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is the most fundamental example. Much like in the classical case of the Lebesgue di erentiation theorem, pointwise convergence almost everywhere of the inverse Fourier transform to f ∈ Lp (R) can be reduced to Lp bounds for the maximal operator C. Weak type L2 bounds were rst obtained by Lennart Carleson in 1966 [8], thus providing a surprising a rmative solution to the question of pointwise convergence of Fourier series of L2 functions. Carleson’s L2 theorem was almost immediately followed by Hunt’s extension to Lp , 1 < p < ∞. The Hilbert transform H , whose Lp boundedness controls Lp-norm convergence of the inverse partial Fourier transforms, commutes with dilations and translations of the input function; these properties are shared by the class of Calderón-Zygmund singular integral operators, for which H is a paradigm. The action of operators from this class can be e ciently represented in terms, loosely speaking, of an orthonormal basis exhibiting the same dilation and translation invariances; this is often referred to as a Littlewood-Paley basis. A non-smooth analogue for such an object is the Haar basis of L2 (R), which, in fact, is invariant under dyadic dilations and translations. In addition to commuting with dilations and translations, the Carleson operator is modulation invariant, in the sense that