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Abstract. The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableaustyle deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results.

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We examine the averaging operator corresponding to the manifold in R 2d of pairs of points (u, v) satisfying |u| = |v| = |u − v| = 1, so that {0, u, v} is the set of vertices of an equilateral triangle. We establish L p × L q → L r boundedness for T for (1/p, 1/q, 1/r) in the convex hull of the set of points {(0, 0, 0) , (1, 0, 1) , (0, 1, 1) , (1/p d , 1/p d , 2/p d )}, where p d = 5d 3d−2 .

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