Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Let G(D) be a linear partial differential operator on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}, with constant coefficients. Moreover let Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^n$$\end{document} be open and F∈Lloc1(Ω,CN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in L^1_{\text {loc}} (\Omega , {\mathbb {C}}^N)$$\end{document}. Then any set of the form Af,F:={x∈Ω|(G(D)f)(x)=F(x)},withf∈Wlocg,1(Ω,Ck)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{f,F}:= \{ x\in \Omega \, \vert \, (G(D)f)(x)=F(x)\}, \text { with }f\in W^{g,1}_{\text {loc}}(\Omega , {\mathbb {C}}^k) \end{aligned}$$\end{document}is said to be a G-primitivity domain of F. We provide some results about the structure of G-primitivity domains of F at the points of the (suitably defined) G-nonintegrability set of F. A Lusin type theorem for G(D) is also provided. Finally, we give applications to the Maxwell type system and to the multivariate Cauchy-Riemann system.


Introduction
Let: • G = [G jl ] be a matrix of polynomials in C[ξ 1 , . . . , ξ n ] of dimension N ×k, with deg G := max j,l (deg G jl ) = g ≥ 1; • (x 1 , . . . , x n ) be the standard coordinates of R n and G(D) denote the system [G jl (D)], where G jl (D) is the linear partial differential operator with constant coefficients obtained by replacing each ξ q in G jl (ξ 1 , . . . , ξ n ) with −i∂/∂x q ; • Ω ⊂ R n be open and F ∈ L 1 loc (Ω, C N ); • m be a positive integer and Σ m denote the family of all matrices S of polynomials in C[ξ 1 , . . . , ξ n ], with N columns, satisfying deg S ≤ m and SG = 0.
Then any set of the form 39 Page 2 of 29 S. Delladio MJOM is said to be a G-primitivity domain of F and the following simple fact holds: If F ∈ W m,1 loc (Ω, C N ) and there is an open ball B ⊂ Ω such that almost all of B is covered by a G-primitivity domain A f,F (i.e., L n (B\A f,F ) = 0), with f ∈ W g+m,1 loc (Ω, C k ), then one has S(D)F = 0 a.e. in B for all S ∈ Σ m . This property, which can be readily extended to the case of f ∈ W g,1 loc (Ω, C k ) (cf. Proposition 3.1), has naturally led us to expect that the structure of the G-primitivity domains of F may be somewhat singular at the points of (1) If F ∈ W m,p (R n , C N ) and f ∈ W g+m,p (R n , C k ), with p ∈ (1, +∞), then one has L n (A f,F ∩ Υ m F ) = 0; (2) If F ∈ W m+1,p (R n , C N ) and f ∈ W g+m+1,p (R n , C k ), with p ∈ (1, n), then the set A f,F ∩ Υ m F is (n − 1)-rectifiable (cf. [13,17]), so that its Hausdorff dimension is less or equal to n − 1.
Things can obviously improve if we consider a wider class of functions f . For example, if F ∈ W m,1 loc (Ω, C N ) then it may very well happen to come across f ∈ W g,p (R n , C k ) such that L n (A f,F ∩ Υ m F ) > 0 (cf. (4) below). However, even in this case, the structure of A f,F at points of Υ m F is significantly affected by the G-nonintegrability properties. In particular, the following fact holds (cf. Corollary 3.5): (3) Let F ∈ W m,p loc (Ω, C N ) and f ∈ W g,p loc (Ω, C k ), with p ∈ (1, +∞). Then, at a.e. point of Υ m F , the set A f,F has density lower than n + pm/(p − 1). In Sect. 4 we provide a Lusin type result which extends [2, Theorem 1] to a certain class of linear partial differential operators with constant coefficients (cf. Theorem 4.1). The assumptions that define this class are quite stringent. In particular, it is required that k = 1 and that the components of G be different from each other. Moreover the following cohercivity condition is required: there exist a nonnegative integer l ≤ g and a positive real number c * such that G(D)ϕ ∞,Ω ≥ c * max MJOM Some Results About the Structure...

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Thus, under suitable conditions, there are G-primitivity domains of F arbitrarily close in measure to Ω, even if F ∈ W m,1 loc (Ω, C N ) and L n (Υ m F ) > 0 (even if Υ m F = Ω, which is the least favorable case for the "G-integrability of F "!).

General Notation
is the open ball in R n with center x and radius r. The open cube of side 2r centered at x in R n , that is (−r, r) n + x, is denoted by Q r (x). For z = (z 1 , . . . , z N ) ∈ C N , we set |z| := (|z 1 | 2 + · · · + |z N | 2 ) 1/2 . The Lebesgue outer measure and the s-dimensional Hausdorff outer measure in R n will be denoted by L n and H s , respectively. If E ⊂ R n is a Lebesgue measurable set and u j , v j : E → R (j = 1, . . . , N) are Lebesgue measurable functions, we say that (u 1 + iv 1 , . . . , u N + iv N ) : E → C N is Lebesgue measurable. If f : E → C N is a Lebesgue measurable function and p ∈ [1, +∞), then we define while f ∞,E is defined as the infimum (which is actually a minimum) of the numbers M ∈ [0, +∞] satisfying If Ω ⊂ R n is open and u, v : Ω → R are Lebesgue integrable (resp. psummable, locally p-summable; p ∈ [1, +∞)) on Ω, then we say that u + iv is Lebesgue integrable (resp. p-summable, locally p-summable) on Ω and define (omitting for simplicity to specify explicitly the measure, which is obviously the Lebesgue measure L n ) The space of p-summable functions on Ω and the space locally p-summable functions on Ω will be denoted by L p (Ω, C) and L p loc (Ω, C), respectively. If f 1 , . . . , f k : Ω → C are Lebesgue integrable (resp. p-summable, locally psummable) on Ω, then we say that f = (f 1 , . . . , f k ) t is Lebesgue integrable (resp. p-summable, locally p-summable) on Ω and define We also set The coordinates of R n are denoted by (x 1 , . . . , x n ) and we set for simplicity ∂ j := ∂/∂x j . For α = (α 1 , . . . , α n ) ∈ N n , define Similarly, if (ξ 1 , . . . , ξ n ) ∈ R n then we write The closure of C ∞ c (Ω, C k ) in (C m (Ω, C k ); · C m (Ω,C k ) ) will be denoted by C m 0 (Ω, C k ). For simplicity, we will write C(Ω, C k ), C c (Ω, C k ) and C 0 (Ω, C k ) in place of C 0 (Ω, C k ), C 0 c (Ω, C k ) and C 0 0 (Ω, C k ), respectively. If m is a positive integer and p ∈ [1, +∞), then we set loc (Ω, C) and α ∈ N n with |α| ≤ m, then ∂ α f will denote the precise representative of the α th weak derivative of f (cf. [12,18]). In particular, denotes the Bessel capacity of E (cf. Section 2.6 in [21]). Recall that B 0,p = L n .

Linear Partial Differential Operators Let
If c α = 0 for some α ∈ N n with |α| = d, then the number d is said to be the total degree of P and is denoted by deg P . As usual (cf. [5,15]), P (D) is the differential operator obtained by replacing each variable ξ j with −i∂ j , namely Observe that if P, Q ∈ C[ξ 1 , . . . , ξ n ] then these identities holds: Then (P (D)ψ)ϕ and (P * (D)ϕ)ψ are obviously Lebesgue summable on Ω and a trivial computation shows that For all (j, l) ∈ {1, . . . , N} × {1, . . . , k} the polynomial P jl can be written as follows where c where C α is the matrix of dimension N × k whose entries are the numbers c

Distributions
Let Ω be an open subset of R n . We recall that a linear functional T : C ∞ c (Ω, C) → C is said to be a distribution on Ω if one has lim j→∞ T (ϕ j ) = T (ϕ) for every sequence {ϕ j } ∞ j=1 ⊂ C ∞ c (Ω, C) and ϕ ∈ C ∞ c (Ω, C) such that (i) There exists a compact set K ⊂ Ω such that supp ϕ j ⊂ K, for all j; (ii) One has lim j→∞ ∂ α ϕ j − ∂ α ϕ ∞,Ω = 0, for all α ∈ N n . If conditions (i) and (ii) are satisfied we say that the sequence {ϕ j } ∞ j=1 converges to ϕ in C ∞ c (Ω, C). The set of all distributions on Ω, denoted by D (Ω), is obviously a vector space with addition and scalar multiplication defined by We recall that, if P ∈ C[ξ 1 , . . . , ξ n ], T ∈ D (Ω) and set Hence, in the special case when u ∈ C m (Ω, C) with m = deg P , recalling (2.4), we find the following regularity identity We shall use the weak topology in D (Ω), according to which The map loc (Ω, C) and

Superdensity
The set of all m-density points of E is denoted by E (m) .
where equality holds if E is L n -measurable. Thus: A remarkable family of superdense sets is the class of finite perimeter sets. Indeed Theorem 1 in [12, Section 6.1.1] states that almost every point in a set E ⊂ R n (with n ≥ 2) of finite perimeter is a m 0 -density point of E, with The number m 0 is also the maximum order of density common to all sets of finite perimeter. More precisely one has this result, cf.

A class of cut-off functions
Consider r > 0, ρ ∈ (0, 1) and a function ψ ∈ C ∞ (R) such that then one obviously has Moreover, a standard computation yields for all α ∈ N n and x = (x 1 , . . . , x n ) ∈ R n , hence for all α ∈ N n , where C(α) is a number depending only on α (and n).

Moreover, for any couple of integers
We shall refer to Υ m F as the " G-nonintegrability set of F ".
Indeed, let S ∈ Σ 0 and observe that it must coincide with a matrix M whose entries are all in C. Then, for all x ∈ A f,F , one has Then S → ||S|| is a norm in M m,h . Obviously Σ m,h is a closed vector subspace of (M m,h , || · ||), normed by the restriction of || · || to Σ m,h . In particular it is separable, i.e., it has a countable subset Σ * m,h which is dense with respect to the norm topology. Observe that for all F ∈ W m,1 From (2.9) and (3.2) we get at once the following property, that The next result extends such a property and will be further generalized in Corollary 3.2 and Corollary 3.5 below (cf. Remark 3.5).
in Ω, i.e., L n (Ω\A f,F ) = 0. In this special case, Proposition 3.1 leads to L n (Υ m F ) = 0, i.e., the obvious compatibility condition S(D)F = 0 a.e. in Ω, for all S ∈ Σ m . Remark 3.4. In general, the problem of determining S such that SG = 0 is not easy and for an account about its resolution we refer the reader to algebraic analysis literature, e.g., [5] (and the references therein), where it is addressed also through the use of specific software. In this regard it must be remembered that a particularly significant case is when S is the matrix The Case of f ∈ W g +m +d,p In paper [10] we have proved the following result. Then the set A * f,F is covered by a finite family of (n−1)-dimensional regularly imbedded C 1 submanifolds of R n .
Then, also recalling (3.1), we obtain and the conclusion follows from Theorem 3.1.
From Corollary 3.1 we get, quite easily, this result in the context of Sobolev functions.
The following facts hold: [13,17]), so that its Hausdorff dimension is less or equal to n − 1.
Proof. Let d ∈ {0, 1} and p ∈ (1, +∞) be such that pd < n. Then, recalling a well known Lusin-type approximation result for Sobolev functions (cf. Theorem 3.10.5 in [21]), we can find Now consider an arbitrary S ∈ Σ * m , define for simplicity Then
The following facts hold: Observe that deg S = g. Moreover, by assumption (3.5), one has (1) and (2) follow at once from Corollary 3.2.

Structure of A f,F at Points of the G-nonintegrability Set of F ∈ W m ,p :
The Case of f ∈ W g,p In Sect. 3.1 we have proved that if F ∈ W m,p (R n , C N ) then every Gprimitivity domain A f,F with f ∈ W g+m,p (R n , C k ) intersects the Gnonintegrability set of F in a set of Lebesgue measure zero. Things change if one considers f ∈ W g,p (R n , C k ). In fact, as we will see, it can happen to come across functions f ∈ W g,p (R n , C k ) such that L n (A f,F ∩Υ m F ) > 0 (cf. Theorem 4.1 and Corollary 4.1 in the next section). However, as Corollary 3.4 below shows, even in this case the G-nonintegrability properties strongly shape the structure of A f,F at points of Υ m F . More precisely: if S ∈ Σ m then, at a.e. point of Υ F,S , the set A f,F has density lower than n+p deg S/(p−1). Consequently, at a.e. point of Υ m F , the set A f,F has density lower than n + pm/(p − 1), cf. Corollary 3.5.
Then there exists a null measure set Z ⊂ Ω such that for all S ∈ Σ m , where δ S := p deg S/(p − 1).

in E. Then there is no function
(3.14) On the other hand, Corollary 3.4 yields Moreover, since p ≥ n, one has n + p/(p − 1) ≤ n + n/(n − 1) so that  Proof. Let S = [S jl ] be the matrix considered in the proof of Corollary 3.3. Since S ∈ Σ g , the conclusion follows at once from Corollary 3.4.

A Lusin Type Result for a Class of Linear Partial Differential Operators
The proofs of Lemma 4.1 and Theorem 4.1 below go along the lines of those of Lemma 4.1 and Theorem 4.1 in [10], respectively. Several steps are actually the same, but the intertwining of these replicas with the new arguments, as well as the complexity of the proof, make it (in our opinion) impossible to cut the presentation without compromising clarity. For this reason we have decided to provide them in full.  D), . . . , G N (D)) t . Assume that there exist α (1) , . . . , α (N ) ∈ N n such that Moreover consider an open set Ω ⊂ R n such that L n (Ω) < +∞, a bounded Proof. According to the first steps in the proof of [2, Lemma 7], we can find δ ∈ (0, 1) and a compact set K ⊂ Ω with the following properties: • The estimate (1) holds and where {Q j } j∈J is a finite family of closed cubes of side (1 − ε/2n)δ, whose centers y j belong to the lattice (δZ) n ; • For j ∈ J, let T j be the closed cube of side δ centered at y j . Then, for all j ∈ J, one has T j ⊂ Ω and Now, for all j ∈ J and x ∈ R n , set and observe that by (2.11). Moreover One obviously has v ∈ C ∞ c (Ω, C), by (4.3). To prove (2) and (3), we need the explicit expressions of the polynomials G r , that is where the coefficients c (r) α are assumed to be zero when |α| exceeds the degree of G r . Recalling (2.1), we find (for x ∈ Ω) where, for suitable integer coefficients k (α) β (which coincide with 1 for β = 0 and β = α), one has It follows that (for x ∈ Ω) (4.5) In the formulae below, C 1 , C 2 , . . . will denote constants which do not depend on f, ε, η, p. From the previous identity, we obtain (for all j ∈ J and x ∈ Ω) for all x ∈ Ω. Since δ ∈ (0, 1), ε ∈ (0, 1/2) and (4.1) holds, it follows that for all x ∈ Ω. Thus Moreover, by Jensen's inequality, one has Analogously (recalling that 1−nt ≤ (1−t) n , for all t ≤ 1,  Consider an open set Ω ⊂ R n with finite measure, F ∈ L ∞ (Ω, C N ) such that F ∞,Ω > 0 and recall that lim q→+∞ F q,Ω = F ∞,Ω (cf. Theorem 2.8 in [1]). Then, as it is outlined in [2], an easy argument shows that the function q → L n (Ω) 1/q / F q,Ω has a finite positive upper bound on [1, +∞).
Step 1: Let us define f 0 := F and show that there exist two sequences of functions of compact subsets of Ω satisfying the following properties, for all j ≥ 1: Recall that 0 < s < +∞, by Remark 4.2; (iii) G(D)v j p,Ω ≤ 2 j(g−1/p) Cε 1/p−g f j−1 p,Ω , for all p ∈ [1, +∞), where C is the constant of (3) in Lemma 4.1; Kj . Such a statement is proved by the following induction argument: for all p ∈ [1, +∞).
Step 2: If F is a Borel function.
Let ε > 0 be fixed arbitrarily. Then, proceeding as in the proof of Theorem 1 in [2], we can find Letting p tend to +∞ in (4.19), we also find

Examples of Application
In this section we apply the theory developed above to three contexts already considered in [10,Section 5], where we dealt with the case of smooth functions. Some basic facts established in [10], including presentations of contexts, will also be useful here and will therefore be recalled for the convenience of the reader.

Alberti's Theorem
Given a positive integer k, let T k denote the set of n-tuples α ∈ N n such that |α| = k and set N k := #T k . Moreover let j → α (j) be an arbitrarily chosen bijection from {1, . . . , N k } to T k . Then, by the same arguments as in Section 5 of [10] with Theorem 4.1 in place of [10, Theorem 4.1], we obtain the following well known result (cf. [2,14,16]).

Maxwell Type System
Let us recall that the electromagnetic field is characterized by the system where E, B, ρ and j are the electric field, the magnetic field, the electric charge density and the electric current density, respectively. The symbol of this system is the following matrix of polynomials in C[ξ 1 , ξ 2 , ξ 3 , ξ 4 ] where ξ 1 , ξ 2 , ξ 3 are the symbols of the spatial differential operators −i∂ x1 , −i∂ x2 , −i∂ x3 , while ξ 4 is the symbol of the time differential operator −i∂ x4 (for consistency with the notation introduced in the previous sections, we denote the time variable with x 4 ). In this case, a remarkable example of matrix in Σ 1 is the one associated to the first syzygies (cf. (3) In the special case when F ∈ W l,p loc (Ω, C 8 ) with l ≥ 1 and p ∈ (1, +∞), one has Hence, in particular, Moreover: Statements (1) and (2) 1, . . . , N).
Then G = (G 1 , . . . , G N ) t is the symbol of the Cauchy-Riemann system in N complex variables z j = x 2j−1 + ix 2j (j = 1, . . . , N), namely Observe that deg G = 1. Analogously as we have done for the Maxwell type system, we can consider the matrix associated to the first syzygies, namely the one of dimension N (N −1) 2 × N used in the proof of Corollary 3.3. Also in this case we denote such a matrix by S and observe that deg S = 1.  -If d = 0 then L 2N (A g,F ∩ Υ l F ) = 0, hence L 2N (A g,F ∩ Υ F,S ) = 0; -If d = 1 and p < 2N then A g,F ∩ Υ l F is (2N − 1)-rectifiable, hence A g,F ∩ Υ F,S is (2N − 1)-rectifiable.
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