Operator symbols

TL;DR

This paper investigates elliptic operators on manifolds with boundary and singular points, analyzing their Fredholm properties through a decomposition into simpler operators to understand their solvability.

Contribution

It introduces a method to analyze Fredholm properties of complex elliptic operators by decomposing them into sums of simpler operators on manifolds with boundary and singularities.

Findings

Fredholm property depends on each component operator in the sum

Decomposition aids in understanding solvability conditions

Applicable to elliptic operators on singular manifolds

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 an each operator from this sum.

Equations

$$(P_{W}u)(x)=\{\begin{array}{ll}u(x),& \text{if x∈W;}\\ 0,& \text{if x∈/W.}\end{array}$$

$$P_{U}A{P}_V$$

$$(Au)(x)=\sum _{|k|=0}^n{a}_k(x)D^{k}u(x),\text{leavevmode}\text{leavevmode}\text{leavevmode}{D}^{k}u=\frac{{\partial }^{k}u}{\partial x_1^{k_1}\cdots \partial x_m^{k_m}},$$

$$(Au)(x)=v.p.\underset{{\mathbb{R}}^m}{\int }K(x,x-y)u(y)dy.$$

$$||u|{|}_s={\left(\underset{{\mathbb{R}}^m}{\int }|\tilde{u}(\xi ){|}^2(1+|\xi |{)}^{2s}d\xi \right)}^{1/2},$$

$$||u|{|}_p={\left(\underset{{\mathbb{R}}^m}{\int }|u(x){|}^{p}dx\right)}^{1/p}.$$

$$|u(x)-u(y)|\le c|x-y{|}^{\alpha },\text{leavevmode}\text{leavevmode}\text{leavevmode}\forall x,y\in {\mathbb{R}}^m,$$

$$||u|{|}_{\alpha }=\inf \{c\},$$

$$f(x)=\sum _{j=1}^k{\phi }_j(x)f(x)$$

$$||f|{|}_{H^s(M)}=\sum _{j=1}^k||{\hat{f}}_j|{|}_s$$

$$||f|{|}_{L_p(M)}=\sum _{j=1}^k||{\hat{f}}_j|{|}_p$$

$$||f|{|}_{C^{\alpha }(M)}=\sum _{j=1}^k||{\hat{f}}_j|{|}_{\alpha }$$

$$A=\sum _{j=1}^n{f}_j\cdot A\cdot g_j+T,$$

$$\sum _{j=1}^n{f}_j(x)\equiv 1,\text{leavevmode}\text{leavevmode}\text{leavevmode}\forall x\in M,$$

$$A=\sum _{j=1}^n{f}_j\cdot A=\sum _{j=1}^n{f}_j\cdot A\cdot g_j+\sum _{j=1}^n{f}_j\cdot A\cdot (1-g_j),$$

$$|||A|||\equiv \inf ||A+T||,$$

$$(S_{\omega }u)(y)=u({\omega }^{-1}(y)),\text{leavevmode}\text{leavevmode}\text{leavevmode}y\in D,\text{leavevmode}\text{leavevmode}\text{leavevmode}(S_{\omega }u)(y)=0,\text{leavevmode}\text{leavevmode}\text{leavevmode}y\notin D.$$

$$|||g_{j}A{f}_j-S_{{\omega }_j^{-1}}{\hat{g}}_j\tilde{A}{\hat{f}}_j{S}_{{\omega }_j}|||<\epsilon .$$

$$\sum _{|k|=0}^n{a}_k(x){\xi }^k,$$

$$RA=I_1+T_1,\text{leavevmode}\text{leavevmode}\text{leavevmode}AR=I_2+T_2$$

$$f\cdot A=f\cdot A\cdot g+f\cdot A\cdot (1-g)=f\cdot A\cdot g+T_1,$$

$$A\cdot f=g\cdot A\cdot f+(1-g)\cdot A\cdot f=g\cdot A\cdot f+T_2,$$

$$f\cdot A\cdot g=f\cdot g\cdot A\cdot g=f\cdot h,\text{leavevmode}\text{leavevmode}\text{leavevmode}g\cdot A\cdot f=g\cdot A\cdot g\cdot f=h\cdot f,$$

$$A=\sum _{j=1}^n{f}_j\cdot A\cdot g_j+T,$$

$$A=\sum _{k=0}^m\left(\sum _{j=1}^{n_k}{f}_{jk}\cdot A\cdot g_{jk}\right)+T,$$

$$\sum _{j=1}^{n_k}{f}_{jk}\cdot A\cdot g_{jk}$$

$$A=\sum _{k=0}^m\left(\sum _{j=1}^{n_k}{f}_{jk}\cdot A_{x_{jk}}\cdot g_{jk}\right)+T.$$

$$R=\sum _{k=0}^m\left(\sum _{j=1}^{n_k}{g}_{jk}\cdot A_{x_{jk}}^{-1}{f}_{jk}\right),$$

$$RA=\left(\sum _{k=0}^m\left(\sum _{j=1}^{n_k}{g}_{jk}{A}_{x_{jk}}^{-1}{f}_{jk}\right)\right)\cdot A=$$

$$\sum _{k=0}^m\sum _{j=1}^{n_k}{g}_{jk}\cdot A_{x_{jk}}^{-1}\cdot (A-A_{x_{jk}}+A_{x_{jk}})\cdot f_{jk}+T_1=$$

Full text

1. Introduction

This paper is devoted to describing 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 [3] and explains why we obtain distinct theories for pseudo-differential equations and boundary value problems and distinct index theorems for such operators.

1.1. Operators of a local type

In this section we will give some preliminary ideas and definitions from [3].

Let $B_{1},B_{2}$ be Banach spaces consisting of functions defined on compact $m$-dimensional manifold $M$, $A:B_{1}\rightarrow B_{2}$ be a linear bounded operator, $W\subset M$, and $P_{W}$ be a projector on $W$ i.e.

[TABLE]

Definition 1. An operator $A$ is called an operator of a local type if the operator

[TABLE]

*is a compact operator for arbitrary non-intersecting compact sets $U,V\subset M$. *

1.2. Simple examples

These are two simplest examples for the illustration.

Example 1. If $A$ is a differential operator of the type

[TABLE]

*then $A$ is an operator of a local type. *

Example 2. If $A$ is a Calderon–Zygmund operator with variable kernel $K(x,y)\in C^{1}(\mathbb{R}^{m}\times(\mathbb{R}^{m}\setminus\{0\})$ of the following type

[TABLE]

Everywhere below we say “an operator” instead of “an operator of a local type”.

1.3. Functional spaces on a manifold

1.3.1. Spaces $H^{s}(\mathbb{R}^{m}),L_{p}(\mathbb{R}^{m}),C^{\alpha}(\mathbb{R}^{m})$

It is possible working with distinct functional spaces [1, 6].

Definition 2. [2] The space $H^{s}(\mathbb{R}^{m}),s\in\mathbb{R},$ is a Hilbert space of functions with the finite norm

[TABLE]

*where the sign $\sim$ over a function means its Fourier transform. *

Definition 3. [1] The space $L_{p}(\mathbb{R}^{m}),1<p<+\infty,$ is a Banach space of measurable functions with the finite norm

[TABLE]

Definition 4. [1] The space $C^{\alpha}(\mathbb{R}^{m}),0<\alpha\leq 1,$ is a space of continuous on $\mathbb{R}^{m}$ functions $u$ satisfying the Hölder condition

[TABLE]

with the finite norm

[TABLE]

*where infimum is taken over all constants $c$ from above inequality. *

1.3.2. Partition of unity and spaces $H^{s}(M),L_{p}(M),C^{\alpha}(M)$

If $M$ is a compact manifold then there is a partition of unity [4]. It means the following. For every finite open covering $\{U_{j}\}_{j=1}^{k}$ of the manifold $M$ there exists a system of functions $\{\varphi_{j}(x)\}_{j=1}^{k},\varphi_{j}(x)\in C^{\infty}(M),$ such that

• •

$0\leq\varphi_{j}(x)\leq 1$,

• •

$supp\leavevmode\nobreak\ \varphi_{j}\subset U_{j}$,

• •

$\sum\limits_{j=1}^{k}\varphi_{j}(x)=1$.

So we have

[TABLE]

for arbitrary function $f$ defined on $M$.

Since an every set $U_{j}$ is diffeomorphic to an open set $D_{j}\subset\mathbb{R}^{m}$ we have corresponding diffeomorphisms $\omega_{j}:U_{j}\rightarrow D_{j}$. Further for a function $f$ defined on $M$ we compose mappings $f_{j}=f\cdot\varphi_{j}$ and as far as $supp\leavevmode\nobreak\ f_{j}\subset U_{j}$ we put $\hat{f}_{j}=f_{j}\circ\omega_{j}^{-1}$ so that $\hat{f}_{j}:D_{j}\rightarrow\mathbb{R}$ is a function defined in a domain of $m$-dimensional space $\mathbb{R}^{m}$. WE can consider the following spaces [2, 1, 6].

Definition 5. A function $f\in H^{s}(M)$ if the following norm

[TABLE]

is finite.

A function $f\in L_{p}(M)$ if the following norm

[TABLE]

is finite.

A function $f\in C^{\alpha}(M)$ if the following norm

[TABLE]

*is finite. *

2. Operators on a compact manifold

On the manifold $M$ we fix a finite open covering and a partitions of unity corresponding to this covering $\{U_{j},f_{j}\}_{j=1}^{n}$ and choose smooth functions $\{g_{j}\}_{j=1}^{n}$ so that $supp\leavevmode\nobreak\ g_{j}\subset V_{j},$ $\overline{U_{j}}\subset V_{j}$, and $g_{j}(x)\equiv 1$ for $x\in supp\leavevmode\nobreak\ f_{j},supp\leavevmode\nobreak\ f_{j}\cap(1-g_{j})=\emptyset$.

Proposition 1. The operator $A$ on the manifold $M$ can be represented in the form

[TABLE]

where $T:B_{1}\rightarrow B_{2}$ is a compact operator.

Proof.

The proof is a very simple. Since

[TABLE]

then we have

[TABLE]

and the proof is completed. ∎

Remark 1. *It is obviously such operator is defined uniquely up to a compact operators which do not influence on an index. *

By definition for an arbitrary operator $A:B_{1}\rightarrow B_{2}$

[TABLE]

where infimum is taken over all compact operators $T:B_{1}\rightarrow B_{2}$.

Let $B^{\prime}_{1},B^{\prime}_{2}$ be Banach spaces consisting of functions defined on $\mathbb{R}^{m}$, $\widetilde{A}:B^{\prime}_{1}\rightarrow B^{\prime}_{2}$ be a linear bounded operator.

Since $M$ is a compact manifold, then for every point $x\in M$ there exists a neighborhood $U\ni x$ and diffeomorphism $\omega:U\rightarrow D\subset\mathbb{R}^{m},\omega(x)\equiv y$. We denote by $S_{\omega}$ the following operator acting from $B_{k}$ to $B_{k}^{\prime},k=1,2$. For every function $u\in B_{k}$ vanishing out of $U$

[TABLE]

Definition 6. A local representative of the operator $A:B_{1}\rightarrow B_{2}$ at the point $x\in M$ is called the operator $\widetilde{A}:B_{1}^{\prime}\rightarrow B_{2}^{\prime}$ such that $\forall\varepsilon>0$ there exists the neighborhood $U_{j}$ of the point $x\in U_{j}\subset M$ with the property

[TABLE]

3. Algebra of symbols

Definition 7. *Symbol of an operator $A$ is called the family of its local representatives $\{A_{x}\}$ at each point $x\in\overline{M}$. *

One can show like [3] this definition of an operator symbol conserves all properties of a symbolic calculus. Namely, up to compact summands

• •

product and sum of two operators corresponds to product and sum of their local representatives;

• •

adjoint operator corresponds to its adjoint local representative;

• •

a Fredholm property of an operator corresponds to a Fredholm property of its local representative.

4. Operators with symbols. Examples.

It seems not every operator has a symbol, and we give some examples for operators with symbols.

Example 3. Let $A$ be the differential operator from example 1, and functions $a_{k}(x)$ be continuous functions on $\mathbb{R}^{m}$. Then its symbol is an operator family consisting of multiplication operators on the function

[TABLE]

*where $\xi^{k}=\xi_{1}^{k_{1}}\cdots\xi_{m}^{k_{m}}$. *

Example 4. *Let $A$ be the Calderon–Zygmund operator from Example 2 and $\sigma(x,\xi)$ be its symbol in sense of [1], then its symbol is an operator family consisting of multiplication operators on the function $\sigma(x,\xi)$. *

More important point is that symbol of an operator should be more simple to verify its Fredholm properties. For two above examples a Fredholm property of an operator symbol is equivalent to its invertibility.

5. Stratification of manifolds and operators

5.1. Sub-manifolds

The above definition of an operator on a manifold supposes that all neighborhoods $\{U_{j}\}$ have the same type. But if a manifold has a smooth boundary even then there are two types of neighborhoods related to a place of neighborhood, namely inner neighborhoods and boundary ones. For inner neighborhood $U$ such that $\overline{U}\subset\mathring{M}$ we have the diffeomorphism $\omega:U\rightarrow D$, where $D\in\mathbb{R}^{m}$ is an open set. For a boundary neighborhood such that $U\cap\partial M\neq\emptyset$ we have another diffeomorphism $\omega_{1}:U\rightarrow D\cap\mathbb{R}^{m}_{+}$, where $\mathbb{R}^{m}_{+}=\{x\in\mathbb{R}^{m}:x=(x_{1},\cdots,x_{m}),x_{m}>0\}$. May be this boundary $\partial M$ has some singularities like conical points and wedges. The conical point at the boundary is a such point, for which its neighborhood is diffeomorphic to the cone $C^{a}_{+}=\{x\in{\mathbb{R}}^{m}:\leavevmode\nobreak\ x_{m}>a|x^{\prime}|,\leavevmode\nobreak\ x^{\prime}=(x_{1},...,x_{m-1}),\leavevmode\nobreak\ a>0\},$ the wedge point of codimension $k,1\leq k\leq m-1,$ is a such point for which its neighborhood is diffeomorphic to the set $\{x\in\mathbb{R}^{m}:x=(x^{\prime},x^{\prime\prime}),x^{\prime\prime}\in\mathbb{R}^{m-k},x^{\prime}=(x_{1},\cdots,x_{m-k-1}),x_{m-k-1}>a|x^{\prime\prime\prime}|,x^{\prime\prime\prime}=(x_{1},\cdots,x_{m-k-2}),a>0\}$. So if the manifold $M$ has such singularities we suppose that we can extract certain k-dimensional sub-manifolds, namely $(m-1)$-dimensional boundary $\partial M$, and $k$-dimensional wedges $M_{k},k=0,\cdots,m-2$; $M_{0}$ is a collection of conical points.

5.2. Enveloping operators

If the family $\{A_{x}\}_{x\in M}$ is continuous in operator topology, then according to Simonenko’s theory there is an enveloping operator, i.e. such an operator $A$ for which every operator $A_{x}$ is the local representative for the operator $A$ in the point $x\in M$.

Example 5. *If $\{A_{x}\}_{x\in M}$ consists of Calderon–Zygmund operators in $\mathbb{R}^{m}$ [1] with symbols $\sigma_{x}(\xi)$ parametrized by points $x\in M$ and this family smoothly depends on $x\in M$ then Calderon–Zygmubd operator with variable kernel and symbol $\sigma(x,\xi)$ will be an enveloping operator for this family. *

Example 6. *If $\{A_{x}\}_{x\in M}$ consists of null operators then an enveloping operator is a compact operator [3]. *

Theorem 1. *Operator $A$ has a Fredholm property iff its all local representatives $\{A_{x}\}_{x\in M}$ have the same property. *

This property was proved in [3]. 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.

5.3. Hierarchy of operators

We will remind here the following definition and Fredholm criteria for operators [5].

Definition 8. Let $B_{1},B_{2}$ be Banach spaces, and $A:B_{1}\rightarrow B_{2}$ be a linear bounded operator. The operator $R:B_{2}\rightarrow B_{1}$ is called a regularizer for the operator $A$ if the following properties

[TABLE]

*hold, where $I_{k}:B_{k}\rightarrow B_{k}$ is identity operator, $T_{k};B_{k}\rightarrow B_{k}$ is a compact operator, $k=1,2$. *

Proposition 2. *The operator $A:B_{1}\rightarrow B_{2}$ has a Fredholm property iff there exists a linear bounded regularizer $R:B_{2}\rightarrow B_{1}$. *

Lemma 1. *Let $f$ be a smooth function on the manifold $M,U\subset M$ be an open set, and $supp\leavevmode\nobreak\ f\subset U$. Then the operator $f\cdot A-A\cdot f$ is a compact operator. *

Proof.

Let $g$ be a smooth function on $M,supp\leavevmode\nobreak\ g\subset V\subset M$, moreover $\overline{U}\subset V,g(x)\equiv 1$ for $x\in supp\leavevmode\nobreak\ f$. Then we have

[TABLE]

[TABLE]

where $T_{1},T_{2}$ are compact operators. Let us denote $g\cdot A\cdot g\equiv h$ and write

[TABLE]

and we obtain the required property. ∎

Definition 9. *The operator $A$ is called an elliptic operator if its operator symbol $\{A_{x}\}_{x\in M}$ consists of Fredholm operators. *

Now we will show that each elliptic operator really has a Fredholm property. Our proof in general follows the book [3], 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

[TABLE]

where $T$ is a compact operator. Without loss of generality we can assume that there are $n$ points $x_{k}\in U_{k}\subset V_{k},k=1,2,...n$. Moreover, we can construct such coverings by balls in the following way. Let $\varepsilon>0$ be enough small number. First, for every point $x\in M_{0}$ we take two balls $U_{x},V_{x}$ with the center at $x$ of radius $\varepsilon$ and $2\varepsilon$ and construct two open coverings for $M_{0}$ namely $\mathfrak{U}_{0}=\cup_{x\in M_{0}}U_{x}$ and $\mathfrak{V}_{0}=\cup_{x\in M_{0}}V_{x}$ . Second, we consider the set $L_{1}=\overline{M}\setminus\mathfrak{V}_{0}$ and construct two coverings $\mathfrak{U}_{1}=\cup_{x\in L_{1}\cap M_{1}}U_{x}$ and $\mathfrak{V}_{1}=\cup_{x\in L_{1}\cap M_{1}}V_{x}$. Further, we introduce the set $L_{2}=\overline{M}\setminus(\mathfrak{V}_{0}\cup\mathfrak{V}_{1})$ and two coverings $\mathfrak{U}_{2}=\cup_{x\in L_{2}\cap M_{2}}U_{x}$ and $\mathfrak{V}_{2}=\cup_{x\in L_{2}\cap M_{2}}V_{x}$. Continuing these actions we will come to the set $L_{m-1}=\overline{M}\setminus(\cup_{k=0}^{m-2}\mathfrak{U}_{k})$ which consists of smoothness points of $\partial M$ and inner points of $M$, construct two covering $\mathfrak{U}_{m-1}=\cup_{x\in L_{m-1}\cap\partial M}U_{x}$ and $\mathfrak{V}_{m-1}=\cup_{x\in L_{m-1}\cap\partial M}V_{x}$. Finally, the set $L_{m}=\overline{M}\setminus(\cup_{k=0}^{m-1}\mathfrak{U}_{k})$ consists of inner points of the manifold $M$ only. We finish this process by choosing the covering $\mathfrak{U}_{m}$ for the latter set $L_{m}$. So, the covering $\cup_{k=0}^{m}\mathfrak{U}_{k}$ will be a covering for the whole manifold $M$. According to compactness property we can take into account that this covering is finite, and centers of balls which cover $M_{k}$ are placed at $M_{k}$.

Now we will rewrite the formula (1) in the following way

[TABLE]

where coverings and partitions of unity $\{f_{jk}\}$ and $\{g_{jk}\}$ are chosen as mentioned above. In other words the operator

[TABLE]

is related to some neighborhood of the sub-manifold $M_{k}$; this neighborhood is generated by covering of the sub-manifold $M_{k}$ by balls with centers at points $x_{jk}\in M_{k}$. Since $A_{x_{jk}}$ is a local representative for the operator $A$ at point $x_{jk}$ we can rewrite the formula (2) as follows

[TABLE]

Let us denote $S_{\omega_{j}^{-1}}\hat{g}_{j}\equiv\tilde{g}_{j},\hat{f}_{j}S_{\omega_{j}}\equiv\tilde{f}_{j}$. Further, we can assert that the operator

[TABLE]

will be the regularizer for the operator $A^{\prime}$; here $A^{-1}_{x_{jk}}$ is a regularizer for the operator $A_{x_{jk}}$

Indeed,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

because $f_{jk}\cdot A_{x_{jk}}=A_{x_{jk}}\cdot f_{jk}+\text{compact summand}$, and $f_{jk}\cdot g_{jk}=f_{jk}$, and

[TABLE]

as the partition of unity. The same property

[TABLE]

[TABLE]

is verified analogously. ∎

6. Piece-wise continuous operator families

Given operator $A$ with the symbol $\{A_{x}\}_{x\in\overline{M}}$ 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 $\{A_{x}\}_{x\in\overline{M}}$.

Theorem 2. If the symbol $\{A_{x}\}_{x\in\overline{M}}$ is a piece-wise continuous operator function then there are $m+1$ operators $A^{(k)},k=0,1,\cdots,m$ such that the operator $A$ and the operator

[TABLE]

*have the same symbols, where the operator $A^{(k)}$ is an enveloping operator for the family $\{A_{x}\}_{x\in\overline{M}_{k}},T$ is a compact operator. *

Proof.

We will use the constructions from proof of Lemma 2, namely the formula (3). We will extract the operator

[TABLE]

which “serves” the sub-manifold $M_{k}$ and consider it in details. 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 $x_{jk}\in M_{k}$ of radius $\varepsilon>0$, therefore $f_{jk},g_{jk},n_{k}$ depend on $\varepsilon$.

According to Simonenko’s ideas [3] we will construct the component $A^{(k)}$ in the following way. Let $\{\varepsilon_{n}\}_{n=1}^{\infty}$ be a sequence such that $\varepsilon_{n}>0,\forall n\in\mathbb{N},\lim\limits_{n\to\infty}\varepsilon_{n}=0$. Given $\varepsilon_{n}$ we choose coverings $\{U_{jk}\}_{j=1}^{n_{k}}$ and $\{V_{jk}\}_{j=1}^{n_{k}}$ as above with partition of unity $\{f_{jk}\}$ and corresponding functions $\{g_{jk}\}$ such that

[TABLE]

we remaind that $U_{jk},V_{jk}$ are balls with centers at $x_{jk}\in\overline{M_{k}}$ of radius $\varepsilon$ and $2\varepsilon$. This requirement is possible according to continuity of family $\{A_{x}\}$ on the sub-manifold $\overline{M_{k}}$. Now we will introduce such constructed operator

[TABLE]

and will show that the sequence $\{A_{n}\}$ is Cauchy sequence with respect to norm $|||\cdot|||$. We have

[TABLE]

where the operator $A_{l}$ is constructed for given $\varepsilon_{l}$ with corresponding coverings $\{u_{ik}\}_{i=1}^{l_{k}}$ and $\{v_{ik}\}_{j=1}^{i_{k}}$ with partition of unity $\{F_{ik}\}$ and corresponding functions $\{G_{ik}\}$ so that

[TABLE]

here $u_{ik},v_{ik}$ are balls with centers at $y_{ik}\in\overline{M_{k}}$ of radius $\tau$ and $2\tau$.

We can write

[TABLE]

[TABLE]

and the same we can write for $A_{l}$

[TABLE]

[TABLE]

Let us consider the difference

[TABLE]

Obviously, summands with non-vanishing supplements to the formula (5) are those for which $U_{jk}\cap u_{ik}\neq\emptyset$. A number of such neighborhoods are finite always for arbitrary finite coverings, hence we obtain

[TABLE]

[TABLE]

[TABLE]

where $K$ is a universal constant.

Thus, we have proved that the sequence $\{A_{n}\}$ is a Cauchy sequence hence there exists $\lim\limits_{n\to\infty}A_{n}=A^{(k)}$. ∎

Corollary 1. *The operator $A$ has a Fredholm property iff all operators $A^{(k)},k=0,1,\cdots,m$ have the same property. *

Remark 2. *The constructed operator $A^{\prime}$ 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. *

7. 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.