Fredholm Property of Non-Smooth Pseudodifferential Operators

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a H\"older space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.


INTRODUCTION
Fredholm operators are often called nearly invertible operators. They admit dimension formulae similar to linear operators between finite dimensional spaces. Because of this they play an important role in the field of partial differential equations in order to get existence and uniqueness results. Great effort was already spent to get conditions for the Fredholmness of smooth pseudodifferential operators with symbols in the Hörmander-class , (ℝ × ℝ ) ∶= ⋂ ∈ℕ , (ℝ × ℝ ; ), where 0 ≤ , ≤ 1 and ∈ ℝ. Here the symbol-class , (ℝ × ℝ ; ) consists of all -times continuous differentiable functions ∶ ℝ × ℝ → ℂ that are smooth with respect to the spatial variable such that for all ∈ ℕ 0 | | ( ) ∶= max For every symbol ∈ , (ℝ × ℝ ; ) we define the associated pseudodifferential operator via ellipticity conditions are satisfied if is uniformly elliptic in the sense that for some , > 0. Here (ℝ ) denotes a Bessel Potential Space for ∈ (1, ∞) and ∈ ℝ, defined in Section 2. Moreover, ∈ , (ℝ × ℝ ) is slowly varying if for all , ∈ ℕ 0 with ≠ 0 we have In [18] Schrohe extended the result of Kumano-go as follows: Smooth pseudodifferential operators with slowly varying symbols of the order zero are Fredholm operators on the weighted Sobolev spaces (ℝ ), see [18] for the definition, if and only if its symbol is uniformly elliptic. In applications (e.g. to non-linear PDEs) also non-smooth pseudodifferential operators appear naturally. Therefore we are interested in sufficient conditions for non-smooth pseudodifferential operators to become a Fredholm operator from (ℝ ) to (ℝ ), where ∈ ℝ, 1 < < ∞. For non-smooth differential operators the Fredholm property can be characterized by the uniform ellipticity of its symbol. This was announced by Cordes in [3], completed by Illner in [8] and partially recovered by Fan and Wong in [5]. This characterization of the Fredholm property was extended to the matrix-valued case in [6] for = 2 and in [19] for general ∈ (1, ∞). In the case = 2 an alternative proof by means of the tool of * -algebras, was given by Taylor in [20]. The goal of this paper is to give sufficient conditions for the Fredholm property of pseudodifferential operators ( , ) with a symbol in the non-smooth symbol-class̃, , For the definition of the Hölder spacẽ, of the order̃∈ ℕ 0 with Hölder regularity 0 < ≤ 1 we refer to Section 2 below. A function ∶ ℝ × ℝ → ℂ is an element of the symbol-class̃, , (ℝ × ℝ ; ), ∈ ℝ, if the following properties hold for all , ∈ ℕ 0 with | | ≤̃and | | ≤ : is an element of the symbol-class̃, , if and only if , ∈̃, , (ℝ × ℝ ; ) for all , = 1, … , , where we identify ∈ ℒ ( ℂ ) with a matrix ( , ) , =1 ∈ ℂ × in the standard way. For a given symbol we define the associated pseudodifferential operator as in the smooth case, cf. (1.1). We remark that in the literature there are also some results concerning the Fredholm property of pseudodifferential operators on compact manifolds, see e.g. [7], [15]. Nistor even gave some criteria for the Fredholmness of pseudodifferential operators on non-compact manifolds in [16].
In the present paper we proceed as follows: We give a short summary of all notations and function spaces needed in Section 2. Moreover we introduce the space of amplitudes and the oscillatory integrals. In Section 3 we define all symbol-classes of pseudodifferential operators needed later on and present their properties. In particular we extend the concept of symbol-smoothing presented in [21,Section 1.3]. Together with the extension of the symbol reduction result of [2] for non-smooth double symbols, see Subsection 3.2 below, the symbol-smoothing becomes the main ingredient in order to verify the main result of our paper: Additionally we choose an arbitrary ∈ (0; min{(̃+ )( − ); 1}) and̃∈ (0, min{( − ) ; ( − )(̃+ ) − ; )}). Moreover let ∈̃,̃, be a symbol fulfilling the following properties for some > 0 and 0 > 0: Then for all ≥ ( + 2) + ⋅ max{1∕2, 1∕ } and ∈ ℝ with is a Fredholm operator.
As in the smooth case, we restrict ourselves to the case of slowly varying symbols in order to show the Fredholm property. As Schrohe already wrote in [18] for a parameter construction of non-classical smooth symbols more than invertibility of the symbol is needed and the parametrix can differ from the Fredholm inverse. We see, that many conditions are needed in Theorem 1.1 to show the Fredholm property of a non-smooth pseudodifferential operator. Hence the question arises which of them are of technical nature and which of them are really necessary. In the smooth case Schrohe showed in [18] that the uniform ellipticity of a zero order symbol is a necessary condition for ( , ) being a Fredholm operator. By means of the composition with order reducing operators one easily obtains that the uniform ellipticity of a smooth symbol of arbitrary order is also a necessary condition for ( , ) being a Fredholm operator. Uniform ellipticity for systems is equivalent to condition 1). For non-smooth differential operators this condition is also necessary, cf. [19]. Therefore 1) is necessary at least if is smooth or ( , ) is a differential operator. Additionally in the smooth case, also the condition 0 ≤ < ≤ 1 arises. Since each Fredholm operator in order to apply the known results on mapping properties of non-smooth pseudodifferential operators. In order to prove the claim of Theorem 1.1, we need to strengthen condition ) due to technical reasons. Finally, also condition 2) is of technical nature. Theorem 1.1 will be proved in Section 4. For the definition of the symbol-class̃,̃, we refer to Definition 3.5 in Subsection 3.1.
On account of the definition of the Hölder spaces and the Leibniz-rule we obtain: Lemma 2.1. Let̃∈ ℕ 0 , 0 < < 1 and , ∈̃, (ℝ ). Then The Bessel Potential space (ℝ ), ∈ ℝ and 1 < < ∞, will play a central role in this paper. The set (ℝ ) is defined by . For the convenience of the reader we mention an interpolation result needed in this paper: Proof. For all ∈ [1, ∞] we denote the real interpolation spaces by ( 0 (ℝ ), +1 (ℝ ) ) , , cf. e.g. [12]. An application of the reiteration theorem, c.f. [13,Theorem 1.2.15], and of Proposition 1.20 in [12] provides This yields the claim. For more details we refer to [17,Lemma 2.41]. □ Since this paper deals with the Fredholm property of pseudodifferential-operators, we finally add the definition of an Fredholm operator: Definition 2.3. Let , be Banach spaces and let ∈ ℒ( , ). Then is called a Fredholm operator if ( ) is finite dimensional and ℛ( ) is closed and has finite co-dimension, i.e., there is a finite dimensional subspace ⊆ such that = ℛ( ) ⊕ .
The following characterization is fundamental for our purposes.
where respectively are the identity operators on respectively .
The proof can e.g. be found in [4,Theorem 3.15].

Space of amplitudes and oscillatory integrals
The aim of the present paper is to define and discuss some properties of oscillatory integrals for all elements of the space of amplitudes where all derivatives are well defined in the sense of distributions. For all elements ∈ , , (ℝ × ℝ ), , ∈ ℕ 0 ∪ {∞}, , ∈ ℝ the oscillatory integral is defined by we can extend some properties of the oscillatory integral proved in Section 2.3 of [2] as follows: Theorem 2.5. Let , ∈ ℝ and , ∈ ℕ 0 ∪ {∞} with > + . Moreover let , ′ ∈ ℕ with ≥ ′ > + and ≥ > + . Then the oscillatory integral (2.2) exists for all ∈ , , (ℝ × ℝ ) and we have for all 1 , 2 ∈ ℕ 0 with 1 ≤ and 2 ≤ : Proof. The claim can be shown similarly to [2,Corollary 2.13].

PSEUDODIFFERENTIAL OPERATORS AND THEIR PROPERTIES
Throughout this section we summarize all properties of pseudodifferential operators needed later on. Additionally we define all symbol-classes of pseudodifferential operators needed in this paper. On account of Lemma 2.2 with ∶=̃+ +1 if < 1 and by means of +1 (ℝ ) ⊆ ,1 (ℝ ) else we can show for all 0 < ≤ 1,̃∈ ℕ 0 , ∈ ℝ, ∈ ℕ 0 ∪ {∞} and 0 ≤ , ≤ 1. For more details see, [17,Remark 4.2]. Additionally we get by means of interpolation, c.f. Lemma 2.2, the next estimate for non-smooth symbols: . Then we get for all ∈ ℕ 0 with | | ≤ and ∈ ℕ 0 with ≤̃: Pseudodifferential operators are bounded as maps between several Bessel Potential spaces. For the proof we refer to [2, Theorem 3.7].

Symbol-smoothing
A well-known tool for proving some properties of non-smooth pseudodifferential operators of the symbol class 1, (ℝ × ℝ ) for certain Banach spaces is the symbol-smoothing, see e.g. [21,Section 1.3]. In order to prove the Fredholm property of nonsmooth pseudodifferential operators, we now generalize the tool of symbol-smoothing for pseudodifferential operators which are non-smooth with respect to the second variable and for ≠ 1. To this end we fix two functions , 0 ∈ ∞ 0 (ℝ ) till the end of this section with the following properties: • 0 ≥ 0, 0 ( ) = 1 for all | | ≤ 1 and 0 ( ) = 0 for all | | ≥ 2.
Then we define for all ∈ ℕ the functions via Using that for any ∈ ℝ there are 1 , 2 > 0 such that we can show the following properties of the functions for all ∈ ℕ 0 : Additionally we define for all > 0 the operator by Note, that for each ∈ ℕ 0 : The operator has the following properties: Proof. On account of [21, Lemma 1.3C] the claims i), ii) and claim iv) in the case | | = 0 hold true. An application of the case | | = 0 on ∶= ∈̃− | |, (ℝ ) provides the general case of claim iv). Because of [21, Lemma 1.3.A] we additionally obtain claim iii) for the case | | = 0. It remains to verify claim iii) for general ∈ ℕ 0 with | | ≤̃. This can be done similarly to the proof of the case | | = 0. For the convenience of the reader we give a short proof of claim iii) for arbitrary we get the boundedness of Since ⟨ ⟩ − and commute, we obtain claim iii) in the general case. □ Our aim is to verify useful properties of the functions ♯ and needed later on. To this end two new symbol-classes are needed, which we define, now.
is an element of the symbol-class̃,̇, The properties of the functions ♯ and are summarized in the next three lemmas: . Moreover let ∈ ( , ). Then we have for̃∈ (0, ( − ) ): Proof. We begin with the proof of ). We choose an arbitrary ∈ ℝ and set ., ) ( ) and the Leibniz rule yields for all , ∈ ℕ 0 with | | ≤ and | | ≤| An application of Lemma 3.3 iv), (3.2) and (3.3) to the previous estimate provides: Similarly we get by means of (3.4), the Leibniz rule, Lemma 3.3 iii) and (3.3) for all , ∈ ℕ 0 with | | ≤ and | | ≤̃: for all ∈ ℝ . On account of (3.6) and (3.5) claim i) holds. Our next goal is show ) and ). In order to prove the claim, we assume ∈̃,̇, (ℝ × ℝ ; ) or ∈̃,̃, (ℝ × ℝ ; ). Additionally we fix some arbitrary , We choose an arbitrary > 0. As before we fix an arbitrary ∈ ℝ and set Moreover we define for all ∈ ℕ 0 the functions By means of integration by parts and the Theorem of Fubini, we obtain for each ∈ ℕ Since we can change the order of the two operators and Our task is to use the previous equality in order to show for̃∈ (0, ( − ) ): Then a combination of (3.6), (3.5) and (3.9) yields claim ) and ). It remains to verify (3.9). The properties of the Fourier for all ∈ . On account of the choice of we get using (3.3): for all ∈ we can choose an > 1 such that for In addition we choose an Then we obtain for all ∈ ℝ by means of Lemma 3.3 iv), (3.10) and (3.11): On account of the properties of the Fourier transform and due to the definition of we get using ∈ (ℝ ): where 1 is independent of ∈ ℕ. The choice of the symbol and the multi-index gives us the existence of añ> 0 such that for all | | ≥̃+ we have (3.14) Using (3.8) we obtain for all ∈ ℝ with | | ≥̃+ : Now let̃be as in the assumptions. Setting ∶= ( − )(̃−| |+ )−( − )(̃−| |+ ) we get by means of interpolation with (3.5) and (3.15), that estimate (3.9) holds: Hence the lemma is proved. □ Lemma 3.7. Let 0 ≤ < ≤ 1,̃∈ ℕ 0 , 0 < < 1, ∈ ℕ ∪ {∞}, ∈ ℝ and ∈̃, , (ℝ × ℝ ; ). Moreover let ∈ ( , ). Then we have for all ∈ ℕ 0 with | | ≤̃: ) .
For each double symbol ∈̃, we define the associated pseudodifferential operator by In the smooth case, i.e. if 1 , 2 = ∞, the symbol-reduction is well-known, cf. e.g.
be bounded. If we define for each ∈ ℬ and ∈ [0, 1] the function we get with ∶= 1 + 2 that ∈̃, , ( ℝ × ℝ ;̃) for all ∈ ℬ and ∈ [0, 1] and the existence of a constant , independent of ∈ ℬ and ∈ [0, 1], such that for all , We combine the ideas of the smooth symbol reduction in [11,Lemma 2.4] and that one in [2,Section 4.2] in order to get the boundedness of additionally some new arguments are needed. Unfortunately one looses some regularity with respect to the second variable of the order + 1 in the proof. The ability to treat the even and odd space dimensions in the same way is based on the next remark: Remark 3.11. Let ∈ ℕ be arbitrary. Then We additionally have for all ∈ ℕ 0 : Definition 3.12. Let ∈ ℕ be arbitrary. Then we define if is even, and In order to improve the symbol reduction, we need the next result: Proposition 3.13. Let 0 ≤ ≤ ≤ 1 with ≠ 1, 0 < < 1,̃∈ ℕ 0 and 1 , 2 ∈ ℝ. Additionally let 1 , 2 ∈ ℕ 0 ∪ {∞} be such that there is an ∈ ℕ with < ≤ 1 . Moreover, let ∈̃, . Considering an for all , , , ∈ ℝ . Then we have ( , , , ) Proof. First of all we prove the claim for even 0 and use 2 0 instead of 0 . Let , ∈ ℝ be arbitrary. We define ∶= 1 + 2 .

FREDHOLM PROPERT Y OF NON-SMOOTH PSEUDODIFFERENTIAL OPERATORS
The present section serves to show the main goal of this paper: The Fredholm property of non-smooth pseudodifferential operators fulfilling certain properties. For the proof of that statement we use the following compactness properties of non-smooth pseudodifferential operators verified by Marschall. They are special cases of Theorem 3 and Theorem 4 of [14].
is compact.
Proof. An application of the Taylor expansion formula to the second variable of around and integration by parts provides Next we need to exchange the oscillatory integral with the integral in the second term of the right side of the previous equality. Hence we choose an arbitrary ∈ (ℝ ) with (0) = 1 and let ∈ ℕ 0 with | | = . Now let = + 1 and̃= 1 + ⌈ 1 + 1− ⌉. Then we obtain due to the Theorem of Fubini and integration by parts (2.4) for the definition of ( . , .), for each > 0: Here the assumptions of the Theorem of Fubini and of integration by parts can be verified. Since ∈ (ℝ ), ( ) → 0 for → 0 if | | ≠ 0. Hence we get by interchanging the limit and the integration on account of (4.2) and since the integrand has an 1 -majorant: where the last equality holds because of Theorem 2.5. Hence (4.1) holds. The rest of the claim is a consequence of Theorem 3.10. □ As a consequence of the previous theorem, we obtain: and for all ∈ ℕ with ≤ , ∈ ℕ 0 with | | = and ∈ [0, 1] we set } we obtain In particular we have ( , ) ∈̃1 , 1 Proof.
Remark 4.8. If we weaken the condition for the second symbol in the previous theorem to 2 ∈̃2 , 2 2 , , we can show in the same way as in the proof of Theorem 4.7, the compactness of for some > 0.