a ces e ntical n orphic 6], e nd ls. ). J. Math. Anal. Appl. 299 (2004) 465–493 www.elsevier.com/locate/jma Estimates for the Poisson kernel and Hardy spa on compact manifolds L.A. Carvalho dos Santosa,∗,1, J. Hounieb,2 a Departamento de Matemática, UNESP, Presidente Prudente, SP, Brazil b Departamento de Matemática, UFSCAR, São Carlos, SP, Brazil Received 30 March 2004 Available online 23 September 2004 Submitted by Steven G. Krantz Abstract We study Hardy spaces on the boundary of a smooth open subset orRn and prove that they can b defined either through the intrinsic maximal function or through Poisson integrals, yielding ide spaces. This extends to any smooth open subset ofRn results already known for the unit ball. As a application, a characterization of the weak boundary values of functions that belong to holom Hardy spaces is given, which implies an F. and M. Riesz type theorem.  2004 Elsevier Inc. All rights reserved. 0. Introduction The real Hardy spaceHp(RN), 0< p � ∞, introduced in 1971 by Stein and Weiss [1 is equal toLp(RN) for p > 1, is properly contained inL1(RN) for p = 1 and is a spac of not necessarily locally integrable distributions for 0< p < 1. Forp � 1, Hp(RN) is an advantageous substitute forLp(RN) [15], as the latter is not a space of distributions a has trivial dual ifp < 1 while for p = 1, L1(RN) is not preserved by singular integra * Corresponding author. E-mail addresses:luis@prudente.unesp.br (L.A. Carvalho dosSantos), hounie@dm.ufscar.br (J. Hounie 1 Research partially supported by CAPES. 2 Research partially supported by CNPq, FAPESP and IM-AGIMB. 0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.03.080 466 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 ifolds, hand, plies 5 a . - d ell ifolds Let us choose a functionΦ ∈ S(RN), with ∫ Φ dz �= 0 and writeΦε(z) = ε−NΦ(z/ε), z ∈ RN , and MΦf (z) = sup 0<ε<∞ ∣∣(Φε ∗ f )(z) ∣∣. Then [15] Hp(RN) = { f ∈ S ′(RN): MΦf ∈ Lp(RN) } . An obstacle to the localization of the elements ofHp(RN), 0< p � 1, is thatψu may not belong toHp(RN) for ψ ∈ C∞ c (RN) andu ∈ Hp(RN). In particular,Hp(RN), 0< p � 1, is not preserved by pseudo-differential operators and is not well defined on man a fact that hinders applications to PDE with variable coefficients. On the other Hp(RN) is preserved by singular integrals with sufficiently smooth kernels, which im that it is locally preserved by pseudo-differential operators of order zero (and typeρ = 1, δ = 0). This fact was used by Strichartz in 1972 [17] who definedH 1(Σ) for a compact smoothN -dimensional manifoldΣ as the space of allf ∈ L1(Σ) such thatTf ∈ L1(Σ) for all pseudo-differential operatorsT of order zero. Then Peetre [14] proposed in 197 more elementary definition ofHp(Σ), p > 0, in terms of an intrinsic maximal function More precisely, he set Hp(Σ) = { f ∈D′(Σ): Msf ∈ Lp(Σ) } , Msf (x) = sup φ∈Ks(x) ∣∣〈f,φ〉∣∣, whereKs(x) is the space of smooth functionsφ ∈ C∞(Σ) such that there is anh > 0 such that suppφ ⊂ B(x,h) and sup0�k�s hN+k‖φ‖k � 1. HereD′(Σ) is the space of dis tributions inΣ , suppφ denotes the support ofφ, B(x,h) is the Riemannian ball centere at x of radiush (assume a Riemannian metric is given onΣ), ‖φ‖k denotes the norm in Ck(Σ) ands is a conveniently large integer that depends onp. It turns out that forp = 1 the spacesH 1(Σ) defined by Strichartz and Peetre coincide. A way around the problem thatHp(RN) is not localizable for 0< p � 1 is the definition of localizable Hardy spaceshp(RN) [9,15] by means of the truncated maximal function mΦf (z) = sup 0<ε�1 ∣∣(Φε ∗ f )(z) ∣∣, hp(RN) = { f ∈ S ′(RN): mΦf ∈ Lp(RN) } . It follows that the spacehp(RN) is stable under multiplication by test functions as w as by change of variables that behave well at infinity and also thathp(RN) = Lp(RN) for 1 < p � ∞. This opens the doorway to a definition of Hardy spaces on smooth man through localization. Namely, if{Uα,Φα} is a family of local charts and{ϕi} a partition of the unity subordinated to the coveringUα then we say thatf ∈ hp(Σ) if, and only if, f ϕi ◦ Φ−1 α ∈ hp(Rn). It is known thathp(Σ) = Hp(Σ) (see, e.g., [4]). Consider now a bounded open subsetΩ ⊂ Rn with smooth boundary∂Ω = Σ and givenf ∈D′(Σ) let u ∈ C∞(Ω) be the solution of the Dirichlet problem{ ∆u = 0 onΩ, (0.1) u|Σ = f. L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 467 - a lution t if ts l n n this ces - ase Pois- he about es. In n op- mmas the the th the dis- Thenu gives rise to two maximal functions: (i) If νx is the outer normal unit vector field defined atx ∈ Σ, the normal maximal func tion is u⊥(x) = sup 0 1, the nontangential maximal function is u∗ a(x) = sup z∈Γa(x) ∣∣u(z) ∣∣, x ∈ Σ, whereΓa(x) = {z ∈ Ω : |z − x| < a dist(z,Σ)} is then-dimensional analogue of Stolz region, here| · | denotes the Euclidean norm inRn and dist(z,Σ) the distance from z to Σ . WhenΩ = B ⊂ Rn, n � 2, is the unit ball andΣ = Sn−1, it is known thatf ∈ Hp(Sn−1), 0 < p � ∞, if and only if u∗ a ∈ Lp(Sn−1) or, equivalently, if and only ifu⊥ ∈ Lp(Sn−1). This is classic for the unit circle [7] and due to Colzani [5] forn � 3. A relevant fact in the proof is that explicit formulas are known for the Poisson kernel that furnishes the so of the boundary problem (0.1) whenΩ is a ball. In particular, these formulas show tha P(z, x) :Ω × ∂Ω → R is the Poisson kernel of the domainΩ then there exist constan Cαβ > 0 for every multi-indexesα,β ∈ Z n+ such that ∣∣Dα z Dβ x P (z, x) ∣∣ � Cαβ |x − z|n−1+|α|+|β| , (z, x) ∈ Ω × ∂Ω, (Kαβ ) at least whenΩ = B. For α = 0 andβ = 0 estimate(K0) is well known for genera smoothly bounded domains (actually, classC2 suffices). A proof of this fact was give by Kerzman in an unpublished set of notes [12] and can be found in [13, p. 332]. I work we prove (Kαβ ) for all α andβ . This is the key to the characterization of the spa Hp(∂Ω), 0 < p � ∞, in terms of the maximal functionsu⊥ andu∗ a . Since this charac terization is well known forp > 1, we are mainly concerned in this paper with the c 0 < p � 1 although the proofs work as well for anyp. The paper is organized as follows. In Section 1 we prove estimates (Kαβ ) by locally flattening the boundary and constructing a pseudo-differential approximation of the son operator following the method of Treves[19] to construct a parameterization of t heat equation. The pseudo-differential approximation gives a wealth of information the Poisson kernel and in particular shows the required estimates for its derivativ Section 2 we study approximations of the identity that are obtained from the Poisso erator but converge faster to the identity. In Section 3 we prove several technical le about these approximations that are instrumental in the proof of the equivalence ofLp “norms” of the different maximal functions defined in terms of Poisson integrals— equivalence of different Poisson’s maximal functions is discussed in Section 4—wi intrinsic maximal function, which is the subject of Section 5. Finally, in Section 6, we cuss holomorphic Hardy spacesHp(Ω), Ω ⊂ Cn, and prove that everyf ∈ Hp(Ω) has a 468 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 an f to ce, s, n- e by line shall m- f is t weak boundary valuebf ∈ Hp(∂Ω) of which it is its Poisson integral. This establishes isomorphism of topological vector spaces betweenHp(Ω) and the subspace ofHp(∂Ω) of distributions that are boundary value of some holomorphic function inΩ . We also prove an “F. and M. Riesz theorem,” showing that if a measure in∂Ω is the boundary value o a holomorphic function defined onΩ it must be absolutely continuous with respect Lebesgue measure. We use the standard notation for distributional spaces, soLp denotes a Lebesgue spa S(Rn) denotes the Schwartz space, its dualS ′(Rn) denotes the tempered distribution D′(Σ) denotes the space of distributions on a manifoldΣ , Cr denotes the space of co tinuous functions with continuous derivatives up to orderr if r is a positive integer and th corresponding Hölder space ifr > 0 is not integral. Different Hardy spaces are denoted Hp, Hp andhp . We also denote byC a positive constant that may change from one to the next. 1. Pointwise estimates for the Poisson kernel The following theorem is the main result of this section. It gives estimates that we later need to characterize Hardy spaces on the boundary of a smooth domain ofRn. Theorem 1.1. LetP(z, x) be the Poisson kernel of a bounded domainΩ ⊆ Rn with smooth boundaryΣ . For every multi-indexesα ∈ Z n+ andβ ∈ Z n−1+ there exist a constantCαβ = Cαβ(Ω) > 0 such that ∣∣Dα z Dβ x P (z, x) ∣∣ � Cαβ |x − z|n−1+|α|+|β| , (z, x) ∈ Ω × Σ, (Kαβ ) Proof. Fix a > 1 and consider the nontangential region insideΩ with vertex atx given by Γa(x) = { z ∈ Ω : |z − x| < a dist(z,Σ) } . For fixedr0 > 0 consider the set X = { (z, x) ∈ Ω × Σ: |z − x| � r0 } and observe that|z−x|n+|α|+|β|−1Dα z D β x P (z, x) is continuous, thus bounded, on the co pact setX, becauseP(z, x) is smooth onΩ × Σ \ Σ × Σ . Therefore, there is no loss o generality if we prove (Kαβ ) assuming that|z − x| < r0 and we shall do so. The proof divided into two cases. Case1. z /∈ Γa(x) By the compactness ofΣ it is enough to prove the estimate whenx is in a small neigh- borhood of an arbitrary pointx0 ∈ Σ . Since|z − x| < r0 we may assume that bothx andz belong to a small neighborhood ofx0. The initial step is to flatten the boundary in tha neighborhood. Thus we consider a diffeomorphism that takes a neighborhoodW of x0 onto a neighborhood of the closure of the cubeQ ⊂ Rn−1 x × Rt given by|x| < 1, |t| < 1 so thatx0 is mapped to(0,0), Ω ∩ W is mapped toQ+ = {(x, t) ∈ Q: t > 0} andΣ is L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 469 ssed y s [19, ry he f ernel, flattened to{t = 0}. Using(x, t) as new coordinates the Poisson kernel may be expre as P̃ (y, t, x) with z = (y, t) ∈ Q+ andx = (x1, . . . , xn−1) ∈ ∂Q+ ∩ {t = 0}. If a > 0 is large enough, the conditionz /∈ Γa(x) implies |x − y| > t . Notice that for|x − y| > t , (|x − y|2 + t2)1/2 is comparable to|x − y|. Thus, it will be enough to prove that for an |x|, |y| < 1 and 0< t < |x − y|∣∣Dα x Dβ y Dk t P̃ (x, t, y) ∣∣ � Cαβk |x − y|n−1+|α|+|β|+k . (1.1) Recall that in the original coordinates u(z) =Pφ(z) = ∫ Σ P(z, y)φ(y) dσ(y), (1.2) wheredσ indicates the volume element inΣ , solves the Dirichlet problem{ ∆u(z) = 0, z ∈ Ω, u(x) = φ(x), x ∈ Σ. (1.3) In the new coordinates, (1.3) becomes, with some abuse of notation,{ L(t, x,Dx,Dt )u = 0, u(x,0) = φ(x), (1.4) where L(t, x,Dx,Dt) = ∂2 ∂t2 + 2 n−1∑ j=1 bj (x, t) ∂2 ∂xj ∂t + ∑ j,k cjk(x, t) ∂2 ∂xj∂xk + · · · (1.5) is an elliptic differential operator with real coefficients and principal symbol σL(t, x, τ, ξ) = −τ2 − 2 n−1∑ j=1 τξj bj (x, t) − ∑ j,k cj,k(x, t)ξj ξk, and the dots in (1.5) denote terms of order one. We now follow the approach of Treve Chapter 3] to construct parameterization of the heat equation. We will apply the machine of pseudodifferential operators to find a family of pseudodifferential operatorsH(t, x,Dx), acting on the variablex and depending smoothly ont > 0 as a parameter, that solves t problem{ L ◦ H ∼ 0 modulo a smooth kernel, H(0, x,Dx) = I. (1.6) The symbolσH (t, x, ξ) of H is identically equal to 1 fort = 0 and has order−∞ for t > 0; furthermore, ⋃ 0 0. Thus,d1 is an elliptic homogeneous symbol of degree one. Even thoughd1 is not a symbol inS1 1,0 because it fails to be smooth at the origin, we proceed as usual and after multiplication by a c function that vanishes for|ξ | < 1/2 and is identically equal to 1 for|ξ | > 1, we can obtain a symbol inS1 1,0 that we still denote byd1. If D1 = Op(d1) then we want to check that L ∼ ( ∂t + n∑ j=1 bj ∂ ∂xj − D1 )( ∂t + n∑ j=1 bj ∂ ∂xj + D1 ) modL1 1,0. (1.8) SetR1 = L− (∂t +∑n j=1 bj ∂ ∂xj −D1)(∂t +∑n j=1 bj ∂ ∂xj +D1). Then, the symbolic calcu lus of pseudo-differential operators shows after a simple computation that the symbσR1 of R1 belongs toS1 1,0. The next step consists in finding a symbold0 ∈ S0 1,0 such that the operatorD0 = Op(d0) satisfies L ∼ ( ∂t + � − (D1 + D0) )( ∂t + � + (D1 + D0) ) modL0 1,0, where we have written � = n∑ j=1 bj ∂ ∂xj . Now let Q1 andQ2 denote respectively∂t + � − D1 and∂t + � + D1 and setR0 = L − (Q1 − D0)(Q2 + D0). Then,R0 = L − Q1Q2 + (Q2 − Q1)D0 + D0D0 + [D0,Q2], and observing thatQ2 − Q1 = 2D1 andL ∼ Q1Q2 modL1 because of (1.8) we haveR0 = D0D0 + 2D1D0 + R1 for someR1 ∈ L1. Then, if r1 is the symbol ofR1 andd1 is the symbol ofD1, we may takeD0 with symbol d0(t, x, τ, ξ) = −1 2 r1(t, x, τ, ξ) d1(t, x, τ, ξ) and obtain thatR0 has order zero. Keeping up this process we may define a seque symbols d−j = −1 2 r1−j d1 + d0 + · · · + d1−j ∈ S −j 1,0 so that their associated operatorsDk = Op(dk), k = 1,0, . . . ,−j , satisfy L ∼ (∂t + � − D1 − D0 − · · · − D−j )(∂t + � + D1 + D0 + · · · + D−j) modL−j 1,0. Since the order ofd−j goes to−∞ asj → ∞, we may find a symbold ∈ S1 1,0 such that d(t, x, ξ) ∼ ∞∑ d1−j (t, x, ξ) modS−∞ j=0 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 471 s ss (see, rove sion to is in the sense thatd − ∑k j=0 d1−j ∈ S−k 1,0 or anyk = 0,1,2 . . . . Hence, the operatorD = Op(d) satisfies (1.7). If we call A1 = n∑ j=1 bj ∂ ∂xj − D and A = n∑ j=1 bj ∂ ∂xj + D, we may rewrite (1.7) as L ∼ (∂t + A1)(∂t + A) modL−∞. Hence, in order to obtain (1.6) it will suffice to find a family of operatorsH(t, x,Dx, ), 0 � t < 1, such that (∂t + A) ◦ H(t, x,Dx) ∼ 0 modulo a smooth kernel, (1.9) with the additional propertyH(0, x,Dx) = identity. Note that the symbola(t, x, ξ) of A has principal symbola1(t, x, ξ) = d1(t, x, ξ) + i ∑n−1 j=0 bj (x, t)ξj . To construct H(t, x,Dx) with symbolσH (t, x, ξ) = h(t, x, ξ) we propose h(t, x, ξ) ∼ e− ∫ t 0 a1(s,x,ξ) ds ( 1+ κ−1(t, x, ξ) + κ−2(t, x, ξ) + · · ·) (1.10) with κ−j ∈ S −j 1,0. An important point here is that, becaused1(t, x, ξ) � c > 0 for |ξ | > 1, e(t, x, ξ) = exp(− ∫ t 0 a1(s, x, ξ) ds) satisfies the following estimates∣∣Dα x D β ξ Dk t e(t, x, ξ) ∣∣ � Cαβk ( 1+ |ξ |)k−|β| expressing the fact thatDk t e ∈ Sk 1,0 uniformly in t . The proof of [19, Theorem 1.1] show thatκ−1, κ−2, . . . satisfyingκ−j (0, x, ξ) = 0 may be inductively determined by a proce similar to the construction ofD so that ifh(t, x, ξ) is given by (1.10) thenH = Op(h) satisfies (1.9). Furthermore,∣∣Dα x D β ξ Dk t h(t, x, ξ) ∣∣ � Cαβk ( 1+ |ξ |)k−|β| . (1.11) Consider the kernel of the pseudo-differentialH(t, x,Dx ) h(t, x, y) = 1 (2π)n−1 ∫ ei(x−y)ξh(t, x, ξ) dξ. It follows from standard estimates for the kernel of pseudo-differential operators e.g., [1,18]) that estimates (1.11) imply the estimates∣∣Dα x Dβ y Dk t h(t, x, y) ∣∣ � Cαβk |x − y|n−1+k+|α|+|β| . (1.12) Notice that estimates (1.12) forh are analogous to the estimates (1.1) that we wish to p for P̃ . Thus, to obtain (1.1) it will be enough to find smooth functionsµ(t, x, y), ρ(t, x, y) defined for|x|, |y| < 1, 0� t < 1 such thatP̃ (x, t, y) = µ(h + ρ)(t, x, y). Let p(x, t, y) be the kernel of the integral operatorP expressed in the new coordinates: its expres is readily obtained from (1.2) (which givesP in the original coordinates) by reverting the new coordinates. We then see thatp(x, t, y) = P̃ (x, t, y)/µ(y) whereµ−1(y) dy is the expression of the area elementdσ of Σ in the new coordinates, in particularµ > 0 and is smooth. Therefore, we need only show thatρ = p − h is smooth up to the boundary. Th 472 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 s nsfer ill of e - ely t follows from the fact that the operatorsP andH , whose kernels are respectivelyp andh, satisfyL ◦ (P −H) ∼ 0 modulo smoothing operators and(P −H)|t=0 = 0. That this is so is already a consequence of the “uniqueness”part of [19, Theorem 1.1] but here it seem simpler to give a direct argument. By returning to the original coordinates, let us tra the operatorH to the initial neighborhoodW � x0 ∈ Σ obtaining an operator that we st call H . Using a cut-off functionχ that is identically 1 in a neighborhoodω of x0 such ω ⊂ W , it is easy to construct an operatorH̃ = χHχ :C∞(Σ) → C∞(Ω) such that (a) ∆H̃ is regularizing when acting on distributions compactly supported inω, (b) H̃φ|Σ = φ if φ is supported inω ∩ Σ . If φ ∈D′(Σ), we have that∆(Pφ − H̃φ) = ψ , whereψ is smooth inΩ ∩ω in view of (a). Furthermore, (b) implies thatPφ − H̃φ vanishes onω∩Σ . By boundary elliptic regularity we conclude thatPφ − H̃φ ∈ C∞(Ω ∩ ω) and since this holds for any distributionφ ∈ D′(Σ) we conclude that the kernel ofP − H̃ is smooth when restricted to(ω ∩ Ω) × (ω ∩ Ω), which proves, as we wished, thatp − h is smooth up to the boundary. The proof Case 1 is complete. Case2. |x − y| � t For |x − y| � t , (|x − y|2 + t2)1/2 is comparable tot . Thus, it will be enough to prov that for any|x|, |y| < 1 and 0< |x − y| � t < 1 ∣∣Dα x Dβ y Dk t P̃ (x, t, y) ∣∣ � Cαβk tn−1+|α|+|β|+k . (1.13) As before, it is enough to prove analogous estimates for the kernelh(t, x, y) of the pseudo differential approximationH of the Poisson kernel. This leads us to look more clos to its symbolh(t, x, y) = exp(− ∫ t 0 a1(s, x, ξ) ds)κ(t, x, ξ) given by (1.10). We recall tha a1 is defined bya1(t, x, ξ) = d1(t, x, ξ) + i ∑n−1 j=0 bj (x, t)ξj whered1(t, x, ξ) > c|ξ | for |ξ | > 1 andκ(t, x, ξ) has order zero uniformly int , furthermore the functionsbj (x, t) are real. Thus∣∣h(t, x, y) ∣∣ � C exp (−tc|ξ |), ξ ∈ R n−1, 0 < t < 1, and h(t, x, y) = 1 (2π)n−1 ∫ ei(x−y)ξh(t, x, ξ) dξ (1.14) is easily seen to satisfy the estimate ∣∣h(t, x, y) ∣∣ � C ∫ Rn−1 exp (−tc|ξ |)dξ � C′ tn−1 . Similarly, we see that forξ ∈ Rn−1 and 0< t < 1∣∣Dα x Dk t h(t, x, ξ) ∣∣ � Cαk ( 1+ |ξ |)|α|+k exp (−tc|ξ |) which implies, after differentiation of (1.14), that L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 473 get ent d for od at en f the s, they do r ∣∣Dα x Dβ y Dk t h(t, x, y) ∣∣ � Cαβk ∫ Rn−1 ( 1+ |ξ |)|α|+|β|+k exp (−tc|ξ |)dξ � Cαβk tn−1+|α|+|β|+k . (1.15) Reasoning as in Case 1, we see that the estimates (1.15) forDα x D β y Dk t h imply the analogous estimates (1.13). The proof of Theorem 1.1 is complete.� Corollary 1.2. Let P(z, x) be the Poisson kernel of a bounded domainΩ ⊆ Rn with smooth boundaryΣ . There exist a constantC = C(Ω) > 0 such that for all(z, x) ∈ Ω ×Σ ∣∣P(z, x)| � C dist(z,Σ)min ( 1 dist(z,Σ)n , 1 |x − z|n ) . (1.16) Proof. It is enough to prove (1.16) when|z − x| is small using local coordinates(x, t). As in the proof of the theorem, we work in a thin tubular neighborhood ofΣ . We first point out that ifz = (y, t), (1.13) for α = β = k = 0 gives |P(z, x)| � Ct1−n � C′ dist(z,Σ)1−n, sincet ∼ dist(z,Σ). This gives (1.16) whenz ∈ Γa(x), in which case |z − x| ∼ dist(z,Σ). When z = (y, t) /∈ Γa(x) we use the mean value theorem to |P(z, x)| � sup|∇zP (z, x)|dist(z,Σ), where the supremum is taken along the segm that joinsz to the pointζ ∈ Σ such that|z − ζ | = dist(z,Σ). Using (Kαβ ) with α = 0, |β| = 1 we obtain|P(z, x)| � C|x − z|−n dist(z,Σ). The corollary easily follows. � Remark 1.3. The proof of Theorem 1.1 shows that the Laplacian may be replace any second order elliptic operator with smooth real coefficients defined in a neighborho of Ω . In this case, the Poisson kernel must be replaced by the kernel of the operator th solves the Dirichlet problem. Remark 1.4. Estimates for the Poisson kernel can beobtained from estimates on the Gre function. However, we point out that classical estimates for the Green function o Laplace–Beltrami operator on compact manifolds with boundary are interior estimate in the sense that constants blow up when approaching the boundary [2, p. 112], so not seem to imply Theorem 1.1. 2. Approximations of the identity Assume without loss of generality thatΩ is such thatzt = x − tνx ∈ Ω and t = dist(zt ,Σ) if x ∈ Σ , 0 < t � 1. We may then shrinkΩ along the normal direction fo 0 < t < 1 and obtain the open set Ωt = { z ∈ Ω : dist(z,Σ) > t } with boundary Σt = ∂Ωt = { x − tνx(x): x ∈ Σ } . 474 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 s tity. e at Clearly, ⋃ 0 0 is an integer andr = k + γ , for some 0< γ < 1, Cr(Σ) is defined as Cr(Σ) = { u ∈ Ck(Σ), ∥∥Q(x,D)u ∥∥ γ < ∞} for all differential operatorsQ(x,D) of order� k with smooth coefficients defined onΣ . Sinceu = Pφ solves the Dirichlet problem with boundary valueφ, it turns out that if φ ∈ Cr(Σ), 0 < r < 1, thenPφ ∈ Cr(Ω) [8]. It follows that |φ(x) − Pt φ(x)| = O(tr ) uniformly in x ∈ Σ , so this gives an estimate of approximation speed ofPtφ to φ for 0 < r < 1 which increases withr. However, if we taker > 1 the exponent will not increas beyond 1. Thus, it is convenient to replacePt by another approximation of the identity th yields a faster approximation. This can be obtained by linear combinations ofPt evaluated at different timest . If f (s), s ∈ R, we recall that the difference operator with stept > 0 is defined as∆tf (s) = f (s + t) − f (s). Then∆2 t f (s) = ∆t(∆tf )(s) = f (s + 2t) − 2f (s + t) + f (s) and ifL � 1 is an integer ∆L t f (s) = L∑ j=0 (−1)L−j ( L j ) f (s + j t). Let f (L) denote the derivative of orderL of f . Taylor’s formula for∆Lf (s) whenf (L) is continuous is given by ∆L t f (s) = tL ∫ [0,1]L f (L) ( s + t (τ1 + · · · + τL) ) dτ. (2.1) If f ∈ Cr(R), r = L + γ , 0< γ < 1, we have ∆L+1 t f (s) = tL ∫ L ∆tf (L) ( s + t (τ1 + · · · + τL) ) dτ [0,1] L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 475 or or which gives the estimate∣∣∆L+1 t f (s) ∣∣ = tr‖f (L)‖γ � tr‖f ‖r . (2.2) Next we define SL t = L∑ j=0 (−1)L−j ( L j ) Pj t , 0 < t � 1/L, where it is understood thatP0 = I = identityoperator. Fort = 0 we see thatSL 0 = (1− 1)LI = 0, so Γ L t = L∑ j=1 (−1)j+1 ( L j ) Pj t = I − (−1)LSL t is an approximation of the identity. The next lemma shows thatΓ L t φ approximatesφ faster thanPt φ is φ is sufficiently regular. Lemma 2.1. Let 0 < r < L not be an integer. There existsC > 0 such that for every φ ∈ CL(Σ) andα ∈ Z n−1+ , 0 � |α| < r, sup x∈Σt ∣∣Dα x ( Γ L t φ(x) − φ(x) )∣∣ = ∥∥Dα x SL t φ ∥∥ L∞ � Ctr−|α|‖φ‖r . (2.3) Proof. Consider first the caseα = 0, so we wish to show that|SL t φ(x)| � Ctr‖φ‖r . Let u = Pφ, sou is harmonic inΩ and has the boundary valueφ. Sinceφ ∈ Cr(Σ), standard Hölder boundary estimates [8] imply thatu ∈ Cr(Ω) and‖u‖r � C‖φ‖r . Then SL t φ(x) = L∑ j=0 (−1)L−j ( L j ) u(x − tνx) = tL−1 ∫ [0,1]L−1 ∆tD L−1 t u ( x − t (τ1 + · · · + τL−1)νx ) dτ. (2.4) We may majorize|∆tD L−1 t u| by tγ ‖DL−1 t u‖γ whereγ = r − L + 1. Writing DL−1 t u ( x − t (τ1 + · · · + τL−1)νx ) = ∑ |α|=L−1 (−t (τ1 + · · · + τL−1)νx )α Dα z u ( x − t (τ1 + · · · + τL−1)νx ) as a sum of derivatives ofu of orderL− 1 we have‖DL−1 t u‖γ � C‖u‖L−1+γ = C‖u‖r � C′‖φ‖r , so we get|∆tD L−1 t u| � Ctγ ‖φ‖r . Plugging this estimate in (2.4) yields (2.3) f α = 0. For |α| = 1 we writeDα x SL t φ = SL t Dα x φ + [Dα x ,SL t ]φ. The estimate already proved f α = 0 with r − 1 in the place ofr gives∥∥SL t Dα x φ ∥∥ ∞ � Ctr−1‖Dαφ‖r−1 � Ctr−1‖φ‖r (2.5) L 476 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 but d ial in local- ve and we need only show a similar estimate for the commutator term[Dα x ,SL t ]φ. We saw in the proof of Theorem 1.1 how to find family of pseudo-differential operatorsHt depending on a parametert differing fromP by a smooth kernel. That was a local construction using a finite partition of unity in a tubular neighborhood ofΣ we can make it global an find a family of pseudo-differentialHt ∈ L0 1,0(Σ), 0< t < 1, such that (i) for anyk = 0,1, . . . , the set{Dk t Ht }0 1 the region Γa(x) = { z ∈ Ω : |z − x| < ad(z,Σ), d(z,Σ) < 1 } is then-dimensional analogue of a truncated Stolz angle. 478 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 e te l tity often l fol- , if (3) The tangential maximal function: u∗∗ m (x) = sup z∈Ω d(z,Σ)<1 ∣∣u(z) ∣∣(d(z,Σ) |z − x| )m , wherem is a positive integer. We will make use inΣ of both the geodesic distanced(x, y) and the Euclidean distanc |x −y|. Sincec1|x −y| � d(x, y) � c2|x −y|, x, y ∈ Σ , for some positive constantsc1, c2, switching from one distance to the other inan inequality will cause no trouble. We deno by B(x, r) the ball of centerx and radiusr > 0 in Rn and byBΣ(x, r) the geodesic bal in Σ . We shall also consider a special family of smooth functions defined onΣ . Definition 3.1. For everys ∈ Z+ andx ∈ Σ let Ks(x) denote the set of allφ ∈ C∞(Σ) such that for someh > 0 the conditions below are satisfied: (i) suppφ ⊆ BΣ(x,h), (ii) sup0�k�s hn−1+k‖φ‖k � 1. Definition 3.2. Forf ∈ D′(Σ) we define the grand maximal function by Msf (x) = sup φ∈Ks(x) ∣∣〈f,φ〉∣∣. The spaceHp(Σ), p > 0, is the subspace ofD′(Σ) of thosef such thatMsf ∈ Lp for s � [(n − 1)/p] + 2. Although the definition ofHp(Σ) seems to depend ons it does not as long ass is sufficiently large (s > (dimΣ)/p suffices). That this is so fors � [(n − 1)/p] + 2 follows also from Theorem 5.1 below. If T :C∞(Σ) → C∞(Σ) is a continuous linear operator, we denote byt T :D′(Σ) → D′(Σ) the transpose operator, defined by〈t T v,φ〉 = 〈v,T φ〉, v ∈ D′(Σ), φ ∈ C∞(Σ). In particular, we denote bytΓh, 0< h < 1, the transpose of the approximation of the iden discussed in the previous section (from now on, in order to alleviate the notation, we write Γh rather thanΓ L h unless there is a need to stress the role ofL). By Lemma 2.1, Γhφ → φ in C∞(Σ) if φ ∈ C∞(Σ). Furthermore,t Γh is bounded inCr(Σ) for every nonintegralr > 0. Indeed, this is clearly so if we replaceΓh by its pseudo-differentia approximationΓ̃h, which is a pseudo-differential of order zero, and the conclusion lows because the difference between the two operators has a smooth kernel. Hence φ ∈ C∞(Σ), t ΓhΓhφ → φ in C∞(Σ) ash ↘ 0. Thus, ifφ ∈ C∞(Σ) and 0< h < 1 we have the following representation: φ = tΓhΓhφ + ∞∑ j=0 ( tΓ2−j−1hΓ2−j−1h − tΓ2−j hΓ2−j h ) φ. (∗) L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 479 ds Definition 3.3. Let m, L be positive integer numbers andf ∈ D′(Σ). We define onΣ a tangential maximal function associated to the approximation of identityΓt as Γ ∗∗ m f (x) = sup y∈Σ 0�t�1/L ∣∣Γtf (y) ∣∣(1+ d(x, y) t )−m , whered(x, y) denotes the geodesic distance inΣ . Lemma 3.4. For all f ∈D′(Σ), the following pointwise inequality holds: Γ ∗∗ m f (x) � Cu∗∗ m (x), x ∈ Σ. (3.2) Proof. We recall thatu∗∗ m (x) is given by u∗∗ m (x) = sup z∈Ω ∣∣u(z) ∣∣(d(z,Σ) |z − x| )m , whereu(z) = 〈f,P (z, ·)〉. We have Γ ∗∗ m f (x) = sup y∈Σ 0�t�1/L ∣∣Γtf (y) ∣∣(1+ d(x, y) t )−m � L∑ j=1 ( L j ) sup y∈Σ 0�t�1/L ∣∣u(y − j tνy) ∣∣(1+ d(x, y) t )−m . Notice that forzj = y − j tνy we have, becausej t = d(zj ,Σ) whentj � 1,( 1+ d(x, y) t )−m � ( 1+ d(x, y) j t )−m � ( d(zj ,Σ) |zj − x| )m so that Γ ∗∗ m f (x) � L∑ j=1 ( L j ) sup z∈Ω ∣∣u(z) ∣∣(d(z,Σ) |z − x| )m � 2Lu∗∗ m (x) which proves (3.2). � We now recall the operator St = L∑ j=0 (−1)L−j ( L j ) Pj t defined in Section 2 and denote byσt (x, y), x, y ∈ Σ , its kernel. The next lemma depen strongly on the estimates (Kαβ ) proved in Theorem 1.1. Lemma 3.5. There existsCα > 0 depending only ofL andn such that for allx �= y in Σ ∣∣Dα x σt (x, y) ∣∣ � CαtL |x − y|n−1+L+|α| , 0 � t � 1/L. 480 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 d ith Proof. Notice that the first term in the sum that definesSt is ±I so its kernel is supporte in the diagonalx = y. Hence, in the proof of the lemma we need only look at terms w j � 1. We prove the estimate forα = 0, the proof for|α| > 0 is similar and will be left to the reader. We consider two cases. Case1. Letx, y ∈ Σ be such thatx �= y and|y − x| < 2Lt . Then( 1 2 )n−1+L � tn−1+L |x − y|n−1+L Ln−1+L (3.3) sinceL � 1. Taking account of (Kαβ ) with α = β = 0 we get ∣∣σt (x, y) ∣∣ � L∑ j=1 ( L j ) P(x − j tνx, y) � L∑ j=1 ( L j ) C |x − j tνx − y|n−1 � L∑ j=1 ( L j ) C (jt)n−1 � 1 tn−1 C2L � C′tL|x − y|−n+1−L, where the last inequality is a consequence of (3.3). Case2. Letx, y ∈ Σ such that|x − y| > 2Lt. Then, using Taylor’s formula, we get ∣∣σt (x, y) ∣∣ � ∣∣∣∣∣(Lt)L ∑ |α|=L ∫ [0,1]L ∂α ∂zα P ( x − t (s1 + · · · + sL)νx, y ) ds ∣∣∣∣∣ � (Lt)L ∑ |α|=L ∫ [0,1]L ∣∣∣∣ ∂α ∂zα P ( x − t (s1 + · · · + sL)νx, y )∣∣∣∣ds � tL ∫ [0,1]L C(L,n) |x − t (s1 + · · · + sL)νx − y|n−1+L ds, where we used (Kαβ ) to obtain the last inequality. Since,|x − y| > 2Lt and|t (s1 + · · · + sL)νx | < Lt it follows that 1 2 |x − y| � ∣∣x − t (s1 + · · · + sL)νx − y ∣∣ which gives the desired estimate also in this case.� Lemma 3.6. Let 1 � m < L be integers and0 < s < L. Letφ ∈ C∞(Σ) be such that (1) suppφ ⊆ BΣ(x,h) and (2) ‖φ‖L∞ � h1−n and‖φ‖CL � h1−n−L. There exists a positive constantc = c(n,Σ,L,m, s) independent ofh andφ such that ifh andth ∈ (0,1/L)∫ ∣∣Sthφ(y) ∣∣(1+ d(x, y) h )m dσ(y) � cts (3.4) Σ L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 481 f and ∫ Σ ∣∣Γhφ(y) ∣∣(1+ d(x, y) h )m dσ(y) � c. (3.5) Proof. Since (3.4) for a given value ofs implies the same estimate for smaller values os we may assume thatL−1 < s < L. Interpolating the estimates (2) we have, withρ = s/L, ‖φ‖s � C‖φ‖1−ρ L∞ ‖φ‖ρ CL � Ch1−n−s . (3.6) Note that (2) also implies that‖φ‖L1 � C. Let B = { y ∈ Σ: d(x, y) < h } , B∗ = { y ∈ Σ: d(x, y) < 2h } . Let us write the integralI on the left hand side of (3.4) asI = I1 + I2 where I1 = ∫ B∗ ∣∣Sthφ(y) ∣∣(1+ d(x, y) h )m dσ(y) and I2 = ∫ Σ\B∗ ∣∣Sthφ(y) ∣∣(1+ d(x, y) h )m dσ(y). To estimateI1 note that for everyy ∈ B∗ we have(1+ d(x, y)/h)m � 3m so∫ B∗ ∣∣Sthφ(y) ∣∣(1+ d(x, y) h )m dσ(y) � 3m ∫ B∗ ∣∣Sthφ(y) ∣∣dσ(y). (3.7) Recalling estimate (2.3) in Lemma 2.1 with|α| = 0 and (3.6) we have∫ B∗ ∣∣Sthφ(y) ∣∣dσ(y) � C ∫ B∗ (th)s‖φ‖s dσ (y) � C ∫ B∗ (th)sh1−n−s dσ (y) � Ctsh1−n ∫ B∗ dσ(y) � Cts, (3.8) with the constantC depending only ofs,L,n,Σ . Thus, (3.7) and (3.8) giveI1 � Cts . To estimateI2 observe that( 1+ d(x, y) h )m � 2mh−md(x, y)m, y ∈ Σ \ B∗. Hence, using Lemma 3.5 withα = 0 to estimate the kernelσth of Sth, we get I2 � 2mh−m ∫ Σ\B∗ ( C(th)L ∫ B |φ(w)| d(y,w)n−1+L dw ) d(x, y)m dσ(y) � CtLhL−m‖φ‖L1(Σ) ∫ ∗ d(x, y)m−n+1−L dσ(y) Σ\B 482 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 � CtLhL−mhm−L = c′′tL � c′′ts , which concludes the proof of (3.4). To prove (3.5) we note that|Γhφ(y)| � |Shφ(y)| + |φ(y)|, so (3.4) witht = 1 implies∫ Σ ∣∣Γhφ(y) ∣∣(1+ d(x, y) h )m dσ(y) � c + ∫ Σ ∣∣φ(y) ∣∣(1+ d(x, y) h )m dσ(y) and the last integral is majorized by|BΣ(x,h)|h1−n2m � c. � In the next lemma we recall a standard majorization ofu⊥(x) by the Hardy–Littlewood maximal function off Mf (x) = sup r>0 1 |BΣ(x, r)| ∫ BΣ(x,r) ∣∣f (y) ∣∣dσ(y). Lemma 3.7. There existsC > 0 such that u⊥(x) � CMf (x), f ∈ L1(Σ). (3.9) Proof. We decompose the integral u(x − tνx) = ∫ Σ P(x − tνx, y)f (y) dσ(y) as ∫ Σ = ∫ BΣ(x,2t ) + ∫ BΣ(x,4t )\BΣ(x,2t ) + ∫ BΣ(x,8t )\BΣ(x,4t ) +· · · = I1 + I2 + I3 + · · · . By Corollary 1.2,|P(x − tνx, y)| � Ct1−n so |I1| � Ct1−n ∫ BΣ(x,2t ) ∣∣f (y) ∣∣dσ(y) � C′Mf (x). To estimateIj , j � 2, we recall that∣∣P(x − tνx, y) ∣∣ � C t d(x, y)n , again by Corollary 1.2, which implies |Ij | � Ct ∫ BΣ(x,2j t ) |f (y)| (t2j )n dσ(y) � C′2−jMf (x). Hence, ∣∣u(x − tνx) ∣∣ � ∑ j |Ij | � CMf (x) ∞∑ j=1 2−j � CMf (x) and taking the supremum int we get (3.9). � L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 483 isson Let re old Since the Hardy–Littlewood maximal operator is bounded inL2 we have Corollary 3.8. If f ∈ L2(Σ), ‖u⊥‖L2(Σ) � C‖f ‖L2(Σ). (3.10) 4. Equivalence of Poisson’s maximal functions In this section we show that the three maximal functions defined through the Po integral—the normal, nontangential and tangential maximal functionsu⊥, u∗ α andu∗∗ m — have equivalentLp norms for fixedm > (n−1)/p and apertureα > 1. As before,Ω ⊂ Rn is a bounded open set with a smooth boundary∂Ω denoted byΣ . Theorem 4.1. Let f ∈ D′(Σ) be a distribution and let u denote its Poisson integral. 0 < p � ∞, 1 < α < ∞ andm > (n − 1)/p be an integer. The following conditions a equivalent: (i) u⊥ ∈ Lp(Σ); (ii) u∗ α ∈ Lp(Σ); (iii ) u∗∗ m ∈ Lp(Σ). Moreover, theLp-norms of the three maximal functions involved are equivalent. Proof. The proof has three steps. (i) ⇒ (ii) We recall the generalized mean value property for harmonic functions due to Hardy and Littlewood [10]: Lemma 4.2. LetB ⊂ Rn be a ball centered atz and suppose thatu is harmonic onB and continuous on its closureB. Then for allq > 0 there exists a constantC = C(n,q) such that ∣∣u(z) ∣∣q � C |B| ∫ B ∣∣u(w) ∣∣q dw. (4.1) Following [6], we apply (4.1) to obtain for a fixedq > 0 the estimate u∗ α(x) � CM [ u⊥q] (x)1/q, x ∈ Σ, whereM denotes Hardy–Littlewood maximal function. It is a classical result thatM is bounded inL2(Rn). This remains true ifRn is replaced by a compact Riemannian manif such asΣ . Choosingq = p/2 we have∫ Σ ∣∣u∗ α(x) ∣∣p dσ(x) � C ∫ Σ { M [ u⊥q]}p/q (x) dσ(x) � C ∫ Σ ∣∣u⊥(x) ∣∣p dσ(x), which shows that (i) implies (ii). 484 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 e with ty (ii) ⇒ (iii) The control ofu∗∗ m by u∗ α follows from very generalarguments. Consider th sets A0 = { z ∈ Ω : |z − x| � αd(z,Σ) } and Aj = { z ∈ Ω : 2j−1αd(z,Σ) � |z − x| < 2jαd(z,Σ) } , j = 1,2, . . . . Since the collection{Aj }∞j=0 is a covering ofΩ we have u∗∗ m (x) � sup A0 {∣∣u(z) ∣∣(d(z,Σ) |z − x| )m } + ∞∑ j=1 sup Aj {∣∣u(z) ∣∣(d(z,Σ) |z − x| )m } . It now follows from the definition ofu∗ α that u∗∗ m (x)p � u∗ α(x)p + ∞∑ j=1 2(1−j)mpu∗ 2j α (x)p. Integrating overΣ we obtain∫ Σ u∗∗ m (x)p dσ(x) � ∫ Σ u∗ α(x)p dσ(x) + ∞∑ j=1 2(1−j)mp ∫ Σ u∗ 2jα (x)p dσ(x). (4.2) We now use a variation of a useful lemma that relates maximal functions defined respect to different apertures [15, p. 62]: Lemma 4.3. There exists a positive constantC depending onΣ such that ifa andb are real numbers satisfying0 < b � a then the following estimate holds:∫ Σ u∗ a(x)p dσ(x) � C ( 1+ 2a b )n−1 ∫ Σ u∗ 1+b(x)p dσ(x). Invoking Lemma 4.3 witha = 2jα andb = α − 1 we get∫ Σ u∗ 2jα (x)p dσ(x) � C(Σ,α)2j (n−1) ∫ Σ u∗ α(x)p dσ(x), which combined with (4.2) yields ∥∥u∗∗ m ∥∥p Lp(Σ) � ∥∥u∗ α ∥∥ Lp(Σ) + C ∞∑ j=1 2j (n−1−mp) ∥∥u∗ α ∥∥p Lp(Σ) � C(Σ,n,p,m,α) ∥∥u∗ α ∥∥p Lp(Σ) becausen − 1− mp < 0. (iii) ⇒ (i) This implication follows trivially from the obvious pointwise inequali u⊥(x) � u∗∗ m (x). � L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 485 . As n the m- y 1.2, 5. Comparison with the intrinsic maximal function We now compare Poisson’s maximal functions with the intrinsic maximal function always,Ω ⊂ Rn is a bounded open set with a smooth boundary∂Ω denoted byΣ with the property that for any 0< t � 1 andx ∈ Σ the pointzt = x − tνx belongs toΩ and d(zt ,Σ) = t . First we see that the normal maximal functionu⊥(x) is dominated pointwise by Msf (x) for anys ∈ Z+. We may find a finite partition of unity{ρk(x)}Nk=1 in Σ such that for j = 1, . . . ,N , suppρj is contained in a ball with radiusrj < rΣ , whererΣ is the injectivity radius ofΣ . If f ∈ D′(Σ) we write f = ∑N j=1 ρjf = ∑N j=1 fj . Since Ms(ρj f )(x) � CMsf (x), it is enough to prove thatu⊥(x) � CMsf (x) when suppf is contained in a ball with a radius smaller thanrΣ . That means that in the definitio of Msf (x) = supφ∈Ks(x) |〈f,φ〉| we may consider test functions inKs(x) supported in BΣ(x, rΣ) and we shall do so. To estimateu⊥(x) we find zt = x − tνx ∈ Ω , 0 < t < 1, such thatu⊥(x) < 2|u(zt )|. Next we splitP(zt , x) as a sum of elements ofKs(x). Consider the finite covering ofBΣ(x, rΣ), BΣ(x,2t) ∪ ( BΣ(x,4t) \ B̄Σ (x, t) ) ∪ ( BΣ(x,8t) \ B̄Σ (x,2t) ) ∪ · · · , and find a partition of unity{ψj (x)}Nj=0 subordinated to this covering that satisfies estimates‖Dα x ψj‖L∞ � C(t2j )−|α|, α ∈ Z n−1+ , |α| � s. We know from Theorem 1.1 that ∣∣Dα y P(zt , y) ∣∣ � C |zt − y|n−1+|α| � C (t + |x − y|)n−1+|α| , |α| � s. Hence,∣∣Dα y ( ψ0(y)P (zt , y) )∣∣ � C tn−1+|α| , |α| � s, since|x −y| � 2t on the support ofψ0, showing thatC−1ψ0(y)P (zt , y) belongs toKs(x). For j � 1, |x − y| � t2j−1 on suppψj , which leads to ∣∣Dα y ( ψj (y)P (zt , y) )∣∣ � C (2j t)n−1+|α| , |α| � s + 1. This already shows thatC−1ψj (y)P (zt , y) ∈ Ks(x) but we need a better estimate to co pensate for the possibly large number of terms in the sum. Thus, invoking Corollar we have∣∣ψj (y)P (zt , y) ∣∣ � Ct (2j t)n = C2−j (2j t)n−1 . Summing up, we know that∥∥ψj (·)P (zt , ·) ∥∥ L∞ � C2−j (2j t)−n+1 and ∣∣Dβ y ( ψj(y)P (zt , y) )∣∣ � C(2j t)−n+1(2j t)−L−1 for |β| = L + 1. For 0� k � L and|α| = k we derive by interpolation 486 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 ∣∣Dα y ψj (y)P (zt , y) ∣∣ � C ∥∥ψj (·)P (zt , ·) ∥∥1−ρk L∞ ( max |β|=L+1 ∥∥Dβ y ψjP (zt , y) ∥∥ L∞ )ρk � C(2−j )1−ρk (2j t)−(n−1)(2j t)−(L+1)ρk � C(2−j )1−ρk (2j t)−(n−1+k) � C(2−j )δ(2j t)−(n−1+k), whereρk = k/(L + 1) andδ = 1/(L + 1). The estimates show that onBΣ(x, rΣ) we may write P(zt , x) = C N∑ j=0 2−jδφj (x), φj (x) ∈ Ks(x). It follows that ∣∣u(zt ) ∣∣ = ∣∣〈f,P (zt , x) 〉∣∣ � C N∑ j=0 2−jδ ∣∣〈f,φj (x) 〉∣∣ � C′Msf (x). Thus u⊥(x) � CMsf (x) (5.1) which may be viewed as a sharper version of Lemma 3.7. The next step is the control ofMsf (x) by u∗∗ m (x) whens > m. We will need to write the identity φ = tΓhΓhφ + ∞∑ j=0 ( tΓ2−j−1hΓ2−j−1h − tΓ2−j hΓ2−j h ) φ, φ ∈ C∞(Σ), (∗) already discussed in Section 3, in a convenient way. HereΓt = Γ L t , 0< t < 1, is chosen with L > s. Setting Γ + t = Γt/2 + Γt , Γ − t = Γt/2 − Γt , 0< t < 1, we may write Γt/2 = 1 2 ( Γ + t + Γ − t ) , Γt = 1 2 ( Γ + t − Γ − t ) , 0 < t < 1. Substitution of these formulas in (∗) gives after some simplifications φ = tΓhΓhφ + 1 2 ∞∑ j=1 tΓ − 2−j−1h Γ + 2−j−1h φ + tΓ + 2−j−1h Γ − 2−j−1h φ (∗∗) for anyφ ∈ C∞(Σ). Thus, iff ∈ D′(Σ) we have 〈f,φ〉 = 〈Γhf,Γhφ〉 + 1 2 ∞∑ j=1 〈 Γ − 2−j−1h f,Γ + 2−j−1h φ 〉 + 〈 Γ + 2−j−1h f,Γ − 2−j−1h φ 〉 . We must estimate each term in the expression above whenφ ∈ Ks(x). Assuming that suppφ ⊂ BΣ(x,h) and that 0< h < 1/L, we have 〈Γhf,Γhφ〉 = ∫ Γhf (y)Γhφ(y) dσ(y). Σ L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 487 e same Taking account of Definition 3.3 and Lemma 3.4 we get ∣∣Γhf (y) ∣∣ � Cu∗∗ m (x) ( 1+ d(x, y) h )m so ∣∣〈Γhf,Γhφ 〉∣∣� Cu∗∗ m (x) ∫ Σ ( 1+ d(x, y) h )m∣∣Γhφ(y) ∣∣dσ(y) � Cu∗∗ m (x), (5.2) in view of (3.5) in Lemma 3.6. To study the terms〈Γ + 2−j−1h f,Γ − 2−j−1h φ〉 we point out that sinceΓ L t = I − (−1)LSL t we have∣∣Γ − 2−j−1h φ ∣∣ � |S2−j−1hφ| + |S2−j−2hφ|. Then, (3.4) in Lemma 3.6 gives∫ Σ ( 1+ d(x, y) h )m∣∣Γ − 2−j−1h φ(y) ∣∣dσ(y) � C2−js . On the other hand, using once more Lemma 3.4, we have ∣∣Γ + h2−j−1f (y) ∣∣ � Cu∗∗ m (x) ( 1+ d(x, y) 2−j−1h )m � Cu∗∗ m (x)2jm ( 1+ d(x, y) h )m so ∣∣〈Γ + 2−j−1h f,Γ − 2−j−1h φ 〉∣∣ � C2jmu∗∗ m (x) ∫ Σ ( 1+ d(x, y) h )m∣∣Γ − h2−j−1φ(y) ∣∣dσ(y) � C2j (m−s)u∗∗ m (x) � C2−j u∗∗ m (x). (5.3) To handle the terms〈Γ − 2−j−1h f,Γ + 2−j−1h φ〉 we writeΓ + 2−j−1h φ = 2φ − (−1)L(S2−j−1hφ + S2−j−2hφ) which leads to〈 Γ − 2−j−1h f,Γ + 2−j−1h φ 〉 = 2 〈 Γ − 2−j−1h f,φ 〉 + Rj (5.4) with |Rj | � ∣∣〈Γ − 2−j−1h f,S2−j−1hφ 〉∣∣ + ∣∣〈Γ − 2−j−1h f,S2−j−2hφ 〉∣∣. The terms on the right hand side can be estimated using Lemmas 3.4 and 3.6—in th fashion used to obtain (5.3)—in order to get |Rj | � C2−ju∗∗ m (x). (5.5) Since 2 ∞∑ j=1 〈 Γ − 2−j−1h f,φ 〉 = 2〈f − Γh/2f,φ〉 = 2(−1)L〈Sh/2f,φ〉, we obtain, in view of (5.2)–(5.5), that∣∣〈f,φ〉∣∣ � Cu∗∗ m (x) + ∣∣〈Sh/2f,φ〉∣∣. 488 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 small l 6). In the proof of this inequality we assumed for simplicity and ease of notation the restriction thatφ was supported in a ball of radiush < 1/L, which was useful for technica reasons (the relevant operators in all terms of (∗∗) satisfied the hypotheses of Lemma 3. In the general case, ifd is the diameter ofΣ , we may findλ > 0 so thatλd < 1/L and replace (∗∗) by φ = tΓλhΓλhφ + 1 2 ∞∑ j=0 tΓ − 2−j−1λh Γ + 2−j−1λh φ + tΓ + 2−j−1λh Γ − 2−j−1λh φ. Carrying out the proof with this representation we get∣∣〈f,φ〉∣∣ � C(λ)u∗∗ m (x) + ∣∣〈f, tSλh/2φ〉∣∣ for anyφ ∈ Ks(x), so taking the supremum inφ ∈ Ks(x) we obtain Msf (x) � Cu∗∗ m (x) + ∣∣〈f, tSλh/2φ〉∣∣. (5.6) The operatort Sh has similar properties toSh. In fact, the latter is related to L = ∂2 ∂t2 + 2 n−1∑ j=1 bj (x, t) ∂2 ∂xj ∂t + ∑ j,k cjk(x, t) ∂2 ∂xj∂xk + · · · , in the same wayt Sh is related to L̃ = ∂2 ∂t2 − 2 n−1∑ j=1 bj (x, t) ∂2 ∂xj ∂t + ∑ j,k cjk(x, t) ∂2 ∂xj∂xk + · · · , which is elliptic if and onlyL is. From the analog of Lemma 3.5 for the kernelσ̃t (x, y) of t Sh (notice thatσ̃t (x, y) vanishes fort = 0 andx �= y) we get for allx �= y in Σ ∣∣Dασ̃t (x, y) ∣∣ � C tL |x − y|n−1+L+|α| , 0 � t � 1/L, (5.7) for someC > 0 depending only onL andn. SetΦ = t Sλh/2φ. By Lemma 2.1 ‖DαΦ‖L∞ � C(λh)r−|α|‖φ‖r � C′λh1−n−|α| (5.8) for |α| < r < L. Assuming without loss of generality thath is smaller that the injectivity radiusrΣ we find a partition of unity{ψj(x)}Nj=0 subordinated to the covering BΣ(x,2h) ∪ ( BΣ(x,4h) \ B̄Σ(x,h) ) ∪ ( BΣ(x,8h) \ B̄Σ (x,2h) ) ∪ · · · of BΣ(x, rΣ) that satisfies the estimates‖Dα x ψj‖L∞ � C(h2j )−|α|, α ∈ Z n−1+ , |α| � s and write Φj (y) = ψj(y)Φ(y) = ψj(y) ∫ σ̃λh/2(y, y ′)φ(y ′) dσ(y ′) soΦ = ∑N j=0 Φj . Choosingr > s, (5.8) and the fact that suppψ0 ⊂ BΣ(x,2h) allows us to write Φ0 = Cλ�0 with � ∈ Ks(x). For j � 1, y ∈ suppψj andy ′ ∈ suppφ, d(y, y ′) ∼ d(y, x) ∼ 2jh, so for those values ofy, y ′ we have∣∣Dα y σ̃λh/2(y, y ′) ∣∣ � C(λ2−j )L(2−jh−1)n−1+|α| L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 489 Let which shows thatΦj = Cλ2−j�j with �j ∈ Ks(x). Thus, takingλ sufficiently small we may assume thatt Sλh/2φ = (1/2) ∑N j=0 2−j−1�j with �j ∈ Ks(x). Now (5.6) gives Msf (x) � Cu∗∗ m (x) + 1 2 Msf (x) which shows that Msf (x) � Cu∗∗ m (x) (5.9) if Msf (x) < ∞. For arbitraryx ∈ Σ we may reason with an approximation ofMsf (x). Set Mε sf (x) = sup φ∈Kε s (x) ∣∣〈f,φ〉∣∣, whereKε s (x) is the space of smooth functionsφ ∈ C∞(Σ) such that there is anh > ε such that suppφ ⊂ B(x,h) and sup0�k�s hN+k‖φ‖k � 1. Reasoning as before withMε sf (x) which is always finite in the place ofMsf (x) we get Mε sf (x) � Cu∗∗ m (x) and lettingε → 0 we obtain (5.9) in general. Summing up, (5.1) and (5.6) show that fors > m we have the pointwise inequalities u⊥(x) � CMsf (x) � C2u∗∗ m (x) which trivially imply ‖u⊥‖Lp(Σ) � C‖Msf ‖Lp(Σ) � C2 ∥∥u∗∗ m ∥∥ Lp(Σ) . However, if m > (n − 1)/p, Theorem 4.1 asserts that theLp norms ofu⊥ andu∗∗ m are comparable. We have proved Theorem 5.1. Let f ∈ D′(Σ) be a distribution and let u denote its Poisson integral. 0 < p � ∞, 1 < α < ∞, and assumes > m > (n − 1)/p are integers. The following conditions are equivalent: (i) f ∈ Hp(Σ); (ii) Msf ∈ Lp(Σ); (iii ) u⊥ ∈ Lp(Σ); (iv) u∗ α ∈ Lp(Σ); (v) u∗∗ m ∈ Lp(Σ). Moreover, theLp-norms of all maximal functions involved are comparable. 6. Complex Hardy spaces In this sectionΩ will denote a bounded open subset of complex spaceCn with smooth boundary∂Ω = Σ . Denote byρ a smooth real function that vanishes precisely onΣ 490 L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 ucture o- g s ry, s such thatdρ �= 0 on Σ andρ > 0 on Ω . For ε > 0 sufficiently small the setΣρ ε = {z ∈ Ω : ρ(z) = ε} is a smooth embedded orientable hypersurface with a Riemannian str inherited fromCn � R2n. For 0< p < ∞, the complex Hardy spaceHp(Ω) is defined as the space of all hol morphic functionsf defined onΩ such that sup 0<ε<ε0 ∫ Σ ρ ε ∣∣f (z) ∣∣p dσρ ε (z) < ∞. (6.1) Since|f |p is subharmonic, the independence of condition (6.1) from the particular definin functionρ follows from the following lemma of Stein [13]. Lemma 6.1. Let ρ andρ′ be two defining functions forΩ and letu be a positive subhar- monic function onΩ . Then sup 0<ε<ε0 ∫ Σ ρ ε u(z) dσρ ε (z) < ∞ if and only if sup 0<ε<ε0 ∫ Σ ρ′ ε u(z) dσρ′ ε (z) < ∞. We may take asρ the functionΩ � x − tνx �→ t , defined forx ∈ Σ and 0< t < t0, and set ‖f ‖p Hp = sup 0 0 are the submanifold Σt already considered in Section 2. We recall that iff (z) is holomorphic onΩ and has tempered growth at the bounda i.e., |f (z)| � C dist(z,Σ)−N for some positive constantsC andN , thenf (z) has a weak boundary valuebf ∈ D′(Σ) [11, p. 66]. This means that if we regard the restrictionsft = f |Σt as distributions defined onΣ via the identificationΣt � x − tνx �→ x ∈ Σ , then 〈ft ,φ〉 → 〈bf,φ〉 for any φ ∈ C∞(Σ) as t → 0. Conversely, ifbf exists,f must have tempered growth at the boundary. We denote byHb(Ω) the space of holomorphic function onΩ with tempered growth at the boundary. Theorem 6.2. LetΩ ⊂ Cn be a bounded open subset with smooth boundary∂Ω = Σ . Let 0 < p � ∞ and letf (z) be a holomorphic function onΩ . The following properties are equivalent: (i) f ∈ Hp(Ω); (ii) |f |p has a harmonic majorant onΩ ; (iii ) f ∈ Hb(Ω) andbf ∈ Hp(Σ); L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 491 sume e g f (iv) f is the Poisson integral of somef0 ∈ Hp(Σ). Proof. Whenp = ∞ (ii) has to be understood as “‖f ‖L∞ has a harmonic majorant onΩ”; the proof of this case is simpler and will be left to the reader, so from now on we as that 0< p < ∞ and prove the theorem in four steps. (i) ⇒ (ii) Let G(z,w) be the Green function ofΩ . Fix a pointz0 ∈ Ω and consider the function w �→ G(z0,w). Since the normal derivative−νw(D)G(z0,w) = P(z0,w) > 0 for w ∈ Σ andG(z0,w) > 0 for w ∈ Ω , we may use a defining functionρ(w) for Ω that coincides withG(z0,w) whenw ∈ Ω is close toΣ . For ε > 0 and small setΩε = {z ∈ Ω : ρ(z) > ε}. Then the Green functionGε(z,w) of Ωε is given byGε(z,w) = G(z,w)−ε and writing the Poisson kernelPε(z0,w) of Ωε, w ∈ ∂Ωε, as the normal derivative of th Green function it follows thatPε(z0,w) → P(z0,w) uniformly onw ∈ Σ asε → 0 if we identify points in∂Ωε with their normal projections ontoΣ . Let f ∈ Hp(Ω). Then the restriction of|f |p to ∂Ωε belongs toL1(∂Ωε) uniformly in ε ↘ 0. By our identification we may think of|f |p∂Ωε as bounded subset ofL1(Σ) and find a sequenceεj such thatvj = |f |p∂Ωεj converges weakly to a positive Radon measureµ ∈ M(Σ). Let uj be harmonic function onΩεj with boundary valuevj = |f |p∂Ωεj . Since|f |p is subharmonic,vj � uj onΩεj , so for largej we have|f (z0)|p � uj (z0). We may write uj (z0) = ∫ ∂Ωεj Pεj (z0, y)vj (y) dσεj (y) and letεj → 0 to get∣∣f (z0) ∣∣p � 〈 µ,P(z0, ·) 〉 .= u(z0) sou(z), the Poisson integral ofµ, is the required harmonic majorant. (ii) ⇒ (iii) Since |f |p has a harmonic majorant,|f |p/2 � 1+|f |p also does. Reasonin as before, we may find a functionv ∈ L2(Σ) which is the weak limit of the restrictions o |f |p/2 to ∂Ωεj . Hence, ifu(z) is the Poisson integral ofv we have|f |p/2(z) � u(z), z ∈ Ω . Then f ⊥(x) .= sup 0 0 small we have|f (x − tνx)| = |U(x − tνx)| � U⊥(x) so∫ Σt ∣∣f (z) ∣∣p dσt (z) � C ∫ Σ U⊥(x)p dσ(x) < ∞ which shows thatf ∈Hp(Ω). � Corollary 6.3. If f ∈Hp(Ω), the “norms” ‖f ‖Hp , ‖f ⊥‖Lp , ‖f ∗ a ‖Lp are all comparable. Although the usual product of functions cannot be extended to distributions pres the associative property, it is possible to define the product of two distributionsf0, g0 ∈ Hb(Ω) asf0g0 = b(fg) wherebf = f0 andbg = g0, turningHb(Ω) into an associative algebra. A more precise version of this fact is given by Corollary 6.4. If f0, g0 ∈ Hp(Σ), the productf0g0 ∈ Hp/2(Σ). Proof. Let f andg be respectively the Poisson integrals off0 andg0, sof,g ∈ Hp(Ω) by Theorem 6.2. Hence, Schwarz inequality gives ∫ Σt ∣∣fg(z) ∣∣p/2 dσt (z) � (∫ Σt ∣∣f (z) ∣∣p dσt (z) )1/2(∫ Σt ∣∣g(z) ∣∣p dσt (z) )1/2 � C showing thatfg ∈Hp/2(Ω). Thus,f0g0 = b(fg) ∈ Hp/2(Σ). � A simple consequence of the Poisson representation forH1(Ω) functions is the follow- ing version of the F. and M. Riesz theorem. L.A. Carvalho dos Santos, J. Hounie / J. Math. Anal. Appl. 299 (2004) 465–493 493 s a ct - ue to rs, ana 5 Car, DM, mática, . 32) ton 1. Theorem 6.5. LetΩ ⊂ Cn be a bounded open subset with smooth boundary∂Ω = Σ and assume thatf (z) is holomorphic inΩ with tempered growth at the boundary and ha measureµ ∈ M(Σ) as weak boundary value. Thenµ is absolutely continuous with respe to dσ . Proof. SinceM(Σ) ⊂ Hp(Σ) for p < 1, Theorem 6.2 shows thatf is the Poisson inte gral of its boundary valueµ. Moreover, the Poisson representationf (z) = 〈µ(y),P (z, y)〉 shows that|f (z)| � 〈µ|(y),P (z, y)〉 where|µ| is the variation ofµ. Interchanging the order of the integration we see that∫ Σt ∣∣f (z) ∣∣dσt (z) � 〈 |µ|(y), ∫ Σt P (z, y) dσt (z) 〉 � C|µ|(Σ). Thusf ∈ H1(Ω) and Theorem 6.2 implies thatµ = bf ∈ H 1(Σ) ⊂ L1(Σ), as we wished to prove. � Remark 6.6. Theorem 6.5 also follows from an analogous and stronger local result d Brummelhuis [3] according to which if a measureµ is defined on an open subsetV of Σ and is the boundary value of a holomorphic functionf defined on one side ofΣ thenµ is absolutely continuous with respect todσ onV . References [1] J. Álvarez, J. Hounie, Estimates for the kernel and continuity properties of pseudo-differential operato Ark. Mat. 28 (1990) 1–22. [2] T. Aubin, Nonlinear Analysis on manifolds.Monge–Ampère Equations, Springer, New York, 1982. [3] R.G.M. Brummelhuis, A microlocal F. and M. Riesz theorem with applications, Rev. Mat. Iberoameric (1989) 21–36. [4] L.A. Carvalhos dos Santos, Espaços de Hardy em variedades compactas, Tese de Doutorado, UFS 2002. [5] L. Colzani, Hardy spaces on unit spheres, Boll. Un. Mat. Ital. 4 (1985) 219–244. [6] C. Fefferman, E.M. Stein,Hp spaces of Several variables, Acta Math. 129 (1972) 137–193. [7] J. García-Cuerva, R. de Francia, Weighted Norm Inequalities and Related Topics, Notas de Mate vol. 104, 1985. [8] D. Gilbarg, M.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1977 [9] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979) 27–42. [10] G.H. Hardy, J.H. Littlewood, Some properties of conjugate functions, J. Reine Angew. Math. 167 (19 405–423. [11] L. Hörmander, The Analysis of Linear Partial Differential Operators I, seconded., Springer, Berlin, 1990. [12] N. Kerzman, Topics in Complex Analysis, unpublished notes. [13] S.G. Krantz, Function Theory of Several Complex Variables, 1992. [14] J. Peetre, Classes de Hardy sur les varietés, C. R. Acad. Sci. Paris Sér. A-B 280 (1975) A439–A441. [15] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Prince Univ. Press, 1993. [16] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 197 [17] R. Strichartz, The Hardy spaceH1 on manifolds and submanifolds, Can. J. Math. 24 (1972) 915–925. [18] M.E. Taylor, Pseudodifferential Operators, Princeton Univ. Press, 1981. [19] F. Treves, Introduction to Pseudodifferential andFourier Integral Operators, Plenum, New York, 1980.