L log L to L, and is bounded on L0 . Then T is bounded on L, 1 < p < p0 with an operator norm of O((p − 1) ). This theorem is false without the assumption of translation invariance, since L is not an interpolation space between L log L and L0 . For a concrete counterexample, take E and F be subsets of X of measure 2 and N rp ′ 02 respectively, where N is a large number. Then the operator
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