We consider the Friedrichs self-adjoint extension for a differential
operator A of the form A=A0+q(x)·, which is defined on a bounded
domain Ω⊂ℝn, n≥1 (for n=1 we assume that Ω=(a,b) is a finite
interval). Here A0=A0(x,D) is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth
coefficients and a potential q(x) is a real-valued integrable function
satisfying the generalized Kato condition. Under these assumptions
for the coefficients of A and for positive λ large enough we obtain the
existence of Green's function for the operator A+λI and its estimates
up to the boundary of Ω. These estimates allow us to prove the absolute and uniform convergence up to the boundary of Ω 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.

1. Introduction

Let Ω be a bounded domain in ℝn (n≥1) with smooth boundary. We consider on Ω an elliptic differential operator of the form

A=A0(x,D)+q(x)·,
where

A0(x,D)=∑|α|≤2maα(x)Dα
is a formally self-adjoint differential operator of order 2m,m=1,2,…. Here Dα=D1α1·D2α2·⋯·Dnαn with Dj=-i(∂/∂xj). The coefficients aα(x) of the operator A0 are assumed to be the complex-valued (in general) bounded smooth functions on the domain Ω for all |α|≤2m such that aα(x) are real valued for |α|=2m and this operator A0 satisfies the uniform ellipticity condition

∑|α|=2maα(x)ξα≥ν|ξ|2m
with some constant ν>0, for all x∈Ω and all ξ∈ℝn. We assume that the potential q(x) is a real-valued L1(Ω)-function satisfying the generalized Kato condition, that is,

supx∈Ω∫Ω|q(y)|ωn(|x-y|)dy<∞,
where function ωn(t) for t>0 is defined by

ωn(t)={t2m-n,2m<n,1+|logt|,2m=n,1,2m>n.

For s>0 we denote by W2s(Ω) the L2-based Sobolev space, where s indicates the “degree” of the smoothness; by W∘2s(Ω) we denote the closure of C0∞(Ω) in W2s(Ω). We denote also by B2,ps(Ω),1≤p≤∞, Besov space (where s indicates the smoothness) with the same notation for B∘2,ps(Ω) 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].

Due to (1.4)-(1.5) the function |q(x)|1/2 satisfies all conditions of Theorem 7.3 of [3] with s=m and therefore for any δ>0 we have the following inequality:

|(qf,f)L2(Ω)|≤∥|q|1/2f∥L2(Ω)2≤C0Nδ(|q|1/2)∥f∥W2m(Ω)2+CδN1(|q|1/2)∥f∥L2(Ω)2,
where the constant C0 depends only on m and n, the constant Cδ depends only on m, n and δ, and the value Nδ(|q|1/2) is defined by

Nδ(|q|1/2):=supx∈Ω∫y∈Ω,|x-y|<δ|q(y)|ωn(|x-y|)dy,
where ωn is as in (1.5).

Since the domain Ω is bounded then Nδ(|q|1/2) tends to 0 as δ→0. It immediately implies that there is a constant C>0 such that

(Af,f)L2(Ω)≥ν2∥f∥W2m(Ω)2-C∥f∥L2(Ω)2
for all f∈C0∞(Ω). Since Aμ=A+μI is positive for sufficiently large μ it has a positive self-adjoint Friedrichs extension (Aμ)F such that

𝒟((Aμ)F)⊂W∘2m(Ω).
We define the Friedrichs extension of A=Aμ-μI to be AF=(Aμ)F-μI such that

𝒟(AF)⊂W∘2m(Ω).
The domain of AF is given by

𝒟(AF)={f∈W∘2m(Ω)∣Af∈L2(Ω)}.

It is also well known that this extension has a purely discrete spectrum {λk}k=1∞ of finite multiplicity having the only one accumulation point at infinity (λk→+∞) and a complete orthonormal system {uk(x)}k=1∞ of eigenfunctions in L2(Ω).

To each function f∈L2(Ω) we can assign the formal series

f=∑k=1∞fkuk(x),
where fk=(f,uk)L2(Ω) are the Fourier coefficients of f with respect to the system {uk(x)}k=1∞.

The study of elliptic differential operators with smooth coefficients on a bounded domain Ω⊂ℝn 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 2m 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 [10], Krasovskiĭ [11, 12], Schechter [3] 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 2m whose coefficients may have singularities in Lp, similar results have been mainly obtained for the Schrödinger operators -Δ+q(x) on ℝ3 with q from L2 or in any dimensions but with q which may have given singularity at one point. For such results, see Alimov and Joó [16], Ashurov [17], Ashurov and Faiziev [18], Khalmukhamedov [19, 20], Serov [21, 22], Serov and Buzurnyuk [23], and others.Some survey of resent results concerning Lp theory of elliptic differential operators of order 2m 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 q satisfies condition (1.4), then there exist constants C>0, δ>0 and λ0>0 such that for all λ≥λ0 the Green function G(x,y,λ) of the operator AF+λI exists and satisfies the following estimates:

2m<n,

|G(x,y,λ)|≤C|x-y|2m-ne-δ|x-y|λ1/2m,

2m=n,

|G(x,y,λ)|≤C(1+|log(|x-y|λ1/2m)|)e-δ|x-y|λ1/2m,

2m>n,

|G(x,y,λ)|≤Cλ(n-2m)/2me-δ|x-y|λ1/2m

for all x,y∈Ω and λ≥λ0.

Without loss of generality, in the following theorem we assume that AF is positive.

Theorem 1.2.

The Fourier series (1.12) converges absolutely and uniformly on the domain Ω for any function f from the domain of the operator AFσ for σ>n/4m.

One of the main results of the present paper is Theorem 1.1 which concerns the estimates up to the boundary of the domain Ω for the Green's function of an elliptic differential operator of order 2m 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 Ω 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 Ω 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 [26]) 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., [14]).

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 AF+λI when λ is positive and sufficiently large.

Definition 2.1.

For λ>0 and y∈Ω, a locally integrable function F(·,y,λ) on Ω is called a fundamental solution for an operator A+λI if and only if
(A+λI)F(x,y,λ)=δ(x-y).

Equation (2.1) holds in the sense of distributions, that is,

∫ΩF(x,y,λ)(A0′+q(x)+λI)φ(x)dx=φ(y)
for all φ∈C0∞(Ω), where

A0′·=∑|α|≤2m(-1)|α|Dα(aα(x)·)
is the transpose of A0.

We will use the following result.

Proposition 2.2.

There exists λ0>0 such that for any λ≥λ0, the differential operator A0+λI has a fundamental solution F0(x,y,λ). Furthermore, for any multi-index α, 0≤|α|≤2m-1, there are constants C0>0, δ>0 such that the following estimates hold:

We will look for the fundamental solution F(x,y,λ) of the operator A0(x,D)+q(x)+λI, for λ positive and large enough, as a solution of the integral equation

F(x,y,λ)=F0(x,y,λ)-∫ΩF0(x,u,λ)q(u)F(u,y,λ)du,
where F0(x,y,λ) is the fundamental solution of the operator A0+λI. By Proposition 2.2, F0(·,y,λ) exists and belongs (at least) to L1(Ω) uniformly with respect to y from Ω.

We need the following lemma, which may have interest of its own right.

Lemma 2.3.

Assume that q satisfies condition (1.4), then there is λ0>0 such that for all λ≥λ0 the fundamental solution F(x,y,λ) exists as a solution of the integral equation (2.4) and satisfies the following estimates:

with some positive constant C, where δ is as in Proposition 2.2 and x,y∈Ω.Proof.

We solve the integral equation (2.4) by iterations. For any j≥1, we denote
Fj(x,y,λ)=-∫ΩF0(x,u,λ)q(u)Fj-1(u,y,λ)du.
We will prove by induction that there is λ0>0 such that for all λ≥λ0 and for each j=0,1,2,…|Fj(x,y,λ)|≤C02jVn(|x-y|)e-(δ/2)|x-y|λ1/2m,
where x,y∈Ω, C0 is as in the Proposition 2.2 and Vn is defined as
Vn(|x-y|)={|x-y|2m-n,2m<n,1+|log(|x-y|λ1/2m)|,2m=n,λ(n-2m)/2m,2m>n.
It is clear that for j=0 estimate (2.6) holds. And it is also clear that (2.6) holds for the case when 2m>n for each j=1,2,…, by choosing λ0 large enough.

In the case 2m≤n (considering two possibilities |x-u|≥|u-y| and |x-u|≤|u-y|) in order to prove (2.6) it is enough to prove that there exists λ0>0 such that
C0∫ΩVn(|x-u|)|q(u)|e-(δ/2)|x-u|λ1/2mdu≤12
for all x,y∈Ω and λ≥λ0, where constant C0 is as in Proposition 2.2.

Indeed, since for 2m<n we have
C0∫ΩVn(|x-u|)|q(u)|e-(δ/2)|x-u|λ1/2mdu=C0∫Ωωn(|x-u|)|q(u)|e-(δ/2)|x-u|λ1/2mdu,
where ωn is as in (1.4), then we can estimate the left-hand side of (2.8) as follows. If |q(u)|>R then the integrals in the latter equality tend to zero as R→∞. The reason is due to that condition (1.4) the measure of the set {u∈Ω:|q(u)|>R} tends to zero as R→∞. If |q(u)|<R then this integral can be estimated by
R∫ΩVn(|x-u|)e-(δ/2)|x-u|λ1/2mdu≤CRλ,
where some positive constant C depends only on δ and dimension n.

In the case n=2m (considering two possibility |x-y|≤λ-1/2m and |x-y|≥λ-1/2m) it can be proved thatC0∫ΩVn(|x-u|)|q(u)|e-(δ/2)|x-u|λ1/2mdu≤C0′(∫Ωωn(|x-u|)|q(u)|e-(δ/2)|x-u|λ1/2mdu+logλ∫Ω|q(u)|e-(δ/2)|x-u|λ1/2mdu).
Then, instead of estimate (2.10) we obtain
R∫ΩVn(|x-u|)e-(δ/2)|x-u|λ1/2mdu≤CRlogλλ.

Combining these two facts (including (2.10) and (2.12)), (2.8) and choosing appropriately R (with respect to λ) we may conclude that inequality (2.6) is proved. Since the solution F(x,y,λ) of the integral equation (2.4) is given by the series
F(x,y,λ)=∑j=0∞Fj(x,y,λ)
the estimates (2.6) prove also Lemma 2.3.

As a consequence of Lemma 2.3 and Proposition 2.2 we can obtain the estimates for the derivatives of order |α|≤2m-1 of the fundamental solution F(x,y,λ). For the derivatives of F(x,y,λ) we use the following representation:

DxαF(x,y,λ)=DxαF0(x,y,λ)-∫ΩDxαF0(x,u,λ)q(u)F(u,y,λ)du.
The following corollary holds.

Corollary 2.4.

Assume that q satisfies the condition (1.4), then for the derivatives of the fundamental solution F(x,y,λ) of order |α|≤2m-1 the following estimates hold:

for some constant C>0, for all x,y∈Ω and λ≥λ0 (where λ0 and δ are as in Lemma 2.3).

The proof of the corollary follows immediately from the integral representation for the derivatives of F(x,y,λ), estimates for the derivatives of F0(x,y,λ) in Proposition 2.2, estimates for F(x,y,λ) in Lemma 2.3, and estimates for the kernels with weak singularities.

Let us note that the fundamental solution F(·,y,λ), which was obtained in Lemma 2.3, belongs (at least) to L1(Ω) uniformly with respect to y from Ω.

Now we are in the position to introduce the Green's function of the operator AF+λI. If λ is sufficiently large then the operator AF+λI is positive and its inverse

(AF+λI)-1:L2(Ω)→L2(Ω)
is a bounded operator. It is also an integral operator with kernel denoted by G(x,y,λ). If we use for this integral operator the symbol Ĝ(λ) then we have

(AF+λI)Ĝ(λ)=I,Ĝ(λ)(AF+λI)=I,G(x,y,λ)=G(y,x,λ)¯.

Definition 2.5.

The kernel G(x,y,λ) of the integral operator Ĝ(λ) is called the Green's function of the operator AF+λI.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

For τ>0, let Ωτ and Ωτ/2 be compact sets, each of them having a smooth boundary, with Ωτ⊂Ωτ/2⊂Ω such that
d(Ωτ,∂Ω)=τ,d(Ωτ/2,∂Ω)=τ2,d(Ωτ,∂Ωτ/2)=τ2.
Here d(X,Y) denotes the distance between the sets X and Y.

Let F(x,y,λ) be a fundamental solution of the operator AF+λI for x,y∈Ω and λ sufficiently large. We choose the function χ∈C0∞(Ω) such that
χ(x)={1,x∈Ωτ,0,x∈Ω∖Ωτ/2,
and set
E(x,y,λ)=χ(x)F(x,y,λ).
By this equation the function E(x,y,λ) is well defined for all x,y∈Ω. Clearly, E(x,y,λ)=F(x,y,λ) for x∈Ωτ, y∈Ω. We will show that E(x,y,λ) is a parametrix for AF+λI. To prove this, let us introduce the function
Q(x,y,λ):=G(x,y,λ)-E(x,y,λ)
and corresponding integral operator with kernel Q(x,y,λ)Q̂(λ):=Ĝ(λ)-Ê(λ),
where Ê(λ) and Ĝ(λ) are integral operators in L2(Ω) with kernels E(x,y,λ) and G(x,y,λ), respectively. Then it follows from (2.20) that
(AF+λI)Ê(λ)=I+P1̂(λ),
where
P1̂(λ)=-(AF+λI)Q̂(λ),Q̂(λ)=-Ĝ(λ)P1̂(λ).
If we denote by P1(x,y,λ) the kernel of the integral operator P1̂(λ), then it follows from (2.24) that for any f∈L2(Ω),
Q̂(λ)f(x)=-∫Ω(∫ΩG(x,u,λ)P1(u,y,λ)du)f(y)dy
and the kernel Q(x,y,λ) (see (2.20)) has the form
Q(x,y,λ)=-∫ΩG(x,u,λ)P1(u,y,λ)du,
where x,y∈Ω. As a matter of fact we cannot characterize and estimate the kernel P1(x,y,λ) 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:
(Ax(x,D)+λI)E(x,y,λ)=χ(x)δ(x-y)+P(x,y,λ),
where x∈Ω (y∈Ω is considered here as a parameter) and λ sufficiently large. The function P(x,y,λ) in (2.27) will be of the form
P(x,y,λ)=∑α>0Dαχ(x)α!A0(α)(x,D)F(x,y,λ)
with the differential operator A0(α)(x,D) having the symbol A0(α)(x,-iξ)=∂ξαA0(x,-iξ). It is the polynomial in ξ∈ℝn of order ≤2m-1 and therefore the differential operators A0(α)(x,D) are of order ≤2m-1. This fact allows us to estimate the function P(x,y,λ) (in comparison with P1). Indeed, by the choice of χ, Dαχ≠0 only on the set Ωτ/2∖Ωτ and therefore the representation (2.28) and Corollary 2.4 imply that the following estimate holds:
|P(x,y,λ)|≤C|x-y|1-ne-(δ/2)|x-y|λ1/2m
for all x,y∈Ω and with δ>0 as in Corollary 2.4.

Now we need the following lemma.

Lemma 2.6.

For all x,y∈Ωχ(y)G(x,y,λ)=χ(x)F(x,y,λ)-∫ΩG(x,u,λ)P(u,y,λ)du,
where P is as in (2.28) and χ is defined as in (2.19).

Proof.

We can rewrite (2.27) in the operator form as
(AF+λI)Ê(λ)=χI+P̂(λ)
or (using (2.20))
P̂(λ)=(1-χ)I-(AF+λI)Q̂(λ).
The latter equation implies
Q̂(λ)=Ĝ(λ)((1-χ)I)-Ĝ(λ)P̂(λ),
and therefore (using (2.20) again)
Ĝ(λ)(χI)=Ê(λ)-Ĝ(λ)P̂(λ).
But this is equivalent to (2.30). Thus, this lemma is proved.

In order to finish the proof of Theorem 1.1 let us introduce new functions F̃ and G̃ which are obtained from F and G multiplying by

e(δ/4)|x-y|λ1/2m{|x-y|n-2m,2m<n,(1+|log(|x-y|λ1/2m)|)-1,2m=n,λ(2m-n)/2m,2m>n,
respectively, where δ 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 2m<n, the cases 2m≥n can be considered similarly)

supx,y∈Ω|χ(y)G̃(x,y,λ)|≤supx,y∈Ω|F̃(x,y,λ)|+supx,y∈Ω|G̃(x,y,λ)|×supx,y∈Ω(|x-y|n-2m∫Ω|x-u|2m-n|u-y|1-ne-(δ/4)|u-y|λ1/2mdu).
Considering two possibilities |x-u|≤|u-y| and |x-u|≥|u-y| the value in the latter brackets can be estimated from above by

C∫Ω|u-y|1-ne-(δ/4)|u-y|λ1/2mdu≤Cλ1/2m.
This estimate allows us to get from (2.36) that

supx,y∈Ω|G̃(x,y,λ)|≤supx,y∈Ω|F̃(x,y,λ)|+Cλ1/2msupx,y∈Ω|G̃(x,y,λ)|.
Since

supx,y∈Ω|F̃(x,y,λ)|<∞,
then for λ large enough (2.38) yields

supx,y∈Ω|G̃(x,y,λ)|<∞.
Thus, Theorem 1.1 is completely proved.

3. Convergence of Fourier Series

Without loss of generality, we assume in this section that AF is positive. Then by the J. von Neumann spectral theorem for AF+μI, where μ≥λ0 with λ0 as in Theorem 1.1, the following representation holds:

(AF+μI)σ=∫0∞(λ+μ)σdEλ,
where σ is real and {Eλ}λ=0∞ is the spectral resolution corresponding to the self-adjoint operator AF. The domain of the operator (3.1) is defined by

D(AFσ)={f∈L2(Ω):∫0∞λ2σd(Eλf,f)L2(Ω)<∞}.
In our case (in the case of pure discrete spectrum), the spectral projector Eλ has the form

Eλf(x)=∑λk<λfkuk(x),
where fk=(f,uk)L2(Ω) are the Fourier coefficients of f with respect to the system {uk(x)}k=1∞. Hence relations (3.1) and (3.2) become

AFσf(x)=∑k=1∞λkσfkuk(x),D(AFσ)={f∈L2(Ω):∑k=1∞λk2σ|fk|2<∞}.
In addition, we need a special representation of the negative fractional powers of AF. If we assume that 0<τ<1 then using well-known properties of Euler beta-function, one can obtain

(AF+μI)-τ=sinτππ∫0∞t-τĜ(μ+t)dt,
where Ĝ(μ+t) is the integral operator with kernel G(x,y,μ+t) from Section 2. This representation shows that operator (3.6) is also integralwith kernel denoted by Gτ(x,y,μ). Using Theorem 1.1 of present work and well-known technique (see, e.g., [6]) it is not so difficult to prove the following estimates

|Gτ(x,y,μ)|≤Ce-(δ/2)|x-y|μ1/2m{|x-y|2mτ-n,2mτ<n,1+|log|x-y||,2mτ=n,1,2mτ>n,
where x,y∈Ω, δ is as in Theorem 1.1 and the constant C>0 depends on Ω.

The following main lemma holds.

Lemma 3.1.

For any function f∈L2(Ω) and for σ>n/4m∥(AF+μI)-σf∥L∞(Ω)≤Cμn/4m-σ∥f∥L2(Ω),
where μ≥λ0 with λ0 as in Theorem 1.1.

Proof.

For any σ>n/4m, we write
σ=τ1+τ+⋯+τm,0<τj<1,j=1,2,…,m
and represent the operator (AF+μI)-σ as the product
(AF+μI)-σ=(AF+μI)-τ1·(AF+μI)-τ2·…·(AF+μI)-τm.
Then, by applying the estimates (3.7) and Lemma 1 in [3], we arrive at (3.8). This completes the proof.

Corollary 3.2.

Assume that σ>n/4m. There is a constant C>0 depending only on Ω, such that the estimate
∑k=1∞|uk(x)|2(λk+μ)2σ≤Cμn/2m-2σ
holds uniformly in x∈Ω and μ≥λ0.

Proof.

By the spectral theorem and relation (3.4), we can rewrite inequality (3.8) in the form
|∑k=1∞(λk+μ)-σfkuk(x)|≤Cμn/4m-σ(∑k=1∞|fk|2)1/2,
where fk are the Fourier coefficients of f with respect to the system {uk(x)}k=1∞. 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
∑λ≤λk<2λ|uk(x)|2≤Cλn/2m
which holds uniformly in x∈Ω and λ large enough. Indeed, from (3.12) we have
∑2λ≤λk+μ<3λ|uk(x)|2(λk+μ)2σ≤Cμ(n/2m)-2σ
uniformly in x∈Ω. If we chose now μ=λ≥λ0 then one can immediately obtain (3.13).

Now we are ready to prove Theorem 1.2.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.2</xref>.

Using the representation (3.4), the inequality (3.11), and the Cauchy-Schwarz-Bunyakovsk 2 inequality, we obtain
∑k=1∞|fkuk(x)|≤(∑k=1∞|fk|2(λk+μ)2σ)1/2(∑k=1∞|uk(x)|2(λk+μ)-2σ)1/2≤Cμn/4m-σ(∑k=1∞|uk(x)|2(λk+μ)2σ)1/2≤C(∑k=1∞|fk|2λk2σ)1/2
uniformly in x∈Ω and for any fixed μ≥λ0. Now the desired assertion follows from (3.5). Theorem 1.2 is completely proved.

The estimate (3.13) allows us to obtain a bit more precise result than in Theorem 1.2. Namely, the following corollary holds.

Corollary 3.4.

Assume that the function f satisfies the condition
∑l=0∞(∑2l≤λk<2l+1|fk|2λkn/2m)1/2<∞,
where fk=(f,uk)L2(Ω) are the Fourier coefficients of f with respect to the system {uk(x)}k=1∞, then the Fourier series (1.12) converges absolutely and uniformly on Ω.

Let us assume now that the potential q(x) satisfies the conditions

q(x)∈L2(Ω),4m>n,q(x)∈Lp(Ω),p>2,4m=n,q(x)∈Ln/2m(Ω),4m<n,
then it is not so difficult to see that for the case 4m≥n conditions (3.17) imply the condition (1.4). For the case 4m<n this condition (3.17) for q must be considered in addition to the condition (1.4). The following result is valid.

Theorem 3.5.

Suppose that the potential q(x) satisfies conditions (3.17), then for any function f∈W∘22m(Ω),
limλ→+∞∥f-Eλf∥W22m(Ω)=0,
where Eλ is given by (3.3).

Proof.

Using the Sobolev embedding theorem we easily conclude that conditions (3.17) imply the following inclusion:
W∘22m(Ω)⊂D(AF).
And for any f(x)∈W∘22m(Ω) the following inequality holds:
∥AFf∥L2(Ω)≤C∥f∥W22m(Ω).
Moreover, we may assert that the operator AF+μI is invertible for μ large enough. Indeed, since the function
h(x):=((A0)F+μI+q(x))f(x),f(x)∈W∘22m(Ω),
belongs to L2(Ω), we have the representation for f(x)f(x)=-((A0)F+μI)-1(qf)(x)+((A0)F+μI)-1(h)(x),
where (A0)F+μI denotes the Friedrichs self-adjoint extension for A0+μI in L2(Ω). Using again the Sobolev embedding theorem and conditions (3.17) we may conclude that the functions h and qf belong to L2(Ω). The results of [6] yield that the operator (A0)F+μI is invertible with small norm for its inverse operator (if μ is large enough). This fact and the latter equality imply that for μ large enough the operator AF+μI is also invertible and for any h(x)∈L2(Ω) we have the following inequality:
∥(AF+μI)-1h∥W22m(Ω)≤C∥h∥L2(Ω).

Now let f(x)∈W∘22m(Ω). Then (3.23) implies
∥f-Eλf∥W22m(Ω)=∥(AF+μI)-1((AF+μI)f-Eλ(AF+μI)f)∥W22m(Ω)≤C∥h-Eλh∥L2(Ω)→0,λ→+∞,
where h(x)=(AF+μI)f(x) belongs to L2(Ω) 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 4m>n. Then for any f(x)∈W∘2s(Ω) with s>n/2limλ→+∞Eλf(x)=f(x)
holds uniformly in x∈Ω.

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 [2], we use the symbol B̃2,ps(Ω)={f:f∈B2,ps(Rn),suppf⊂Ω¯}.

Theorem 3.7.

Assume that the potential q(x) belongs to Sobolev space W22ml(Ω), where l=[n/4m] is an entire part of n/4m, then for any function f from Besov space B̃2,1n/2(Ω) the Fourier series (1.12) converges absolutely and uniformly on the domain Ω.

Proof.

Using the Sobolev embedding theorem and the following representation:
AFl+1f(x)=∑r=0l+1Cr,l(A0)Fr(ql+1-r(x)f(x)),
where Cr,l are some constants, we can conclude that the condition for the potential q(x) implies the following inclusion:
W∘22m(l+1)(Ω)⊂D(AFl+1).
Then using the results of [2] (see Theorem 4.3.2/1) we may conclude that
B̃2,22m(l+1)(Ω)⊂W∘22m(l+1)(Ω)⊂D(AF(l+1)).
Consequently, by Theorem 4.3.2/2 of [2] and by Peetre's method of real interpolation (see, e.g., [2]), we have
B̃2,1n/2(Ω)=(L2(Ω),B̃2,22m(l+1)(Ω))n/4m(l+1),1⊂(L2(Ω),D(AF(l+1)))n/4m(l+1),1.
But the latter space is the interpolation space Dn/4m,1 of Peetre (see [26]) for the elliptic differential operator of order 2m. Since estimate (3.13) for the spectral function holds in our case, we can apply the results of [26] and conclude that the proof of this theorem is complete.

Remark 3.8.

If n is even then the statement of this theorem holds for any function f(x) from Besov space B∘2,1n/2(Ω) due to the equality (see Theorem 4.3.2/1 of [2])
B̃2,1n/2(Ω)=B∘2,1n/2(Ω).
And the Sobolev embedding theorem for Besov spaces (see, e.g., [2]) implies that this theorem also holds for any function f(x) from Sobolev space W∘2s(Ω) with s>n/2 and arbitrary integer n.

Acknowledgment

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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