Proposition 2. A is restricted weak-type (pd, qd). Proof (Proposition 2 ⇒ Theorem 1). Proposition 2 corresponds to an estimate at the vertex Pd of the trapezium. Since A and the adjoint operator A ∗ have the same mapping properties, one concludes there is a restricted weak-type estimate for the exponents corresponding to the vertex P ′ d. By localisation we know A is type (1, 1) and type (∞,∞) ...
