Distributional Inequalities for Non-harmonic Functions

The relationship between the non-tangential maximal function and convenient versions of the area function of a general (non harmonic) function in a upper-half space are studied.


INTRODUCTION
Let u be a harmonic function in the upper half space R n+1 + = {(w, y) : w ∈ R n , y > 0} of R n .Given α > 0, we define the area function of u at a point x ∈ R n as the integral of |∇u| 2 over the cone Γ α (x) in the x n+1 -direction: |∇u(w, y)| 2 y 1−n dw dy and the non-tangential maximal function of u at x as |u(w, y)|.
We say that u is non-tangentially bounded at x if (N α,0,s u)(x) < ∞ for some α > 0 and some (all) ∞ > s > 0. Calderón ( [7]) proved that, if u is a harmonic function in R n+1 + and E is the set of points in R n on which u is non-tangentially bounded, then for almost every point x ∈ E, (A α,0,s u)(x) < ∞ for all α > 0, s > 0. Stein [24] proved the converse: if F ⊂ R n is a set with the property that for every x ∈ F , there exists α = α(x) > 0, s = s(x) > 0 such that (A α,0,s u)(x) < ∞, then for almost every point x ∈ F , the function u is nontangentially bounded at x.These results were first proved in dimension 1 by Marcinkiewicz and Zygmund [20] and Spencer [23].Calderón ( [8]) also showed that a harmonic function in R n+1 + has a non-tangential limit at almost every point where it is non-tangentially bounded.Summarizing: if u is a harmonic function in the upper half space R n+1 + , the sets {x ∈ R n : u has non-tangential limit at x}, {x ∈ R n : there exist s = s(x), α = α(x) such that (A α,0,s u)(x) < ∞}, can only differ in a set of measure 0. The results of Calderón and Stein above on the connection between the area integral and the non-tangential maximal function extend to the L p -setting.By a result of Fefferman and Stein, for all α, β, p > 0, the L p -norm of A α is dominated by the L p -norm of N β .The converse estimate holds if one in addition assumes that lim y→∞ u(x, y) = 0 for each x ∈ R n .For an Orlicz-norm extension of this result see the paper [5] by Burkholder and Gundy.The proofs are based on inequalities relating the distribution functions of the non-tangential maximal function and the area function.These inequalities, which came to be known as good-λ inequalities, were sharpened by Murai and Uchiyama [22] and Bañuelos and Moore [3].The result of Murai and Uchiyama can be stated as follows: If u is a harmonic function in the upper half space R n+1 + and 0 < β < α, then there exist constants C 1 , C 2 > 0 (depending on α, β, n) such that for any M > 1, λ > 0 one has and The proof involves the construction of certain "sawtooth" regions and estimates on the BMO-norm of truncated versions of the area function in such regions.Although both estimates have an exponential decay, observe that (1.2) does not have the whole subgaussian decay.These results have been extended to other contexts.
In [21], [28], [17] area integrals of subharmonic functions are considered and an L p -estimate is proved.Area integrals for solutions of second order elliptic equations were considered by Dahlberg, Jerison and Kenig [13].See also [17].The result of Bañuelos-Moore is deeper than (1.2), and using the preceding notation can be stated as: The proof is based on reducing the desired estimate to its dyadic martingale analog, which was proved by Chang, Wilson and Wolff ( [9]).It is worth mentioning that Bañuelos and Moore ( [3]) extended the Murai-Uchiyama result (1.1) to harmonic functions in Lipschitz domains, while the estimate (1.3) is not known in this generality (see [2, p. 98, p.113]).For an exposition and extension of these ideas see [2, Chapter IV] by Bañuelos and Moore.
Yet another result in similar vein is the Law of the Iterated Logarithm for harmonic functions that we next describe.Suppose that u is harmonic in the upper half plane and that |∇u(x, y)| ≤ C/y for each (x, y).Then, according to a result of Makarov ( [19]), (1.4) lim sup for almost all x ∈ R.This result was extended to harmonic functions in the upper half space in [1] and to Lipschitz domains in [18].It can be considered as a Fatou type theorem: for each x ∈ R n , the estimate |∇u(x, y)| ≤ C/y gives a logarithmic upper bound on the growth of u when y tends to zero, but a substantially improved estimate holds for almost all x.The following area integral version of this Law of the Iterated Logarithm was established by Bañuelos, Klemeš and Moore [1], [2, Chapter III] by reducing it to the dyadic martingale setting.Fix 0 < β < α and 0 < γ < 1.There is a constant The results of Calderón and Stein cited above state that the sets where u is non-tangentially bounded and where the area function is finite can only differ on a set of Lebesgue measure zero.In the complement of this set, that is, at almost every point where the function is not non-tangentially bounded, this Law of the Iterated Logarithm (LIL) measures the relative growth of these quantities.
It is clear that the results above do not hold for all smooth functions.Very recently two of us showed in [14] that the result of Makarov on the growth of harmonic functions has the following analog for solutions to the Poisson equation ∆u = f in the unit ball B n of R n under a growth condition on f : Let u be a C 2 function in the unit ball B n that satisfies |∇u(x)| ≤ C/(1 − |x|) and assume in addition that for almost all ζ ∈ S n−1 .Here c depends only on C and n.
Notice that the growth order of |u(x)∆u(x)| above is at most C(1 − |x|) −2 , which is no more than the worst possible growth of |∇u(x)| 2 .Because of this, the contribution of the Laplacian can be embedded in the "gradient" estimates and the indicated version of the LIL can be proven.
In this paper, we continue the analysis of the relation between the non-tangential maximal function and the area function of non-harmonic functions.To control the non-tangential maximal function by the area function, very light assumptions on the function are needed.More precisely, we say that a C 2 -function u in R n+1 + has ϕ-controlled oscillation if there exists a constant 0 ≤ η < 1 and an increasing positive function + , the estimate: holds, where (1 + η)B is the ball with the same center as B and radius We will often refer to this condition as condition ( * ).
A harmonic function u in R n+1 + satisfies condition ( * ) for any 0 < η < 1 with ϕ(t) = Ct, where C is a constant depending on η.This is a simple consequence of the subharmonicity of |∇u|.Condition ( * ) holds for many functions.For instance, we will prove that a function + , for a fixed constant C, satisfies condition ( * ) with the function ϕ(t) = At, where A = A(C) is a constant.In the general context of functions u satisfying ( * ), we consider the area function S α u defined as Clearly, for functions satisfying (1.6), the area function (S α u) is comparable to the usual area function (A α u).Also, if x ∈ R n and y > 0, we denote by (S α u)(x, y) (resp.(N α u)(x, y)) the area function (resp.the non tangential maximal function), obtained by replacing the cone Γ α (x) by its translation Γ α (x, y) = y + Γ α (x) so that its vertex is at (x, y).
We prove a good-λ inequality analogous to (1.2), relating the non-tangential maximal function and the area function S α of a function satisfying condition ( * ).We do not know if a subgaussian estimate of the type of (1.3) holds in this more general setting.The good-λ inequality we prove leads to a Fatou-type theorem, to L p -estimates and to a certain LIL, which we collect in the following result.
For 0 < p < ∞, there exists a constant C depending on p, α, β, n, ϕ, m such that (c) There exists a constant C depending on α, β, ϕ, n such that As mentioned before, in the harmonic case one can take ϕ(t) = ct.So, in this case, comparing (c) with the LIL (1.5) which holds for harmonic functions, there is a square root missing in the term log log S α .This is due to the fact that the good-λ inequality we prove does not have the whole subgaussian decay.On the other hand, our results hold on Lipschitz domains, while the LIL and the good-λ inequality with subgaussian decay for harmonic functions on Lipschitz domains are open problems.See [2, p. 98].Also, for solutions of second order elliptic equations an estimate analogous to (1.3) is only known in very concrete cases.See [27] and [2, p. 49].
The control of the area function by the non-tangential maximal function requires more specific assumptions on the function u.Let u ∈ C 2 (R n+1 + ).We say that the function u satisfies condition (1.7) if there exists 0 < θ < 1 such that (1.7) for any (x, y) ∈ R n+1 + .As before, for functions u satisfying condition (1.7), the area functions A α u and S α u are comparable.We prove a good-λ inequality which is analogous to the estimate (1.1) of Murai and Uchiyama.Again, the good-λ inequality leads to a Fatou type result, an L p -estimate and to a Law of the Iterated Logarithm.
+ ) be a function satisfying condition (1.7) for a constant 0 < θ < 1.Let 0 < α < β and assume there exists there exists a constant C 1 depending on p, α, β, n, θ such that (c) There exists a constant C 2 depending on α, β, n, θ such that Local versions of Theorems 1.1 and 1.2 also hold.For instance, one can replace A α , N α in (a) in both results by A α,0,1 , N α,0,1 .
We do not have an example to show that the condition 0 ≤ θ < 1 is essential, but we see no real hope in relaxing this condition because of the following: There is a bounded C 2 -function u so that the area integral of u is infinite at every point, but still (1.6)holds in the following averaged sense for some constant C ≥ 1: for each ball B = B((w, t), (1 − ε)t), where ε > 0 is a small positive constant.This indicates that perhaps one should replace (1.7) by an averaged integral, and similarly replace u by the average of u.We have not been able to do this, but the reason might well be only technical.
The proofs of Theorems 1.1 and 1.2 are based on stopping time arguments and on Green's formula on sawtooth regions.In this sense, the work of Murai and Uchiyama [22] and of Bañuelos and Moore [3] provide not only an outline for our proof, but also important techniques.However, the fact that our function is not harmonic causes several difficulties, which are solved by a careful choice of the domains where we apply Green's formula, by stopping time arguments and Caccioppoli inequalities.Even if one only wishes to prove the results in the upper half space, the use of stopping time arguments and sawtooth regions leads to Lipschitz domains, and one needs to study the corresponding results in this context.It is actually on the setting of Lipschitz domains that we prove our results.The work of Dahlberg ([11]) provides the necessary estimates for the harmonic measure and Green's functions on such domains.
The paper is organized as follows.Section 2 contains notation and background material on Lipschitz domains.Section 3 is devoted to the results on the control of the non-tangential maximal function by the area function S. Section 4 is devoted to the converse results, that is, to the control of the area function by the non-tangential maximal function.Finally, Section 5 contains the proof that condition (1.6) implies ( * ) with ϕ(t) = At.
We are indebted to the referee who pointed out several errors and proposed alternative arguments.

NOTATION, DEFINITIONS AND BACKGROUND ON LIPSCHITZ DOMAINS
From now on, we will consider domains of the form where φ : R n → R is a Lipschitz function with Lipschitz constant M, that is, we also introduce the truncated cones: with special attention to the cases t = 0 or s = ∞.
We define the non-tangential maximal function of u by: (N α u)(x) = sup Γ α (x) |u|, and also its truncated version: (N α,t,s u)(x) = sup Γ α,t,s (x) |u|.Furthermore, we define the area function associated to the density f by: , and also the truncated version: We are specially interested in the case f = |∇u|2 .Then, we will simply denote, and we will refer to them as the (truncated) area functions of u.When u is harmonic, these forms of the area functions were already used in [3].
We also need to consider a new square function (S α u)(x), in spirit very similar to the usual area function, (A α u)(x).In fact, they both coincide when u is harmonic.So, following the notation above, we define and its truncated version If Ω, φ are as above, then, whenever As in the case of R n , it is useful to deal with dyadic decompositions in Ω. Denote by F k the family of all dyadic cubes of the generation k, that is, all cubes Finally, we close this preliminary section with some properties of Green's function and the harmonic measure in Lipschitz domains that will be needed later.
Suppose that and let g, ω be Green's function and the harmonic measure in Q, with respect to p Q .We also denote by ω * the projection of ω, restricted to the graph The following theorem proved by Dahlberg [11] collects the central properties of ω.
In particular, ω * is mutually absolutely continuous with respect to the Lebesgue measure in Q.(b) The density dω * /dx is an A ∞ -weight.Actually, there exist positive constants, C, α, β depending only on M (the Lipschitz constant of Ω), such that , where 1 2 Q is the cube with the same center as Q and half its side length.Then

CONTROL OF THE NON-TANGENTIAL MAXIMAL FUNCTION BY THE AREA FUNCTION
Throughout this section we will consider functions u ∈ C 2 (Ω) whose oscillation on hyperbolic balls is controlled by a quantity similar to the one defining (S α u)(x).By a hyperbolic ball centered at a point w 0 ∈ Ω, we will understand an euclidean ball centered at w 0 whose radius is cδ(w 0 ), where 0 < c < 1, and δ(w 0 ) = dist(w 0 , ∂Ω).Since Ω is a Lipschitz domain, these balls are actually comparable to the ones induced by the hyperbolic metric in Ω.
We recall the notation given at the introduction.We say that a function u satisfies condition ( * ) if there exists a constant η, 0 ≤ η < 1, and an increasing function where (1 + η)B is the ball with the same center as B and radius Note that, if u is harmonic in Ω, then it satisfies condition ( * ) for any 1 > η > 0 and ϕ(t) = Ct, where C is a constant depending on η.This is simply a consequence of the subharmonicity of the gradient.Actually, given w 1 , w 2 ∈ B, let L be the line that joins them.Then, using that |∇u| is subharmonic, Fubini's Theorem and Hölder's inequality, we get for with comparison constants only depending on η.From now on, B f will denote the average of the function f over the ball B.
We proceed now to state and prove a lemma that will become our main auxiliary result, but before that let us fix the notation. Let and by w Q the point w Q = (x 0 , φ(x 0 )+ (Q)), which is contained in the boundary of T ( Q).Also recall that the usual dyadic decomposition of Q gives a dyadic decomposition of Q ⊂ Ω.

Then there exist constants
Proof.The idea is similar to the one given in [25] to prove the local version of Fatou's Theorem for harmonic functions.We will build a new Lipschitz domain R ⊂ Ω and we will apply Green's formula in R to the functions (u − u(w Q )) 2  and g(z), where g(z) is some Green's function to be specified later on.In the harmonic case, the boundary terms in Green's formula are controlled using properties of the Poisson Kernel.Since this tool is not available in our setting, we will need to choose very carefully the hyperplanes in the construction of ∂R, in order to overcome the technical difficulties that arise in estimating such boundary terms.
We will often use dm and dσ to denote the Lebesgue measure and the surface measure, respectively.Also by A B, (A B) we will mean that A ≤ CB (A ≥ CB), where C is a constant.Moreover, A B means that A B and A B. Without loss of generality we can assume that Since T ( Q) can be covered by a fixed number of cones and (S α u)(x) ≤ a for all x ∈ Q, we get with comparison constant depending on α, M and n.By Chebychev's inequality, we can choose a "horizontal hyperplane" L such that (3.1) 1  2 < y 0 < 1. Observe that condition ( * ) and the bound on the function S α give that osc(u, T ( Q)) < C(α, M, n)ϕ(a).Hence, if the constants c, A are chosen sufficiently large, one may assume that Next, consider the domain Let p 0 = (x 0 , φ(x 0 ) + 3 2 y 0 ) and let g(•) = g(•, p 0 ) be Green's function of D with pole at p 0 .Denote by ω(•) = ω(•, p 0 , D) the harmonic measure in D with respect to p 0 .Even though D is not a Lipschitz domain, it is clear that the results concerning the harmonic measure and behavior of Green's function stated in Theorem 2.1 hold on D as well.
Applying now Green's formula in the region to the functions (u − u(w Q )) 2 and g, we obtain Because g vanishes in ∂D ∩ ∂R, the integrals on ∂R can be written as We proceed now to estimate the integrals on L ∩ T ( Q). Observe that in the hyperbolic metric the diameter of T ( Q k ), where Q k is any dyadic cube, is bounded from above and below by constants depending only on M and n.In particular, T ( Q) can be covered by k balls B i of center w i and radius ρδ(w i ), where k depends on ρ and on n.Fix ρ 0 = ρ 0 (α) so that each such hyperbolic ball is covered by a bounded number of cones.Now condition ( * ) implies that for any Observe that one can assume that δ(w i ) is comparable to δ(w) for all w ∈ (1 + η)B i .Thus for some constant c = c(ρ 0 ) > 1 and for any w ∈ L ∩ T ( Q), Also, g(z) 1 and |∂ n g| y −n 0 on L ∩ T ( Q) by the estimates related to Green's function in Theorem 2.1, therefore 2 |∂ n g| dσ ϕ 2 (ca) (3.4) and aϕ(ca).(3.5)Note that the last inequality is a consequence of (3.1).
Next we estimate the right hand side term in (3.2).Consider g * the Green's function for the domain D ∪ ( Q j ) with pole at the point p 0 .Then by the maximum principle, g * ≥ g in D. Thus, changing the order of integration and using the left inequality in Theorem 2.1(c), one has where The argument below will indicate how large A must be chosen.
To estimate the last integral in (3.6) we will first assume that |u(w Q )| ≥ 2Nϕ(ca) and consider the general case afterwards.So, let us assume that |u(w Therefore, going back to (3.2) and using (3.4) and (3.5), we obtain The cubes Q j are chosen so that |u(w) − u(w Q )| > Nϕ(ca) for some w ∈ T ( Q j ).Since the oscillation on hyperbolic balls is controlled by ϕ(ca), arguing as above we get that for all w ∈ T ( Q j ) (3.7) where c 0 = c 0 (ρ 0 , α, M, n).We deduce then that (3.8) 204 M.J. GONZ ÁLEZ, P. KOSKELA, J.G.

Now we use Theorem 2.1(b) with
Taking roots and using the assumption (Q) = 1, this ends the proof in the case , replace N by N/8 and apply the previous argument.If Nϕ(ca) > 4|u(w Q )| we consider an intermediate family of cubes.Denote by {Q (1) } the maximal dyadic cubes contained in Q satisfying sup w∈T ( Q (1) ) Because of the maximality of Q (1) , if we denote by w the point w Q , where Q is the smallest dyadic cube which properly contains Q (1) , we have Since w and w Q (1) are at some fixed hyperbolic distance, the same argument as before gives: (3.9) Hence, and we can apply the previous case, that is, for each Q (1) , choose the maximal dyadic cubes {Q (2)  k } contained in Q (1) such that sup So, proceeding as before we obtain 1 Finally, note that if w ∈ Q \ k Q (2)  k , because of the left inequality in (3.9) and the maximality of {Q (2)   k }, if N is big enough.Hence j Q j ⊂ ∪Q (2)  k and the proof is completed.
Ë Next, we shall prove a good-λ inequality between the distribution functions of the non-tangential maximal function N β u and the area function S α u, where u is a function having ϕ-controlled oscillation in an unbounded Lipschitz domain This type of a distribution inequality will lead to an L p -inequality comparing both quantities, to a certain Law of the Iterated Logarithm and to a version of Fatou's Theorem in this setting.Fix α and β such that 0 < α < β < 1/M.Then there exist constants c 0 , c 1 and c 2 depending only on (α, β, M, n, ϕ) such that, for any λ ≥ 1 and any γ > c 0 , one has Proof.Fix λ > 1.We may assume that the set {x ∈ R n : (N α u)(x) > λ} has a finite measure.Let ε 0 > 0 be a small positive number, to be fixed later.Let Q be a maximal dyadic cube such that Also, observe that, by maximality, on the parent of Q, call it Q, we have the Hence, if ε 0 = ε 0 (n, M) is chosen sufficiently small, the inequality |u(w)| < λ holds for every point Since the volumes of Q and Q ω are comparable, replacing Q by Q w in (3.10) if necessary, it can be assumed that |u(w Q )| < λ.So the proof consists on showing (3.10) under the additional assumption that We will mainly follow the proof in Lemma 3.1, instead of directly applying its conclusions.
Define the set E = {x ∈ Q : (S β u)(x) ≤ ϕ −1 (λ)} and the Lipschitz domain , where β is chosen so that α < β < β.Then Ω is a Lipschitz domain contained in Ω with Lipschitz constant 1/β , and we have (S β u)(x) ≤ ϕ −1 (λ) for all x ∈ E. Hence the area function of u (in Ω ) is bounded by a fixed multiple of ϕ −1 (λ) at the points of (∂Ω ) ∩E.A technical difficulty arises because we do not know such estimate in the whole ∂Ω .In the harmonic setting, it holds (see Lemma 4.2.9 in [2]), but this lemma does not seem to hold in our situation.
We consider now the dyadic decomposition of Q with respect to the domain Ω .Denote the dyadic cubes in Ω by { Q }.Since Q was chosen to be big enough, we can assume as well that |u| ≤ λ on T ( Q ).The idea is to run a stopping time process in Ω .
We are essentially in the setting of Lemma 3.1 with Ω replacing Ω, except for the fact that the condition (S β u)(x) < ϕ −1 (λ) involves the distance to ∂Ω which could be quite different from the one to ∂Ω .This will create some technical difficulties that can be solved by adapting the proof of Lemma 3.1 to this situation.
Proceeding as in Lemma 3.1, consider the maximal dyadic cubes (with respect to for some constant C 0 to be chosen later.Define the corresponding regions D and R and apply Green's formula to u 2 and g (Green's function in D ).Hence Since δ(w) = dist(w, ∂Ω) and δ (w) = dist(w, ∂Ω ) for w ∈ Ω , do not need to be comparable, we need to consider a new Green's function to estimate this last integral.Let g β (•) be Green's function in Ω β = x∈E Γ β (x) with the same pole as g .Note that Ω β ⊃ Ω , so by the maximum principle, g β (w) ≥ g (w) for all The point now is that, for points w ∈ R , δ β (w) = dist(w, ∂Ω β ) is comparable to δ(w) = dist(w, ∂Ω), with constants depending on (β, β , n).So changing the order of integration as in Lemma 3.1 we can bound (3.12) by Recall that (w) for w ∈ R , we deduce that To estimate the corresponding integral for points in Q \ E, we use an argument in [22], which uses the different apertures α < β < β, (x 0 , t 0 ) = P ∈ ∂Ω β and x ∈ E such that P is in the closure of Γ β (x).Since P ∉ Ω β , the vertical cone (in the negative direction) of aperture β with vertex at P , Therefore (3.13) The rest of the argument is exactly the same as in Lemma 3.1.Just note that the oscillations of u on tops of the cubes Q will be controlled by the oscillation on hyperbolic balls in Ω .The quantity that controls such oscillations depends on the euclidean radius of the balls.That might be very small compared to the distance of the ball to ∂Ω.So, we need to consider some bigger balls that will help us control the oscillation in the smaller ones.Let B be a hyperbolic ball in Ω , that is, B = B(w 0 , ρ 0 δ (w 0 )) ⊂ Ω , where ρ 0 < 1 2 .Define B to be the ball Then obviously B ⊃ B and the euclidean radius of B is comparable to δ(w) for all w ∈ B. Fix ρ 0 = ρ 0 (β) so that and the same argument which leads to (3.3) implies that osc(u, B) ≤ ϕ(cϕ −1 (λ)) λ for some c = c(β, β , n).This is where the assumption ϕ(2t) < Cϕ(t) is used.
Next, since lim inf λ→∞ ϕ(λ)/λ = c ϕ > 0, we can choose c 1 such that So the statement (3.8) (with a = ϕ −1 (λ), N = C 0 ) becomes Choosing C 0 big enough, the proof of Lemma 3.1 gives: Changing the notation now, we set Q j = Q (1)  j .For each j, we repeat the construction, that is, we consider {Q (2)  i }, the collection of the maximal dyadic subcubes (with respect to Ω ) of Q (1)  j satisfying sup Repeating the same process n 0 times, where n 0 = n 0 (γ) will be chosen later, we obtain nested families for any k = 1, . . ., n 0 .Moreover, the same argument as in (3.9) yields and by the maximality of the families {Q (n) j : j}, we have that . Thus, if n 0 γ/C 0 , then one has x ∈ E : sup where C is a constant depending on α, n, M. For n 0 [γ/C 0 ], we can then deduce which concludes the proof.

Ë
Remark.Assume that the function u satisfies condition ( * ) for a function ϕ such that for a fixed constant c > 0.Then, the conclusion of Theorem 3.2 holds for any λ > lim sup |w|→∞ u(w).
Once this type of distributional inequalities is established, standard arguments lead to the following L p -inequalities for functions u that vanish at infinity.Then for 0 < p < ∞ and 0 < α < β there exists a constant C = C(p, α, β, n, M, ϕ), such that We can also obtain a Law of the Iterated Logarithm in this setting, but some technical difficulties arise because no version of Lemma 4.2.9 in [2] seems to hold in our setting.
Theorem 3.4.Under the hypothesis of Theorem 3.2, and assuming that Proof.It is enough to show the result for points x ∈ Q, where Q is any sufficiently large cube in R n .We may assume that there exist x ∈ Q and t > 0 such that (S β u)(x, t) < ∞.Given x ∈ Q, consider the set are Lipschitz domains with Lipschitz constant depending only on α, β, M, ϕ, but not on k.We will apply the proof of Theorem 3.2 in the domain Ω k .The main difficulty is that we only know that As before, it is worth mentioning that the analogue of Lemma 4.2.9 in [2] does not seem to hold in our situation.However, observe that for any point P ∈ ∂Ω k (β )   (3.14) Actually, if P is of the form P = (x, η k (x)), we already know it.For general Hence, (3.13) holds.
Observe that, if we would know that (S β u)(P ) ϕ −1 (2 k ) for any point P ∈ ∂Ω k (β ), we could directly apply Theorem 3.2 to obtain To deduce (3.15) from (3.14), we follow the proof of Theorem 3.2.We apply Green's formula to the functions u 2 and g in a subdomain R of Ω k (α).Here g is Green's function of a convenient subdomain of Ω k (α).As in the proof of Theorem 3.2, the key estimate (3.13) follows easily from (3.14).These considerations accomplish the first step in the proof of Theorem 3.2.Successive steps run in the same way as in Theorem 3.2. Choose Then k e −c 2 γ k < ∞, and the Borel-Cantelli lemma implies that almost all x ∈ Q are, at most, in a finite number of the sets eventually, that is, there exists an integer k 0 that may depend on x, such that for all k ≥ k 0 , we have N α,η k (x),∞ u(x) ≤ γ k 2 k .Consider such points x which in addition satisfy (S β u)(x) = ∞.For these points η k (x) → 0 as k → ∞.So, for any t < η k 0 (x), choosing k such that η k+1 (x) < t < η k (x), we get and Thus, 2 k < ϕ((S β,t,∞ u)(x)) and as we wanted to prove.

Ë
We end this section applying these techniques to obtain a Fatou-type result.

Theorem 3.5. Under the hypotheses of Theorem 3.2 and assuming that
(ii) The function u has a finite non-tangential limit, that is, lim w→(x,φ(x)) u(w) exists, where the limit is taken when Proof.It is quite easy to deduce (i) from Theorem 3.2 and a standard point of density argument, therefore we omit its proof.Assume now that (ii) does not hold.Since (N α u)(x) is finite a.e.x ∈ {x : S β u(x) < ∞}, there must exist δ > 0 and a set where the limits are taken when w tends to (x, φ(x)), w ∈ Γ α (x).Choose ε > 0. Then a point of density argument provides a dyadic cube Q ⊂ R n , and a set F ⊂ Q such that |F |/|Q| > 3  4 , and for all x ∈ F, (S β,0, (Q) )(x) < ε and (3.16) holds.Consider the Lipschitz domains As before, we only know that the area function of u is bounded (by ⊂ ε) at points P ∈ ∂Ω of the form P = x ∈ F .If we knew that, for any point in ∂Ω, then, applying Lemma 3.1 to Ω ∩ Ω, we would get a contradiction, since ϕ(0) = 0 and the constant ε can be taken arbitrarily small.As before, to overcome this difficulty we will take profit of the two different angles α < β.Arguing as in the proof of Theorem 3.2, one obtains that for any P ∈ ∂Ω, Now, one can apply the proof of Lemma 3.1 to obtain a contradiction.
In particular, Proof.We will only prove (i), the proof of (ii) being similar.We start with the right hand side.If δ = δ(ε, M) is conveniently chosen, then, by Proposition 4.1 and Fubini's Theorem: f (z, y) dz dy dz dy where C is some positive constant depending only on M, α, ε.For the left-hand side, we also get: is a cube of side length and x ∈ Q, consider the truncated cones Γ α,0, (x), Γ α, ,∞ (x) and the corresponding area functions (A α,0, f )(x), (A α, ,∞ f )(x) associated to f .The following two results were proved in the harmonic setting by Bañuelos and Moore [3].Lemma 4.3.Assume that f ≥ 0 is measurable in Ω and satisfies the uniform estimate for some A > 0 and any (z, y) ∈ Ω.Then if Q ⊂ R n is a cube of side length , 0 < α < 1/M and A α, ,∞ (f ) is as above, we have for any x 1 , x 2 ∈ Q, where C is a constant depending only on M, α, A, n.
Proof.Take = 2M √ n .An elementary commutation shows that for any x ∈ R n .Therefore, it is enough to prove the lemma with instead of .Now, We claim that (I) and (II) are bounded by some constant C = C(M, n, α, A).To estimate (I), note that, from the choice of it follows, so, by Proposition 4.1, (1): To estimate (II), note that where σ n is the Lebesgue measure in R n .As in [3], this last inequality, together with Fubini's Theorem, gives (II) ≤ C(M, n, α, A).Ë Now, for a cube Q ⊂ R n , centered at x 0 ∈ R n and of side length , denote by Q * the cube also centered at x 0 with side length 4 , and let g, ω be Green's function and the harmonic measure in Q * , with respect to p Q * = (x 0 , φ(x 0 ) + /2).We follow the notation introduced in Section 2, that is, for The following lemma is a standard consequence of Fubini's Theorem, Theorem 2.1 (c) and the elementary estimates in Proposition 4.1.

Lemma 4.4. For any
where C is a constant depending on α, M and n.
In what follows, we will be interested in the class of functions u ∈ C 2 (Ω) that satisfy for some θ, 0 < θ < 1.As mentioned in the introduction, if 0 ≤ t ≤ s ≤ +∞, we will define The following lemma is a Caccioppoli inequality for this class.This is where our assumption 0 < θ < 1 gets used.
Proof.The proof goes as in the usual Caccioppoli inequality.Fix B and let , where C only depends on n.Then by Green's formula applied to u where C on M, n, α and θ, the constant in ( * * ).
Proof.We use the notation of Lemma 4.4.By Green's Theorem, we remind that f ε denotes the density introduced at the beginning of this section.
Proof.Choose ε > 0 so that α ≤ β − with the notation of Lemma 4.2.If Q is any cube in R n , centered at x 0 , with side length , it follows by Lemma 4.3 that (A α, ,∞ f ε )(x) < ∞ for any x ∈ Q.On the other hand, (A β,0, u)(x) < ∞ for a.e. (ω * ) x ∈ Q, by Corollary 4.6, so (A α,0, f ε )(x) < ∞ for a.e.(w * )x ∈ Q.Since Q is arbitrary, it follows that (A α f ε )(x) < ∞ for a.e.x ∈ R n (recall that the harmonic measure and the surface measure are mutually absolutely continuous on the boundary of any Lipschitz domain).Now as in [3] we show that (A α f ε ) 2 ∈ BMO(R n ).Indeed, let Q be a cube in R n of side length .Corollary 4.6 gives Since ω * satisfies the A ∞ -condition of Theorem 2.1, we deduce for some b > 0. On the other hand, Lemma 4.5 gives the necessary estimates for the hypothesis of Lemma 4.3, with the function f where f = |∇u| 2 .Actually, the fact that the expressions in Lemma 4.5 are averages is the reason to introduce functions.Hence, applying Lemma 4.3, and one deduces that (A α f ε ) 2 ∈ BMO(R n ).

Ë
Now, the John-Nirenberg inequality gives the following result.Corollary 4.8.Under the hypothesis of Theorem 4.7, for any cube Q ⊂ R n there exists a constant a Q such that for every t > 0, where C 1 , C 2 depend on M, n, α, β, θ.In particular, Constructing suitable Lipschitz domains, we will show that Theorem 4.7 leads to good-λ inequalities relating the size of the area function and the non-tangential maximal function.
Ë By means of the change of variables, (z, t) → (z, t+φ(x)), which transforms where h(z, t) = f (z, t+φ(z)).This shows that we can reduce ourselves to the general setting considered in [10].In particular, from Proposition 4 there, adapted to our situation, we get A α u p A β u p whenever 0 < α < β < 1/M, where the comparison constant only depends on α, β, p.This observation, together with a well-known standard argument, shows that the weak form of Theorem 4.9 is enough to get the usual comparison of the L p -norms of the area function and the non-tangential maximal functions, as follows.

Theorem 3 . 2 .
Let Ω be an unbounded Lipschitz domain with Lipschitz constant M. Let u ∈ C 2 (Ω) be a function having ϕ-controlled oscillation in Ω for an increasing function ϕ satisfying