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Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space
We consider the Friedrichs self-adjoint extension for a differential operator of the form , which is defined on a bounded domain , (for we assume that is a finite interval). Here is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth coefficients and a potential is a real-valued integrable function satisfying the generalized Kato condition. Under these assumptions for the coefficients of and for positive large enough we obtain the existence of Green's function for the operator and its estimates up to the boundary of . These estimates allow us to prove the absolute and uniform convergence up to the boundary of of Fourier series in eigenfunctions of this operator. In particular, these results can be applied for the basis of the Fourier method which is usually used in practice for solving some equations of mathematical physics.
Let be a bounded domain in () with smooth boundary. We consider on an elliptic differential operator of the form
is a formally self-adjoint differential operator of order . Here with . The coefficients of the operator are assumed to be the complex-valued (in general) bounded smooth functions on the domain for all such that are real valued for and this operator satisfies the uniform ellipticity condition
with some constant , for all and all . We assume that the potential is a real-valued -function satisfying the generalized Kato condition, that is,
where function for is defined by
For we denote by the -based Sobolev space, where indicates the “degree” of the smoothness; by we denote the closure of in . We denote also by Besov space (where indicates the smoothness) with the same notation for as for Sobolev space. The definition of Sobolev and Besov spaces as well as the embedding theorems for these spaces can be found in [1, 2].
where the constant depends only on and , the constant depends only on , and , and the value is defined by
where is as in (1.5).
Since the domain is bounded then tends to as . It immediately implies that there is a constant such that
for all . Since is positive for sufficiently large it has a positive self-adjoint Friedrichs extension such that
We define the Friedrichs extension of to be such that
The domain of is given by
It is also well known that this extension has a purely discrete spectrum of finite multiplicity having the only one accumulation point at infinity () and a complete orthonormal system of eigenfunctions in .
To each function we can assign the formal series
where are the Fourier coefficients of with respect to the system .
The study of elliptic differential operators with smooth coefficients on a bounded domain with smooth boundary has a long history. We restrict the bibliographical remarks to the works that are of interest from the viewpoint of the present article.
The estimates for the Green's function and convergence of spectral expansions of a general elliptic differential operator of order with smooth coefficients on a bounded domain have been studied by many authors. We refer to a four-volume monograph of Hörmander [4, 5], the works of Alimov [6–9], Gårding , Krasovskiĭ [11, 12], Schechter  and others. We mention also the papers [13–15] of the author of the present which deal with the operators whose coefficients may have local singularities of specific order on an arbitrary smooth surface whose dimension is strictly less than that of the original domain. As to elliptic operators of order whose coefficients may have singularities in , similar results have been mainly obtained for the Schrödinger operators on with from or in any dimensions but with which may have given singularity at one point. For such results, see Alimov and Joó , Ashurov , Ashurov and Faiziev , Khalmukhamedov [19, 20], Serov [21, 22], Serov and Buzurnyuk , and others.Some survey of resent results concerning theory of elliptic differential operators of order can be found in the articles of Davies [24, 25].
The aim of this paper is to prove the following results.
Theorem 1.1. Suppose that satisfies condition (1.4), then there exist constants , and such that for all the Green function of the operator exists and satisfies the following estimates:for all and .
Without loss of generality, in the following theorem we assume that is positive.
Theorem 1.2. The Fourier series (1.12) converges absolutely and uniformly on the domain for any function from the domain of the operator for .
One of the main results of the present paper is Theorem 1.1 which concerns the estimates up to the boundary of the domain for the Green's function of an elliptic differential operator of order with singular potential from the generalized Kato space. In all previous publications, as far as we know, the estimates for the Green function are proved on an arbitrary compact subset from the domain and for the case when the coefficients of operator are either smooth or have some special type of singularities.
Another main result of this paper is Theorem 1.2. It gives some sufficient conditions which provide the absolute convergence up to the boundary of of Fourier series in eigenfunctions for functions from the domain of some power of this operator. In addition to Theorem 1.2, we would like to take into consideration Theorem 3.7 (see Section 3 of the paper) which is the generalization of the well-known result of Peetre (see ) to the operators with singular coefficients. It can be mentioned also here that in the scale of the spaces associated with some powers of our operator the results of Theorems 1.2 and 3.7 are sharp (see, e.g., ).
This paper is organized such that Theorem 1.1 is proved in Section 2 and Theorem 1.2 in Section 3. Some additional theorems about the absolute convergence of Fourier series are also proved in Section 3.
2. Green's Function
In this section we obtain the estimates for the Green's function of the operator when is positive and sufficiently large.
Definition 2.1. For and , a locally integrable function on is called a fundamental solution for an operator if and only if
Equation (2.1) holds in the sense of distributions, that is,
for all , where
is the transpose of .
We will use the following result.
Proposition 2.2. There exists such that for any , the differential operator has a fundamental solution . Furthermore, for any multi-index , , there are constants , such that the following estimates hold: for all and .
We will look for the fundamental solution of the operator , for positive and large enough, as a solution of the integral equation
where is the fundamental solution of the operator . By Proposition 2.2, exists and belongs (at least) to uniformly with respect to from .
We need the following lemma, which may have interest of its own right.
Lemma 2.3. Assume that satisfies condition (1.4), then there is such that for all the fundamental solution exists as a solution of the integral equation (2.4) and satisfies the following estimates: with some positive constant , where is as in Proposition 2.2 and .
Proof. We solve the integral equation (2.4) by iterations. For any , we denote
We will prove by induction that there is such that for all and for each
where , is as in the Proposition 2.2 and is defined as
It is clear that for estimate (2.6) holds. And it is also clear that (2.6) holds for the case when for each by choosing large enough.
In the case (considering two possibilities and ) in order to prove (2.6) it is enough to prove that there exists such that for all and , where constant is as in Proposition 2.2.
Indeed, since for we have where is as in (1.4), then we can estimate the left-hand side of (2.8) as follows. If then the integrals in the latter equality tend to zero as . The reason is due to that condition (1.4) the measure of the set tends to zero as . If then this integral can be estimated by where some positive constant depends only on and dimension .
In the case (considering two possibility and ) it can be proved that Then, instead of estimate (2.10) we obtain
Combining these two facts (including (2.10) and (2.12)), (2.8) and choosing appropriately (with respect to ) we may conclude that inequality (2.6) is proved. Since the solution of the integral equation (2.4) is given by the series the estimates (2.6) prove also Lemma 2.3.
The following corollary holds.
Corollary 2.4. Assume that satisfies the condition (1.4), then for the derivatives of the fundamental solution of order the following estimates hold: for some constant , for all and (where and are as in Lemma 2.3).
The proof of the corollary follows immediately from the integral representation for the derivatives of , estimates for the derivatives of in Proposition 2.2, estimates for in Lemma 2.3, and estimates for the kernels with weak singularities.
Let us note that the fundamental solution , which was obtained in Lemma 2.3, belongs (at least) to uniformly with respect to from .
Now we are in the position to introduce the Green's function of the operator . If is sufficiently large then the operator is positive and its inverse
is a bounded operator. It is also an integral operator with kernel denoted by . If we use for this integral operator the symbol then we have
Definition 2.5. The kernel of the integral operator is called the Green's function of the operator .
Proof of Theorem 1.1. For , let and be compact sets, each of them having a smooth boundary, with such that
Here denotes the distance between the sets and .
Let be a fundamental solution of the operator for and sufficiently large. We choose the function such that and set By this equation the function is well defined for all . Clearly, for , . We will show that is a parametrix for . To prove this, let us introduce the function and corresponding integral operator with kernel where and are integral operators in with kernels and , respectively. Then it follows from (2.20) that where If we denote by the kernel of the integral operator , then it follows from (2.24) that for any , and the kernel (see (2.20)) has the form where . As a matter of fact we cannot characterize and estimate the kernel from (2.22)—(2.24). That is why we will proceed a little bit differently, as follows. Equality (2.19) implies that in the sense of distributions the following representation holds: where ( is considered here as a parameter) and sufficiently large. The function in (2.27) will be of the form with the differential operator having the symbol . It is the polynomial in of order and therefore the differential operators are of order . This fact allows us to estimate the function (in comparison with ). Indeed, by the choice of , only on the set and therefore the representation (2.28) and Corollary 2.4 imply that the following estimate holds: for all and with as in Corollary 2.4.
Now we need the following lemma.
In order to finish the proof of Theorem 1.1 let us introduce new functions and which are obtained from and multiplying by
respectively, where is as in Corollary 2.4. Then (2.30) (see Lemma 2.6) and estimate (2.29) formally yield the following estimate (for simplicity let us consider here only the case , the cases can be considered similarly)
Considering two possibilities and the value in the latter brackets can be estimated from above by
This estimate allows us to get from (2.36) that
then for large enough (2.38) yields
Thus, Theorem 1.1 is completely proved.
3. Convergence of Fourier Series
Without loss of generality, we assume in this section that is positive. Then by the J. von Neumann spectral theorem for , where with as in Theorem 1.1, the following representation holds:
where is real and is the spectral resolution corresponding to the self-adjoint operator . The domain of the operator (3.1) is defined by
In our case (in the case of pure discrete spectrum), the spectral projector has the form
In addition, we need a special representation of the negative fractional powers of . If we assume that then using well-known properties of Euler beta-function, one can obtain
where is the integral operator with kernel from Section 2. This representation shows that operator (3.6) is also integralwith kernel denoted by . Using Theorem 1.1 of present work and well-known technique (see, e.g., ) it is not so difficult to prove the following estimates
where , is as in Theorem 1.1 and the constant depends on .
The following main lemma holds.
Lemma 3.1. For any function and for where with as in Theorem 1.1.
Corollary 3.2. Assume that . There is a constant depending only on , such that the estimate holds uniformly in and .
Proof. By the spectral theorem and relation (3.4), we can rewrite inequality (3.8) in the form where are the Fourier coefficients of with respect to the system . Now inequality (3.11) follows by duality. The proof is complete.
Remark 3.3. The inequality (3.11) has an independent interest since it gives the "bundle" estimate of the eigenfunctions in the form which holds uniformly in and large enough. Indeed, from (3.12) we have uniformly in . If we chose now then one can immediately obtain (3.13).
Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2. Using the representation (3.4), the inequality (3.11), and the Cauchy-Schwarz-Bunyakovsk 2 inequality, we obtain uniformly in and for any fixed . Now the desired assertion follows from (3.5). Theorem 1.2 is completely proved.
Corollary 3.4. Assume that the function satisfies the condition where are the Fourier coefficients of with respect to the system , then the Fourier series (1.12) converges absolutely and uniformly on .
Let us assume now that the potential satisfies the conditions
then it is not so difficult to see that for the case conditions (3.17) imply the condition (1.4). For the case this condition (3.17) for must be considered in addition to the condition (1.4). The following result is valid.
Proof. Using the Sobolev embedding theorem we easily conclude that conditions (3.17) imply the following inclusion:
And for any the following inequality holds:
Moreover, we may assert that the operator is invertible for large enough. Indeed, since the function
belongs to , we have the representation for
where denotes the Friedrichs self-adjoint extension for in . Using again the Sobolev embedding theorem and conditions (3.17) we may conclude that the functions and belong to . The results of  yield that the operator is invertible with small norm for its inverse operator (if is large enough). This fact and the latter equality imply that for large enough the operator is also invertible and for any we have the following inequality:
Now let . Then (3.23) implies where belongs to and the convergence to zero in the last term follows from the J. von Neumann spectral theorem. The proof is complete.
The Sobolev embedding theorem gives the immediate corollary.
Corollary 3.6. Let . Then for any with holds uniformly in .
The next theorem gives us some sufficient conditions which provide the absolute and uniform convergence of Fourier series (1.12) in the classical Besov and Sobolev spaces. Following , we use the symbol .
Theorem 3.7. Assume that the potential belongs to Sobolev space , where is an entire part of , then for any function from Besov space the Fourier series (1.12) converges absolutely and uniformly on the domain .
Proof. Using the Sobolev embedding theorem and the following representation: where are some constants, we can conclude that the condition for the potential implies the following inclusion: Then using the results of  (see Theorem ) we may conclude that Consequently, by Theorem of  and by Peetre's method of real interpolation (see, e.g., ), we have But the latter space is the interpolation space of Peetre (see ) for the elliptic differential operator of order . Since estimate (3.13) for the spectral function holds in our case, we can apply the results of  and conclude that the proof of this theorem is complete.
Remark 3.8. If is even then the statement of this theorem holds for any function from Besov space due to the equality (see Theorem of ) And the Sobolev embedding theorem for Besov spaces (see, e.g., ) implies that this theorem also holds for any function from Sobolev space with and arbitrary integer .
This work was supported by the Academy of Finland (application no. 213476, Finnish Programme for Centres of Excellence in Research 2006–2011).
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