1 Historical origins of the Hardy spaces Hp The Hardy spaces in one variable have their original setting in complex analysis. They first appeared as spaces of holomorphic functions and were introduced with the aim of characterizing boundary values of holomorphic functions on the unit disk D := fjzj < 1g. Namely, let us look at the following problem: what are the possible functions S1 ! C arising as boundary values of some holomorphic function F : D ! C? This question, as just stated, is too vague: due to the lack of compactness of D, holomorphic functions defined on D could exhibit a wild behaviour as we approach the boundary (for instance, we can prescribe arbitrary values of F on any discrete subset of D). In order to obtain a meaningful notion of boundary value, it is natural to impose integrability conditions on our functions F . As a motivation of the forthcoming definitions, let us make a heuristic remark: if the trace of F on @D = S1 is some complex function f 2 Lp(S1), for some 1 ≤ p ≤ 1, then F (which is holomorphic and thus harmonic) is given by the Poisson integral of f. In polar coordinates we have the formula Z iθ i(θ−η) iη F (re ) = Pr(e )f(e ) dη: S1 R iη The Poisson kernel is everywhere positive and satisfies S1 Pr(e ) dη = 1 for any r, so (by Young's inequality on the group S1) kF (r·)kLp(S1) ≤ kfkLp(S1) and in particular all the norms in the left-hand side remain bounded as r " 1. In 1915 Hardy observed that, for any holomorphic function F : D ! C, the map r 7! kF (r·)kLp is nondecreasing (for an arbitrary 0 < p ≤ 1). These observations lead us to define the space p H (D) := F : D ! C holomorphic with lim kF (r·)kLp(S1) < +1 : r"1 When p > 1, using the weak* compactness of Lp(S1), it is not difficult to show that any F 2 Hp(D) is given by the Poisson integral of a complex-valued function f 2 Lp(S1) satisfying fb(k) = 0 for all k < 0. Conversely, given any such f, its Poisson integral lies in Hp(D). Moreover, one can show that F (r·) ! f in Lp(S1) and lim F (reiθ) = f(eiθ) for a.e. θ; r"1 so that f deserves to be regarded as the set of boundary values of F (we mention that for a.e. θ one has even a nontangential convergence of F to f(eiθ)). This settles the problem for 1 < p ≤ 1. Let us also remark that the condition fb(k) = 0 (for all k < 0) amounts to saying that =(f), up to constants, equals the Hilbert-Riesz transform of −<(f). When 1 < p < 1 the Hilbert-Riesz transform maps Lp(S1) 1 into itself, so any function in Lp(S1) arises as the real part of the trace of some element of Hp(D). The case of 0 < p ≤ 1 is more difficult. F. Riesz, in a paper published in 1923, introduced the notation Hp(D) for these spaces of holomorphic function (the letter H stands of course for Hardy) and proved many interesting properties, such as the following factorization theorem. Theorem 1. Any F 2 Hp(D) can be written as F = BG, for suitable holomorphic functions B; G : D ! C such that jBj ≤ 1, G 6= 0 everywhere and G 2 Hp(D)(B is the so-called Blaschke product associated to the zeros of F ). This theorem enabled him to prove the existence of a trace f 2 Lp(S1) such that we have again all the convergence results mentioned before for the case p > 1: the trick 1 is that, as G 6= 0 everywhere, one can take a k-th root of G (for an arbitrary k > p ) and, observing that G1=k 2 Hkp(D) and B 2 H1(D), we get back to the previous case. Again, when p = 1, it can be shown that the possible traces are precisely the complex-valued functions f 2 L1(S1) satisfying fb(k) = 0 for all k < 0. As before, the possible real values of traces of functions form the set H1(S1) = f 2 L1(S1): Rf 2 L1(S1) (here R denotes the Hilbert-Riesz transform). But R does not map L1(S1) into itself any longer, so this set is a proper subspace of L1(S1). 2 These results still hold in the upper half-plane R+ := fz = x + iy : y > 0g, replacing S1 with R (see [2]), leading to the definition of H1(R). Later, in 1960, Stein and Weiss introduced the systems of conjugate harmonic func- tions in several variables, inspiring the correct definition of Hardy spaces in higher dimension. The first characterization avoiding conjugate functions was provided by Burkholder, Gundy and Silverstein in 1971: they proved that a holomorphic function belongs to Hp if and only if the nontangential maximal function of its real part lies in Lp. The importance of this result lies in the fact that it allows to decide the membership of a function f to Hp by looking just at f itself. In 1972 Fefferman and Stein, in a single pioneering paper, provided new real char- acterizations of the Hardy spaces, introducing different useful maximal functions and showing that the Poisson kernel can be replaced essentially by any other ker- nel. In this paper Fefferman and Stein also proved that singular integrals map Hardy spaces to themselves (and in particular H1 to L1), as well as the duality (H1)∗ = BMO. The Littlewood{Paley characterization of these spaces was first given by Peetre, while the atomic decomposition was obtained by Coifman (in one dimension) and Latter (in arbitrary dimension). 2 2 Equivalent definitions and basic properties of H1(Rn) We now introduce the Hardy space H1(Rn), with a strong emphasis on the modern real-variable point of view outlined in the last part of [.] This important space can be characterized in many useful ways: indeed, H1(Rn) is the space of all functions in L1(Rn) satisfying one of the equivalent definitions provided by Theorem 2 ((8) being the closest to the historical one). Before stating the theorem, let us recall that the Schwartz space S(Rn) is a Fr´echet space, with the following increasing sequence of (semi)norms: jαj 2 k=2 X @ k kk := sup (1 + jxj ) α (x); k ≥ 0: x2 n @x R jα|≤k n R Theorem 2. Fix any ' 2 S(R ) with '(x) dx 6= 0. There exists an N ≥ 0 (independent of ') such that the following are equivalent, for a function f 2 L1(Rn): v 1 n (1) the vertical maximal function M'f(x) := supt>0 j't ∗ fj (x) lies in L (R ); (2) the conical maximal function c M'f(x) := sup j't ∗ fj (y) t>0; y2Bt(x) lies in L1(Rn); (3) the tangential maximal function −n−1 t jy − xj M'f(x) := sup j't ∗ fj (y) 1 + t>0; t n y2R lies in L1(Rn); (4) the grand maximal function n GMf(x) := sup fj t ∗ fj (x) j t > 0; 2 S(R ); k kN ≤ 1g lies in L1(Rn); (5) the similar grand maximal function 0 1 GM f(x) := sup fj't ∗ fj (x) j t > 0; 2 Cc (B1(0)); kr kL1 ≤ 1g lies in L1(Rn); (6) there exists an atomic decomposition, namely there exist λk ≥ 0 and 1-atoms ak (see Definition 4) such that 1 1 X 1 n X f = λkak in L (R ); λk < +1; k=0 k=0 3 v 1 n (7) the vertical maximal function with the Poisson kernel, i.e. MP f, lies in L (R ) (notice that P (x) = F −1(e−tjξj) = cn 62 S( n); (1+jxj2)(n+1)=2 R −1 ξj 1 n (8) the tempered distributions Rjf := F −i jξj fb(ξ) belong to L (R ); _ 0 n (9) f belongs to the homogeneous Triebel-Lizorkin space F1;2(R ). Each of the preceding statements defines also a norm on H1(Rn): (1) defines the v norm M'f 1 (and similarly for (2), (3), (4), (5), (7)), (6) induces the norm P L inf k λk (the infimum ranging among all the possible decompositions), (8) provides Pn 1 n the norm kfk 1 + kRjfk 1 and (9) defines kfk _ 0 (which is a norm on L ( )\ L j=1 L F1;2 R _ 0 n F1;2(R )). The proof of this theorem is scattered across the next sections. We will show the following diagram of implications (with the corresponding bounds on the induced norms): (7) (8) ' (4) (6) for a suitable (1) (2) (3) for a suitable ' (5) We left out (9) in the diagram, since its equivalence with the other definitions is slightly involved and invokes the vector-valued space H1(`2), which will be intro- duced in Section 8. Similarly, as we will see in Section 4, the proof of (1) ) (2) is quite circuitous and uses a particular refinement of the proofs of (2) ) (3) ) (4). Let us rely on definition (1) as for now, i.e. let H1(Rn) denote the space of functions f 2 L1( n) satisfying (1) and let kfk := Mv f . A first basic question is R H1 ' L1 1 n 1 n whether Cc (R ) is included in H (R ). Surprisingly, this property (which holds for most of the common functional spaces) fails for H1(Rn), as the next proposition shows.