Causal sparse domination of Beurling maximal regularity operators

We prove boundedness of Calderön–Zygmund operators acting in Banach function spaces on domains, defined by the L1 Carleson functional and Lq (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.


Introduction
We prove estimates of Calderón-Zygmund operators (CZOs) acting on functions f (t, x) defined on a half space R 1+n + = {x = (t, x) : t > 0, x ∈ R n }, in function spaces defined by L p (R n ) norms of the classical Carleson and non-tangential maximal functionals (Cf )(x) and (Nf )(x) respectively.We recall their definitions in (3) and (2) below.For CZOs to be even locally bounded inside R 1+n + , it is necessary to modify C and N since these build on L 1 (R 1+n + ) and L ∞ (R 1+n + ) norms respectively.Following Kenig and Pipher [19], Auscher and Axelsson [3], Hytönen and Rosén [17] and Huang [14], we use Whitney L q averaging W q f (t, x) with 1 < q < ∞, as defined in (4) below, and consider function spaces defined by norms C(W q f ) p and N(W q f ) p , which encode interior local L q regularity, transversal L 1 or L ∞ regularity, and L p regularity along the boundary R n .It was shown in [17] that the norms N(W q ′ f ) p ′ and C(W q f ) p are dual, 1/p + 1/p ′ = 1, 1/q + 1/q ′ = 1.In applications to elliptic partial differential equations (PDEs) as in [19,3], the case q = 2 is particularly important, since for gradients f of weak solutions to elliptic equations, in general we do not have pointwise interior bounds but only local L 2 estimates on Whitney regions.
A general CZO on R 1+n + Sf (x) = R 1+n + k(x, y)f (y)dy, x = (t, x), y = (s, y), fails to be bounded in any of the norms above, even with Whitney averaging.A key observation that we make is that for causal CZOs, bounds in some of the norms hold.By causal we mean that we consider either an upward mapping CZO, denoted S + , where k = 0 for t < s, or a downward mapping CZO, denoted S − , where k = 0 for t > s.In our main Theorem 5.1, we prove that downward mapping CZOs S − are bounded in Whitney modified Carleson norms C(W q f ) p and that, dually, upward mapping CZOs S + are bounded in Whitney modified non-tangential maximal norms N(W q ′ f ) p ′ .
In Section 6, we discuss motivating examples: maximal regularity operators arising from integration of elliptic PDEs in the variable transversal to the boundary as in [3].At the end of the present paper, we explain how our Theorem 5.1, in the presence of pointwise kernel bounds, sharpens the estimate of half of the maximal regularity operators in [3].In the case of the Laplace equation in R 2  + , the maximal regularity operators S ± appearing are the two causal parts of the Beurling transform, in which case one of the key estimates from [3] independently have been proved by Astala and González [2].We recall that the kernel k(x, y) = (x − y) −2 of this Beurling transform is symmetric and as a consequence its causal parts S ± are L 2 bounded.Hence they are examples of rough CZOs: their kernels are discontinuous across the hypersurface t = s.With a slight abuse of notation we refer to such rough CZOs simply as CZOs below.
Our Carleson estimate for S − uses the method of sparse domination of CZO by A. Lerner.In Section 4 we adapt the proof for general CZO from Lerner [22] and prove a domination of causal CZOs by certain causal sparse operators.Here we take a direction somewhat against the mainstream of recent activity, where the trend has been to establish sparse domination for ever more general classes of operators; in contrasts, we deal with operators having additional structure (the causality), and the point is to preserve at least part of this structure in the dominating sparse operators as well.The one-dimensional special case, n = 0, of our Theorem 4.1 reads where the sum is over a sparse collection D f of dyadic intervals (a, a + ℓ), but the average is over the right half and a right neighbourhood of this interval.
There are a number of previous results dealing with causal (also known as onesided) operators and related weight classes.Due to the intimate connections of sparse domination and weighted norm inequalities, which have been explored in the recent literature, we briefly comment on these works.Sawyer [28] found that the one-sided maximal operator on the line satisfies weighted estimates if and only if the weight satisfies a one-sided A p condition.Later, Aimar et al. [1] showed that the one-sided A p condition is also the right condition for the boundedness of one-sided singular integrals on the line.Versions of these results for operators on a half-line (case n = 0 of our setting in R 1+n + ), including extensions to operator-valued kernels and applications to maximal regularity of Cauchy problems, are recently due to Chill and Krol [11].We refer to this paper for an extensive bibliography of other related works.
A sharp form of the one-sided maximal function estimates was found by Martín-Reyes and de la Torre [25].The corresponding problem (a "one-sided A 2 conjecture") for one-sided singular integrals remains open, but the analogous result for one-sided martingale transforms has been achieved in Chen and Lacey [10].At the time of writing, the established methods of reducing continuous singular integrals to their dyadic models (including [20,21,22,23,24] in the case of regular CZOs and [9,13] for some rough extensions, to name but a selection of the extensive literature) are not available in a form that would respect the one-sided structure.In the present work we obtain a result that may be seen as a version of sparse domination for one-sided operators.While this version serves our present purposes, it is not strong enough to make progress on the mentioned weighted questions.The problem is that our dominating sparse operators are not strictly one-sided (for that, the integral in (1) should be over (a + ℓ, a + 2ℓ) only) and cannot possibly be estimated in terms of one-sided A p weights.
As mentioned above, one-sided weighted estimates for vector-valued singular integrals have been applied to maximal regularity operators for Cauchy problems in [11].However, our Carleson estimates cannot be viewed as such weighted estimates, but rather correspond to the end-point norms r = 1 and r = ∞ in the scale of L r tent spaces from Coifman, Meyer and Stein [12].Estimates for maximal regularity operators in tent spaces based on the L 2 area functional are in Auscher, Kriegler, Monniaux and Portal [5].Tent space estimates of horizontally mapping, that is, acting in the R n variable, CZOs are in Auscher and Prisuelos-Arribas [6].In Section 2, we include counterexamples that show that for non-causal R 1+n + CZOs, as well as horizontally mapping R n CZOs, the Carleson estimates considered in this paper fail in general.

CZOs on non-tangential and Carleson spaces
We denote cubes on the boundary R n by Q, R, . . ., and cubes in the half-space R 1+n + by boldface Q, R, . ... We denote n or n + 1 dimensional measure by | • | depending on the dimension of the cube, and we denote by cQ, c > 0, the cube with same center as Q and sidelength ℓ(cQ) = cℓ(Q).For Q ⊂ R 1+n + we form cQ in R 1+n as above, and define cQ ⊂ R 1+n + as the intersection with R 1+n + , which may not be a cube if where Q is a cube in R n with sidelength ℓ(Q).The corresponding Whitney region is the top half We denote points in R n by x, y, . . ., and points in R 1+n + by boldface x = (t, x), y = (s, t), . . .Indicator functions of cubes and more general sets E are denoted by 1 E .Subsets, not necessarily strict, are denoted A ⊂ B.
In particular we use dyadic cubes, and let D = j∈Z D j denote a system of dyadic cubes in R n , with D j being the cubes of sidelength ℓ(Q) = 2 −j , such that the dyadic cubes in D form a connected tree under inclusion.Let D = j∈Z D j denote the associated dyadic system for R 1+n + , where D j consists of dyadic cubes of the form Q = (k2 −j , (k + 1)2 −j ) × Q, Q ∈ D j , k = 0, 1, 2, . ... We note that also D form a connected tree under inclusion where the dyadic Carleson cubes, the cubes touching R n , will play the special role.
We consider CZOs acting on functions defined on the half space R 1+n + , n ≥ 1, and belonging to endpoint spaces in the scale of tent spaces from [12], whose definitions use the non-tangential maximal and Carleson functionals where the second supremum is over all cubes Q ⊂ R n containing x, and the Whitney averaging operator (4) where Q ⊂ R n is the cube with center x and sidelength t.More precisely, we consider the norms , and predual norms C(W q f ) Lp(R n ) considered in [17].We exclude p = 1, where the former norms are ∞ except for f = 0, while the latter norms are W q f ∞ .To avoid technicalities we also exclude p = ∞, although this endpoint case sometimes is of interest and some of our results hold for p = ∞, mutatis mutandis.We recall from [17,18] that up to constants these two scales of norms are independent of the choice of aperture α > 0 and the precise shape of the Whitney regions and Carleson cubes.Moreover, for the dyadic functionals acting on sequences (f Q ) Q∈D , and dyadic Whitney L q averages Finally, we recall that if 1 < p < ∞, then Cf p ≈ Af p , where but the estimate breaks down as p → ∞, and breaks down as p → 1.We are interested in boundedness of CZOs in the norms C(W q f ) p or equivalently, by duality, in the norms N(W q ′ f ) p ′ .As discussed in the Introduction, it is necessary to require 1 < q < ∞, and main applications concern q = 2.For the Carleson functional, which builds on L 1 integrals, it is natural to make use of the recent technology of sparse domination of CZOs.We recall the following estimate of A. Lerner [22].Assume given a standard CZO S on R 1+n + and a function f ∈ L 1 (R 1+n + ) with bounded support.Then there exists a collection D f ⊂ D of cubes, which is η-sparse, for some 0 < η < 1 independent of f , such that |f (y)|dy holds pointwise almost everywere.Here η-sparse means that there exists subsets The method of sparse domination has proven very successful in proving optimal bounds for CZOs on weighted L p spaces.We apply in this paper the technique to the scales of Banach function spaces described above.Since N and C involve an L ∞ and L 1 norm respectively however, it is not surprising that in general we have the following counterexamples.
Example 2.1.A sparse operator, that is a sublinear operator of the form with a fixed η-sparse collection D independent of f , is not in general bounded on C(W q f ) p for any p, q.An example is as follows.Let n = 1 and fix a dyadic Carleson cube and let D be the collection of all dyadic Carleson cubes contained in Q 0 .Then It follows that S fails to be bounded in the norm C(W q f ) p for any 1 < p, q < ∞, since q → W q is increasing.
Example 2.2.Although domination of CZOs by sparse operators is considered to be quite sharp, Example 2.1 does not disprove that CZOs are bounded in these Carleson norms.And indeed, by direct calculations one can show that the Beurling transform on the upper half plane ( 6) (5) boundedly in the Carleson norms.To find a counterexample for the Beurling transform, it is convenient to consider a weak limit (7) f (t, x) = g(x)δ 0 (t), of functions like f N , to be modulated by a function g(x) supported on Q 0 = (0, 1) that remains to be chosen.In this limit we have and for real-valued g we note from Cauchy-Riemann's equations that |Sf (z)| = |∇u(z)|, where u is the Poisson extension of g.As discussed in [18,Intro.],by contructing g via a lacunary Fourier series, it is known that the bound (8) C(∇u) p g p cannot hold uniformly for g, for any fixed 1 < p < ∞.Replacing δ 0 in (7) by 2 N 1 (0,2 −N ) , it can be shown that S fails to be bounded in the norm C(W q f ) p for any 1 < p, q < ∞.The technical details of this counterexample are found in the Appendix of this paper.
Example 2.3.We next demonstrate that R n CZOs acting horizontally, that is pointwise in t, in general are not bounded either in the norms N(W q ′ f ) p ′ or C(W q f ) p .A concrete counterexample is the following.Let x − y dy be the Hilbert transform acting horizontally in R 2 + , and consider the function with support in [0, 1] 2 .The terms in this double sum are supported on thin rectangles, with integrals equal to their lengths in the x-direction.Consider a Carleson square Q with sidelength 2 −m .The terms for a given 0 , since the thin rectangles are unions of Whitney regions.We conclude that However, an explicit computation reveals that so C(Hf N ) N 2 on (0, 1).It follows that H fails to be bounded in the norm C(W q f ) p for any 1 < p, q < ∞, since q → W q is increasing.

Causal Calderón-Zygmund operators
We consider a Calderón-Zygmund (CZ) kernel k(x, y) on R 1+n + .More precisely, we assume kernel bounds and regularity We consider a linear operator operator We also consider a linear operator operator Thus S ± are (rough) CZOs, where S + is upward mapping away from R n and S − is downward mapping towards R n .We refer to S ± as causal operators.They are simple examples of singular integral with rough kernels, in that k ± ((t, x), (s, y)) may be discontinuous on the hyperplane t = s in R 2(1+n) , with a simple jump discontinuity when x = y and t = s.Lemma 3.1.The kernels k ± satisfy the Hörmander regularity condition (10) for t < a and t > b, and using (9) for a < t < b.
Denote by Mf the standard Hardy-Littlewood maximal function of f .As in [22], we also require a weak L 1 estimate of Lerner's maximal singular integral with supremum over dyadic cubes Q ⊂ R 1+n + , into which S ± maps from the complement of the enlarged (non-dyadic) cube 3Q.
Proposition 3.2.The causal CZOs S ± and the maximal singular integrals The weak L 1 estimate for S ± itself follows from the standard proof employing the Calderón-Zygmund decomposition, since this only requires the Hörmander regularity estimate from Lemma 3.1.
(2) Following the standard proof of Cotlar's lemma, see for example [26,Sec. 7.7], we write f 1 = 1 3Q f and f 2 = 1 (3Q) c f, for a cube Q with center x 0 .We estimate S ± (1 (3Q) c f ) = S ± f 2 at each fixed x 1 ∈ Q, by comparing it to the value at a variable point x ∈ Q, writing Raising the terms to power 1/p, p > 1, and taking the average over Q, terms II and III are estimated in the usual way: and (Mf (x 0 )) 1/p by Kolmogorov's inequality and weak L 1 estimate (1) above.

Causal sparse domination of S −
To state our causal sparse domination, we need the following subsets of the neighbourhood 3Q.For Q = (a, a + ℓ) × Q ∈ D, Q ∈ D, we define the upper and lower halves of the parts of (a, a + 2ℓ) × 3Q above and below t = a + ℓ/2 The following sparse estimate is a causal adaptation of the estimate in [22].We specifically note that the very argument of [22] seemed more amenable to this adaptation than either its predecessors or successors in the sparse domination literature.
We remark that an essential point in Theorem 4.1 is that the sum uses the same dyadic system as we start out with.If we allow us to replace D by a finite number of other dyadic systems, then the result is immediate from [22], but cannot be used to prove our Carleson bounds.
We also remark that the one-dimensional result, n = 0, is somewhat cleaner.In this case D is the standard dyadic intervals and for Q = (a, a + ℓ), the average in the sparse sum is over (a + ℓ/2, a + 2ℓ), independent of x ∈ Q, as in (1).The reason why Q T , which is the direct analogue of (a + ℓ/2, a + 2ℓ) for n ≥ 1, must be replaced by the larger set Q ⊓ x in Theorem 4.1, is the term II 1 appearing in its proof.The even larger set Q ⊓ is too large, since it cannot control the term II 1 appearing in the later proof of Theorem 5.1: it is important that these sets do not touch t = a.However, we have found that Theorem 5.1 is valid when Q ⊓ x is enlarged to a set, to be denoted by Q x , which goes down to the t coordinate of x also inside Q.
Proof.(1) Fix Q ∈ D and assume that the closure of 3Q contains the support of f .Write f T := 1 Q T f and define the set . .be an enumeration of all the maximal subcubes of Q u such that (11) |Q j ∩ E| > α|Q j |.
Here c > 0 and α ∈ (0, 1) are parameters to be fixed below.We obtain a family Hence, splitting the indicator 1 Q outside S − in the first step and the function f inside S − in the second one, Using causality for S − , we have replaced f by f T in terms II 2 and II 3 .
(2) We next show that terms II are bounded by Q ⊓ x |f |dy, for almost every x ∈ Q.For II 1 , this follows from (9) and the downward mapping property of S − since |x − y| −(1+n) |Q ⊓ x | −1 for all y ∈ (3Q) \ (3R i ) and x ∈ R i .For II 2 , it follows from Lebegue's differentiation theorem that modulo a set of measure zero.The estimate of II 2 then follows from the definition of E.
For II 3 , consider one of the stopping cubes Q j .By definition, we have |Q j ∩ E| > α|Q j |.We require an upper estimate of the measure of E. It follows from Proposition 3.2 that Moreover, if Q p j denotes the dyadic parent of Q j (the minimal dyadic cube such that Q p j Q j ), then by maximality of Q j , its parent does not satisfy (11) and we have Let α = 2 −2−n .Then in particular there exists x ∈ Q j \E since |Q j \E| ≥ |Q j |/2 > 0 with this choice of α.From the definition of E it now follows that on almost all of Q j .This proves the estimate of II 3 .
(3) We have shown that for f supported on 3Q, we have at almost all x ∈ Q.To conclude the proof, we iterate this estimate for all the subcubes R i and Q j .To this end, we note that and, since Finally consider a disjoint union modulo zero sets, where Q k ∈ D are constructed as follows.Since supp (f ) is bounded and D is connected, we can choose Q 1 ∈ D containing supp (f ).Then let P 1 = Q 1 and recursively define P j+1 to be the dyadic parent of P j , that is, the smallest dyadic cube such that P j+1 P j .The siblings of P j are the other dyadic subcubes of P j+1 of same size as P j .Now choose Q 2 , Q 3 , . . . to be an ordering of all the siblings of all P j , j = 1, 2, . ... Then Q 1 ⊂ 3Q k for all k, so that and the estimates above apply to each Q = Q k .We define the family of dyadic cubes D f to be Q 1 , Q 2 , . . .along with, for each of Q k , all generations of stopping cubes Q j and subcubes R i starting from Q = Q k , constructed as in (1) above.It follows that D f is 1/4 sparse and that we have the stated sparse bound of S − f .

Bounds for causal CZOs
The Carleson bounds established in this section apply to slightly larger causal sparse operators where we have replaced the region Q ⊓ x appearing in the sparse operator in Theorem 4.1, by the slightly larger region for the union of Q T and Q L , modulo a set of measure zero.The following theorem is the main result of this paper.Its proof uses regions R for dyadic cubes R ⊂ Q w , where Q w is a dyadic Whitney region.We define enlarged auxiliary Whitney regions Theorem 5.1.Let S + and S − be causal Calderón-Zygmund operators as in Section 3 and 1 < p, q < ∞.Then we have the estimates More precisely, the estimate holds pointwise on R n for the causal sparse operator S from (12), for any η-sparse collection D ⊂ D with 0 < η < 1.Here W D,q (f ) denotes the Whitney L q average over Q w .
We remark that interior regularity 1 < q < ∞ is necessary for boundedness of CZOs, and is used to ensure boundedness of the Hardy-Littlewood maximal operator in the proof below, while p > 1 is necessary for the Carleson and non-tangential maximal norms to be non-degenerate.The restriction p < ∞ is less essential, but is made so that the following lemma hold.Lemma 5.2.Fix 1 < p, q < ∞ and define Banach spaces + ) ∩ Z with bounded support, and let X 0 denote the subspace of functions f ∈ L q ′ (R 1+n + ) ∩ X with bounded support.Then Z 0 is dense in Z, in the norm CW q (•) p , and X 0 is dense in X, in the norm NW q ′ (•) p ′ .Moreover, Z is non-reflexive and X equals the dual space Z * , using the standard L 2 (R 1+n + ) pairing.Proof.The density of Z 0 in Z follows from [17, Lem.2.5], which proves that even compactly supported L q functions are dense in Z. Also the non-reflexivity of Z and that X = Z * was shown in [17,Thm. 3.2].
To see the density of X 0 in X, let ǫ > 0 and f ∈ X so that N D W D,q ′ f p ′ < ∞.Recall that equivalences of norms allow us to use the dyadic versions of the norms.Since p > 1 and D is connected, there exists
To estimate f 1 , let Q 1 denote the dyadic parent of Q 0 .Then sup and we conclude that it follows that f 2 ∈ X 0 .This shows that X 0 is dense in X.
Proof of Theorem 5.1.We first show that it suffices to prove the estimate (15).Indeed, combining this and Theorem 4.1 yields the bound for S − on Z 0 , using the notation from Lemma 5.2 and equivalences of norms between the different versions of Carleson functionals and Whitney averages from [17,Sec. 3].By Lemma 5.2, we have a unique extension of S − by continuity to all Z.
Next consider S + on L q ′ (R 1+n + ) ⊃ X 0 , with adjoint (S + ) * on L q (R 1+n + ) ⊃ Z 0 .For f ∈ X 0 and g ∈ Z 0 , we obtain the bound ( 16) by applying the duality to f and g sgn((S + f )g).By Fatou's lemma and the density of Z 0 in Z, (16) continues to hold for all g ∈ Z.Let h = S + f and let K ⊂ R 1+n + be a compact set, so that NW q ′ (h1 K ) p ′ < ∞.It follows from [17, Thm 3.2] and ( 16) that h1 K X f X .Exhausting R 1+n + with K and using Fatou's lemma shows that S + f ∈ X with S + f X f X , for all f ∈ X 0 .By Lemma 5.2, we have a unique extension of S + by continuity to all X.
It remains to prove (15).Fix Q 0 ∈ D and consider for a.e.x ∈ Q w , where we split the sum according to the cases R Q ca , R ⊃ Q ca and R∩Q ca = ∅, noting that R, Q ca ∈ D, with Q w being the top half of Q ca .Clearly III = 0, for I we have R ⊂ Q w since R ∋ x, and for II we have that R = R ca are Carleson boxes since Q ca , and hence R, touches R n .The local terms I do not require causality and we may replace R x by the larger set R .Standard sparse estimates via duality apply as follows.Let g ≥ 0 be supported on Q w with g q ′ dx = 1, 1/q + 1/q ′ = 1.Then We have used that ) on all E R , Hölder's inequality, and L q ′ and L q boundedness of the Hardy-Littlewood maximal operator M. Taking supremum over g gives the pointwise bound W D,q (I) W D,q (f ).Integration over Q ca 0 yields the desired estimate of I by C D W D,q (f ).The non-local terms II do not require sparseness, and we may increase the sum over R ca ∈ D to a sum over R ∈ D. Note that the term II is essentially constant on Q w , and we replace the Whitney L q average by the supremum norm on Q w , and increase We have where we split the outer sum according to R ⊂ Q 0 and R Q 0 .For the tail terms II 2 we increase (R ca ) Q ⊂ (R ca ) = 3R ca , where we recall that 3R ca = 3R ca ∩ R 1+n + according to our definition, and get for the k'th ancestor of Q 0 , and since |Q ca 0 |/ℓ(Q 0 ) = |Q 0 |.For the mid-range terms II 1 , we estimate For the second last estimate, note that at a point (s, y) ∈ 3Q ca 0 we have 1 3R ca = 0 if 2 −k < s/3, where ℓ(R) = 2 −k , while at each scale 2 −k ≥ s/3 at most 3 n cubes R contribute the the sum.In total we obtain a bound of II by C(f ), which completes the proof.

Motivations from elliptic PDEs
We end this paper by discussing motivations for the estimates in Theorem 5.1 coming from maximal regularity estimates for PDEs.A natural point of departure for the discussion is the parabolic problem for u t (x) = u(t, x), t > 0, x ∈ R n , with u 0 = 0.Here L = −div x A(x)∇ x , where A ∈ L ∞ (R n , L(C n )) is accretive.The maximal regularity problem for a space H of functions in R 1+n + , is whether a source g ∈ H yields a solution u with Lu (or equivalently ∂ t u) in H.The parabolic maximal regularity operator is (17) Lu which is upward mapping like the operators S + considered in this paper.However, the operator g → Lu does not have the CZO kernel bounds ( 9), even for L = −∆ x .Indeed the estimate |∂ t (t −n/2 e −|x| 2 /(4t) )| (t + |x|) −(1+n) fails, and non-tangential estimates are not natural for this problem due to the parabolic scaling t ∼ |x| 2 .
Instead our motivation comes from the analogous considerations for elliptic divergence form equations (18) div t,x A(t, x)∇ t,x u(t, x) = 0, )) is accretive.Following [4,3], we consider the associated first order generalized Cauchy-Riemann system (19) ∂ t f t + DB t f t = 0 for the conormal gradient f t = a∂ t u + b∇ x u ∇ x u T of u, consisting of the conormal derivative and tangential gradient.This uses the self-adjoint first order differential operator D = 0 div x −∇ x 0 and transformed accretive matrix acting in L 2 (R n ; C 1+n ) for each t > 0. For each t > 0, the conormal gradient belongs to the closure of the range of D since curl x (∇ x u) = 0, and it was demonstrated in [4,Sec. 3] that ( 18) is equivalent to (19), under this tangential curl free constraint on the tangential part of f t .Now fix bounded and accretive coefficients B = B(x), which are independent of t, and consider the equation ( 20) where f and g are functions of t > 0 and take values in The t-independent case when B t = B and g = 0 was studied in [4], whereas perturbation results for t-dependent coefficients B t were obtained in [3] via Duhamel's principle with g t = D(B − B t )f t .As in these works, we make use of the holomorphic functional calculus of the bisectorial operator DB in H: the symbols ) yields the sectorial operator Λ = |DB| and the symbols e −t|z| yield the Poisson semigroup e −tΛ generated by Λ.Following [3], we integrate (20) with boundary and decay conditions lim t→0 E + f t = 0 = lim t→∞ E − f t and a general source term g t , and obtain which uses both an upward mapping maximal regularity operator and a downward mapping maximal regularity operator (23) We saw above that the parabolic maximal regularity problem concerned an upward mapping operator (17), using the heat semigroup exp(−tL), which is not of the form considered in this paper.The elliptic maximal regularity problem involves both upward and downward mapping operators T + and T − .Both T ± use the Poisson semigroup exp(−tΛ), and T ± generalize the causal parts of the Beurling singular integral as the following example from [27] shows.
Example 6.1.Consider the Cauchy-Riemann equations for a holomorphic function f (z) in the region above the graph γ of a Lipschitz function y = φ(x).Parametrizing z = x + i(φ(x) + t), these equations read when written in the real basis {1 + iφ ′ (x), i}.Here B = (1 + iφ ′ (x)) −1 is an accretive, t-independent complex multiplier, and D = −i∂ x is the first order self-adjoint derivative.
Consider now the associated Poisson semigroups and maximal regularity operators in (21), with DB replaced by the similar operator BD = B(DB)B −1 .As observed in [27,Lem. 3.1], by computing the kernels of resolvents and operators in the functional calculus of BD, the causal operators from ( 22) and ( 23) are ( 24) We parametrize w = y + iφ(y), y ∈ R, and consider the bounded and invertible multiplier dw/dy = 1 + iφ ′ (y).It follows that T ± = S ± (1 + iφ ′ ), where S ± are defined as in ( 24) and ( 25), but replacing dw by dy.
The operators T ± , and hence also the operators S ± , are bounded on L 2 (R).See [3, Thm.6.5], which also applies to the more general operators ( 22) and ( 23) in any dimension.Moreover, we see that the kernels k ± for S ± are the causal truncations of Since φ is a Lipschitz function, it is readily verified that the CZ bounds ( 9) and (10) hold for this k, with γ = 1.
For general elliptic equations (18), the estimate (9) of the distribution kernels of the associated operators T ± holds only in an average L 2 sense.Global L 2 bounds Λe −|t−s|Λ E ± L 2 →L 2 1/|t − s| follow from the Kato quadratic estimates, more precisely [8,Thm. 3.1].Also L 2 off-diagonal estimates for Λe −|t−s|Λ E ± can be derived from such estimates for the resolvents of DB (see [8,Prop. 5.2]).As for pointwise kernel estimates, we saw in Example 6.1 that for general non-smooth coefficients B, only the first estimate in (10) can hold for DB, and similarly only the second estimate can hold for BD.More importantly, even (9) may fail for n ≥ 2. For n = 1, at least for real coefficients, the pointwise kernel bound (9) follows from the interior regularity estimates for solutions to (18) of De Giorgi, Nash and Moser (which hold for real coefficients in any dimension).Indeed, f t = e −t|BD| f 0 solves the first order equation ∂ t f t + BDf t = 0, where the first component of f satisfies a second order equation ( 18) and hence have pointwise bounds.See [3,Sec. 3.2].In general this is not true for the remaining components (conjugate functions) in f , except when n = 1, in which case also the second component/conjugate function does satisfy an equation (18), with some conjugate coefficients.See [7,Lem. 5.3].To summarize, the operators T ± from ( 22) and ( 23) are causal CZOs, as in Section 3, at least if n = 1, if A is real and if we factor out an invertible multiplicative factor.
Coming to estimates of T ± , we consider conormal gradients f solving the Cauchy-Riemann system (19) in R 1+n + .It is known from [3] that on the one hand the function space X in R 1+n + with norm along with the subspace Y * = L 2 (R 1+n + ; t −1 dtdx), are natural for boundary value problems with boundary topology L 2 (R n ) for lim t→0 f t .On the other hand, the function space Y = L 2 (R 1+n + ; tdtdx), along with the subspace Z with norm f Z = C(W 2 f ) 2 , are natural for boundary value problems with boundary topology H −1 (R n ) for lim t→0 f t .(Corresponding to boundary function space L 2 (R n ) for the potential u.)It was shown in [17,Thm. 3.1] that Z * = X , but that Z is not reflexive.
Previously known bounds for the causal operators T ± in ( 22) and ( 23) from [3, Prop.7.1], for general accretive coefficients B and not assuming pointwise kernel bounds, can be summarized in the following diagrams.
shows that Z ⊂ Y, and hence Y * ⊂ X .It was shown in [3,Lem. 5.5] that a multiplication operator E maps X → Y * if it satisfies the Carleson condition ( 27) Moreover, it follows from [17] that E maps as in the diagrams if and only if (27) holds.
We solve PDEs ∂ t f t + DB t f t = 0 with t-dependent coefficients by freezing the coefficients B t to B. Writing with B = lim t→0 B t , the PDE becomes ∂ t f t + DBf t = DBE t .To prove the existence of boundary values of solutions f t and representation formulas as in [3,, it is needed that (28) (T + + T − )E is a bounded operator (with small norm to obtain a Cauchy type representation formula of f in terms of the boundary trace f 0 ).This will yield equivalences of norms f 0 L 2 (R n ) ≈ f X for solutions f ∈ X to (19), and (19).Thus two fundamental estimates in the study of boundary problems for elliptic divergence form equations (18) are to bound the operators T ± E in the X and Y norms, under as weak as possible hypothesis on E, which measures the t-variation of the coefficients.
• L 2 (R n ) boundary traces of solutions f : Assuming that pointwise kernel bounds hold as discussed above, so that T + is a CZO modulo a trivial multiplicative factor, it follows from Theorem 5.1 that already T + : X → X is bounded.We obtain the new result that only E ∈ L ∞ (R 1+n + ), which is weaker than (27), is needed for boundedness of T + E : X → X .However, this does not yield any new estimate for (T + + T − )E as a whole, needed for the PDE application.Indeed, for the downward mapping maximal regularity operator, the best known bound is that T − : Y * → X .Similar to below, we conjecture that the Carleson condition (27) is in general necessary for the boundedness of T − E : X → X .• H −1 (R n ) boundary traces of solutions f : It is known from [3,17] that already The new result from Theorem 5.1 that T − : Z → Z is bounded, is not relevant for the boundedness of T − E : Y → Y.
For the upward mapping maximal regularity operator, the best known bound is that T + : Z → Y.This follows by duality from [3,Thm. 6.8].Furthermore, the Carleson condition (27) on multipliers E is in general necessary for the boundedness of T + E : Y → Y.To see this, by duality and the above estimate T − : Y → Y, we equivalently consider the boundedness of E * (T + + T − ) : Y * → Y * .In the case of the Beurling transform it was proved in Astala-Gonzalez [2, Thm.1] that this latter boundedness holds if and only if E satisfies (27), but without the Whitney factor W ∞ (due to the pointwise kernel bounds present for the Beurling transform).

6.1.
Summary and open problems.Causal maximal regularity operators T ± are fundamental in the study of elliptic boundary value problems, as evidenced by [3].Such operators T ± are close to being causal CZOs S ± and we have investigated boundedness of such S ± in this paper.In regards to this motivation from elliptic PDEs, our results are meager: assuming pointwise kernel bounds, it follows from Theorem 5.1 that the X norm of T + E is bounded by E ∞ .However, we believe that the techniques developed in this paper are important and that future work in this direction may result in more substantial applications.We end by formulating some open problems.
• Can the estimates in Theorem 5.1, with p = q = 2 at least, be shown for the more general operators T ± appearing in ( 22) and ( 23), without assuming pointwise kernel bounds?A positive result in this direction are the weighted norm estimates through sparse domination for such generalized CZOs by Bernicot, Frey and Petermichl [9].• Consider the L r based tent spaces of Coifman, Meyer and Stein [12], with 1 < r < ∞ and in particular r = 2.The present paper concerns the end point cases r = 1 and r = ∞, and the duality results [17] have been extended to the scale of tent spaces by Huang [14].Can sparse estimates, possibly using L r averages, prove tent space estimates of CZOs, or more general maximal regularity operators as in [5] we derive a series of consequences, eventually leading to an obvious impossibility.By [17,Prop. 3.7], we may replace C and W q by their dyadic versions, but we continue to denote them by the same symbols for simplicity.Since W q (Sf ) ≥ W 1 (Sf ) and C(W 1 (Sf )) = C(Sf ), our first derived bound will be (30) As suggested in Example 2.2, we apply (30) to f N (t, x) = g(x)φ N (t), where φ N = 2 N 1 (0,2 −N ) , and investigate the limit N → ∞.
We first consider the right-hand side.For a Whitney cube R w , we have the L q (dyadic) maximal operator restricted to cubes of side-length ℓ(R) ≤ 2 −N .Thus, for all x ∈ Q ∈ D, we have and hence C(W q f N ) ≤ M(M q,2 −N g) and If p > q, we could further dominate this by g p , uniformly in N, but we do not wish to impose this restriction.Instead, if g is, say, a continous compactly supported function (and this will be enough for our eventual counterexample), then M q,2 −N g(x) → |g(x)| pointwise, as N → ∞, and M q,2 −N g is dominated by a bounded and compactly supported function, uniformly in N. It hence follows from dominated convergence that (31) lim sup We then turn to the left-hand side of (30).If Im z > 2 −N , there is no singularity in (6) for f = f N , and hence Since any z ∈ C with Im z > 0 satisfies Im z > 2 −N when N is large enough, we have lim From repeated applications of Fatou's lemma it then follows that In combination with (31), this shows that (30) implies the estimate If g is real-valued, by a direct computation one checks that |v| = |∇u|, where u is the Poisson extension of g, and the previous estimate can be rewritten as (8).By splitting into real and imaginary parts, it is immediate that if this holds for real g, it also holds for complex g.It remain to show that estimate (8) cannot hold.Using CF (x) ≥ 1 Q (x) Q ca |F |dsdy with Q = (0, 1) and |∇u| ≥ |∂ t u|, we find that (8) would in particular imply that (32) Let us consider random functions of the form g = k ε k g k , where the ε k are independent random signs with P (ε k = −1) = P (ε k = +1) = 1 2 .Denoting by u k the Poisson extension of g k , (32) would then imply, using basic properties of random sums (Kahane's contraction principle in the first step, and Khintchine's inequality in the last one; see [15,Prop. 3.2.10]and [16,Thm. 6.1.13,Prop. 6.3.3]), that where E, appearing only in the intermediate steps, denotes the mathematical expectation.(The advantage of (33) in view of reaching a contradiction is that we have eliminated interactions between different functions g k ; in particular, on the left, we only need to check that individual terms are large for appropriate parameter values, and we do not need to worry about this largeness being destroyed by the largeness of some other term of opposite sign.) We will now construct appropriate g k to see the failure of this derived estimate (33).Let Ṗt g k L 1 (0,1) ≥ 2 −2πt2 k (2π2 k P t φ L 1 (0,1) − Ṗt φ L 1 (0,1) ).

Theorem 4 . 1 .
Let S − be a downward mapping causal Calderón-Zygmund operator as in Section 3. Let f ∈ L 1 (R 1+n ) have bounded support.Then there exists a 1/4-sparse family of cubes D f ⊂ D such that

∂∂
t u k (t, •) L 1 (0,1) dt ≥ t u k (t, •) L 1 (0,1) dt k 0 +K j=k 0 +1 1 = K,while, on the other hand, we havek 0 +K k=k 0 +1 |g k | 2 1 (6))concrete PDE application would be to reprove the bound T + : Z → Y from [3, Prop.7.1], using sparse estimates.Note that Z and Y are L 1 and L 2 based tent spaces respectively.•WestartedthisSectionbyconsidering the more well known upward mapping maximal regularity operator(17)for parabolic initial value problems, but this does not have the CZ kernel bounds.Can sparse estimates be adapted to show a parabolic analogue of the non-tangential bound (14)?We supply the details omitted in Example 2.2, which show that no Carleson bound can hold for the Beurling transform S defined in(6).