A Sharp Divergence Theorem with Nontangential Traces

Dorina Mitrea, Irina Mitrea, and Marius Mitrea

1. A Brief Historical Perspective functions on a compact interval [푎, 푏], the Fundamental The Fundamental Theorem of Calculus, one of the most Theorem of Calculus reads: spectacular scientific achievements, stands as beautiful, 푏 powerful, and relevant today as it did more than three ∫ 퐹′(푥) 푑푥 = 퐹(푏) − 퐹(푎) (1.1) centuries ago when it first emerged onto the mathemati- 푎 cal scene. Typically, Isaac Newton and Gottfried Leibniz for every 퐹 ∈ AC([푎, 푏]). are credited with developing much of the mathematical It is a stark example of how local information, encoded in machinery associated with this result into a coherent the- the instantaneous rate of change (aka derivative) 퐹′, can ory for infinitesimal quantities (the bedrock of modern cal- be pieced together via integration to derive conclusions of culus), a mathematical landscape within which the Fun- a global nature about the variation of 퐹 over [푎, 푏], a fundamental Theorem of Calculus stands out as the crown- damental paradigm in calculus. ing achievement. In its sharp one-dimensional version, Intriguingly, while (1.1) is essentially optimal, dealing involving the class AC([푎, 푏]) of absolutely continuous with higher-dimensional versions of the Fundamental The- Dorina Mitrea is a professor of mathematics at Baylor University. Her email orem of Calculus remains an active area of research in con- 푛 address is [email protected]. temporary mathematics. In its standard version, with ℒ Irina Mitrea is a professor of mathematics at Temple University. Her email denoting the 푛-dimensional Lebesgue measure in ℝ푛, the address is [email protected]. Divergence Theorem asserts that Marius Mitrea is a professor of mathematics at Baylor University. His email 푛 1 address is [email protected]. if Ω ⊆ ℝ is a bounded domain of class 풞 , Communicated by Notices Associate Editor Daniela De Silva. with outward unit normal 휈 and surface mea- ⃗ 푛 ⃗| (1.2) For permission to reprint this article, please contact: sure 휎, then ∫Ω div 퐹 푑ℒ = ∫휕Ω 휈 ⋅ (퐹|휕Ω) 푑휎 푛 [email protected]. for each vector field 퐹⃗ ∈ [풞1(Ω)] . DOI: https://doi.org/10.1090/noti2149

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1295 ′ 푛 Since the divergence of a continuously differentiable vec- where ퟏΩ is the characteristic function of Ω and 풟 (ℝ ) de- tor field 퐹⃗ may be computed pointwise as (div 퐹)(푥)⃗ = notes the space of distributions in ℝ푛. Conversely, since 푦−푥 푛 푛 −1 ⃗ 푛−1 both ∇ퟏΩ and −휈휎 are vector distributions in ℝ of order lim ℒ (퐵(푥, 푟)) ∫휕퐵(푥,푟) ( ) ⋅ 퐹(푦) 푑ℋ (푦), where 푟→0+ 푟 ≤ 1, their actions canonically extend to vector fields from ℋ 푛−1 stands for the (푛 − 1)-dimensional Hausdorff mea- 푛 [풞1(ℝ푛)] , in which scenario we recover (1.2). Hence, ℝ푛 (div 퐹)(푥)⃗ 푐 sure in , it follows that the quantity is em- (1.3) amounts to an equivalent reformulation of the classi- ⃗ blematic of the tendency of a vector field 퐹 to collect (sink cal Divergence Theorem (1.2), which has a purely geomet- effect) or disperse (source effect) at a point 푥. In view of ric measure theoretic nature. In particular, (1.3) brings this feature, the Divergence Formula in (1.2) may be re- into focus the fact that the distributional gradient of the garded as a conservation law, asserting that the solid inte- characteristic function of a bounded 풞1 domain is a locally gral of all such sources and sinks associated with a given finite Borel vector-valued measure in ℝ푛. vector field is equal to the net flow of said vector field As far as the latter property is concerned, R. Caccioppoli, 1 through the solid’s boundary. Hence, in complete anal- E. De Giorgi, and H. Federer registered a decisive leap for- ogy to the Fundamental Theorem of Calculus mentioned ward by considering the largest class of Euclidean subsets earlier, the Divergence Theorem describes how the infini- enjoying said property, i.e., the class of sets of locally fi- tesimal sink/source effects created by a vector field may be nite perimeter. It turns out that this consists of Lebesgue pieced together inside a given domain to produce a global, 푛 measurable subsets Ω of ℝ with the property that ퟏΩ is macroscopic effect along the boundary. 푛 푛 of locally bounded variation in ℝ , i.e., ퟏΩ ∈ BVloc(ℝ ). The classical result recorded in (1.2) is usually associ- 푛 In turn, membership to BVloc(ℝ ) is conceived in such a ated with the names of J.-L. Lagrange, who first established way that the Riesz Representation Theorem can naturally a special case of the Divergence Theorem in 1762 working be applied to the functional on the propagation of sound waves (cf. [14]); C. F. Gauss, who independently considered a particular case in 1813 푛 1 푛 푛 Λ(퐹)⃗ ∶= ∫ div 퐹⃗ 푑ℒ for all 퐹⃗ ∈ [풞푐 (ℝ )] , (1.4) (cf. [9]); M. V. Ostrogradsky, who gave the first proof of the Ω general theorem in 1826 (cf. [20]); G. Green, who used a to conclude that there exist a Borel measure 휎 in ℝ푛, related formula in 1828 (cf. [10]); A. Cauchy, who in 1846 ∗ which is actually supported on 휕Ω, and a 휎 -measurable first published, without proof, the nowadays familiar form ∗ vector-valued function 휈 ∶ ℝ푛 → ℝ푛 satisfying |휈| = 1 at of Green’s theorem (cf. [1]); B. Riemann, who provided a 휎 -a.e. point in ℝ푛, such that proof of Green’s formula in his 1851 inaugural disserta- ∗ tion (see [21]); Lord Kelvin, who in 1850 discovered a spe- 1 푛 푛 Λ(퐹)⃗ = ∫ 휈 ⋅ 퐹⃗ 푑휎∗ for all 퐹⃗ ∈ [풞푐 (ℝ )] . (1.5) cial version of Stokes’ theorem (in the three-dimensional ℝ푛 setting, also known as the curl theorem); and E.´ Cartan, 휈 who first published the general form of Stokes’ theorem (in The function is referred to as the geometric measure the- Ω 휎 the language of differential forms on manifolds) in 1945, oretic outward unit normal to . Bearing in mind that ∗ 휕Ω among others. However, a precise attribution is fraught is actually supported on , the following version of the with difficulty since the Divergence Theorem in its mod- Divergence Theorem emerges from (1.4) and (1.5): ern format has undergone successive waves of reformula- ⃗ 푛 ⃗ tions and generalizations, as well as more rigorous proofs, ∫ div 퐹 푑ℒ = ∫ 휈 ⋅ 퐹 푑휎∗ Ω 휕Ω (1.6) with inputs from a multitude of sources (general historical 푛 for each vector field 퐹⃗ ∈ [풞1(ℝ푛)] . accounts may be found in [12] and [22]). 푐 Specializing the Divergence Formula in (1.2) to the The real achievement of De Giorgi and Federer is further case when 퐹⃗ is the restriction to Ω of vector fields from refining (1.6) by establishing that actually 푛 [풞 ∞(ℝ푛)] (where 풞 ∞(ℝ푛) denotes the space of smooth, 푐 푐 휎 = ℋ 푛−1⌊휕 Ω, compactly supported functions in ℝ푛) yields the state- ∗ ∗ (1.7) ment: where 휕∗Ω denotes the measure theoretic boundary of Ω, if Ω ⊆ ℝ푛 is a bounded domain of class 풞1, defined as with outward unit normal 휈 and surface mea- (1.3) ℒ푛(퐵(푥, 푟) ∩ Ω) 푛 푛 ′ 푛 휕∗Ω ∶= {푥 ∈ ℝ ∶ lim sup 푛 > 0 and sure 휎, then ∇ퟏΩ = −휈휎 in [풟 (ℝ )] , 푟→0+ 푟 ℒ푛(퐵(푥, 푟) ⧵ Ω) lim sup 푛 > 0}. 푟→0+ 푟 1For instance, imagining 퐹⃗ as the velocity field of an incompressible fluid flow occupying a fixed region Ω, this informally states that “what goes in must come Hence, near points in 휕∗Ω there is enough mass (relative to out.” the scale) both in Ω and in ℝ푛 ⧵Ω. Let us also note that, in

1296 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 principle, 휕∗Ω can be a much smaller set than the topolog- Lipschitz domains, and also for a class of Reifenberg-flat ical boundary 휕Ω. Substituting (1.7) back into (1.6) then domains (cf. [3], [13]), but to go beyond this one needs yields the following result: genuinely new techniques. Progress in this regard has been registered in [11], which treats a much larger class of do- Theorem 1.1 (De Giorgi–Federer’s Divergence Theorem [4, mains than Lipschitz, without any flatness assumptions. 5,7,8]). Let Ω ⊆ ℝ푛 be a set of locally finite perimeter. Let 휈 be However, the version of the Divergence Theorem estab- the geometric measure theoretic outward unit normal to Ω, and 푛 lished in [11] requires that the (nontangential) trace of the let 휎 ∶= ℋ 푛−1⌊휕Ω. Then for each vector field 퐹⃗ ∈ [풞1(ℝ푛)] 푐 vector field 퐹⃗ on the boundary is 푝th power integrable for one has some 푝 > 1. This requirement is an artifact of the proof, ∫ (div 퐹⃗)| 푑ℒ푛 = ∫ 휈 ⋅ (퐹⃗| ) 푑휎. (1.8) which relies on the boundedness of the Hardy–Littlewood Ω 휕∗Ω 푝 Ω 휕∗Ω maximal function on 퐿 with 푝 > 1. The goal of this article is to describe a new brand of In a nutshell, one of the key results of the De Giorgi– Divergence Theorem developed in [17] exhibiting the fol- Federer theory is the identity lowing features (all of which are absent from De Giorgi– 푛−1 ∇ ퟏΩ = −휈 ℋ ⌊휕∗Ω Federer’s version of the Divergence Theorem recorded in as distributions in ℝ푛, whenever Theorem 1.1): • when 푛 = 1 and Ω is a finite interval on the real line, Ω ⊆ ℝ푛 , is a set of locally finite perimeter our theorem reduces precisely to the sharp version of the which may then readily be reinterpreted as the Divergence Fundamental Theorem of Calculus formulated in (1.1); Formula (1.8) simply by untangling jargon. A timely expo- • the vector field 퐹⃗ is intrinsically defined in Ω, and may sition may be found in [6, §5.8, Theorem 1, p. 209]. For lack continuity, or even local boundedness; the original work see [4], [5], [7], [8], as well as [2] for • the divergence of 퐹⃗ is computed in the sense of distribu- additional comments and references. tions and is allowed to exhibit certain types of singulari- The nature of the Divergence Theorem is such that ties; the smoother the category of vector fields considered, the • the only quantitative aspect not directly associated with rougher the class of domains which may be allowed in the the ability of writing the two integrals making up the Diver- formulation of said theorem. While the De Giorgi–Federer gence Formula in a meaningful way is an integrability con- version of the Divergence Theorem applies to a large class dition imposed on the nontangential maximal function of of domains (i.e., sets of locally finite perimeter), the vector the vector field 퐹⃗; 1 푛 fields involved are assumed to have components in 풞푐 (ℝ ). • the trace of 퐹⃗ on the boundary is considered in a point- Thus, the vector fields in the De Giorgi–Federer version of wise nontangential sense (i.e., considering the limit of 퐹⃗ the Divergence Theorem belong to a very restrictive class, from within certain nontangential approach regions with are exceedingly regular, as well as completely unrelated to vertices at points on 휕Ω). the underlying domain. Moreover, when specialized to Compared with the classical results of De Giorgi– the case 푛 = 1, for a finite interval of the real line, the Federer, our work brings into focus the role of the nontan- De Giorgi–Federer version of the Divergence Theorem for- gential maximal operator and the nontangential bound- mulated in Theorem 1.1 fails to yield the sharp version of ary trace in the context of the Divergence Theorem. The the Fundamental Theorem of Calculus, recorded in (1.1). idea of imposing an integrability condition on the nontan- While formula (1.8) has been successfully used in many gential maximal operator and then using this to prove the branches of mathematics, Theorem 1.1 is not adequate for existence of nontangential boundary limits originates in a variety of problems in partial differential equations, scat- the classical work of Fatou. In particular, the class of func- tering, and harmonic analysis, since in many fundamen- tions for which such a nontangential boundary trace exists ⃗ tal instances 퐹 is not continuous up to and including the serves as a natural enlargement of the category of functions boundary, but rather the trace of 퐹⃗ to 휕Ω is considered in which are continuous up to, and including, the topological a pointwise nontangential sense. As such, one needs a di- boundary of the underlying domain. In a broader perspec- vergence formula for rough integrands and rough bound- tive, describing the qualitative and quantitative bound- aries that can handle these cases. Of course, any signifi- ary behavior of a function via its nontangential bound- cant weakening of the assumptions on the vector field 퐹⃗ ary trace and its nontangential maximal operator is a nat- in Theorem 1.1 should be accompanied by a correspond- ural point of view which has been adopted in a multitude ing strengthening of the assumptions on the underlying of branches of analysis. Concrete examples of this flavor, domain Ω. Ad hoc techniques, based on approximating highlighting the adequacy and appropriateness of taking the original set Ω by a suitable sequence of subdomains boundary traces in a nontangential pointwise sense, and Ω푗 ↗ Ω, have sufficed for continuous vector fields in

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1297 imposing integrability conditions on the nontangential (ii) Σ is said to be upper Ahlfors regular if there exists maximal operator, include: Hardy space of holomor- a constant 퐶 ∈ (0, ∞) with the property that phic functions, singular integral operators of Calder´on– ℋ 푛−1(퐵(푥, 푟) ∩ Σ) ≤ 퐶 푟 푛−1 Zygmund-type, boundary value problems in rough do- (2.3) mains, and Fatou-type results. for each 푥 ∈ Σ and 푟 > 0. (iii) Σ is said to be Ahlfors regular3 if it is both lower 2. Divergence Theorems with Nontangential and upper Ahlfors regular. Pointwise Traces Definition 2.3. Given an arbitrary nonempty open proper Our first main result pertains to the Divergence Theorem 푛 in its standard format, as the equality between the solid subset Ω of ℝ , define its nontangentially accessible integral of the divergence of a given vector field and the boundary as boundary integral of the inner product of said field with 휕nta Ω ∶= {푥 ∈ 휕Ω ∶ 푥 ∈ Γ휅(푥) for each 휅 > 0}. (2.4) the geometric measure theoretic outward unit normal to A basic result proved in [17] states that if Ω ⊆ ℝ푛 is the underlying domain. In order to state it, a number of an open set with a lower Ahlfors regular boundary and definitions and preliminary results are discussed. such that the measure 휎 ∶= ℋ 푛−1⌊휕Ω is doubling,4 then Given an open set Ω ⊆ ℝ푛 and an aperture parameter 휎(휕 Ω ⧵ 휕 Ω) = 0; that is, the set 휕 Ω covers 휕 Ω up to 휅 ∈ (0, ∞), define the nontangential approach regions ∗ nta nta ∗ a 휎-nullset. Γ휅(푥) ∶= {푦 ∈ Ω ∶ |푦 − 푥| < (1 + 휅) dist (푦, 휕Ω)} Hypothesis 2.4. Fix 푛 ∈ ℕ and let Ω be a nonempty, proper, for each 푥 ∈ 휕Ω. In turn, these regions are used to define open subset of ℝ푛, with a lower Ahlfors regular boundary such the nontangential maximal operator 풩휅, acting on each that 휎 ∶= ℋ 푛−1⌊휕Ω is a doubling measure. Throughout, the 푛 2 ℒ -measurable function 푢 defined in Ω according to symbol 휈 is reserved for the geometric measure theoretic outward

∞ 푛 unit normal to Ω. (풩휅푢)(푥) ∶= ‖푢‖퐿 (Γ휅(푥),ℒ ) for each 푥 ∈ 휕Ω. Note that if we work (as one usually does) with equiva- In the context of Hypothesis 2.4, Ω is a set of locally fi- lence classes, obtained by identifying functions which co- nite perimeter, so 휈 is defined 휎-a.e. on 휕∗Ω. Also, if 휕Ω is incide ℒ푛-a.e., the nontangential maximal operator is in- actually Ahlfors regular, then automatically 휎 is doubling. dependent of the specific choice of a representative ina Here is the actual statement of the theorem alluded to ear- given equivalence class. More generally, if 푢 ∶ Ω → ℝ lier: is a Lebesgue measurable function and 퐸 ⊆ Ω is an ar- Theorem 2.5. Assume Hypothesis 2.4. Fix 휅 ∈ (0, ∞) and bitrary ℒ푛-measurable set, we denote by 풩퐸푢 the non- 휅 assume that 퐹⃗ = (퐹 , … , 퐹 ) ∶ Ω → ℂ푛 is a vector field with tangential maximal function of 푢 restricted to 퐸, that is, 1 푛 퐸 Lebesgue measurable components, satisfying 풩휅 푢 ∶ 휕Ω → [0, +∞] defined as 휅−n.t. 퐸 푛 ∞ 푛 퐹⃗| exists (in ℂ ) at 휎-a.e. point on 휕 Ω, (풩휅 푢)(푥) ∶= ‖푢‖퐿 (Γ휅(푥)∩퐸,ℒ ) for each 푥 ∈ 휕Ω. 휕Ω nta 1 Definition 2.1. Fix a background parameter 휅 > 0 and 풩휅퐹⃗ belongs to the space 퐿 (휕Ω, 휎), and (2.5) let 푢 be a complex-valued Lebesgue measurable function 1 푛 div 퐹⃗ ∶= 휕1퐹1 + ⋯ + 휕푛퐹푛 ∈ 퐿 (Ω, ℒ ), defined ℒ푛-a.e. in an open set Ω ⊂ ℝ푛. Consider a point where all partial derivatives are considered in the sense of distri- 푥 ∈ 휕Ω such that 푥 ∈ Γ (푥). The 휅-nontangential limit of 휅 butions in Ω. 푢 at 푥 from within Γ휅(푥) is said to exist, and its value is the 휅 ′−n.t. ′ ⃗| number 푎 ∈ ℂ, provided Then for any 휅 > 0 the nontangential trace 퐹|휕Ω exists 휎-a.e. on 휕 Ω and is actually independent of 휅 ′. When for every 휀 > 0 there exists 푟 > 0 so that nta 푛 (2.1) regarding it as a function defined 휎-a.e. on 휕∗Ω (which, up |푢(푦) − 푎| < 휀 for ℒ -a.e.푦 ∈ Γ휅(푥) ∩ 퐵(푥, 푟). to a 휎-nullset, is contained in 휕 Ω), this nontangential trace Whenever the 휅-nontangential limit of 푢 at 푥 from within nta 1 푛 휅−n.t. belongs to [퐿 (휕∗Ω, 휎)] . Also, with the dependence on the pa- Γ (푥) exists, its value is denoted by (푢| )(푥). ′ 휅 휕Ω rameter 휅 dropped, one has 푛 Definition 2.2. Let Σ ⊆ ℝ be a closed set. n.t. ∫ div 퐹⃗ 푑ℒ푛 = ∫ 휈 ⋅ (퐹⃗ | ) 푑휎 (2.6) (i) Σ is said to be lower Ahlfors regular provided there 휕Ω Ω 휕∗Ω exists a constant 푐 ∈ (0, ∞) such that when either Ω is bounded, or 휕Ω is unbounded and 푛 ≥ 2. In 푛−1 푛−1 푐 푟 ≤ ℋ (퐵(푥, 푟) ∩ Σ) the remaining cases, i.e., when Ω is unbounded, and either 휕Ω (2.2) for each 푥 ∈ Σ and 푟 ∈ (0, 2 diam (Σ)). 3Often, this is referred to as Ahlfors–David regular, or ADR for short. 2 4 We make the convention that (풩휅푢)(푥) ∶= 0 whenever 푥 ∈ 휕Ω is such that That is, there exists some 퐶 ∈ [1, ∞) such that 0 < 휎(퐵(푥, 2푟) ∩ 휕Ω) ≤ Γ휅(푥) = ∅. 퐶휎(퐵(푥, 푟) ∩ 휕Ω) < +∞ for all 푥 ∈ 휕Ω and 푟 ∈ (0, ∞).

1298 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 is bounded or 푛 = 1, formula (2.6) continues to hold under the flavor, we start by observing that, given a nonempty open additional assumption that there exists 휆 ∈ (1, ∞) such that set Ω ⊆ ℝ푛, one has the injective embeddings5 ′ ′ ⃗ 푛 2 ℰ (Ω) ↪ 풟 (Ω) and ∫ |푥 ⋅ 퐹(푥)| 푑ℒ (푥) = 표(푅 ) as 푅 → ∞, (2.13) (2.7) 1 푛 ′ 퐴휆,푅 ∩ Ω 퐿 (Ω, ℒ ) ↪ CBM(Ω) ↪ 풟 (Ω). where 퐴휆,푅 ∶= 퐵(0, 휆 푅) ⧵ 퐵(0, 푅). In view of these embeddings, it makes sense to consider This result refines work in [11], [15], [18], [19], where the subspace ℰ′(Ω) + CBM(Ω) of 풟′(Ω) defined as the col- earlier versions, interesting in their own right, along with lection of those 푢 ∈ 풟′(Ω) for which there exist some proofs and a wealth of applications may be found. The 푤 ∈ ℰ′(Ω) and 휇 ∈ CBM(Ω) such that 푢 = 푤 + 휇 in current, sharp version, appears in [17]. 풟′(Ω). We also introduce the space of smooth, bounded, As expected, Theorem 2.5 contains (1.2) as a special complex-valued functions in Ω, i.e., case. More generally, the scenario in which Ω is a bounded ∞ ∞ 푛 풞 (Ω) ∶= {푓 ∈ 풞 (Ω) ∶ 푓 bounded in Ω}, (2.14) Lipschitz domain in ℝ푛 and 퐹⃗ ∈ [풞 0(Ω)] is a vector 푏 ∗ 푛 ∞ and denote by (풞푏 (Ω)) the algebraic dual of this linear field which is differentiable at every point in Ω and ∑ 휕푗퐹푗 ∗ (⋅, ⋅) 푗=1 space. Throughout, we shall use 푋 푋 to denote the du- (where the partial derivatives are considered in a pointwise, ality pairing between a linear space 푋 and its algebraic dual ∗ classical sense) is continuous and absolutely integrable on 푋 . Ω is also covered by Theorem 2.5. Significantly, Theo- Theorem 2.6. Assume Hypothesis 2.4. Fix 휅 ∈ (0, ∞) and rem 2.5 contains (when 푛 = 1) the sharp form of the Fun- assume that the vector field damental Theorem of Calculus recalled in (1.1). ⃗ ′ 푛 Going further, observe that absolutely integrable func- 퐹 = (퐹1, … , 퐹푛) ∈ [풟 (Ω)] (2.15) tions in an open subset Ω of ℝ푛 may be identified with is such that there exists a compact set 퐾 contained in Ω satisfy- complex Borel measures in Ω (the collection of which is ing 푛 henceforth denoted by CBM(Ω)) via ⃗| 1 푛 퐹|Ω⧵퐾 ∈ [퐿 loc(Ω ⧵ 퐾, ℒ )] and 퐿1(Ω, ℒ푛) ∋ 푓 ⟼ 푓ℒ푛 ∈ CBM(Ω). (2.8) (2.16) Ω⧵퐾 1 풩휅 퐹⃗ ∈ 퐿 (휕Ω, 휎). A useful generalization of Theorem 2.5 is obtained by re- loc In addition, assume that the pointwise nontangential boundary placing the last condition in (2.5) with the requirement trace that 휅−n.t. 휅−n.t. 휅−n.t. 푛 퐹⃗| = (퐹 | , … , 퐹 | ) ⃗ ′ 휕Ω 1 휕Ω 푛 휕Ω the distribution div 퐹 ∶= ∑푗=1 휕푗퐹푗 ∈ 풟 (Ω) (2.9) exists (in ℂ푛) at 휎-a.e. point on 휕 Ω, has the property that extends to a complex Borel measure in Ω. nta 휅−n.t. div 퐹⃗ ⃗| 1 Retaining the symbol for the said extension, formula 휈 ⋅ (퐹|휕Ω ) ∈ 퐿 (휕∗Ω, 휎), (2.17) (2.12) takes the form and the divergence of 퐹⃗, taken in the sense of distributions in n.t. ⃗ ⃗ | Ω, is the sum (in 풟 ′(Ω)) of a compactly supported distribution (div 퐹)(Ω) = ∫ 휈 ⋅ (퐹 |휕Ω) 푑휎. (2.10) 휕∗Ω in Ω and a complex Borel measure in Ω, i.e., We can also dispense with the decay condition (2.7) at div 퐹⃗ ∈ ℰ ′(Ω) + CBM(Ω). (2.18) the expense of an additional term, encoding information 휅 ′−n.t. about the behavior of 퐹⃗ at infinity. Specifically, if one fixes ′ ⃗| Then for any 휅 > 0 the nontangential trace 퐹|휕Ω exists some 휙 ∈ 풞 ∞(ℝ푛) satisfying 휙 ≡ 1 near the origin in ℝ푛, ′ 푐 휎-a.e. on 휕nta Ω and is actually independent of 휅 . In addition, it follows that the limit with the dependence on the parameter 휅 ′ dropped, one has ⃗ 1 ⃗ 푛 n.t. [퐹 ]∞ ∶= lim { ∫(∇휙)(푥/푅) ⋅ 퐹(푥) 푑ℒ (푥)} (2.11) ∗ ⃗ ∞ ⃗ | ⃗ 푅→∞ 풞 ∞(Ω) (div 퐹 , 1)풞 (Ω) = ∫ 휈 ⋅ (퐹 | ) 푑휎 − [퐹 ]∞. 푅 Ω ( 푏 ) 푏 휕Ω 휕∗Ω (henceforth termed the contribution of 퐹⃗ at infinity) exists Also, the contribution at infinity vanishes, i.e., [퐹⃗ ]∞ = 0, and whenever there exists 휆 ∈ (1, ∞) such that n.t. (div 퐹⃗)(Ω) = ∫ 휈 ⋅ (퐹⃗ | ) 푑휎 − [퐹⃗ ] . (2.12) 휕Ω ∞ ∫ |푥 ⋅ 퐹(푥)|⃗ 푑ℒ푛(푥) = 표(푅 2) as 푅 → ∞, 휕∗Ω 퐴휆,푅 ∩ Ω (2.19) Once the possibility of allowing the divergence of the where 퐴휆,푅 ∶= 퐵(0, 휆 푅) ⧵ 퐵(0, 푅). vector field to belong to a class larger than the collection of integrable functions is considered, other natural options 5Recall that ℰ′(Ω) is the subspace of 풟′(Ω) consisting of all compactly sup- present themselves. To discuss an important case of this ported distributions in Ω.

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1299 In such a scenario, the above Divergence Formula reduces to are discussed at length in the monograph [17]. Here we

n.t. limit ourselves to presenting a couple of applications of ∞ ∗ (div 퐹⃗ , 1)풞 ∞(Ω) = ∫ 휈 ⋅ (퐹⃗ | ) 푑휎. (2.20) general appeal. (풞푏 (Ω)) 푏 휕Ω 휕∗Ω First, we consider a versatile version of the integration Finally, condition (2.19) is automatically satisfied when ei- by parts formula for first-order operators. To facilitate its ther Ω is bounded, or statement, given any constant coefficient first-order system 퐷 = ∑ 퐴 휕 + 퐵 휕Ω is unbounded, 푛 ≥ 2, 푗 푗 푗 we agree to define its (real) transpose ⊤ ⊤ ⊤ Ω⧵퐾 (2.21) as the operator 퐷 ∶= − ∑ 퐴 휕 + 퐵 , and its principal and 풩 퐹⃗ ∈ 퐿1(휕Ω, 휎). 푗 푗 푗 휅 symbol as the matrix-valued function associating to each Hence, (2.20) holds in either of these cases. 푛 휉 = (휉푗)푗 ∈ ℝ the matrix Sym (퐷 ; 휉) = ∑푗 휉푗퐴푗. Theorem 2.6 is proved employing a localization argu- Theorem 3.1. Assume Hypothesis 2.4. Let 퐷 be an 푁 × 푀 ment aimed at decomposing the given vector field into a first-order system with constant complex coefficients in ℝ푛, and regular part, to which Theorem 2.5 applies, plus a singu- let 푢 ∶ Ω → ℂ푀 , 푤 ∶ Ω → ℂ푁 be Lebesgue measurable lar part which has compact support and, as such, can be functions satisfying, for some 휅, 휅 ′ > 0, handled directly (using distribution theory; cf. [16]). 풩휅푢 < ∞ and 풩휅 ′ 푤 < ∞ at 휎-a.e. point on 휕Ω, There is also a version of Theorem 2.6 formulated in 1 an open set Ω ⊆ ℝ푛 without imposing the condition that 풩휅푢 ⋅ 풩휅 ′ 푤 belongs to the space 퐿 (휕Ω, 휎), the “surface measure” 휎 ∶= ℋ 푛−1⌊휕Ω is doubling. Re- 휅−n.t. 휅 ′−n.t. 푢| , 푤| exist at 휎-a.e. point on 휕 Ω, markably, there is only a relatively small price to pay in 휕Ω 휕Ω nta this scenario, namely, the loss of flexibility in the choice 1 푛 푁 ⊤ 1 푛 푀 퐷푢 ∈ [퐿 loc(Ω, ℒ )] , 퐷 푤 ∈ [퐿 loc(Ω, ℒ )] , of the aperture parameter 휅 ∈ (0, ∞) used to define the ⟨퐷푢, 푤⟩ ∈ 퐿1(Ω, ℒ푛) and ⟨푢, 퐷⊤푤⟩ ∈ 퐿1(Ω, ℒ푛). nontangential approach regions entering the definition of 휅−n.t. 휅″−n.t. the nontangential boundary trace 퐹⃗| and the nontan- ″ | 휕Ω Then for any 휅 > 0 the nontangential traces 푢|휕Ω , gential maximal function 풩 퐹⃗ for the given vector field 퐹⃗. 휅″−n.t. 휅 푤| exist 휎-a.e. on 휕 Ω and are actually independent of At the same time, we may further relax the demand made 휕Ω nta 휅″. When regarding them as functions defined 휎-a.e. on 휕 Ω in (2.18) on the nature of the distribution div 퐹⃗ by now ∗ and dropping the dependence on the parameter 휅″, the follow- merely asking that this be extended to a functional in the ∗ ing integration by parts formula, involving absolutely convergent 풞 ∞(Ω) algebraic dual ( 푏 ) exhibiting a mild, natural, conti- integrals, holds: nuity property (that is automatically satisfied when (2.18) holds). ∫ ⟨퐷푢, 푤⟩ 푑ℒ푛 = ∫ ⟨푢, 퐷⊤푤⟩ 푑ℒ푛 (3.1) We also wish to note that the results presented here are Ω Ω robust, as they continue to work in other settings of inter- n.t. n.t. | | est. For example, we have a natural version of the classi- + ∫ ⟨Sym (퐷 ; 휈)(푢|휕Ω) , 푤|휕Ω⟩ 푑휎, cal Stokes theorem for differential forms on manifolds in 휕∗Ω which the pullback of the given form to the boundary is when either Ω is bounded, or 휕Ω is unbounded and 푛 ≥ 2. now interpreted as a suitable Radon measure. Furthermore, formula (3.1) also holds if Ω is unbounded, and either 휕Ω is bounded or 푛 = 1, provided there exists 휆 ∈ (1, ∞) 3. Some Applications such that The Divergence Theorems presented earlier share a num- ∫ |푢||푤| 푑ℒ푛 = 표(푅) as 푅 → ∞, ber of common features. For example, all vector fields in- (3.2) volved may lack any type of continuity, and their bound- 퐴휆,푅 ∩ Ω where 퐴 ∶= 퐵(0, 휆 푅) ⧵ 퐵(0, 푅). ary traces are taken in a nontangential pointwise fash- 휆,푅 ion. The rationale for insisting on such characteristics Of course, some of the most familiar choices for 퐷 may be traced back to classical results in harmonic anal- are the gradient, curl, and divergence operators. Also, a ysis, complex analysis, partial differential equations, and closely related version of this theorem holds on Riemann- potential theory (specifically, the theory of Hardy spaces, ian manifolds, where natural choices for 퐷 include the Fatou-type theorems, elliptic boundary value problems, exterior derivative operator, the deformation tensor, the and Calder´on–Zygmundtheory for singular integral oper- Levi-Civita connection, etc. (see the discussion in [18] ators, among others). In turn, this innate affinity with the on this topic). Regarding the optimality of Theorem 3.1, very nature of this body of mathematics makes our brand a glimpse is offered by considering the very special case of Divergence Theorem an effective tool in dealing with when 푛 = 1, Ω = (푎, 푏) ⊂ ℝ, and 퐷 = 푑/푑푥. Then Theo- problems in these areas. A vast number of applications rem 3.1 asserts that for any functions 푢, 푤 ∈ 퐿∞(푎, 푏) with

1300 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 푢 ′, 푤 ′ ∈ 퐿1 (푎, 푏) 푢 ′푤 푢푤 ′ 퐿1(푎, 푏) 휅 ′−n.t. loc , such that and are in , ′ | + + Then for any 휅 > 0 the nontangential trace 푢|휕Ω exists and the limits 푢(푎 ) ∶= lim 푢(푥), 푤(푎 ) ∶= lim 푤(푥), ′ 푥→푎+ 푥→푎+ at 휎-a.e. point on 휕 Ω and is actually independent of 휅 . − − nta 푢(푏 ) ∶= lim 푢(푥), 푤(푏 ) ∶= lim 푤(푥) exist, we have Moreover, with the dependence on the parameter 휅 ′ dropped, 푥→푏− 푥→푏− for ℒ2-a.e. point 푧 ∈ Ω one has (with absolutely convergent 푏 푏 integrals) ∫ 푢 ′푤 푑푥 = 푢(푏−)푤(푏−) − 푢(푎+)푤(푎+) − ∫ 푢푤 ′ 푑푥. 푎 푎 n.t. 1 (푢| )(휁) The special case when 푢 ∈ AC([푎, 푏]) and 푤 = 1 yields the 푢(푧) = ∫ 휕Ω 푑휁 (3.8) 2휋푖 휁 − 푧 sharp Fundamental Theorem of Calculus stated in (1.1). 휕∗Ω

푛 Sketch of proof of Theorem 3.1. If 퐷 = ∑ 퐴 휕 + 퐵, then 1 (휕푢)(휁) 푗=1 푗 푗 − ∫ 푑ℒ2(휁), the idea is to apply Theorem 2.5 to the vector field 휋 Ω 휁 − 푧 퐹⃗ = (퐹푗)1≤푗≤푛, whose 푗th component is 퐹푗 ∶= ⟨퐴푗푢, 푤⟩ provided Ω is bounded, or 휕Ω is unbounded. In the remaining for each 푗 ∈ {1, … , 푛}. Upon checking that div 퐹⃗ = 휅−n.t. case, i.e., when Ω is unbounded and 휕Ω is bounded (that is, ⊤ ′ ⃗| ⟨퐷푢, 푤⟩ − ⟨푢, 퐷 푤⟩ in 풟 (Ω) and that 퐹|휕Ω is given by when Ω is an exterior domain), formula (3.8) holds under the 휅−n.t. 휅−n.t. ⟨Sym (퐷 ; 휈)(푢| ) , 푤| ⟩ 휎 휕 Ω additional assumption that there exists 휆 ∈ (1, ∞) such that |휕Ω |휕Ω at -a.e. point on nta , the formula claimed in (3.1) is implied by (2.6). □ ∫− |푢| 푑ℒ2 = 표(1) as 푅 → ∞, Our second application is a very general version of the 퐴휆,푅 (3.9) Cauchy–Pompeiu representation formula. To state it, we 퐴 ∶= 퐵(0, 휆 푅) ⧵ 퐵(0, 푅). first recall that the Cauchy–Riemann operator 휕 in the where 휆,푅 plane and its conjugate are, respectively, defined as As a corollary, if 휕푢 = 0 in Ω, then one has the Cauchy 1 1 휕 ∶= (휕푥 + 푖 휕푦) and 휕 ∶= (휕푥 − 푖 휕푦), (3.3) 2 2 integral representation formula where 푖 ∶= √−1 ∈ ℂ. Given any Lebesgue measurable set n.t. 1 (푢| )(휁) Ω ⊆ ℝ2 ≡ ℂ of locally finite perimeter, we agree to define 푢(푧) = ∫ 휕Ω 푑휁, ∀ 푧 ∈ Ω, (3.10) 2휋푖 휁 − 푧 the complex arc-length measure on 휕Ω by 휕∗Ω

푑휁 ∶= −2푖 휕ퟏΩ, (3.4) provided Ω is bounded, or 휕Ω is unbounded. In addition, when Ω is unbounded and 휕Ω is bounded, formula (3.10) continues where the derivatives are taken in the sense of distributions to be valid provided (3.9) is also assumed. in ℝ2. Hence, if 휎 ∶= ℋ1⌊휕Ω and 휈 denotes the geometric measure theoretic outward unit normal to Ω (canonically 휅 ′−n.t. 휅 ′ > 0 푢| identified with a complex-valued measure), it follows that Proof. That for any the nontangential trace |휕Ω exists 휎-a.e. on 휕 Ω and is independent of 휅 ′ is proved the complex arc-length measure on 휕Ω is supported on nta in [17]. We make two observations. One is that the first 휕∗Ω and satisfies property in (3.6) entails 푑휁 = 푖 휈(휁) 푑휎(휁) on 휕∗Ω. (3.5) 푢 ∈ 퐿∞ (Ω, ℒ2). (3.11) Theorem 3.2. Let Ω ⊆ ℝ2 ≡ ℂ be an open set with a lower loc 1 Ahlfors regular boundary, such that 휎 ∶= ℋ ⌊휕Ω is a doubling The other one is that (3.7) implies that for ℒ2-a.e. 푧 ∈ ℂ measure on 휕Ω. In this context, suppose 푢 ∶ Ω → ℂ is an we have ℒ2-measurable complex-valued function which, for some 휅 > 0, |(휕푢)(휁)| satisfies ∫ | | 푑ℒ2(휁) < +∞. (3.12) | 휁 − 푧 | (풩 푢)(휁) Ω | | ∫ 휅 푑휎(휁) < +∞, 휕Ω 1 + |휁| (3.6) Indeed, since elementary considerations show that there 휅−n.t. exists a constant 퐶 ∈ (0, ∞) such that for each 푅 ∈ (0, ∞) and 푢| exists 휎-a.e. on 휕 Ω. 휕Ω nta we have In addition, with the Cauchy–Riemann operator 휕 taken in the 푑ℒ2(푧) 푅 2 sense of distributions in Ω, assume that ∫ ≤ 퐶 ⋅ for all 휁 ∈ ℂ, (3.13) 퐵(0,푅) |푧 − 휁| 푅 + |휁| 푑ℒ2(휁) 휕푢 ∈ 퐿1(Ω, ). (3.7) 1 + |휁| for each 푅 ∈ (0, ∞) one can use the Fubini–Tonelli

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1301 Theorem to write point of 푢 with the property that (3.12) holds. In com- |(휕푢)(휁)| bination with (3.16) and (3.19) (and also assuming (3.9) ∫ ( ∫ 푑ℒ2(휁))푑ℒ2(푧) in the case when Ω is an exterior domain), for each such |휁 − 푧| 퐵(0,푅) Ω point 푧 this permits us to write 푑ℒ2(푧) = ∫ |(휕푢)(휁)| ∫ 푑ℒ2(휁) (휕푢)(휁) ( ) 2휋 푢(푧) + 2 ∫ 푑ℒ2(휁) Ω 퐵(0,푅) |휁 − 푧| Ω 휁 − 푧

n.t. |(휕푢)(휁)| 2 = ∞ ∗ (div 퐹⃗ , 1) ∞ = ∫ ⟨휈 , 퐹⃗ | ⟩ 푑휎 ≤ 퐶푅 ∫ 푑ℒ (휁) < +∞ (3.14) (풞푏 (Ω)) 푧 풞푏 (Ω) 푧 휕Ω Ω 1 + |휁| 휕∗Ω for some constant 퐶푅 ∈ (0, ∞). In turn, (3.14) implies that n.t. n.t. 2 (푢| )(휁) (푢| )(휁) (3.13) holds for ℒ -a.e. point 푧 ∈ 퐵(0, 푅), and the desired = ∫ ⟨휈(휁) , ( 휕Ω , 푖 휕Ω )⟩ 푑휎(휁) 푅 휁 − 푧 휁 − 푧 conclusion follows on account of the arbitrariness of . 휕∗Ω

Next, fix a Lebesgue point 푧 ∈ Ω of 푢 with the property n.t. that (3.12) holds, and define the vector field (푢| )(휁) = ∫ 휕Ω 휈(휁) 푑휎(휁) 푢(휁) 푢(휁) 2 휕 Ω 휁 − 푧 퐹푧⃗ (휁) ∶= ( , 푖 ) for ℒ -a.e. 휁 ∈ Ω. (3.15) ∗ 휁 − 푧 휁 − 푧 n.t. 푢| (휁) ⃗ 1 ( |휕Ω) Then from (3.15) and (3.11)one concludes that 퐹푧 belongs = ∫ 푑휁, (3.21) 2 1 2 푖 휕 Ω 휁 − 푧 to [퐿 loc(Ω, ℒ )] . Also, with 훿푧 denoting the Dirac distri- ∗ bution with mass at 푧, we have where the last equality uses (3.5). From this (3.8) follows 푢(휁) at each Lebesgue point 푧 ∈ Ω of 푢 such that (3.12) holds, div 퐹⃗ (휁) = 2휕 [ ] (3.16) 푧 휁 휁 − 푧 hence at ℒ2-a.e. point 푧 ∈ Ω. □

(휕푢)(휁) ′ It turns out that the doubling assumption on the mea- = 2휋 푢(푧)훿푧 + 2 in 풟 (Ω), 휁 − 푧 sure 휎 ∶= ℋ1⌊휕Ω may be relaxed to simply asking that 휎 where the last equality is a slight extension of [16, Exer- be a locally finite measure on 휕Ω. The price to pay is hav- cise 7.47, p. 292]. In particular, (3.16) and (3.12) imply ing to demand that the aperture parameter 휅 be sufficiently that large (depending on Ω) to begin with, and we may lose the flexibility of changing it when considering nontangential div 퐹⃗ belongs to ℰ′(Ω) + 퐿1(Ω, ℒ2). (3.17) 푧 boundary traces. Modulo these nuances, the format of the If we now consider 퐾 ∶= 퐵(푧, dist (푧, 휕Ω)/2), then 퐾 is a main result (i.e., the integral representation formula (3.8)) compact set contained in Ω and (3.15) allows us to esti- remains the same. mate This being said, the lower Ahlfors regularity condition for 휕Ω may not be simply dropped. To see this, consider Ω⧵퐾 ⃗ (풩휅푢)(휁) (풩휅 퐹푧)(휁) ≤ 퐶푧 for all 휁 ∈ 휕Ω. (3.18) the open subset Ω ∶= 퐵(0, 1) ⧵ {0} of ℂ and the function |휁 − 푧| 1 푢 ∶ Ω → ℂ defined as 푢(휁) ∶= for each 휁 ∈ Ω. It is In turn, from (3.18) and the first condition in (3.6) we 휁 Ω⧵퐾 1 clear that 휎 ∶= ℋ1⌊휕Ω is a locally finite measure. Also, conclude that 풩휅 퐹푧⃗ ∈ 퐿 (휕Ω, 휎). Moreover, (3.15) and 휅−n.t. the function 푢 is holomorphic in Ω, and the conditions in the assumptions on 푢 imply that 퐹⃗ | exists 휎-a.e. in n.t. 푧 휕Ω | 1 (3.6) are satisfied for any 휅 > 0. In addition, (푢|휕Ω)(휁) = 휕nta Ω and, in fact, that 휁 휅−n.t. 휅−n.t. for every 휁 ∈ 휕퐵(0, 1). If we now fix a point 푧 ∈ Ω and 휅−n.t. 푢| (휁) 푢| (휁) 1 ( |휕Ω ) ( |휕Ω ) define 푓(휁) ∶= for 휁 ∈ ℂ ⧵ {푧, 0}, we have that 푓 is 퐹⃗ | (휁) = , 푖 휁(휁−푧) ( 푧| ) ( ) (3.19) 휕Ω 휁 − 푧 휁 − 푧 meromorphic with poles of order one at 0 and 푧. Hence,

for 휎-a.e. 휁 ∈ 휕nta Ω. the boundary integral in the right-hand side of (3.8) may Finally, observe that in the case when Ω is an exterior be computed using the Residue Theorem as domain, condition (3.9) implies that ∫ 푓(휁) 푑휁 = 2휋푖[Res(푓, 푧) + Res(푓, 0)] 2 휕퐵(0,1) ∫ |퐹푧⃗ | 푑ℒ = 표(푅) as 푅 → ∞. (3.20) 퐴휆,푅 1 1 = 2휋푖[ − ] = 0. (3.22) 푧 푧 In summary, we have proved that 퐹푧⃗ satisfies all hypotheses in Theorem 2.6 whenever 푧 ∈ Ω is a Lebesgue Given that the solid integral in (3.8) is zero, if (3.8) were

1302 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 1 1 to hold in this case, we would obtain 푢(푧) = 0 for each For example, for ℒ1-a.e. 푥 ∈ ( − , + ) we have 푧 ∈ Ω, a contradiction. 2 2 We also wish to stress that both assumptions in (3.6) 1 lim 푢(푥 ± 푖푦) = ± 푓(푥) are necessary. To see that this is the case, consider the slit 푦→0+ 2 disk Ω ∶= {푧 ∈ 퐵(0, 1) ∶ 푧 ∉ [0, 1)} and bring in the 푥−휀 +1/2 1 1 푓(푡) holomorphic function 푢 ∶ Ω → ℂ given by 푢(휁) ∶= for + lim ( ∫ + ∫ ) 푑푡, 휁 휀→0+ 2휋푖 푧 − 푡 each 휁 ∈ Ω. Then the boundary integral in (3.8) is just as −1/2 푥+휀 in (3.22), hence zero. Thus, (3.8) becomes 푢(푧) = 0 for implying that each 푧 ∈ Ω, a contradiction. In this scenario, Ω satisfies there exists some Lebesgue measurable set all geometric hypotheses stipulated in Theorem 3.2, and 1 1 1 푢 퐴 ⊆ ( − , + ) with ℒ (퐴) > 0 such that satisfies all but the first condition in (3.6). The latter 2 2 (3.27) 휅 > 0 lim 푢(푥 + 푖푦) − lim 푢(푥 − 푖푦) = 푓(푥) ≠ 0 presently fails. Specifically, since for each fixed we 푦→0+ 푦→0+ −1 have (풩휅푢)(푥) ≈ 푥 uniformly for 푥 ∈ (0, 1) ⊆ 휕Ω, it for each point 푥 ∈ 퐴. follows that In the current setting, Ω satisfies all geometric hypothe- 1,∞ 1 풩휅푢 ∈ 퐿 (휕Ω, 휎) but 풩휅푢 ∉ 퐿 (휕Ω, 휎). (3.23) ses stipulated in Theorem 3.2, and 푢 satisfies all but the second condition in (3.6). However, (3.27) proves that This shows that the first condition in (3.6) is indeed neces- 휅−n.t. | sary. the latter fails and, even though 푢|휕Ω does exist at 휎-a.e. As regards the necessity of the second condition in (3.6), point on 휕∗Ω, the failure of the second condition in (3.6) consider the open subset of ℂ described as ultimately invalidates (3.8). As the reader surely suspects by now, Theorems 2.5–2.6 1 1 Ω ∶= {푧 ∈ 퐵(0, 1) ∶ 푧 ∉ [ − , + ] × {0}}, (3.24) have many other fundamental applications, such as sharp 2 2 versions of all Green integral identities for second-order 0 1 1 pick some 푓 ∈ 풞푐 ((− , + )) which is not identically zero, operators (including boundary layer potential representa- 2 2 푛 then define the function tions for solutions of second-order elliptic PDE’s) in ℝ . We, however, shall stay in the complex plane and con- 1/2 1 푓(푡) clude with a very general version of the classical Residue 푢(푧) ∶= ∫ 푑푡 ∀ 푧 ∈ Ω. (3.25) 2휋푖 −1/2 푧 − 푡 Theorem in complex analysis. Then 푢 is holomorphic in Ω, and Theorem 3.3. Let Ω ⊆ ℝ2 ≡ ℂ be an open set with a lower Ahlfors regular boundary, and with the property that 1/2 n.t. 1 푓(푡) 1 | 휎 ∶= ℋ ⌊휕Ω is a doubling measure on 휕Ω. Suppose 푓 is a (푢|휕Ω)(휁) = ∫ 푑푡 2휋푖 −1/2 휁 − 푡 (3.26) meromorphic function in Ω whose poles are contained in some compact set 퐾 ⊂ Ω. For some aperture parameter 휅 > 0, as- for each 휁 ∈ 휕∗Ω = 휕퐵(0, 1). sume that Consequently, Fubini’s Theorem implies that for each Ω⧵퐾 풩 (푓| ) ∈ 퐿1(휕Ω, 휎) and point 푧 ∈ Ω we have 휅 Ω⧵퐾 휅−n.t. (3.28) n.t. 푓| exists (in ℂ) at 휎-a.e. point on 휕 Ω. | 휕Ω nta (푢|휕Ω)(휁) ∫ 푑휁 휅 ′−n.t. 휁 − 푧 ′ 휕∗Ω | Then for any 휅 > 0 the nontangential trace 푓|휕Ω exists ′ 1/2 at 휎-a.e. point on 휕 Ω and is actually independent of 휅 . 1 푓(푡) 푑휁 nta = ∫ ( ∫ 푑푡) When regarding the latter function as being defined 휎-a.e. on 2휋푖 휁 − 푡 휁 − 푧 1 휕퐵(0,1) −1/2 휕∗Ω, this belongs to 퐿 (휕∗Ω, 휎) and, with the dependence on the parameter 휅 ′ dropped, 1/2 1 푑휁 = ∫ 푓(푡)( ∫ ) 푑푡 n.t. 2휋푖 (휁 − 푡)(휁 − 푧) | −1/2 휕퐵(0,1) 2휋푖 ⋅ ∑ Res(푓, 푧) = ∫ (푓|휕Ω)(휁) 푑휁 (3.29) 푧 pole of 푓 휕∗Ω = 0, in the case when either Ω is bounded, or 휕Ω is unbounded. since by the Residue Theorem the last integral on the unit Finally, when Ω is an exterior domain, formula (3.29) contin- circle vanishes. As such, if the integral representation for- ues to hold under the additional assumption that there exists 2 mula (3.8) were to hold, it would presently imply that some 휆 ∈ (1, ∞) such that ∫퐵(0,휆 푅)⧵퐵(0,푅) |푓| 푑ℒ = 표(푅) as 푢(푧) = 0 for each 푧 ∈ Ω. However, this is not the case. 푅 → ∞.

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1303 Proof. Let {푧푗}1≤푗≤푁 ⊆ 퐾 be the poles of the meromorphic then from (3.28) and (3.32) we conclude that function 푓. More specifically, assume that there exists a 휅−n.t. 퐹⃗| exists (in ℂ2) 휎-a.e. on 휕 Ω, family {풪푗}1≤푗≤푁 of mutually disjoint open subsets of Ω 휕Ω nta 푗 ∈ {1, … , 푁} 푧 ∈ 2 with the property that for each we have 푗 ⃗| 1 2 퐹|Ω⧵퐾 ∈ [퐿 loc(Ω ⧵ 퐾, ℒ )] , and 풪푗 and we may find a holomorphic function 푔푗 ∶ 풪푗 → ℂ with 푔 (푧 ) ≠ 0 along with an integer 푚 ∈ ℕ such that Ω⧵퐾 1 푗 푗 푗 풩휅 (퐹⃗| ) ∈ 퐿 (휕Ω, 휎). 푚푗 Ω⧵퐾 푓(푧) = 푔푗(푧)/(푧 − 푧푗) for each point 푧 ∈ 풪푗 ⧵ {푧푗}. Then for each 푗 ∈ {1, … , 푁} it follows that 푧푗 is a pole of order 푚푗 In addition, for 푓 and ′ div 퐹⃗ = 휕푥푢 + 푖휕푦푢 = 2휕푢 in 풟 (Ω); (3.35) 1 휕 푚푗 −1 푚푗 ⃗ ′ Res(푓, 푧푗) = lim ( ) [(푧 − 푧푗) 푓(푧)] hence div 퐹 ∈ ℰ (Ω). Moreover, in the case when Ω is (푚 − 1)! 푧→푧 휕푧 푗 푗 an exterior domain, the decay of 푓 implies that 퐹⃗ satisfies 1 (푚 −1) (2.19). Finally, at 휎-a.e. point on 휕∗Ω we have = 푔 푗 (푧 ). (3.30) (푚 − 1)! 푗 푗 휅−n.t. 휅−n.t. 휅−n.t. 푗 ⃗| | | ⟨휈 , 퐹|휕Ω ⟩ = 휈1(푓|휕Ω ) + 푖휈2(푓|휕Ω ) ∞ Next, extend 푓 ∈ 풞 (Ω ⧵ {푧1, … , 푧푁 }) to a distribution 푢 in 휅−n.t. Ω 푢 풪 | by taking to be in each 푗 the distribution = 휈(푓|휕Ω ). (3.36) 푚 −1 (−1) 푗 휕 푚푗 −1 1 At this stage, (3.29) follows from Theorem 2.6, bearing in 푔푗( ) [ ], (3.31) mind (3.5), (3.35), (3.34), and (3.36). □ (푚푗 − 1)! 휕푧 푧 − 푧푗 where the expression in brackets is regarded as a locally This version of the Residue Theorem is actually more ef- integrable function and the subsequent iterated derivatives ficient than the standard technology (based on choosing in 푧 are taken in the sense of distributions. Hence a suitable contour of integration, evaluating various integrals, and passing to the limit) even in such mundane sce- 푢 ∈ 풟 ′(Ω) and 푢| = 푓| . (3.32) narios as the task of showing that Ω⧵{푧1,…,푧푁 } Ω⧵{푧1,…,푧푁 } +∞ 푖푥 Let us also consider the distribution 푤 ∈ ℰ′(Ω) given by 푒 휋 ∫ 2 푑푥 = . (3.37) −∞ 푥 + 1 푒 푚 −1 푁 푗 푘 푘 휋(−1) 푚 − 1 (푚푗 −1−푘) 휕 2 ∑ ∑ ( 푗 ) 푔 (푧 )( ) 훿 . Specifically, choosing Ω ∶= ℝ+ and considering the mero- 푘 푗 푗 휕푧 푧푗 푗=1 푘=0 (푚푗 − 1)! morphic function 푒푖푧 Since 휕[1/(푧 − 푧 )] = 휋훿 , one may easily check that 휕푢 푓(푧) ∶= for 푧 ∈ Ω, (3.38) 푗 푧푗 푧2 + 1 coincides with 푤 in each 풪 with 1 ≤ 푗 ≤ 푁 (see, e.g., 푗 푓 has a simple pole at 푧 = 푖, with residue 푒−1, the function [16, Theorem 7.43, p. 289] and [16, Exercise 2.45, p. 34]). 푓 may be extended continuously to a neighborhood of 휕Ω, Given that we also know that both 휕푢 and 푤 vanish in the 푑푥 and ∫ |푓| 푑휎 = ∫ < +∞. set Ω ⧵ {푧1, … , 푧푁 }, we ultimately conclude that 휕Ω ℝ 푥2+1 Incidentally, it is not much harder to see that the first 푤 = 휕푢 in 풟′(Ω). (3.33) condition in (3.28) is true in this case. Indeed, if we set 퐾 ∶= 퐵(푖, 1/2), then for each 휅 > 0 fixed we deduce (keep- As seen from (3.30) and the definition of 푤, the pairing ing in mind that |푒푖푧| ≤ 1 for each 푧 ∈ Ω) that ℰ′(Ω)⟨푤, 1⟩ℰ(Ω) equals 1 Ω⧵퐾 | 푁 (풩휅 (푓|Ω⧵퐾 ))(푥) ≈ 2 , 휋 (푚 −1) 푥 + 1 (3.39) ∑ 푔 푗 (푧 ) = 휋 ⋅ ∑ Res(푓, 푧). (푚 − 1)! 푗 푗 uniformly for 푥 ∈ ℝ ≡ 휕Ω, 푗=1 푗 푧 pole of 푓 from which the desired conclusion follows. Thus, (3.29) In concert with (3.33), this implies that holds and this gives (3.37).

∗ ∞ 풞 ∞(Ω) ( 휕푢 , 1)풞 (Ω) = ℰ′(Ω)⟨ 휕푢 , 1⟩ ( 푏 ) 푏 ℰ(Ω) ACKNOWLEDGMENTS. The authors are grateful to the referee for a careful reading. The authors were = 휋 ⋅ ∑ Res(푓, 푧). (3.34) supported in part by the Simons Foundation grants 푧 pole of 푓 426669, 616050, and 637481; a Simons Fellowship at If we now consider the vector field the Isaac Newton Institute; and the NSF DMS grant 2 1900938. 퐹⃗ ∶= (푢 , 푖푢) ∈ [풟 ′(Ω)] ,

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