The Kato Square Root Problem for Divergence Form Operators with Potential

The Kato square root problem for divergence form elliptic operators with potential $V:\mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence statement $\left\Vert \left( \mathcal{L}_{A}^{V} \right)^{\frac{1}{2}}u \right\Vert \simeq \left\Vert \nabla u \right\Vert + \left\Vert V^{\frac{1}{2}} u\right\Vert$, where $\mathcal{L}_{A}^{V} := - \mathrm{div}\left( A \nabla \right) + V$ and the perturbation $A$ is an $L^{\infty}$ complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential satisfying $\left\Vert \left\vert V \right\vert u \right\Vert \lesssim \left\Vert \left( \left\vert V \right\vert - \Delta \right) u \right\Vert$ for all $u \in D \left( \left\vert V \right\vert - \Delta \right)$, with range contained in some positive sector. The class of potentials that will satisfy such a condition is known to contain the reverse H\"{o}lder class $RH_{q}$ for any $q \geq 2$ and $L^{\frac{n}{2}}\left( \mathbb{R}^{n} \right)$ in dimension $n>4$. To prove the Kato estimate with potential, a non-homogeneous version of the framework developed by A. Axelsson, S. Keith and A. McIntosh is developed. In addition to applying this non-homogeneous framework to the scalar Kato problem with zero-order potential, it will also be applied to the Kato problem for systems of equations with zero-order potential and scalar equations with first-order potential.


Introduction
For Hilbert spaces H and K, let L (H, K) denote the space of bounded linear operators from H to K and set L (H) := L (H, H). Fix n ∈ N * and let A ∈ L ∞ (R n ; L (C n )). Consider the sesquilinear form l A : H 1 (R n ) × H 1 (R n ) → C defined by for u, v ∈ H 1 (R n ). Suppose that l A satisfies the Gårding inequality (1) Re (l A [u, u]) ≥ κ A ∇u since the two sides of the above relation will naturally coincide whenever the righthand side makes sense. The operator L A will be a densely defined maximal accretive operator. As such, it is possible to define a square root operator √ L A , with domain D (L A ), that satisfies √ L A · √ L A = L A . A famous conjecture first posed by Kato asks whether the domain of this square root operator extends to all of H 1 (R n ). This is the Kato square root problem. In essence, it amounts to proving that the estimate L A u ∇u is true for any u ∈ D (L A ). This long-standing problem withstood solution until [3] where it was proved using local T (b) methods. Then, in [4] this solution was generalised to elliptic systems. We will be interested in an alternate method of proof that was built from similar principles and appeared a few years later. Let Π := Γ + Γ * be a Dirac-type operator on a Hilbert space H and Π B := Γ + B 1 Γ * B 2 be a perturbation of Π by bounded operators B 1 and B 2 . Typically, Π is considered to be a first-order system acting on H := L 2 R n ; C N for some n, N ∈ N * and the perturbations B 1 and B 2 are multiplication by matrix-valued functions B 1 , B 2 ∈ L ∞ R n ; L C N . In their seminal paper [6], A. Axelsson, S. Keith and A. McIntosh developed a general framework for proving that the perturbed operator Π B possessed a bounded holomorphic functional calculus. This ultimately amounted to obtaining square function estimates of the form where Q B t := tΠ B I + t 2 Π 2 B −1 and u is contained in the range R (Π B ). They proved that this estimate would follow entirely from a set of simple conditions imposed upon the operators Γ, B 1 and B 2 , labelled (H1) -(H8). Then, by checking this list of simple conditions, the Axelsson-Keith-McIntosh framework, or AKM framework by way of abbreviation, could be used to conclude that the particular selection of operators (3) Γ := 0 0 ∇ 0 , B 1 = I, B 2 = I 0 0 A , defined on L 2 (R n )⊕L 2 (R n ; C n ), would satisfy (2) and therefore possess a bounded holomorphic functional calculus. The Kato square root estimate then followed almost trivially from this.
Let V : R n → C be a measurable function that is finite almost everywhere on R n . V can be viewed as a densely defined closed multiplication operator on L 2 (R n ) with domain D (V ) = u ∈ L 2 (R n ) : V · u ∈ L 2 (R n ) . The density of D(V ) follows from the measurability of V . Define the subspace Here the complex square root V 1 2 is defined via the principal branch {z ∈ C : Re (z) < 0}. Let A ∈ L ∞ (R n ; L (C n )) be as before with (1) satisfied for some κ A > 0. Consider the sesquilinear form l V A : for u, v ∈ H 1 V (R n ). Suppose that there exists some κ V A > 0 for which l V A satisfies the associated Gårding inequality (5) Re for all u ∈ H 1 V (R n ). Remark 1.1. If the range of V is contained in some sector S µ + := {z ∈ C ∪ {∞} : |arg (z)| ≤ µ or z = 0, ∞} for some µ ∈ [0, π 2 ), then (5) will follow automatically from (1). Once again, the accretivity of l V A implies the existence of a maximal accretive operator associated with this form denoted by . Define W to be the class of all finite almost everywhere measurable functions V : R n → C for which sup u∈D(|V |−∆) In this paper, our aim is to prove the potential dependent Kato estimate as presented in the following theorem.
Theorem 1.1 (Kato with Potential). Let V ∈ W and A ∈ L ∞ (R n ; L (C n )). Suppose that the Gårding inequalities (1) and (5) are both satisfied with constants κ A > 0 and κ V A > 0 respectively. There exists a constant C V > 0 such that for all u ∈ D L V A . In direct analogy to the potential free case, the Kato problem with potential will be solved by constructing appropriate potential dependent Dirac-type operators and demonstrating that they retain a bounded holomorphic functional calculus under perturbation. In particular, this strategy will be applied to the Dirac-type operator It should be observed that the operator Γ |V | 1 2 will not necessarily satisfy the cancellation and coercivity conditions, (H7) and (H8), of [6] due to the presence of the zero-order potential term. As such, the original framework developed by Axelsson, Keith and McIntosh cannot be directly applied. The key difficulty in proving our result is then to alter the original framework in order to allow for such operators. In particular, a non-homogeneous version of the Axelsson-Keith-McIntosh framework will be developed to handle operators of the form where D and J are, respectively, homogeneous and non-homogeneous first-order differential operators. The technical challenge presented by the inclusion of the non-homogeneous operator J will be overcome by separating our square function norm into components and demonstrating that the non-homogeneous term will allow for the first two components to be bounded while the third component can be bound using an argument similar to the classical argument of [6].
Since the operator Γ J is of a more general form than Γ |V | 1 2 , the non-homogeneous AKM framework that we develop will have applications not confined to zero-order scalar potentials. Indeed, the non-homogeneous framework will also be used to prove Kato estimates for systems of equations with zero-order potential and for scalar equations with first-order potentials. It takes no great leap of imagination to see that our framework could also be applied to a combination of these two situations. That is, it is possible to apply our framework to systems of equations with first-order potentials. This, however, will be left to the readers discretion.
The structure of this paper is as follows. Section 2 is quite classical in nature. It provides a brief survey of the natural functional calculus for bisectorial operators. Section 3 describes the non-homogeneous AKM framework and states the main results associated with it. Section 4 contains most of the technical machinery and is dedicated to a proof of our main result. Section 5 will apply the nonhomogeneous AKM framework to the scalar Kato problem with potential, the Kato problem for systems with zero-order potential and the scalar Kato problem with first-order potential. It is here that a proof of Theorem 1.1 will be completed. Finally, in Section 6, we will provide a meta-discussion on the proof techniques used and compare our work with what has been previously accomplished on nonhomogeneous Kato type estimates. Comparative strengths and weaknesses of our approach will be highlighted.
Notation. Throughout this article, the notation A B and A B will be used to denote that there exists a constant C > 0 for which A ≤ C · B and C −1 · B ≤ A ≤ C · B respectively. 1.1. Acknowledgements. This paper is part of my PhD thesis undertaken at the Australian National University. I am very thankful to my supervisor Pierre Portal for his numerous suggestions and corrections and for the encouragement that made this article possible. I am also grateful to my second supervisor Adam Sikora for his sage advice, in particular with regards to Proposition 5.1. A part of the paper was written while visiting him at Macquarie University While this paper was in its final stages of preparation, it was found that Andrew Morris and Andrew Turner from the University of Birmingham had obtained similar results. After meeting them and discussing their research, it appears that the two approaches differ in their assumptions and, more substantially, their proofs.
Finally, I would like to thank the anonymous referee of a previous version of this article for providing such a detailed and thoughtful critique. Reflecting on their comments led me to several significant improvements.

Preliminaries
Let's outline the construction of the natural functional calculus associated with a bisectorial operator. The treatment of functional calculi found here follows closely to [15] with significant detail omitted. Appropriate changes are made to account for the fact that we consider bisectorial operators instead of sectorial operators. Other thorough treatments of functional calculus for sectorial operators can be found in [16] or [1].
For µ ∈ [0, π) define the open and closed sectors Then, for µ ∈ 0, π 2 , define the open and closed bisectors respectively. Throughout this section we consider bisectorial operators defined on a Hilbert space H. if the spectrum σ (T ) is contained in the bisector S ω and if for any µ ∈ ω, π 2 , there exists C µ > 0 such that the resolvent bound (10) |ζ| (ζI − T ) −1 ≤ C µ holds for all ζ ∈ C \ S µ . T is said to be bisectorial if it is ω-bisectorial for some ω ∈ 0, π 2 . Sectorial operators are defined identically except with the sector S µ+ performing the role of the bisector S µ . An important fact concerning bisectorial operators is the following decomposition result. Let T be an ω-bisectorial operator for ω ∈ 0, π 2 and µ ∈ ω, For any f ∈ H ∞ 0 S o µ , one can associate an operator f (T ) as follows. For u ∈ H, define where the curve γ := ±re ±iν : 0 ≤ r < ∞ for some ν ∈ (ω, µ) is traversed anticlockwise.
is a well-defined algebra homomorphism. Moreover, it is independent of of the value of ν.
Proof. The resolvent bounds of our operator and the size estimates on f imply that the above integral will converge absolutely ensuring that f (T ) is a well-defined bounded operator. An application of the Cauchy integral formula will give us the independence of the definition of f (T ) from the value of ν. For a proof of the homomorphism property refer to [15,Lem. 2.3.1].
Since the functions in H ∞ 0 S o µ approach zero at the origin we should expect that the null space of the newly formed operator will be larger than the null space of the original operator. This is indeed the case as stated in the below proposition.
µ r and e ∈ E S o µ a regularizing function for f . This definition is independent of the chosen regularizer e for f and therefore Φ T is well-defined. We have the following important theorem that establishes the desired properties of a functional calculus for this extension.
The following definition plays a vital role in the solution method to the Kato square root problem using the AKM framework.
An ω-bisectorial operator T : D(T ) ⊂ H → H is said to have a bounded H ∞ S o µ -functional calculus if there exists c > 0 such that T is said to have a bounded holomorphic functional calculus if it has a bounded H ∞ S o µ -functional calculus for some µ.
Remark 2.1. Note that a more intuitive definition for a bounded H ∞ S o µfunctional calculus would be to require that (11) hold for all f ∈ H ∞ S o µ . Unfortunately at this stage it is impossible to ascertain whether H ∞ S o µ ⊂ E S o µ r . When this inclusion does not hold, the operator f (T ) will not be well-defined for all f ∈ H ∞ S o µ . If T so happens to be injective, then each f ∈ H ∞ S o µ is in fact regularizable by z 1 + z 2 and the estimate (11) makes sense for all f ∈ H ∞ S o µ . Fortunately, in this situation the two definitions coincide. That is, (11) will be true For t > 0, let q t denote the function q t (z) := q(tz) for z ∈ S o µ . It is not too difficult to see that q t ∈ H ∞ 0 S o µ for any t > 0 (c.f. [15, pg. 29]).

Definition 2.3 (Square Function Estimates). A bisectorial operator T on a Hilbert
space H is said to satisfy square function estimates if there exists a constant C > 0 such that The above definition is the same as saying that the seminorm on H defined through is norm equivalent to · H when restricted to the Hilbert subspace R (T ).
Remark 2.2. The function q in the above definition of square function estimates is somewhat arbitrary. It can be replaced by any function where ψ t (z) := ψ(tz) for z ∈ S o µ and t > 0. The fact that this square function norm is equivalent up to multiplicative constant to (13) on R(T ) can be found in [15,Thm. 7

Proposition 2.3 (Resolution of the Identity). For any u ∈ H,
where P N (T ) denotes the projection operator onto the subspace N (T ).
Proof. Equality follows from Proposition 2.2 for u ∈ N (T ). For u ∈ R (T ) this is given by Theorem 5.2.6 of [15] in the sectorial case. The bisectorial case can be proved similarly.
Equality will hold if u ∈ R (T ).

Proof.
As T is self-adjoint, if follows from the definition of q t (T ) that it must also be self-adjoint. On expanding the square function norm, The previous proposition then gives Equality will clearly hold in the above if u ∈ R(T ).
A fundamental result due to A. McIntosh is the equivalence of square function estimates with a bounded holomorphic functional calculus. Finally, the following Kato type estimate follows from the previous theorem using a well-known classical argument. This argument can be found, for example, in the proof of Corollary 3.4 in [11].
Corollary 2.2. Suppose that the bisectorial operator T satisfies square function estimates. Then there exists a constant c > 0 such that for any u ∈ D (T ).

Non-Homogeneous Axelsson-Keith-McIntosh
In this section we describe how the Axelsson-Keith-McIntosh framework can be altered to account for non-homogeneous operators of the form (9). Our main results for this framework will also be stated.
3.1. AKM without Cancellation and Coercivity. The operators that we wish to consider, Γ J , will satisfy the first six conditions of [6]. However, they will not necessarily satisfy the cancellation condition (H7) and the coercivity condition (H8). It will therefore be fruitful to see what happens to the original AKM framework when the cancellation and coercivity conditions are removed.
Similar to the original result, we begin by assuming that we have operators that satisfy the hypotheses (H1) -(H3) from [6]. Recall these conditions for operators Γ, B 1 and B 2 on a Hilbert space H.
(H2) B 1 and B 2 satisfy the accretivity conditions (H3) The operators Γ and Γ * satisfy In [6] Section 4, the authors assume that they have operators that satisfy the hypotheses (H1) -(H3) and they derive several important operator theoretic consequences from only these hypotheses. As our operators Γ, B 1 and B 2 also satisfy (H1) -(H3), it follows that any result proved in [6] Section 4 must also be true for our operators and can be used with impunity. In the interest of making this article as self-contained as possible, we will now restate any such result that is to be used in this paper.
The Hilbert space H has the following Hodge decomposition into closed subspaces: . When B 1 = B 2 = I these decompositions are orthogonal, and in general the decompositions are topological. Similarly, there is also a decomposition where and The bisectoriality of Π B ensures that the following operators will be well-defined.
When there is no perturbation, i.e. when B 1 = B 2 = I, the B will dropped from the superscript or subscript. For example, instead of Θ I t or Π I the notation Θ t and Π will be employed.
Remark 3.1. An easy consequence of Proposition 3.2 is that the operators R B t , P B t and Q B t are all uniformly L 2 -bounded in t. Furthermore, on taking the Hodge decomposition Proposition 3.1 into account, it is clear that the operators Θ B t will also be uniformly L 2 -bounded in t.
The next result tells us how the operators Π B and P B t interact with Γ and Γ * B . Lemma 3.1 ([6]). The following relations are true.
The subsequent lemma provides a square function estimate for the unperturbed Dirac-type operator Π. When considering square function estimates for the perturbed operator, there will be several instances where the perturbed case can be reduced with the assistance of this unperturbed estimate. Its proof follows directly from the self-adjointness of the operator Π and Corollary 2.1.
The following result will play a crucial role in the reduction of the square function estimate (2).
holds for all u ∈ R (Γ), together with three similar estimates obtained on replacing The following corollary is proved during the course of the proof of Proposition 4.8 of [6].
From this point onwards, it will also be assumed that our operators satisfy the additional hypotheses (H4) -(H6). These hypotheses are stated below for reference.
(H4) The Hilbert space is H = L 2 R n ; C N for some n, N ∈ N * .
(H5) The operators B 1 and B 2 represent multiplication by matrix-valued functions. That is, (H6) For every bounded Lipschitz function η : for all x ∈ R n and some constant c > 0.
In contrast to the original result, our operators will not be assumed to satisfy the cancellation condition (H7) and the coercivity condition (H8). Without these two conditions, many of the results from Section 5 of [6] will fail. One notable exception to this is that the bounded operators associated with our perturbed Dirac-type operator Π B will satisfy off-diagonal estimates.
Let {U t } t>0 be a family of operators on H = L 2 R n ; C N . This collection is said to have off-diagonal bounds of order M > 0 if there exists C M > 0 such that whenever E, F ⊂ R n are Borel sets and u ∈ H satisfies supp u ⊂ F .

Proposition 3.4 ([6]
). Let U t be given by either u(y) dy for x ∈ R n , t > 0 and u ∈ H, where Q(x, t) is the unique dyadic cube in ∆ t that contains the point x.
For an operator family {U t } t>0 that satisfies off-diagonal bounds of every order, there exists an extension U t : L ∞ R n ; C N → L 2 loc R n ; C N for each t > 0. This is constructed by defining for x ∈ Q ∈ ∆ t and u ∈ L ∞ R n ; C N . The convergence of the above limit is guaranteed by the off-diagonal bounds of {U t } t>0 . Further detail on this construction can be found in [6], [11], [18] or [12]. The above extension then allows us to introduce the principal part of the operator U t . Definition 3.3. Let {U t } t>0 be operators on H that satisfy off-diagonal bounds of every order. For t > 0, the principal part of U t is the operator ζ t : The following generalisation of Corollary 5.3 of [6] will also be true with an identical proof. Finally, the ensuing partial result will also be valid. Its proof follows in an identical manner to the first part of the proof of Proposition 5.5 of [6]. Proposition 3.6. Let {U t } t>0 be operators on H that satisfy off-diagonal bounds of every order. Let ζ t : R n → L C N denote the principal part of U t . Then there exists c > 0 such that for any v ∈ H 1 R n ; C N ⊂ H and t > 0.
3.2. Additional Structure. At this point, further structure will be imposed upon our operators in order to generalise the non-homogeneous operator Γ |V | 1 2 defined in (7). This additional structure will later be exploited in order to obtain square function estimates.
Hilbert spaces. Let P i : C N → C N be the projection operator onto the space V i for i = 1, 2 and 3. Our Hilbert space will have the following orthogonal decomposition The notation P i will also be used to denote the natural projection operator from H onto L 2 (R n ; V i ). For a vector v ∈ H, v i ∈ L 2 (R n ; V i ) will denote the ith component for i = 1, 2 or 3.
Let Γ J be an operator on H of the form where J and D are closed densely defined operators Let B 1 , B 2 ∈ L ∞ R n ; L C N be matrix-valued multiplication operators. The following key assumption will be imposed on our operators throughout the entirety of this article.  (H8) There exists c > 0 such that  (7) and (8) with (1) and (5) satisfied.

Remark 3.2.
Since the operator Γ 0 , together with the perturbations B 1 and B 2 , satisfy all eight conditions (H1) -(H8) of [6], it follows that any result from that paper must be valid for these operators. When there is no perturbation, i.e. when B 1 = B 2 = I, the B will dropped from the superscript or subscript. For example, instead of Θ J,I t the notation Θ J t will be employed.
We now introduce a coercivity condition to serve as a replacement for (H8) for the operators {Γ J , B 1 , B 2 }. This condition will not be automatically imposed upon our operators but, rather, will be taken as a hypothesis for our main results.  becomes trivially satisfied. Furthermore, when this occurs, (H8J) will be equivalent to the condition The main result of the non-homogeneous AKM framework can now be stated.
The proof of this theorem will be reserved for Section 4. For now, let's prove an estimate that serves as a dual to the above estimate. Suppose that B 1 = I. Then for all u ∈ R (Γ * J ).
Proof. As {Γ * J , B 2 , B 1 } satisfy (H1) -(H6), it follows from Remark 3.1 that the operatorsP J,B t are well-defined and uniformly L 2 -bounded. On applying this to the left-hand side of (24), where the inequality Γ J v ≤ Π J v for v ∈ D (Π J ) follows immediately from the three-by-three matrix form of the operators and Lemma 3.2 was used in the last line.
From our main result, Theorem 3.1, and Proposition 3.7, the upper and lower square function estimates for Q J,B t can be proved using the results of [6].  From the upper and lower estimate of the previous theorem, Theorem 2.4 then implies that Π J,B has a bounded holomorphic functional calculus. The following Corollary is then readily deduced from Corollary 2.2.

Square Function Estimates
In this section, a proof of our main result, Theorem 3.1, will be provided. As stated in the introduction, the technical challenge presented by the inclusion of the non-homogeneous operator J will be overcome by separating our square function norm into components. In this manner, where the notation P i , introduced in Section 3, denotes the projection operator onto the subspace L 2 (R n ; V i ) ⊂ H for i = 1, 2 and 3. Notice that for P 1 P J t u = 0 for any u ∈ R (Γ J ) and thus the boundedness of the first component is trivial. The boundedness of the second component relies on the non-homogeneous term J and is given in the following lemma.
Proof. First it will be proved that for u ∈ R (Γ J ) we have P 2 P J t u ∈ D Γ * B,J . Note that this is equivalent to Since P J t and Γ J commute by Lemma 3.1, it follows that Γ * J,B P 2 P J t u Γ * J P J t u for u ∈ R (Γ J ). On applying the uniform L 2 -boundedness of the P J,B t operators, On successively applying Proposition 3.1 and Lemma 3.2 we obtain It remains to bound the third component of our square function estimate, This will be handled in a similar manner to the classical proof in [6] but the effect of the projection P 3 must be accounted for. Introduce the notationΘ J,B for w ∈ C N and x ∈ R n . Evidently we must haveγ J,B t (x)w = γ J,B t (x)P 3 w. Our square function norm can be reduced to this principal part by applying the splitting (28) Since the operator Θ J,B t satisfies the conditions of Proposition 3.6, it follows that . It then follows from Lemma 3.1 that . This allows us apply (H8) for the operator Γ 0 to obtain It is not too difficult to see, simply by expanding out the operators, that (H8J) implies Π 0 Γ Jṽ Π J Γ Jṽ for anyṽ ∈ D (Γ J ). Therefore Our theorem has therefore been reduced to a proof of the following square function estimate On splitting from above using the triangle inequality, To proceed any further, the following result is required.
Proof. The estimate is trivially satisfied for any u ∈ N (Π J ) since for any t > 0. So suppose that u ∈ R (Π J ). On applying the resolution of the identity, equation (17) of [6], The Cauchy-Schwarz inequality leads to Let's estimate the term Next, suppose that t > s. Then the equality P J t Q J s = s t Q J t P J s gives For the second term we have since P 3 S J = 0. On applying Lemma 5.6 of [6] to Υ = Γ 0 , where the inequality Γ 0 v ≤ Π J v for v ∈ D (Π J ), used to obtain the third line of the above equation, follows trivially from the matrix form of the operators Γ 0 and Π J . Putting everything together gives .
This bound can then be applied to (31) to give (30).
Recall from Proposition 3.5 that the uniform estimate γ J,B t A t 1 is true for all t > 0. Furthermore, notice that A 2 t = A t and P 3 A t = A t P 3 for all t > 0. These facts combine together with the above proposition to produce For the second term in (29), apply Carleson's theorem ([20] Theorem 2, page 59) to obtain where µ is the measure on R n+1 defined through dµ(x, t) := γ J,B t (x) 2 dx dt t for x ∈ R n and t > 0, and µ C is the Carleson norm of µ, The proof of our theorem has thus been reduced to showing that the measure µ is a Carleson measure.

4.1.
Carleson Measure Estimates. The aim of this section is to prove the following Carleson measure estimate, Let L 3 denote the subspace (36) By construction, we haveγ J,B t (x) ∈ L 3 for any t > 0 and x ∈ R n sincẽ γ J,B Let σ > 0 be a constant to be determined at a later time. Let V be a finite set consisting of ν ∈ L 3 with |ν| = 1 such that ∪ ν∈V K ν = L 3 \ {0}, where Then, in order to prove our Carleson measure estimate (35), it is sufficient to fix ν ∈ V and prove that Recall the John-Nirenberg lemma for Carleson measures as applied in [6] and [3].
Proof. Fix Q ∈ ∆ and let {Q k1 } k1 be a collection of subcubes as in the hypotheses of the lemma. Apply the bound (38) to the decomposition For each k 1 , let {Q k1,k2 } k2 be a collection of subcubes of Q k1 that satisfy the hypotheses of the lemma. Decompose ρ(Q k1 ) and once again apply (38) to obtain Iterating this process and summing the resulting geometric series gives (39).
With this tool at our disposal, the proof of our theorem can be reduced to the following proposition.
For > 0 the function f w Q, can be defined in an identical manner to [6]. Specifically, let η Q : R N → [0, 1] be a smooth cutoff function equal to 1 on 2Q, with support in 4Q and with ∇η Q ∞ ≤ 1 l where l := l(Q). Then define w Q := η Q · w and for any > 0. Moreover, C will not depend on Q, σ, ν, w or .
Proof. The first claim follows from On recalling that w is zero in the first two components, At this point, apply Lemma 5.6 of [6] to the operator Υ = Γ 0 to obtain where the inequality Γ 0 v ≤ Γ J v for v ∈ D (Γ J ) follows trivially from the matrix form of Γ 0 and Γ J .
Moreover, D will not depend on Q, σ, ν, w or From this point forward, with C as in Lemma 4.3, set := 1 4C 2 and introduce the notation f w Q := f w Q, . With this choice of it must be true that That is, On rearranging we find that In this context, Lemma 5.11 of [6] will take on the below form. for all dyadic subcubes Q ∈ ∆ of Q which satisfy R Q ∩ E * Q,ν = ∅. Moreover, β, c 1 and c 2 are independent of Q, σ, ν and w.
The proof of this statement follows in an identical manner to the argument in [6]. If we set σ = c1 2c2 , then the following pointwise estimate can be deduced.
Proof. First observe that Proof of Proposition 4.2. From the pointwise bound of the previous lemma, Lemma 4.4 states that the final term in the above estimate will be bounded from above by a multiple of |Q|. This reduces the task of proving the proposition to bounding the first term of the above splitting. Recall that f w Q can be expressed in the form Since the first and second components of our main square function estimate have already been proved to be bounded we have

Scalar Kato with
Zero-Order Potential. Theorem 1.1, the promised result of the introductory section, will now be proved. Brand the definition of the operators Γ J , B 1 and B 2 to be as follows. Define our Hilbert space to be for some n ∈ N * . Let V : R n → C be a complex-valued measurable function that is finite almost everywhere on R n . Set J = |V | 1 2 and D = ∇. Our operator Γ J is then given by is as defined in the introductory section. The density of H 1 V (R n ) in L 2 (R n ) follows from the measurability of |V | 1 2 . The adjoint of this operator is given by Let A ∈ L ∞ (R n ; L (C n )) be a matrix-valued multiplication operator and suppose that the Gårding inequalities (1) and (5)  Our perturbed Dirac-type operator then becomes It is straightforward to check that with correct domains. Thus the square of our perturbed Dirac-type operator is It is clear from the form of our operator Γ 0 and the fact that A satisfies (1) that the operators {Γ 0 , B 1 , B 2 } satisfy (H1) -(H8). Similarly, since A and V satisfy (5), it follows that {Γ J , B 1 , B 2 } will satisfy the properties (H1) -(H6). The proof of Theorem 1.1 can now be concluded. Let V ∈ W. Then (H8J) will clearly be satisfied. Theorem 1.1 then follows immediately from Corollary 3.2. 5.1.1. Scalar Potentials that Satisfy the Kato Estimate. At this stage the unperturbed condition V ∈ W is still in quite an abstract form. It will therefore be instructive to unpack this condition and compare W with other commonly used classes of potentials. Recall the definition of the reverse Hölder class of potentials.
Definition 5.1. A non-negative and locally integrable function V : R n → R is said to satisfy the reverse Hölder inequality with index 1 < q < ∞ if there exists C > 0 such that holds for every ball B ⊂ R n . Let RH q denote the class of all potentials that satisfy the reverse Hölder inequality of index q.
Remark 5.1. It is obvious that any potential bounded both from above and below must be contained in RH q for any 1 < q < ∞. It is also well-known that for any polynomial P , |P | will be contained in RH q for any 1 < q < ∞ (this is given as an exercise in [20] on pg. 219 for example).
The reverse Hölder classes RH q have played a very influential role in the development of the harmonic analysis of Schrödinger operators. These potentials form a natural class for the construction of numerous harmonic analytic objects associated with the Schrödinger operator. Indeed, to name a few important results, this development led to the construction of both a Hardy space ( [9]) and a Muckenhoupt weight class ( [7]) associated with V − ∆. The most important result for our purposes is the boundedness of Riesz transforms associated with the Schrödinger operator for reverse Hölder potentials. The following result was first proved by Z. Shen in the seminal paper [19] for dimension n ≥ 3 and q ≥ n 2 . This result was later improved and extended to arbitrary dimension by P. Auscher and B. Ali in [2]. [2]). For any V ∈ RH q with q ≥ 2 there exists a c V > 0 for which Notice that the above inclusion implies, in particular, that the Kato estimate holds for the absolute value of any polynomial potential. The ensuing proposition demonstrates that the inclusion of the reverse Hölder potentials in W is strict, at least in dimension n > 4. Proof. Fix V ∈ L n 2 (R n ). Hölder's inequality gives us It is well-known that the Riesz potential (V − ∆) −1 is bounded from L 2 (R n ) to L 2n n−4 (R n ) (see for example Theorem 4 of [7]). There must then exist some C V > 0 for which Systems with Zero-Order Potential. Fix m ∈ N * and A ∈ L ∞ (R n ; L (C n ⊗ C m )). Let V : R n → L (C m ) be a measurable matrix-valued function with coefficients that are finite almost everywhere. V can be viewed as a densely defined closed multiplication operator on L 2 (R n ; C m ) with domain It will be assumed that there exists some U ∈ L ∞ (R n ; L (C n ⊗ C m )) for which V has the decomposition for each x ∈ R n , where |V (x)| := V (x) * V (x). Similar to the scalar case, one can define forms l A and l V A defined respectively through for u and v contained in Assume that the forms l A and l V A satisfy the Gårding inequalities (1) and (5) with constants κ A > 0 and κ V A > 0 respectively. Then l A and l V A will both have a unique associated maximal accretive operator, L A and L V A . In the below theorem, our non-homogeneous framework will be applied to determine the domain of L V A for a wide class of potentials.
Theorem 5.2. Suppose that there exists c V > 0 such that for all u ∈ D (|V | − ∆). Then there must exist some C V > 0 such that Proof. Set D := ∇ : H 1 (R n ; C m ) ⊂ L 2 (R n ; C m ) → L 2 (R n ; C n ⊗ C m ) and J := |V |  2 ) ⊂ L 2 (R n ; C m ) → L 2 (R n ; C m ) , both defined as operators on L 2 (R n ; C m ). Define the perturbation matrices It is not too difficult to see that the operators {Γ 0 , B 1 , B 2 } will satisfy conditions (H1) -(H8) and {Γ J , B 1 , B 2 } will satisfy (H1) -(H6). Indeed, the only non-trivial condition for both sets of operators is (H2) and this follows from the respective Gårding inequalities (1) and (5). It is also clear from (49) that (H8J) will be satisfied. The Kato estimate then follows from Corollary 3.2.
5.3. First Order Potentials. Let b : R n → C m be measurable and finite almost everywhere and A ∈ L ∞ (R n ; C n ). We will prove two different Kato estimates for first order potentials.

Second Kato Estimate.
For a result of a slightly different flavour, one could alternatively set the Hilbert space to be H := L 2 (R n ) ⊕ L 2 (R n ; C n ) ⊕ L 2 (R n ; C n ) .
Also let B 1 = I as usual and To see that the above theorem is true, simply note that (50) implies (H8J) in this context.

Final Remarks
It is important to note that this is not the first time that Kato type estimates have been studied for non-homogeneous operators. We will now take some time to outline how our article differs in techniques and results from each of these previous forays.
Recently, in [13] and [14], F. Gesztesy, S. Hofmann and R. Nichols studied the domains of square root operators using techniques distinct from those developed in [6]. The article [13] considers potentials in the class L p + L ∞ but is not directly relevant since it considers bounded domains. On the other hand, [14] does not impose a boundedness assumption on the domain and considers the potential class L n 2 + L ∞ . There is already an immediate comparison with our potential class since it was shown in Proposition 5.1 that L n 2 ⊂ W in dimension n > 4. It is not immediately clear whether L ∞ is contained within our class.