Elliptic operators and their symbols

https://doi.org/10.1515/dema-2019-0025 Received January 19, 2019; accepted July 19, 2019 Abstract: We consider special elliptic operators in functional spaces on manifolds with a boundary which has some singular points. Such an operator can be represented by a sum of operators, and for a Fredholm property of an initial operator one needs a Fredholm property for each operator from this sum.


Introduction
This paper is devoted to describing the structure of a special class of linear bounded operators on a manifold with non-smooth boundary. Our description is based on Simonenko's theory of envelopes [1] and explains why we obtain distinct theories for pseudo-differential equations and boundary value problems and distinct index theorems for such operators.

Operators of a local type
In this section we will give some preliminary ideas and definitions from [1]. Let

Simple examples
These are two of the simplest examples for illustration.

Example 1. If A is a differential operator of the type
(Au)(x) = n ∑︁ |k|=0 a k (x)D k u(x), D k u = ∂ k u ∂x k1 1 · · · ∂x km m , then A is an operator of local type. Everywhere below we say "an operator" instead of "an operator of local type".

Functional spaces on a manifold
It is possible to work with distinct functional spaces [2,3].

Definition 2. [4]
The space H s (R m ), s ∈ R, is a Hilbert space of functions with the finite norm where the sign ∼ over a function means its Fourier transform.

Definition 4. [2]
The space C α (R m ), 0 < α ≤ 1, is a space of continuous functions u on R m satisfying the Hölder condition where infimum is taken over all constants c from the above inequality.

Partition of unity and spaces H s (M), Lp(M), C α (M)
If M is a compact manifold then there is a partition of unity [5]. It means the following. For every finite open covering {U j } k j=1 of the manifold M there exists a system of functions So we have for an arbitrary function f defined on M.
Since every set U j is diffeomorphic to an open set D j ⊂ R m we have corresponding diffeomorphisms ω j : U j → D j . Further, for a function f defined on M we compose mappings f j = f · φ j and as long as supp f j ⊂ U j we putf j = f j ∘ ω −1 j so thatf j : D j → R is a function defined on a domain of m-dimensional space R m . We can consider, for example, the following functional spaces [2][3][4].

Definition 5. A function f ∈ H s (M) if the following norm
is finite.

Operators on a compact manifold
On the manifold M we fix a finite open covering and a partition of unity corresponding to this covering {U j , f j } n j=1 . We then choose smooth functions {g j } n j=1 so that supp g j ⊂ V j , U j ⊂ V j , and g j (x) ≡ 1 for x ∈ supp f j , supp f j ∩ (1 − g j ) = ∅.

Proposition 1. The operator A on the manifold M can be represented in the form
Proof. The proof is straightforward. Since and the proof is completed.

Remark 1. Obviously such an operator is defined uniquely up to a compact operators which have no influence on an index.
By definition, for an arbitrary operator A : where infimum is taken over all compact operators T : Let B ′ 1 , B ′ 2 be Banach spaces consisting of functions defined on R m , and let̃︀ A : Since M is a compact manifold, then for every point x ∈ M there exists a neighborhood U ∋ x and a diffeomorphism ω : U → D ⊂ R m , ω(x) ≡ y. We denote by Sω the following operator acting from B k to B ′ k , k = 1, 2. For every function u ∈ B k vanishing out of U Definition 6. A local representative of the operator A :

Definition 7. Symbol of an operator A is called the family of its local representatives {Ax} at each point x ∈ M.
One can show like [1] this definition of an operator symbol conserves all properties of a symbolic calculus. Namely, up to compact summands we have the following: -the product and the sum of two operators corresponds to the product and the sum of their local representatives; -the adjoint operator corresponds to its adjoint local representative; -a Fredholm property of an operator corresponds to a Fredholm property of its local representative.

Operators with symbols. Examples of operators
It seems not every operator has a symbol, and we give some examples for operators with symbols. where ξ k = ξ k1 1 · · · ξ km m .
Example 4. Let A be the Calderon-Zygmund operator from Example 2 and σ(x, ξ ) be its symbol in the sense of [2], then its symbol is an operator family consisting of multiplication operators on the function σ(x, ξ ). The more important point is that the symbol of an operator is simpler than general operator, and it permits to verify its Fredholm properties. For the two above examples a Fredholm property of an operator symbol is equivalent to its invertibility.

Sub-manifolds
The above definition of an operator on a manifold supposes that all neighborhoods {U j } have the same type. But even if a manifold has a smooth boundary then there are two types of neighborhoods related to a placement of neighborhood, namely inner neighborhoods and boundary ones. For an inner neighborhood U such that U ⊂M we have the diffeomorphism ω : U → D, where D ∈ R m is an open set. For a boundary neighborhood such that U ∩ ∂M ≠ ∅ we have another diffeomorphism ω 1 : Maybe this boundary ∂M has some singularities like conical points and wedges. The conical point at the boundary is such a point, for which its neighborhood is diffeomorphic to the cone The wedge point of codimension k, 1 ≤ k ≤ m−1, is such a point for which its neighborhood is diffeomorphic to . So if the manifold M has such singularities we suppose that we can extract certain k-dimensional submanifolds, namely an (m − 1)-dimensional boundary ∂M, and k-dimensional wedges M k , k = 0, · · · , m − 2; M 0 are a collection of conical points.

Enveloping operators
If the family {Ax} x∈M is continuous in the operator topology, then according to Simonenko's theory there is an enveloping operator, i.e. such an operator A for which every operator Ax is the local representative for the operator A in the point x ∈ M. [2] with symbols σx(ξ ) parametrized by points x ∈ M and this family smoothly depends on x ∈ M then the Calderon-Zygmubd operator with variable kernel and symbol σ(x, ξ ) will be an enveloping operator for this family.

Example 6.
If {Ax} x∈M consists of null operators then an enveloping operator is a compact operator [1].

Theorem 1. The operator A has a Fredholm property if and only if its all local representatives {Ax} x∈M have the same property.
This property was proved in [1], but we will give the proof (see Lemma 2) including some new constructions because it will be used below for a decomposition of the operator.

Hierarchy of operators
We will remind the reader here of the following definition and Fredholm criteria for operators [6].

Lemma 1. Let f be a smooth function on the manifold M, U ⊂ M be an open set, and supp f ⊂ U. Then the operator f ·
where T 1 , T 2 are compact operators. Let us denote g · A · g ≡ h and write and we obtain the required property.

Definition 9. The operator A is called an elliptic operator if its operator symbol {Ax} x∈M consists of Fredholm operators.
Now we will show that each elliptic operator really has a Fredholm property. Our proof in general follows the book [1], but our constructions are more stratified and we need such constructions below.

Lemma 2. Let A be an elliptic operator. Then the operator A has a Fredholm property.
Proof. To obtain the proof we will construct the regularizer for the operator A. For this purpose we choose two coverings like Proposition 1 and write the operator A in the form where T is a compact operator. Without loss of generality we can assume that there are n points x k ∈ U k ⊂ V k , k = 1, 2, ...n. Moreover, we can construct such coverings by balls in the following way. Let ε > 0 be a small enough number. First, for every point x ∈ M 0 we take two balls Ux , Vx with the center at x of radius ε and 2ε and construct two open coverings for M 0 namely U 0 = ∪ x∈M0 Ux and V 0 = ∪ x∈M0 Vx . Second, we consider the set L 1 = M\V 0 and construct two coverings U 1 = ∪ x∈L1∩M1 Ux and V 1 = ∪ x∈L1∩M1 Vx. Further, we introduce the set L 2 = M \ (V 0 ∪ V 1 ) and two coverings U 2 = ∪ x∈L2∩M2 Ux and V 2 = ∪ x∈L2∩M2 Vx. Continuing these actions we will come to the set L m−1 = M \ (∪ m−2 k=0 U k ) which consists of smoothness points of ∂M and inner points of M. We then construct two covering U m−1 = ∪ x∈Lm−1∩∂M Ux and V m−1 = ∪ x∈Lm−1∩∂M Vx. Finally, the set Lm = M \ (∪ m−1 k=0 U k ) consists of inner points of the manifold M only. We finish this process by choosing the covering Um for the latter set Lm. So, the covering ∪ m k=0 U k will be a covering for the whole manifold M. According to the compactness property we can take into account that this covering is finite, and the centers of balls which cover M k are placed at M k . Now we will rewrite the formula (1) in the following way where the coverings and partitions of unity {f jk } and {g jk } are chosen as mentioned above. In other words the operator n k ∑︁ j=1 f jk · A · g jk is related to some neighborhood of the sub-manifold M k ; this neighborhood is generated by covering the sub-manifold M k by balls with centers at points x jk ∈ M k . Since Ax jk is a local representative for the operator A at point x jk we can rewrite the formula (2) as follows Let us denote S ω −1 jĝ j ≡g j andf j Sω j ≡f j . Further, we can assert that the operator will be the regularizer for the operator A ′ ; here A −1 x jk is a regularizer for the operator Ax jk . Indeed, because f jk · Ax jk = Ax jk · f jk + compact summand, and f jk · g jk = f jk , and as the partition of unity. The same property is verified analogously.

Piece-wise continuous operator families
Given an operator A with the symbol {Ax} x∈M which generates a few operators in dependence on a quantity of singular manifolds; we consider this situation in the following way. We will assume additionally some smoothness properties for the symbol {Ax} x∈M . Proof. We will use the constructions from the proof of Lemma 2, namely the formula (3). We will extract the operator n k ∑︁ j=1 f jk · Ax jk · g jk which "serves" the sub-manifold M k and consider it in detail. This operator is related to neighborhoods {U jk } and the partition of unity {f jk }. Really, U jk is the ball with the center at x jk ∈ M k of radius ε > 0, and therefore f jk , g jk , n k depend on ε. According to Simonenko's ideas [1] we will construct the component A (k) in the following way. Let {εn} ∞ n=1 be a sequence such that εn > 0 for all n ∈ N, lim n→∞ εn = 0. Given εn we choose coverings {U jk } n k j=1 and {V jk } n k j=1 as above with partition of unity {f jk } and corresponding functions {g jk } such that |||f jk · (Ax − Ax jk ) · g jk ||| < εn , ∀x ∈ V jk ; we remind that U jk , V jk are balls with centers at x jk ∈ M k of radius ε and 2ε. This requirement is possible according to continuity of family {Ax} on the sub-manifold M k . Now we will introduce such a constructed operator f jk · Ax jk · g jk and will show that the sequence {An} is a Cauchy sequence with respect to a norm ||| · |||. We have where the operator A l is constructed for a given ε l with corresponding coverings {u ik } l k i=1 and {v ik } i k j=1 with partition of unity {F ik } and corresponding functions {G ik } so that here u ik , v ik are balls with centers at y ik ∈ M k of radius τ and 2τ.
We can write F ik · f jk · Ax jk · g jk · G k + T 1 , and the same can be done for A l f jk · F ik · Ay ik · G ik · g jk + T 2 .

Corollary 1. The operator A has a Fredholm property if and only if all operators A (k)
, k = 0, 1, ..., m have the same property.

Remark 2.
The constructed operator A ′ generally speaking does not coincide with the initial operator A because they act in different spaces. But for some cases they may be the same.

Conclusion
This paper is a general concept of my vision to the theory of pseudo-differential equations and boundary value problems on manifolds with a non-smooth boundary. The second part will be devoted to applying these abstract results to index theory for such operator families and then to concrete classes of pseudo-differential equations.