The duality of H^1(R^n) and BMO(R^n) with applications

The reason of interest of this thesis project goes back to the theory of Homogeneous Singular Integral operators (SIOs) on R^n. These operators, whose a prototypical example is the Hilbert transform on R, are bounded operators on L^p(R^n) for 1<p<\infty. As L^1 boundedness fails to hold, a natural question is whether there is a proper subspace of L^1 where these operators are bounded on L^1: this is where real Hardy spaces came in. In Chapter 2, Real Hardy spaces H^p(R^n) are dealt with: for 0<p<\infty, these are spaces of bounded tempered distributions on R^n characterized by the L^p boundedness of the Poisson maximal function. We show that H^p coincides with L^p when p>1, while H^1 is a proper subspace of L^1. In the case 0<p <=1, we prove an Atomic Decomposition, meaning that we can find some special distributions, called atoms, that serve as building blocks for H^p. Lastly, we show, as suggested before, that Homogeneous SIOs are bounded from H^1 to L^1 and, conversely, that H^1 can be equivalently defined in terms of the boundedness of the Riesz transforms, the n-dimensional analogues of the Hilbert transform. In Chapter 3, we prove that the dual space of H^1 is the space of functions of bounded mean oscillation BMO and study the main properties of the latter. In Chapter 4, we give two applications of such duality: the first is a characterization of BMO(R^n), meaning that a locally integrable function belongs to BMO(R^n) if and only if the commutator of the pointwise product by b and any Riesz transform is a linear bounded operator on L^p(\R^n), for 1<p<\infty. This result, originally proved by R. Coifman, R. Rochberg and G. Weiss, here is obtained using a strategy highlighted by B.Wick. Finally, by a duality argument, we prove the Div-Curl lemma, which states that the dot product of two square integrable vector fields, that a priori is in L^1, in fact belongs to H^1, as soon as one vector field is divergence-free and the other is curl-free.