WEAK HARDY SPACES
LOUKAS GRAFAKOS AND DANQING HE
Abstract. We provide a careful treatment of the weak Hardy spaces Hp,∞(Rn) for all indices 0 < p < ∞. The study of these spaces presents differences from the study of the Hardy-Lorentz spaces Hp,q(Rn) for q < ∞, due to the lack of a good dense subspace of them. We obtain several properties of weak Hardy spaces and we discuss a new square function characterization for them, obtained by He [16].
Contents 1. Introduction 1 2. Relevant background 2 3. The proof of Theorem 1 5 4. Properties of Hp,∞ 18 5. Square function characterization of Hp,∞ 22 References 24
1. Introduction The impact of the theory of Hardy spaces in the last forty years has been significant. The higher dimensional Euclidean theory of Hardy spaces was developed by Fefferman and Stein [10] who proved a variety of characterizations for them. A deep atomic decomposition characterization of these spaces was given by Coifman [3] in dimension one and by Latter [18] in higher dimensions. The treatments of Hardy spaces in Lu [19], Garc´ıa-Cuerva and Rubio de Francia [11], Grafakos [13], Stein [24], and Triebel [25] cover the main aspects of their classical theory in the Euclidean setting. Among hundreds of references on this topic, the works of Coifman, and Weiss [4], Mac´ıas, and Segovia [22], Duong, and Yan [7], Han, M¨ullerand Yang [15], Hu, Yang, and Zhou [17] contain powerful extensions of the theory of Hardy spaces to the setting of spaces of homogeneous type. A new type of Hardy space, called Herz-type Hardy space was introduced by Lu and Yang [21] to measure the localization fine-tuned on cubical shells centered at the origin. In this work we provide a careful treatment of the weak Hardy space Hp (henceforth Hp,∞) on Rn for 0 < p < ∞. This is defined as the space of all bounded tempered
Date: June 24, 2015. 1991 Mathematics Subject Classification. Primary 42B20. Secondary 46E35. Key words and phrases. Besov spaces, Calder´on-Zygmund operators. Grafakos’ research partially supported by the NSF under grant DMS 0900946. 1 2 LOUKAS GRAFAKOS AND DANQING HE distributions whose Poisson maximal function lies in weak Lp (or Lp,∞). The study of these spaces presents certain crucial differences from the study of the Hardy-Lorentz spaces Hp,q when q < ∞, due to the fact that they lack a good dense subspace of smooth functions such as the Schwartz class; this is explained in Theorem 6 and it was also observed by Fefferman and Soria [9]. As a consequence, certain results concerning these spaces cannot be proved by restricting attention to the Schwartz class. In this article we bypass this difficulty to obtain several maximal characterizations of weak Hardy spaces working directly from the definition and we prove an interpolation result for these spaces working with general bounded distributions. It is well known that function spaces can be characterized in terms of Littlewood- Paley square function expressions. Such characterizations are given in Ding, Lu, Xue [6], Lu, Yang [20], Peetre [23], and Triebel [25], but we discuss here a new square function characterization of Hp,∞ in terms of the Littlewood-Paley operators, a result that was recently obtained by He [16]. Fefferman, Riviere, and Sagher [8], Fefferman and Soria [9], Alvarez [2], Abu- Shammala and Torchinsky [1] have obtained a variety of results concerning the weak Hp spaces. Fefferman, Riviere, and Sagher [8] have studied interpolation between the Hardy-Lorentz spaces Hp,q. Fefferman and Soria [9] carefully investigated the space H1,∞, and they provided its atomic decomposition. Alvarez [2] provided the atomic decomposition of the spaces Hp,∞ and she also studied the action of singular integrals on them. In [8], Fefferman, Riviere and Sagher obtained the spaces Hp,q as an intermediate interpolation space of Hardy spaces, but their proof was only given for Schwartz functions which are not dense in Hp,∞, and thus their proof contains an incomplete deduction in the case q = ∞. He [16] overcomes the technical issues arising from the lack of density of Schwartz functions via a Calder´on-Zygmund type decomposition for general weak Hp distributions. Some results in the literature of weak Hardy spaces are based on the interpolation result in [8] and possibly on the assumption that locally integrable functions are dense in this space. Although the latter is unknown as of this writing, the former is possible and is explained here. Our exposition builds the theory of weak Hardy spaces, starting from the classical definition of the Poisson maximal function. We discuss various maximal characteri- zations of these spaces and we state an interpolation theorem for Hp,∞ from initial p0 p1 strong H and H estimates with p0 < p < p1. Using this interpolation result, the second author [16] has obtained a new square function characterization for the spaces Hp,∞, which is presented here without proof. This characterization is based on a singular integral estimate for vector-valued weak Hp spaces. For this reason, we develop the theory of weak Hardy spaces in the vector-valued setting.
2. Relevant background To introduce the vector-valued weak Hardy spaces we need a sequence of defini- tions given in this section. We denote by `2 the space `2(Z) of all square-integrable sequences and by `2(L) the finite-dimensional space of all sequences of length L ∈ Z+ 2 0 n 2 with the ` norm. We say that a sequence of distributions {fj}j lies in S (R , ` ) if WEAK HARDY SPACES 3 there are constants C, M > 0 such that for every ϕ ∈ S(Rn) we have 1/2 X 2 X β α {hfj, ϕi}j `2 = |hfj, ϕi| ≤ C sup |y ∂ ϕ(y)|. y∈Rn j |α|,|β|≤M
~ n And this sequence of distributions f = {fj}j is called bounded if for any ϕ ∈ S(R ) we have 1/2 X 2 ∞ n (1) {ϕ ∗ fj}j `2 = |ϕ ∗ fj| ∈ L (R ). j Let a, b > 0 and let Φ be a Schwartz function on Rn.
~ n Definition 1. For a sequence f = {fj}j∈Z of tempered distributions on R we define the smooth maximal function of f~ with respect to Φ as ~ M(f ; Φ)(x) = sup {(Φt ∗ fj)(x)}j `2 . t>0 We define the nontangential maximal function with aperture a of f~ with respect to Φ as ∗ ~ M a(f ; Φ)(x) = sup sup {(Φt ∗ fj)(y)}j `2 . t>0 y∈Rn |y−x|≤at We also define the auxiliary maximal function {(Φ ∗ f )(x − y)} ∗∗ ~ t j j `2 M b (f ; Φ)(x) = sup sup −1 b . t>0 y∈Rn (1 + t |y|)
For a fixed positive integer N we define the grand maximal function of f~ as ~ ∗ ~ (2) MN (f ) = sup M 1(f ; ϕ) , ϕ∈FN where
n n o (3) FN = ϕ ∈ S(R ): NN (ϕ) ≤ 1 , and Z N X α NN (ϕ) = (1 + |x|) |∂ ϕ(x)| dx n R |α|≤N+1 is the “norm” of ϕ. More generally, we define the “norm” of ϕ adapted to the pair n + (x0,R) ∈ R × R by setting
Z N x − x0 X |α| α NN (ϕ; x0,R) = 1 + R |∂ ϕ(x)| dx . n R R |α|≤N+1
Note that NN (ϕ; 0, 1) = NN (ϕ). 4 LOUKAS GRAFAKOS AND DANQING HE
If the function Φ is the Poisson kernel, then the maximal functions M(f~ ; Φ), ∗ ~ ∗∗ ~ M a(f ; Φ), and M b (f ; Φ) are well defined for sequences of bounded tempered dis- ~ tributions f = {fj}j in view of (1). We note that the following simple inequalities ~ ∗ ~ b ∗∗ ~ (4) M(f ; Φ) ≤ M a(f ; Φ) ≤ (1 + a) M b (f ; Φ) are valid. We now define the vector-valued Hardy space Hp,∞(Rn, `2). ~ Definition 2. Let f = {fj}j be a sequence of bounded tempered distributions on Rn and let 0 < p < ∞. We say that f~ lies in the vector-valued weak Hardy space Hp,∞(Rn, `2) vector-valued Hardy space if the Poisson maximal function ~ M(f ; P )(x) = sup {(Pt ∗ fj)(x)}j `2 t>0 lies in Lp,∞(Rn). If this is the case, we set 1 2 ~ ~ X 2 f Hp,∞(Rn,`2) = M(f ; P ) Lp,∞(Rn) = sup |Pε ∗ fj| . ε>0 Lp,∞(Rn) j The next theorem provides a characterization of Hp,∞ in terms of different maximal functions. Theorem 1. Let 0 < p < ∞. Then the following statements are valid: (a) There exists a Schwartz function Φ with integral 1 and a constant C1 such that ~ ~ (5) M(f ; Φ) Lp,∞(Rn,`2) ≤ C1 f Hp,∞(Rn,`2) ~ for every sequence f = {fj}j of tempered distributions. n (b) For every a > 0 and Φ in S(R ) there exists a constant C2(n, p, a, Φ) such that ∗ ~ ~ (6) M a(f ; Φ) Lp,∞(Rn,`2) ≤ C2(n, p, a, Φ) M(f ; Φ) Lp,∞(Rn,`2) ~ for every sequence f = {fj}j of tempered distributions. n (c) For every a > 0, b > n/p, and Φ in S(R ) there exists a constant C3(n, p, a, b, Φ) such that ∗∗ ~ ∗ ~ (7) M b (f ; Φ) Lp,∞(Rn,`2) ≤ C3(n, p, a, b, Φ) M a(f ; Φ) Lp,∞(Rn,`2) ~ for every sequence f = {fj}j of tempered distributions. n R (d) For every b > 0 and Φ in S(R ) with Rn Φ(x) dx 6= 0 there exists a constant C4(b, Φ) such that if N = [b] + 1 we have ~ ∗∗ ~ (8) MN (f ) Lp,∞(Rn,`2) ≤ C4(b, Φ) M b (f ; Φ) Lp,∞(Rn,`2) ~ for every sequence f = {fj}j of tempered distributions. (e) For every positive integer N there exists a constant C5(n, N) such that every ~ ~ sequence f = {fj}j of tempered distributions that satisfies MN (f ) Lp,∞(Rn,`2) < ∞ is bounded and satisfies ~ ~ (9) f Hp,∞(Rn,`2) ≤ C5(n, N) MN (f ) Lp,∞(Rn,`2) , WEAK HARDY SPACES 5 that is, it lies in the Hardy space Hp,∞(Rn, `2). ~ We conclude that for bounded distributions f = {fj} the following equivalence of quasi-norms holds ~ ∗∗ ~ ∗ ~ ~ MN (f ) Lp,∞ ≈ Mb (f; Φ) Lp,∞ ≈ Ma (f; Φ) Lp,∞ ≈ M(f; Φ) Lp,∞ with constants that depend only on Φ, a, n, p, and all the preceding quasi-norms are ~ also equivalent with kf kHp,∞(Rn,`2). There is an alternative characterization of the weak Hp quasi-norm via the weak Lp quasi-norm of the associated square function. As usual, we denote by Ψ ∆j(f) = ∆j (f) = Ψ2−j ∗ f −n the Littlewood-Paley operator of f, where Ψt(x) = t Ψ(x/t). Theorem 2. ([16]) Let Ψ be a radial Schwartz function on Rn whose Fourier trans- 1 1 1 form is nonnegative, supported in 2 + 10 ≤ |ξ| ≤ 2 − 10 , and satisfies (40). Let ∆j be the Littlewood–Paley operators associated with Ψ and let 0 < p < ∞. Then there p n exists a constant C = Cn,p,Ψ such that for all f ∈ H (R ) we have
1 2 X 2 (10) |∆j(f)| ≤ C f p,∞ . Lp,∞ H j∈Z Conversely, suppose that a tempered distribution f satisfies
1 X 22 (11) |∆j(f)| < ∞ . Lp,∞ j∈Z Then there exists a unique polynomial Q(x) such that f −Q lies in Hp,∞ and satisfies the estimate 1 1 2 X 2 (12) f − Q p,∞ ≤ |∆j(f)| . C H Lp,∞ j∈Z The following version of the classical Fefferman-Stein vector-valued inequality [10] is useful. This result for upper Boyd indices less than infinity is contained in [5] (page 85). A self-contained proof of the following result is contained in [16].
p,∞ q Proposition 1. If 1 < p, q < ∞, then for all sequences of functions {fj}j in L (` ) we have
kk{M(fj)}k`q kLp,∞ ≤ Cp,q kk{fj}k`q kLp,∞ , where M is the Hardy-Littlewood maximal function.
3. The proof of Theorem 1 The proof of this theorem is based on the following lemma whose proof can be found in [13]. 6 LOUKAS GRAFAKOS AND DANQING HE
+ n R Lemma 1. Let m ∈ Z and let Φ in S(R ) satisfy Rn Φ(x) dx = 1. Then there n exists a constant C0(Φ, m) such that for any Ψ in S(R ), there are Schwartz functions Θ(s), 0 ≤ s ≤ 1, with the properties Z 1 (s) (13) Ψ(x) = (Θ ∗ Φs)(x) ds 0 and Z m (s) m (14) (1 + |x|) |Θ (x)| dx ≤ C0(Φ, m) s Nm(Ψ). Rn We now prove Theorem 1. Proof. (a) Consider the function ψ(s) defined on the interval [1, ∞) as follows: √ √ 1 e 1 − 2 (s−1) 4 2 1 (15) ψ(s) = e 2 sin (s − 1) 4 . π s 2 Clearly ψ(s) decays faster than any negative power of s and satisfies ( Z ∞ 1 if k = 0, (16) sk ψ(s) ds = 1 0 if k = 1, 2, 3,... . We now define the function Z ∞ (17) Φ(x) = ψ(s)Ps(x) ds , 1 where Ps is the Poisson kernel. Note that the double integral Z Z ∞ s −N n+1 s ds dx Rn 1 (s2 + |x|2) 2 R converges and so it follows from (16) and (17) that Rn Φ(x) dx = 1. Furthermore, we have that Z ∞ Z ∞ −2πs|ξ| Φ(b ξ) = ψ(s)Pcs(ξ) ds = ψ(s)e ds 1 1 −2πs|ξ| using that Pcs(ξ) = e . This function is rapidly decreasing as |ξ| → ∞ and the same is true for all the derivatives Z ∞ α α −2πs|ξ| (18) ∂ Φ(b ξ) = ψ(s)∂ξ e ds . 1 Moreover, the function Φb is clearly smooth on Rn \{0} and we will show that it also smooth at the origin. Notice that for all multiindices α we have
α −2πs|ξ| |α| −mα −2πs|ξ| ∂ξ (e ) = s pα(ξ)|ξ| e + for some mα ∈ Z and some polynomial pα(ξ). By Taylor’s theorem, for some function v(s, |ξ|) with 0 ≤ v(s, |ξ|) ≤ 2πs|ξ|, we have L X |ξ|k (−2πs|ξ|)L+1 e−2πs|ξ| = (−2π)k sk + e−v(s,|ξ|) . k! (L + 1)! k=0 WEAK HARDY SPACES 7
Choosing L > mα gives
L k L+1 α −2πs|ξ| X k |ξ| k+|α| pα(ξ) |α| pα(ξ) (−2πs|ξ|) −v(s,|ξ|) ∂ξ (e ) = (−2π) s + s e , k! |ξ|mα |ξ|mα (L + 1)! k=0 which inserted in (18) and in view of (16), yields that when |α| > 0, the derivative ∂αΦ(b ξ) tends to zero as ξ → 0 and when α = 0, Φ(b ξ) → 1 as ξ → 0. We conclude that Φb is continuously differentiable, and hence smooth at the origin, hence it lies in the Schwartz class, and thus so does Φ. Finally, we have the estimate 1/2 ~ X 2 M(f; Φ)(x) = sup |(Φt ∗ fj)(x)| t>0 j Z ∞ 21/2 X = sup ψ(s)(Pts ∗ fj)(x) ds t>0 j 1 Z ∞ 1/2 Z ∞ 1/2 X 2 ≤ |ψ(s)| ds sup |(Pts ∗ fj)(x)| |ψ(s)| ds t>0 1 j 1 Z ∞ 1/2 Z ∞ 1/2 X 2 ≤ |ψ(s)| ds sup |(Pts ∗ fj)(x)| |ψ(s)| ds t>0 1 1 j Z ∞ ≤ |ψ(s)| ds M(f~; P )(x) , 1 R ∞ and the required conclusion follows since 1 |ψ(s)| ds ≤ C1. We have actually ob- ~ ~ tained the pointwise estimate M(f; Φ) ≤ C1 M(f; P ) which clearly implies (5).
(b) We present the proof only in the case when a = 1 since the case of general a > 0 is similar. We derive (6) as a consequence of the estimate
∗ ~ p 00 p ~ p 1 ∗ ~ p (19) M (f; Φ) p,∞ ≤ C (n, p, Φ) M(f; Φ) p,∞ + M (f; Φ) p,∞ , 1 L 2 L 2 1 L ∗ ~ which requires a priori knowledge of the fact that kM1 (f; Φ)kLp,∞ < ∞. This presents a significant hurdle that needs to be overcome by an approximation. For this reason ∗ ~ ε,N we introduce a family of maximal functions M1 (f; Φ) for 0 ≤ ε, N < ∞ such that ∗ ~ ε,N ∗ ~ ε,N ∗ ~ kM1 (f; Φ) kLp < ∞ and such that M1 (f; Φ) ↑ M1 (f; Φ) as ε ↓ 0 and we prove ∗ ~ ε,N ∗ ~ ε,N (19) with M1 (f; Φ) in place of M1 (f; Φ) , i.e., we prove
∗ ~ ε,N p 0 p ~ p 1 ∗ ~ ε,N p (20) M (f; Φ) p,∞ ≤ C (n, p, Φ,N) M(f; Φ) p,∞ + M (f; Φ) p,∞ , 1 L 2 L 2 1 L 0 where there is an additional dependence on N in the constant C2(n, p, Φ,N), but ∗ ~ ε,N there is no dependence on ε. The M1 (f; Φ) are defined as follows: for a bounded 8 LOUKAS GRAFAKOS AND DANQING HE distribution f~ in S0(Rn, `2) such that M(f~; Φ) ∈ Lp we define
1/2 N ∗ ~ ε,N X 2 t 1 M1 (f; Φ) (x) = sup sup (Φt ∗ fj)(y) N . 1 |y−x| ∗ ~ ε,N p n ∞ n We first show that M1 (f; Φ) lies in L (R ) ∩ L (R ) if N is large enough ~ ~ depending on f. Indeed, using that (Φt ∗ fj)(x) = hfj, Φt(x − ·)i and the fact that f 0 n 2 is in S (R , ` ), we obtain constants C = Cf~ and m = mf~, m > n such that: 1 2 X 2 X γ β (Φt ∗ fj)(y) ≤ C sup |w (∂ Φt)(y − w)| w∈Rn j |γ|≤m,|β|≤m X m m β ≤ C sup (1 + |y| + |z| )|(∂ Φt)(z)| z∈Rn |β|≤m m X m β ≤ C (1 + |y| ) sup (1 + |z| )|(∂ Φt)(z)| z∈Rn |β|≤m m (1 + |y| ) X m β ≤ C n n+m sup (1 + |z| )|(∂ Φ)(z/t)| min(t , t ) z∈Rn |β|≤m m (1 + |y|) m X m β ≤ C n n+m (1 + t ) sup (1 + |z/t| )|(∂ Φ)(z/t)| min(t , t ) z∈Rn |β|≤m m −m m −n −n−m ≤ Cf,~ Φ(1 + ε|y|) ε (1 + t )(t + t ) . t N −N 1 Multiplying by ( t+ε ) (1 + ε|y|) for some 0 < t < ε and |y − x| < t yields 1/2 −m −m n−N n+m−N X 2 t N 1 ε (1 + ε )(ε + ε ) (Φt ∗ fj)(y) ≤ C ~ , t + ε (1 + ε|y|)N f,Φ (1 + ε|y|)N−m j 1 00 ~ and using that 1 + ε|y| ≥ 2 (1 + ε|x|), we obtain for some C (f, Φ, ε, n, m, N) < ∞, C00(f,~ Φ, ε, n, m, N) M ∗(f~; Φ)ε,N (x) ≤ . 1 (1 + ε|x|)N−m ∗ ~ ε,N p,∞ n Taking N > m + n/p, we have that M1 (f; Φ) lies in L (R ). This choice of N ~ depends on m and hence on the sequence of distributions f = {fj}j. We now introduce functions 1/2 N ~ ε,N X 2 t 1 U(f; Φ) (x) = sup sup t ∇(Φt ∗ fj)(y) N 1 |y−x| We need the norm estimate 2 ~ ε,N p ∗ ~ ε,N (21) V (f; Φ) Lp,∞ ≤ C(n) M1 (f; Φ) Lp,∞ and the pointwise estimate (22) U(f~; Φ)ε,N ≤ A(n, p, Φ,N) V (f~; Φ)ε,N , where n [ 2n ]+1 X 2n A(Φ, N, n, p) = 2 p C0(∂jΦ,N + [ ] + 1) N 2n (∂jΦ) . p N+[ p ]+1 j=1 To prove (21) we observe that when z ∈ B(y, t) ⊆ B(x, |x − y| + t) we have 1 2 N X 2 t 1 ∗ ε,N (Φt ∗ fj)(y) ≤ M (f; Φ) (z) , t + ε (1 + ε|y|)N 1 j from which it follows that for any y ∈ Rn, 1 2 X 2 t N 1 (Φt ∗ fj)(y) t + ε (1 + ε|y|)N j 2 1 Z p p ∗ ~ ε,N 2 ≤ M1 (f; Φ) (z) dz |B(y, t)| B(y,t) 2n 2 |x − y| + t p 1 Z p p ∗ ~ ε,N 2 ≤ M1 (f; Φ) (z) dz t |B(x, |x − y| + t)| B(x,|x−y|+t) 2n [ ]+1 2 p p |x − y| + t p ≤ M M ∗(f~; Φ)ε,N 2 (x) . t 1 We now use the boundedness of the Hardy–Littlewood maximal operator M from L2,∞ to L2,∞ to obtain (21) as follows: p 2 ~ ε,N ∗ ~ ε,N 2 p V (f; Φ) Lp,∞ = M (M1 (f; Φ) ) L2,∞ 2 p p ∗ ~ ε,N 2 = M (M1 (f; Φ) ) L2,∞ 2 2 p p p ∗ ~ ε,N 2 ≤ C(n) (M1 (f; Φ) ) L2,∞ 2 p ∗ ~ ε,N = C(n) M1 (f; Φ) Lp,∞ In proving (22), we may assume that Φ has integral 1; otherwise we can multiply Φ by a suitable constant to arrange for this to happen. We note that for each x ∈ Rn we have √ n ~ ~ X ~ t ∇(Φt ∗ f )(x) `2 = (∇Φ)t ∗ f (x) `2 ≤ n (∂jΦ)t ∗ f (x) `2 , j=1 10 LOUKAS GRAFAKOS AND DANQING HE and it suffices to work with each partial derivative ∂jΦ. Using Lemma 1 we write Z 1 (s) ∂jΦ = Θ ∗ Φs ds 0 for suitable Schwartz functions Θ(s). Fix x ∈ Rn, t > 0, and y with |y −x| < t < 1/ε. Then we have N ~ t 1 (∂jΦ)t ∗ f (y) 2 ` t + ε (1 + ε|y|)N N Z 1 t 1 (s) ~ (23) = N (Θ )t ∗ Φst ∗ f (y) ds t + ε (1 + ε|y|) 0 `2 N Z 1 Z ~ t −n (s) −1 Φst ∗ f (y − z) `2 ≤ t Θ (t z) N dz ds . t + ε 0 Rn (1 + ε|y|) Inserting the factor 1 written as [ 2n ]+1 N [ 2n ]+1 N ts p ts ts + |x − (y − z)| p ts + ε ts + |x − (y − z)| ts + ε ts ts in the preceding z-integral and using that 1 (1 + ε|z|)N ≤ (1 + ε|y|)N (1 + ε|y − z|)N and the fact that |x − y| < t < 1/ε, we obtain the estimate N Z 1 Z ~ t −n (s) −1 Φst ∗ f (y − z) `2 t Θ (t z) N dz ds t + ε 0 Rn (1 + ε|y|) [ 2n ]+1 Z 1 Z p ~ ε,N N ts + |x − (y − z)| −n (s) −1 ds ≤ V (f; Φ) (x) (1 + ε|z|) t Θ (t z) dz N 0 Rn ts s Z 1 Z ε,N −[ 2n ]−1−N N [ 2n ]+1 (s) ≤ V (f~; Φ) (x) s p (1 + εt|z|) (s + 1 + |z|) p Θ (z) dz ds 0 Rn 2n [ p ]+1 2n ~ ε,N ≤ 2 C0(∂jΦ,N + [ ] + 1) N 2n (∂jΦ) V (f; Φ) (x) p N+[ p ]+1 in view of conclusion (14) of Lemma 1. Combining this estimate with (23), we deduce (22). Estimates (21) and (22) together yield 2 ~ ε,N p ∗ ~ ε,N (24) U(f; Φ) Lp,∞ ≤ C(n) A(n, p, Φ,N) M1 (f; Φ) Lp,∞ . We now set n ~ ε,N ∗ ~ ε,N Eε = x ∈ R : U(f; Φ) (x) ≤ KM1 (f; Φ) (x) WEAK HARDY SPACES 11 for some constant K to be determined shortly. With A = A(n, p, Φ,N) we have ∗ ~ ε,N p 1 ~ ε,N p M1 (f; Φ) p,∞ c ≤ U(f; Φ) p,∞ c L ((Eε) ) Kp L ((Eε) ) 1 ~ ε,N p ≤ U(f; Φ) p,∞ Kp L (25) 2 p C(n) A ∗ ~ ε,N p ≤ M (f; Φ) p,∞ Kp 1 L 1 ∗ ~ ε,N p ≤ M (f; Φ) p,∞ , 2 1 L provided we choose K such that Kp = 2 C(n)p A(n, p, Φ,N)p. Obviously K is a function of n, p, Φ,N and in particular depends on N. p,∞ ∗ ε,N It remains to estimate the weak L quasi-norm of M1 (f; Φ) over the set Eε. We claim that the following pointwise estimate is valid: 1 1 h i q ∗ ~ ε,N 0 q ~ q (26) M1 (f; Φ) (x) ≤ 4 C (n, N, K) M M(f; Φ) (x) 0 for any x ∈ Eε and 0 < q < ∞ and some constant C (n, N, K), where M is the Hardy-Littlewood maximal operator. For the proof of (26) we cite [13] in the scalar case, but we indicate below why the proof also holds in the vector-valued setting. To prove (26) we fix x ∈ Eε and we also fix y such that |y − x| < t. ∗ ε,N n+1 By the definition of M1 (f; Φ) (x) there exists a point (y0, t) ∈ R+ such that 1 |x − y0| < t < ε and N ~ t 1 1 ∗ ~ ε,N (27) (Φt ∗ f )(y0) `2 N ≥ M1 (f; Φ) (x) . t + ε (1 + ε|y0|) 2 ~ ε,N By the definitions of Eε and U(f; Φ) , for any x ∈ Eε we have N ~ t 1 ∗ ~ ε,N (28) t ∇(Φt ∗ f )(ξ) 2 ≤ KM (f; Φ) (x) ` t + ε (1 + ε|ξ|)N 1 1 for all ξ satisfying |ξ − x| < t < ε . It follows from (27) and (28) that N ~ ~ 1 + ε|ξ| (29) t ∇(Φt ∗ f )(ξ) ≤ 2K (Φt ∗ f )(y0) `2 1 + ε|y0| 1 for all ξ satisfying |ξ − x| < t < ε . We let z be such that |z − x| < t and |z − y0| < t. Applying the mean value theorem and using (29), we obtain, for some ξ between y0 and z, ~ ~ ~ (Φt ∗ f )(z) − (Φt ∗ f )(y0) `2 = ∇(Φt ∗ f )(ξ) `2 |z − y0| N 2K ~ 1 + ε|ξ| ≤ (Φt ∗ f )(ξ) `2 |z − y0| t 1 + ε|y0| N+1 2 K ~ ≤ (Φt ∗ f )(y0) 2 |z − y0| t ` 1 ~ ≤ (Φt ∗ f )(y0) 2 , 2 ` 12 LOUKAS GRAFAKOS AND DANQING HE −N−2 −1 provided z also satisfies |z − y0| < 2 K t in addition to |z − x| < t. Therefore, −N−2 −1 for z satisfying |z − y0| < 2 K t and |z − x| < t we have ~ 1 ~ 1 ∗ ~ ε,N (Φt ∗ f )(z) 2 ≥ (Φt ∗ f )(y0) 2 ≥ M (f; Φ) (x) , ` 2 ` 4 1 where the last inequality uses (27). Thus we have 1 Z M M(f~; Φ)q(x) ≥ M(f~; Φ)(w)q dw |B(x, t)| B(x,t) 1 Z ≥ M(f~; Φ)(w)q dw −N−2 −1 |B(x, t)| B(x,t)∩B(y0,2 K t) Z 1 1 ∗ ~ ε,N q ≥ q M1 (f; Φ) (x) dw −N−2 −1 |B(x, t)| B(x,t)∩B(y0,2 K t) 4 |B(x, t) ∩ B(y , 2−N−2K−1t)| 1 ≥ 0 M ∗(f~; Φ)ε,N (x)q |B(x, t)| 4q 1 0 −1 −q ∗ ~ ε,N q ≥ C (n, N, K) 4 M1 (f; Φ) (x) , where we used the simple geometric fact that if |x − y0| ≤ t and δ > 0, then |B(x, t) ∩ B(y , δt)| 0 ≥ c > 0 , |B(x, t)| n,δ the minimum of this constant being obtained when |x − y0| = t. This proves (26). Taking q = p/2 and applying the boundedness of the Hardy–Littlewood maximal operator on L2,∞ yields ∗ ~ ε,N 0 ~ (30) M (f; Φ) p,∞ ≤ C (n, p, Φ,N) M(f; Φ) p,∞ . 1 L (Eε) 2 L Combining this estimate with (25), we finally prove (20). ∗ ~ ε,N Recalling the fact (obtained earlier) that kM1 (f; Φ) kLp,∞ < ∞, we deduce from (20) that 1 ∗ ~ ε,N p 0 ~ (31) M1 (f; Φ) Lp,∞ ≤ 2 C2(n, p, Φ,N) M(f; Φ) Lp,∞ . The preceding constant depends on f~ but is independent of ε. Notice that −N N ∗ ~ ε,N 2 t ~ M1 (f; Φ) (x) ≥ N sup sup (Φt ∗ f )(y) `2 (1 + ε|x|) 0 Keeping this significant observation in mind, we repeat the preceding argument from the point where the functions U(f~; φ)ε,N and V (f~; φ)ε,N are introduced, setting 00 0 ε = N = 0. Then we arrive to (19) with a constant C2 (n, p, Φ) = C2(n, p, Φ, 0) which is independent of N and thus of f~. We conclude the validity of (6) with 1/p 00 C2(n, p, 1, Φ) = 2 C2 (n, p, Φ) when a = 1. An analogous constant is obtained for different values of a > 0. (c) Let B(x, R) denote a ball centered at x with radius R. Recall that k(Φ ∗ f~ )(x − y)k 2 M ∗∗(f~; Φ)(x) = sup sup t ` . b |y| b t>0 y∈Rn t + 1 It follows from the definition ∗ ~ ~ Ma (f; Φ)(z) = sup sup k(Φt ∗ f )(w)k`2 t>0 |w−z| n 1 Z n ~ b ∗ ~ b k(Φt ∗ f )(x − y)k`2 ≤ Ma (f; Φ)(z) dz |B(x − y, at)| B(x−y,at) 1 Z n ∗ ~ b ≤ Ma (f; Φ)(z) dz |B(x − y, at)| B(x,|y|+at) n |y| + at ∗ n ≤ M M (f~; Φ) b (x) at a n −n |y| ∗ n ≤ max(1, a ) + 1 M M (f~; Φ) b (x) , t a from which we conclude that for all x ∈ Rn we have b n n o n ∗∗ ~ −b ∗ ~ b Mb (f; Φ)(x) ≤ max(1, a ) M Ma (f; Φ) (x) . We now take Lp,∞ norms on both sides of this inequality and using the fact that pb/n > 1 and the boundedness of the Hardy–Littlewood maximal operator M from Lpb/n,∞ to itself, we obtain the required conclusion (7). (d) In proving (d) we may replace b by the integer b0 = [b] + 1. Let Φ be a Schwartz function with integral equal to 1. Applying Lemma 1 with m = b0, we write any function ϕ in FN as Z 1 (s) ϕ(y) = (Θ ∗ Φs)(y) ds 0 for some choice of Schwartz functions Θ(s). Then we have Z 1 (s) ϕt(y) = ((Θ )t ∗ Φts)(y) ds 0 14 LOUKAS GRAFAKOS AND DANQING HE for all t > 0. Fix x ∈ Rn. Then for y in B(x, t) we have Z 1 Z ~ (s) ~ k(ϕt ∗ f )(y)k`2 ≤ |(Θ )t(z)| k(Φts ∗ f )(y − z)k`2 dz ds 0 Rn Z 1 Z b0 (s) ∗∗ ~ |x − (y − z)| ≤ |(Θ )t(z)| Mb0 (f; Φ)(x) + 1 dz ds 0 Rn st Z 1 Z |x − y| |z| b0 −b0 (s) ∗∗ ~ ≤ s |(Θ )t(z)| Mb0 (f; Φ)(x) + + 1 dz ds 0 Rn t t 1 Z Z b b0 ∗∗ ~ −b0 (s) 0 ≤ 2 Mb0 (f; Φ)(x) s |Θ (w)| |w| + 1 dw ds 0 Rn Z 1 b0 ∗∗ ~ −b0 b0 ≤ 2 Mb0 (f; Φ)(x) s C0(Φ, b0) s Nb0 (ϕ) ds , 0 where we applied conclusion (14) of Lemma 1. Setting N = b0 = [b] + 1, we obtain for y in B(x, t) and ϕ ∈ FN , b0 ∗∗ ~ 2 ~ k(ϕt ∗ f )(y)k` ≤ 2 C0(Φ, b0) Mb0 (f; Φ)(x) . Taking the supremum over all y in B(x, t), over all t > 0, and over all ϕ in FN , we obtain the pointwise estimate ~ b0 ∗∗ ~ n MN (f )(x) ≤ 2 C0(Φ, b0) Mb0 (f; Φ)(x) , x ∈ R , b0 where N = b0. This clearly yields (8) if we set C4 = 2 C0(Φ, b0). ~ 0 n ~ (e) We fix an f ∈ S (R ) that satisfies kMN (f )kLp,∞ < ∞ for some fixed positive integer N. To show that f~ is a bounded distribution, we fix a Schwartz function ϕ and we observe that for some positive constant c = cϕ, we have that c ϕ is an element ∗ ~ ~ of FN and thus M1 (f; c ϕ) ≤ MN (f ). Then ~ ~ ∗ ~ ~ c k(ϕ ∗ f )(x)k`2 ≤ sup k(cϕ ∗ f )(z)k`2 ≤ M1 (f; cϕ)(y) ≤ MN (f )(y) |z−y|≤1 ~ for |y − x| ≤ 1. So let λ = c k(ϕ ∗ f )(x)k`2 and then the inequality 1 1 p λ λ λ ~ p ~ p,∞ (33) vn 2 ≤ 2 |{MN (f ) > 2 }| ≤ kMN (f )kL < ∞ − 1 ~ p shows that λ is finite and can be controlled by 2kMN (f )kLp,∞ vn . Here vn = |B(0, 1)| n ~ is the volume of the unit ball in R . This implies that kϕ∗f k`2 is a bounded function. We conclude that f~ is a bounded distribution. We now show that f~ is an element of Hp,∞. We fix a smooth function with compact support θ such that ( 1 if |x| < 1, θ(x) = 0 if |x| > 2. WEAK HARDY SPACES 15 We observe that the identity ∞ X P (x) = P (x)θ(x) + θ(2−kx)P (x) − θ(2−(k−1)x)P (x) k=1 n+1 ∞ Γ( 2 ) X −k θ( · ) − θ(2( · )) = P (x)θ(x) + n+1 2 n+1 (x) 2 −2k 2 2 π k=1 (2 + | · | ) 2k is valid for all x ∈ Rn. Setting (k) 1 Φ (x) = θ(x) − θ(2x) n+1 , (2−2k + |x|2) 2 (k) we note that for some fixed constant c0 = c0(n, N), the functions c0 θ P and c0Φ lie in FN uniformly in k = 1, 2, 3,... . Lemma 2. Let f be a bounded distribution on Rn. Then we have n+1 ∞ Γ( 2 ) X −k (k) (P ∗ f)(x) = ((θP ) ∗ f)(x) + n+1 2 (Φ2−k ∗ f)(x) 2 π k=1 for all x ∈ Rn, where the series converges in S0(Rn). Combining this observation with the identity for P (x) obtained earlier, we conclude that n+1 ∞ 1 Γ( 2 ) X −k (k) ~ 2 ~ 2 k ~ sup kPt ∗ f k` ≤ sup k(θP )t ∗ f k` + n+1 sup 2 (c0Φ )2 t ∗ f `2 t>0 t>0 c0 2 t>0 π k=1 ~ ≤ C5(n, N)MN (f ) , which proves the required conclusion (9). We observe that the last estimate also yields the stronger estimate ∗ ~ ~ ~ (34) M1 (f; P )(x) = sup sup |(Pt ∗ f )(y)| ≤ C5(n, N)MN (f )(x) . t>0 y∈Rn |y−x|≤t ∗ ~ It follows that the quasinorm kM1 (f; P )kLp,∞ is at most a constant multiple of ~ ~ kMN (f )kLp,∞ and thus it is also equivalent to kf kHp,∞ . This concludes the proof of Theorem 1 It remains to prove Lemma 2. Proof of Lemma 2. We begin with the identity ∞ X P (x) = P (x)θ(x) + θ(2−kx)P (x) − θ(2−(k−1)x)P (x) k=1 n+1 ∞ Γ( 2 ) X −k θ( · ) − θ(2( · )) = P (x)θ(x) + n+1 2 n+1 (x) 2 −2k 2 2 π k=1 (2 + | · | ) 2k which is valid for all x ∈ Rn. 16 LOUKAS GRAFAKOS AND DANQING HE Fix a function φ ∈ S(Rn) whose Fourier transform is equal to 1 in a neighborhood of zero. Then f = φ∗f +(δ0 −φ)∗f and we also have Pt ∗f = Pt ∗φ∗f +Pt ∗(δ0 −φ)∗f. Given a function ψ in S(Rn) we need to show that n+1 N Γ( 2 ) X −k (k) (θP )t ∗ φ ∗ f, ψ + n+1 2 (Φ )2kt ∗ φ ∗ f, ψ → hPt ∗ φ ∗ f, ψi 2 π k=1 and n+1 N Γ( 2 ) X −k (k) (θP )t ∗(δ0 −φ)∗f, ψ + n+1 2 (Φ )2kt ∗(δ0 −φ)∗f, ψ → hPt ∗(δ0 −φ)∗f, ψi 2 π k=1 as N → ∞. The first of these claims is equivalent to n+1 N Γ( 2 ) X −k (k) φ ∗ f, ψ ∗ (θP )t + n+1 2 φ ∗ f, ψ ∗ (Φ )2kt → hφ ∗ f, ψ ∗ Pti 2 π k=1 as N → ∞. Here φ ∗ f ∈ L∞ and the actions h·, ·i are convergent integrals in all three cases. This claim will be a consequence of the Lebesgue dominated convergence theorem since: n+1 N Γ( 2 ) X −k (k) ψ ∗ (θP )t + n+1 2 ψ ∗ (Φ )2kt → ψ ∗ Pt 2 π k=1 pointwise (which is also a consequence of the Lebesgue dominated convergence the- orem) as N → ∞ and hence the same is true after multiplying by the bounded function φ ∗ f and also n+1 N Γ( 2 ) X −k (k) 1 n |φ ∗ f| ψ ∗ (θP )t + n+1 2 ψ ∗ (Φ )2kt ≤ |φ ∗ f| (|ψ| ∗ Pt) ∈ L (R ). 2 π k=1 We now turn to the corresponding assertion where φ is replaced by δ − φ. Using the Fourier transform, this assertion is equivalent to Γ( n+1 ) N 2 X −k (k) f,b ([θP )t(1 − φb)ψb + n+1 2 f,b (Φ\)2kt(1 − φb)ψb → hf,b Pbt(1 − φb)ψbi 2 π k=1 Since fb ∈ S0(Rn), this assertion will be a consequence of the fact that Γ( n+1 ) N 2 X −k (k) ([θP )t(1 − φb)ψb + n+1 2 (Φ\)2kt(1 − φb)ψb → Pbt(1 − φb)ψb 2 π k=1 in S(Rn). It will therefore be sufficient to show that for all multiindices α and β we have Γ( n+1 ) N α 2 X −k (k) β sup ∂ξ ([θP )t(ξ) + n+1 2 (Φ\)2kt(ξ) − Pbt(ξ) (1 − φb(ξ))ψb(ξ)ξ → 0 ξ∈Rn 2 π k=1 WEAK HARDY SPACES 17 The term inside the curly brackets is equal to the Fourier transform of the function −N (θ(2 x) − 1)Pt(x), which is Z (e−2πt|ξ−ζ| − e−2πt|ξ|)2Nnθb(2N ζ) dζ Rn since θb has integral 1. Note that |ξ| ≥ c > 0 since 1 − φb(ξ) = 0 in a neighborhood of zero. Let εN > 0. First consider the integral Z (e−2πt|ξ−ζ| − e−2πt|ξ|)2Nnθb(2N ζ) dζ . |ξ−ζ|>εN Both exponentials are differentiable in this range and differentiation gives Z γ −2πt|ξ−ζ| −2πt|ξ| Nn N ∂ξ (e − e )2 θb(2 ζ) dζ |ξ−ζ|>εN Z ξ−ζ 1 −2πt|ξ−ζ| ξ 1 −2πt|ξ| Nn N = Qγ |ξ−ζ| , |ξ−ζ| e − Qγ |ξ| , |ξ| e 2 θb(2 ζ) dζ , |ξ−ζ|>εN where Qγ is a polynomial of the following n + 1 variables ξ 1 ξ ξ 1 Q , = Q 1 ,..., n , γ |ξ| |ξ| γ |ξ| |ξ| |ξ| that depends on γ. Note that ξ−ζ 1 −2πt|ξ−ζ| ξ 1 −2πt|ξ| Qγ , e − Qγ , e ≤ C |ζ| . |ξ−ζ| |ξ−ζ| |ξ| |ξ| Thus the integral is bounded by Z C 2Nn|ζ|θb(2N ζ) dζ = C02−N , |ξ−ζ|>εN which tends to zero as N → ∞. Now consider the integral Z γ −2πt|ξ−ζ| −2πt|ξ| Nn N ∂ξ (e − e )2 θb(2 ζ) dζ |ξ−ζ|≤εN Z γ −2πt|ζ| −2πt|ξ| Nn N = ∂ξ (e − e )2 θb(2 (ξ − ζ)) dζ |ζ|≤εN in which |ξ| ≥ c > 0. We obtain another expression which is bounded by n N|γ| n N|α| C εN 2 ≤ C εN 2 , −N|α|/n which tends to zero if we pick εN = 2 /N. Finally α−γn βo sup ∂ξ (1 − φb(ξ))ψb(ξ)ξ < ∞ ξ∈Rn and so the claimed conclusion follows by applying Leibniz’s rule to the expression on α which ∂ξ is acting. Fix a smooth radial nonnegative compactly supported function θ on Rn such that θ = 1 on the unit ball and vanishing outside the ball of radius 2. Set Φ(k)(x) = 18 LOUKAS GRAFAKOS AND DANQING HE