Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space

Abstract

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⊂ℝ^𝑛, 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 ℝ^𝑛 (𝑛≥1) with smooth boundary. We consider on Ω an elliptic differential operator of the form

𝐴=𝐴₀(𝑥,𝐷)+𝑞(𝑥)⋅, (1.1) where

𝐴₀(𝑥,𝐷)=_{|𝛼|≤2𝑚}𝑎_𝛼(𝑥)𝐷^𝛼 (1.2) is a formally self-adjoint differential operator of order 2𝑚,𝑚=1,2,…. Here 𝐷^𝛼=𝐷^𝛼1₁⋅𝐷^𝛼2₂⋅⋯⋅𝐷^𝛼𝑛_𝑛 with 𝐷_𝑗=−𝑖(𝜕/𝜕𝑥_𝑗). The coefficients 𝑎_𝛼(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 𝑎_𝛼(𝑥) are real valued for |𝛼|=2𝑚 and this operator 𝐴₀ satisfies the uniform ellipticity condition

_|𝛼|=2𝑚𝑎_𝛼(𝑥)𝜉^𝛼≥𝜈|𝜉|^2𝑚 (1.3) with some constant 𝜈>0, for all 𝑥∈Ω and all 𝜉∈ℝ^𝑛. We assume that the potential 𝑞(𝑥) is a real-valued 𝐿¹(Ω)-function satisfying the generalized Kato condition, that is,

sup_𝑥∈ΩΩ|𝑞(𝑦)|𝜔_𝑛(|𝑥−𝑦|)𝑑𝑦<∞, (1.4) where function 𝜔_𝑛(𝑡) for 𝑡>0 is defined by

𝜔_𝑛𝑡(𝑡)=^2𝑚−𝑛,2𝑚<𝑛,1+|log𝑡|,2𝑚=𝑛,1,2𝑚>𝑛. (1.5)

For 𝑠>0 we denote by 𝑊^𝑠₂(Ω) the 𝐿²-based Sobolev space, where 𝑠 indicates the “degree” of the smoothness; by 𝑊^∘𝑠₂(Ω) we denote the closure of 𝐶^∞₀(Ω) in 𝑊^𝑠₂(Ω). We denote also by 𝐵^𝑠_2,𝑝(Ω),1≤𝑝≤∞, Besov space (where 𝑠 indicates the smoothness) with the same notation for 𝐵^∘𝑠_2,𝑝(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‖²_𝐿2(Ω)≤𝐶₀𝑁_𝛿(|𝑞|^1/2)‖𝑓‖²_{𝑊𝑚2(Ω)}+𝐶_𝛿𝑁₁(|𝑞|^1/2)‖𝑓‖²_𝐿2(Ω), (1.6) where the constant 𝐶₀ depends only on 𝑚 and 𝑛, the constant 𝐶_𝛿 depends only on 𝑚, 𝑛 and 𝛿, and the value 𝑁_𝛿(|𝑞|^1/2) is defined by

𝑁_𝛿(|𝑞|^1/2)∶=sup_{𝑥∈Ω𝑦∈Ω,|𝑥−𝑦|<𝛿}|𝑞(𝑦)|𝜔_𝑛(|𝑥−𝑦|)𝑑𝑦, (1.7) where 𝜔_𝑛 is as in (1.5).

Since the domain Ω is bounded then 𝑁_𝛿(|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‖²_{𝑊𝑚2(Ω)}−𝐶‖𝑓‖²_𝐿2(Ω) (1.8) for all 𝑓∈𝐶^∞₀(Ω). Since 𝐴_𝜇=𝐴+𝜇𝐼 is positive for sufficiently large 𝜇 it has a positive self-adjoint Friedrichs extension (𝐴_𝜇)_𝐹 such that

𝒟((𝐴_𝜇)_𝐹)⊂𝑊^∘𝑚₂(Ω). (1.9) We define the Friedrichs extension of 𝐴=𝐴_𝜇−𝜇𝐼 to be 𝐴_𝐹=(𝐴_𝜇)_𝐹−𝜇𝐼 such that

𝒟(𝐴_𝐹)⊂𝑊^∘𝑚₂(Ω). (1.10) The domain of 𝐴_𝐹 is given by

𝒟(𝐴_𝐹)={𝑓∈𝑊^∘𝑚₂(Ω)∣𝐴𝑓∈𝐿²(Ω)}. (1.11)

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

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

𝑓=^∞_𝑘=1𝑓_𝑘𝑢_𝑘(𝑥), (1.12) where 𝑓_𝑘=(𝑓,𝑢_𝑘)_𝐿²_(Ω) are the Fourier coefficients of 𝑓 with respect to the system {𝑢_𝑘(𝑥)}^∞_𝑘=1.

The study of elliptic differential operators with smooth coefficients on a bounded domain Ω⊂ℝ^𝑛 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 𝐿^𝑝, 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 𝐿^𝑝 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 𝐴_𝐹+𝜆𝐼 exists and satisfies the following estimates:(1)2𝑚<𝑛,|𝐺(𝑥,𝑦,𝜆)|≤𝐶|𝑥−𝑦|^2𝑚−𝑛𝑒^{−𝛿|𝑥−𝑦|𝜆1/2𝑚},(2)2𝑚=𝑛,|𝐺(𝑥,𝑦,𝜆)|≤𝐶(1+|log(|𝑥−𝑦|𝜆^1/2𝑚)|)𝑒^{−𝛿|𝑥−𝑦|𝜆1/2𝑚},(3)2𝑚>𝑛,|𝐺(𝑥,𝑦,𝜆)|≤𝐶𝜆^{(𝑛−2𝑚)/2𝑚}𝑒^{−𝛿|𝑥−𝑦|𝜆1/2𝑚}for all 𝑥,𝑦∈Ω and 𝜆≥𝜆₀.

Without loss of generality, in the following theorem we assume that 𝐴_𝐹 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 𝐴^𝜎_𝐹 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 𝐴_𝐹+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). (2.1)

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

_Ω𝐹(𝑥,𝑦,𝜆)(𝐴^′₀+𝑞(𝑥)+𝜆𝐼)𝜑(𝑥)𝑑𝑥=𝜑(𝑦) (2.2) for all 𝜑∈𝐶^∞₀(Ω), where

𝐴^′₀⋅=_{|𝛼|≤2𝑚}(−1)^|𝛼|𝐷^𝛼(𝑎_𝛼(𝑥)⋅) (2.3) 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: (1)2𝑚<𝑛+|𝛼|,|𝐷^𝛼𝐹₀(𝑥,𝑦,𝜆)|≤𝐶₀|𝑥−𝑦|^{2𝑚−𝑛−|𝛼|}𝑒^{−𝛿|𝑥−𝑦|𝜆1/2𝑚},(2)2𝑚=𝑛+|𝛼|,|𝐷^𝛼𝐹₀(𝑥,𝑦,𝜆)|≤𝐶₀(1+|log(|𝑥−𝑦|𝜆^1/2𝑚)|)𝑒^{−𝛿|𝑥−𝑦|𝜆1/2𝑚},(3)2𝑚>𝑛+|𝛼|,|𝐷^𝛼𝐹₀(𝑥,𝑦,𝜆)|≤𝐶₀𝜆^{(𝑛+|𝛼|−2𝑚)/2𝑚}𝑒^{−𝛿|𝑥−𝑦|𝜆1/2𝑚},for all 𝑥,𝑦∈Ω and 𝜆≥𝜆₀.

The proof of Proposition 2.2 can be found in [6].

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, (2.4) 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: (1)2𝑚<𝑛,|𝐹(𝑥,𝑦,𝜆)|≤𝐶|𝑥−𝑦|^2𝑚−𝑛𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚},(2)2𝑚=𝑛,|𝐹(𝑥,𝑦,𝜆)|≤𝐶(1+|log(|𝑥−𝑦|𝜆^1/2𝑚)|)𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚},(3)2𝑚>𝑛,|𝐹(𝑥,𝑦,𝜆)|≤𝐶𝜆^{(𝑛−2𝑚)/2𝑚}𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚},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 𝐹_𝑗(𝑥,𝑦,𝜆)=−_Ω𝐹₀(𝑥,𝑢,𝜆)𝑞(𝑢)𝐹_𝑗−1(𝑢,𝑦,𝜆)𝑑𝑢. (2.5) We will prove by induction that there is 𝜆₀>0 such that for all 𝜆≥𝜆₀ and for each 𝑗=0,1,2,…|𝐹_𝑗𝐶(𝑥,𝑦,𝜆)|≤⁰2^𝑗𝑉_𝑛(|𝑥−𝑦|)𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚}, (2.6) where 𝑥,𝑦∈Ω, 𝐶₀ is as in the Proposition 2.2 and 𝑉_𝑛 is defined as 𝑉_𝑛(|𝑥−𝑦|)=|𝑥−𝑦|^2𝑚−𝑛,2𝑚<𝑛,1+|log(|𝑥−𝑦|𝜆^1/2𝑚𝜆)|,2𝑚=𝑛,_{(𝑛−2𝑚)/2𝑚},2𝑚>𝑛. (2.7) 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 𝐶_0Ω𝑉_𝑛(|𝑥−𝑢|)|𝑞(𝑢)|𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}1𝑑𝑢≤2 (2.8) for all 𝑥,𝑦∈Ω and 𝜆≥𝜆₀, where constant 𝐶₀ is as in Proposition 2.2.
Indeed, since for 2𝑚<𝑛 we have 𝐶_0Ω𝑉_𝑛(|𝑥−𝑢|)|𝑞(𝑢)|𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑑𝑢=𝐶_0Ω𝜔_𝑛(|𝑥−𝑢|)|𝑞(𝑢)|𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑑𝑢, (2.9) where 𝜔_𝑛 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_Ω𝑉_𝑛(|𝑥−𝑢|)𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑅𝑑𝑢≤𝐶𝜆, (2.10) 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 that𝐶_0Ω𝑉_𝑛(|𝑥−𝑢|)|𝑞(𝑢)|𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑑𝑢≤𝐶^′₀(_Ω𝜔_𝑛(|𝑥−𝑢|)|𝑞(𝑢)|𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑑𝑢+log𝜆_Ω|𝑞(𝑢)|𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑑𝑢). (2.11) Then, instead of estimate (2.10) we obtain 𝑅_Ω𝑉_𝑛(|𝑥−𝑢|)𝑒^{−(𝛿/2)|𝑥−𝑢|𝜆1/2𝑚}𝑑𝑢≤𝐶𝑅log𝜆𝜆. (2.12)
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=0𝐹_𝑗(𝑥,𝑦,𝜆) (2.13) 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:

𝐷^𝛼_𝑥𝐹(𝑥,𝑦,𝜆)=𝐷^𝛼_𝑥𝐹₀(𝑥,𝑦,𝜆)−_Ω𝐷^𝛼_𝑥𝐹₀(𝑥,𝑢,𝜆)𝑞(𝑢)𝐹(𝑢,𝑦,𝜆)𝑑𝑢. (2.14) 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: (1)2𝑚<𝑛+|𝛼|,|𝐷^𝛼_𝑥𝐹(𝑥,𝑦,𝜆)|≤𝐶|𝑥−𝑦|^{2𝑚−𝑛−|𝛼|}𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚},(2)2𝑚=𝑛+|𝛼|,|𝐷^𝛼_𝑥𝐹(𝑥,𝑦,𝜆)|≤𝐶(1+|log(|𝑥−𝑦|𝜆^1/2𝑚)|)𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚},(3)2𝑚>𝑛+|𝛼|,|𝐷^𝛼_𝑥𝐹(𝑥,𝑦,𝜆)|≤𝐶𝜆^{(𝑛+|𝛼|−2𝑚)/2𝑚}𝑒^{−(𝛿/2)|𝑥−𝑦|𝜆1/2𝑚},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 𝐴_𝐹+𝜆𝐼. If 𝜆 is sufficiently large then the operator 𝐴_𝐹+𝜆𝐼 is positive and its inverse

(𝐴_𝐹+𝜆𝐼)⁻¹∶𝐿²(Ω)→𝐿²(Ω) (2.15) 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 𝐺(𝜆) then we have

(𝐴_𝐹+𝜆𝐼)𝐺(𝜆)=𝐼,𝐺(𝜆)(𝐴_𝐹+𝜆𝐼)=𝐼,𝐺(𝑥,𝑦,𝜆)=𝐺(𝑦,𝑥,𝜆). (2.16)

Definition 2.5. The kernel 𝐺(𝑥,𝑦,𝜆) of the integral operator 𝐺(𝜆) is called the Green's function of the operator 𝐴_𝐹+𝜆𝐼.

Proof of Theorem 1.1. For 𝜏>0, let Ω^𝜏 and Ω^𝜏/2 be compact sets, each of them having a smooth boundary, with Ω^𝜏⊂Ω^𝜏/2⊂Ω such that 𝑑(Ω^𝜏,𝜕Ω)=𝜏,𝑑(Ω^𝜏/2𝜏,𝜕Ω)=2,𝑑(Ω^𝜏,𝜕Ω^𝜏/2𝜏)=2. (2.17) 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 𝐴_𝐹+𝜆𝐼 for 𝑥,𝑦∈Ω and 𝜆 sufficiently large. We choose the function 𝜒∈𝐶^∞₀(Ω) such that 𝜒(𝑥)=1,𝑥∈Ω^𝜏,0,𝑥∈Ω⧵Ω^𝜏/2, (2.18) and set 𝐸(𝑥,𝑦,𝜆)=𝜒(𝑥)𝐹(𝑥,𝑦,𝜆). (2.19) 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∈Ω^𝜏, 𝑦∈Ω. We will show that 𝐸(𝑥,𝑦,𝜆) is a parametrix for 𝐴_𝐹+𝜆𝐼. To prove this, let us introduce the function 𝑄(𝑥,𝑦,𝜆)∶=𝐺(𝑥,𝑦,𝜆)−𝐸(𝑥,𝑦,𝜆) (2.20) and corresponding integral operator with kernel 𝑄(𝑥,𝑦,𝜆)𝑄(𝜆)∶=𝐺(𝜆)−𝐸(𝜆), (2.21) where 𝐸(𝜆) and 𝐺(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 (𝐴_𝐹+𝜆𝐼)𝐸(𝜆)=𝐼+𝑃₁(𝜆), (2.22) where 𝑃₁(𝜆)=−(𝐴_𝐹𝑃+𝜆𝐼)𝑄(𝜆),𝑄(𝜆)=−𝐺(𝜆)₁(𝜆). (2.23)(2.24) If we denote by 𝑃₁(𝑥,𝑦,𝜆) the kernel of the integral operator 𝑃₁(𝜆), then it follows from (2.24) that for any 𝑓∈𝐿²(Ω), 𝑄(𝜆)𝑓(𝑥)=−_Ω(_Ω𝐺(𝑥,𝑢,𝜆)𝑃₁(𝑢,𝑦,𝜆)𝑑𝑢)𝑓(𝑦)𝑑𝑦 (2.25) 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, (2.26) 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: (𝐴_𝑥(𝑥,𝐷)+𝜆𝐼)𝐸(𝑥,𝑦,𝜆)=𝜒(𝑥)𝛿(𝑥−𝑦)+𝑃(𝑥,𝑦,𝜆), (2.27) 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>0𝐷^𝛼𝜒(𝑥)𝐴𝛼!^0(𝛼)(𝑥,𝐷)𝐹(𝑥,𝑦,𝜆) (2.28) with the differential operator 𝐴₀^(𝛼)(𝑥,𝐷) having the symbol 𝐴₀^(𝛼)(𝑥,−𝑖𝜉)=𝜕^𝛼_𝜉𝐴₀(𝑥,−𝑖𝜉). It is the polynomial in 𝜉∈ℝ^𝑛 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, 𝐷^𝛼𝜒≠0 only on the set Ω^𝜏/2⧵Ω^𝜏 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} (2.29) 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, (2.30) 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 (𝐴_𝐹+𝜆𝐼)𝐸(𝜆)=𝜒𝐼+𝑃(𝜆) (2.31) or (using (2.20)) 𝑃(𝜆)=(1−𝜒)𝐼−(𝐴_𝐹+𝜆𝐼)𝑄(𝜆). (2.32) The latter equation implies 𝑄(𝜆)=𝐺(𝜆)((1−𝜒)𝐼)−𝐺(𝜆)𝑃(𝜆), (2.33) and therefore (using (2.20) again) 𝐺(𝜆)(𝜒𝐼)=𝐸(𝜆)−𝐺(𝜆)𝑃(𝜆). (2.34) 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 𝐹 and 𝐺 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|𝜆^1/2𝑚)|)⁻¹𝜆,2𝑚=𝑛,_{(2𝑚−𝑛)/2𝑚},2𝑚>𝑛, (2.35) 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)

sup_{𝑥,𝑦∈Ω}|𝜒(𝑦)𝐺(𝑥,𝑦,𝜆)|≤sup_{𝑥,𝑦∈Ω}|𝐹(𝑥,𝑦,𝜆)|+sup_{𝑥,𝑦∈Ω}|𝐺(𝑥,𝑦,𝜆)|×sup_{𝑥,𝑦∈Ω}(|𝑥−𝑦|^𝑛−2𝑚_Ω|𝑥−𝑢|^2𝑚−𝑛|𝑢−𝑦|^1−𝑛𝑒^{−(𝛿/4)|𝑢−𝑦|𝜆1/2𝑚}𝑑𝑢). (2.36) 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|𝜆1/2𝑚}𝐶𝑑𝑢≤𝜆^1/2𝑚. (2.37) This estimate allows us to get from (2.36) that

sup_{𝑥,𝑦∈Ω}|𝐺(𝑥,𝑦,𝜆)|≤sup_{𝑥,𝑦∈Ω}|𝐶𝐹(𝑥,𝑦,𝜆)|+𝜆^1/2𝑚sup_{𝑥,𝑦∈Ω}|𝐺(𝑥,𝑦,𝜆)|. (2.38) Since

sup_{𝑥,𝑦∈Ω}|𝐹(𝑥,𝑦,𝜆)|<∞, (2.39) then for 𝜆 large enough (2.38) yields

sup_{𝑥,𝑦∈Ω}|𝐺(𝑥,𝑦,𝜆)|<∞. (2.40) Thus, Theorem 1.1 is completely proved.

3. Convergence of Fourier Series

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

(𝐴_𝐹+𝜇𝐼)^𝜎=^∞₀(𝜆+𝜇)^𝜎𝑑𝐸_𝜆, (3.1) where 𝜎 is real and {𝐸_𝜆}^∞_𝜆=0 is the spectral resolution corresponding to the self-adjoint operator 𝐴_𝐹. The domain of the operator (3.1) is defined by

𝐷(𝐴^𝜎_𝐹)={𝑓∈𝐿²(Ω)∶^∞₀𝜆^2𝜎𝑑(𝐸_𝜆𝑓,𝑓)_𝐿2(Ω)<∞}. (3.2) In our case (in the case of pure discrete spectrum), the spectral projector 𝐸_𝜆 has the form

𝐸_𝜆𝑓(𝑥)=_{𝜆𝑘<𝜆}𝑓_𝑘𝑢_𝑘(𝑥), (3.3) where 𝑓_𝑘=(𝑓,𝑢_𝑘)_𝐿²_(Ω) are the Fourier coefficients of 𝑓 with respect to the system {𝑢_𝑘(𝑥)}^∞_𝑘=1. Hence relations (3.1) and (3.2) become

𝐴^𝜎_𝐹𝑓(𝑥)=^∞_𝑘=1𝜆^𝜎_𝑘𝑓_𝑘𝑢_𝑘(𝑥),𝐷(𝐴^𝜎_𝐹)={𝑓∈𝐿²(Ω)∶^∞_𝑘=1𝜆_𝑘^2𝜎|𝑓_𝑘|²<∞}. (3.4)(3.5) In addition, we need a special representation of the negative fractional powers of 𝐴_𝐹. If we assume that 0<𝜏<1 then using well-known properties of Euler beta-function, one can obtain

(𝐴_𝐹+𝜇𝐼)^−𝜏=sin𝜏𝜋𝜋^∞₀𝑡^−𝜏𝐺(𝜇+𝑡)𝑑𝑡, (3.6) where 𝐺(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 𝐺_𝜏(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

|𝐺_𝜏(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, (3.7) 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‖(𝐴_𝐹+𝜇𝐼)^−𝜎𝑓‖_𝐿∞(Ω)≤𝐶𝜇^{𝑛/4𝑚−𝜎}‖𝑓‖_𝐿2(Ω), (3.8) where 𝜇≥𝜆₀ with 𝜆₀ as in Theorem 1.1.

Proof. For any 𝜎>𝑛/4𝑚, we write 𝜎=𝜏₁+𝜏+⋯+𝜏_𝑚,0<𝜏_𝑗<1,𝑗=1,2,…,𝑚 (3.9) and represent the operator (𝐴_𝐹+𝜇𝐼)^−𝜎 as the product (𝐴_𝐹+𝜇𝐼)^−𝜎=(𝐴_𝐹+𝜇𝐼)^−𝜏1⋅(𝐴_𝐹+𝜇𝐼)^−𝜏2⋅…⋅(𝐴_𝐹+𝜇𝐼)^−𝜏𝑚. (3.10) 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|𝑢_𝑘(𝑥)|²(𝜆_𝑘+𝜇)^2𝜎≤𝐶𝜇^{𝑛/2𝑚−2𝜎} (3.11) 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(𝜆_𝑘+𝜇)^−𝜎𝑓_𝑘𝑢_𝑘(𝑥)|≤𝐶𝜇^{𝑛/4𝑚−𝜎}(^∞_𝑘=1|𝑓_𝑘|²)^1/2, (3.12) where 𝑓_𝑘 are the Fourier coefficients of 𝑓 with respect to the system {𝑢_𝑘(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}|𝑢_𝑘(𝑥)|²≤𝐶𝜆^𝑛/2𝑚 (3.13) which holds uniformly in 𝑥∈Ω and 𝜆 large enough. Indeed, from (3.12) we have _{2𝜆≤𝜆𝑘+𝜇<3𝜆}|𝑢_𝑘(𝑥)|²(𝜆_𝑘+𝜇)^2𝜎≤𝐶𝜇^{(𝑛/2𝑚)−2𝜎} (3.14) 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 1.2. Using the representation (3.4), the inequality (3.11), and the Cauchy-Schwarz-Bunyakovsk 2 inequality, we obtain ^∞_𝑘=1|𝑓_𝑘𝑢_𝑘(𝑥)|≤(^∞_𝑘=1|𝑓_𝑘|²(𝜆_𝑘+𝜇)^2𝜎)^1/2(^∞_𝑘=1|𝑢_𝑘(𝑥)|²(𝜆_𝑘+𝜇)^−2𝜎)^1/2≤𝐶𝜇^{𝑛/4𝑚−𝜎}(^∞_𝑘=1|𝑢_𝑘(𝑥)|²(𝜆_𝑘+𝜇)^2𝜎)^1/2≤𝐶(^∞_𝑘=1|𝑓_𝑘|²𝜆_𝑘^2𝜎)^1/2 (3.15) 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}|𝑓_𝑘|²𝜆_𝑘^𝑛/2𝑚)^1/2<∞, (3.16) where 𝑓_𝑘=(𝑓,𝑢_𝑘)_𝐿²_(Ω) are the Fourier coefficients of 𝑓 with respect to the system {𝑢_𝑘(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)∈𝐿^𝑝(Ω),𝑝>2,4𝑚=𝑛,𝑞(𝑥)∈𝐿^𝑛/2𝑚(Ω),4𝑚<𝑛, (3.17) 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^∘₂^2𝑚(Ω), lim_𝜆→+∞‖𝑓−𝐸_𝜆𝑓‖_{𝑊22𝑚(Ω)}=0, (3.18) where 𝐸_𝜆 is given by (3.3).

Proof. Using the Sobolev embedding theorem we easily conclude that conditions (3.17) imply the following inclusion: 𝑊^∘₂^2𝑚(Ω)⊂𝐷(𝐴_𝐹). (3.19) And for any 𝑓(𝑥)∈𝑊^∘₂^2𝑚(Ω) the following inequality holds: ‖𝐴_𝐹𝑓‖_𝐿2(Ω)≤𝐶‖𝑓‖_{𝑊22𝑚(Ω)}. (3.20) Moreover, we may assert that the operator 𝐴_𝐹+𝜇𝐼 is invertible for 𝜇 large enough. Indeed, since the function ℎ(𝑥)∶=((𝐴₀)_𝐹+𝜇𝐼+𝑞(𝑥))𝑓(𝑥),𝑓(𝑥)∈𝑊^∘₂^2𝑚(Ω), (3.21) 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), (3.22) 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 𝐴_𝐹+𝜇𝐼 is also invertible and for any ℎ(𝑥)∈𝐿²(Ω) we have the following inequality: ‖(𝐴_𝐹+𝜇𝐼)⁻¹ℎ‖_{𝑊22𝑚(Ω)}≤𝐶‖ℎ‖_𝐿²_(Ω). (3.23)
Now let 𝑓(𝑥)∈𝑊^∘₂^2𝑚(Ω). Then (3.23) implies ‖𝑓−𝐸_𝜆𝑓‖_{𝑊22𝑚(Ω)}=‖(𝐴_𝐹+𝜇𝐼)⁻¹((𝐴_𝐹+𝜇𝐼)𝑓−𝐸_𝜆(𝐴_𝐹+𝜇𝐼)𝑓)‖_{𝑊22𝑚(Ω)}≤𝐶‖ℎ−𝐸_𝜆ℎ‖_𝐿2(Ω)→0,𝜆→+∞, (3.24) where ℎ(𝑥)=(𝐴_𝐹+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^∘𝑠₂(Ω) with 𝑠>𝑛/2lim_𝜆→+∞𝐸_𝜆𝑓(𝑥)=𝑓(𝑥) (3.25) 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 𝐵^𝑠_2,𝑝(Ω)={𝑓∶𝑓∈𝐵^𝑠_2,𝑝(𝑅^𝑛),supp𝑓⊂Ω}.

Theorem 3.7. Assume that the potential 𝑞(𝑥) belongs to Sobolev space 𝑊₂^2𝑚𝑙(Ω), where 𝑙=[𝑛/4𝑚] is an entire part of 𝑛/4𝑚, then for any function 𝑓 from Besov space 𝐵^𝑛/2_2,1(Ω) the Fourier series (1.12) converges absolutely and uniformly on the domain Ω.

Proof. Using the Sobolev embedding theorem and the following representation: 𝐴_𝐹^𝑙+1𝑓(𝑥)=^𝑙+1_𝑟=0𝐶_𝑟,𝑙(𝐴₀)^𝑟_𝐹(𝑞^{𝑙+1−𝑟}(𝑥)𝑓(𝑥)), (3.26) where 𝐶_𝑟,𝑙 are some constants, we can conclude that the condition for the potential 𝑞(𝑥) implies the following inclusion: 𝑊^∘₂^{2𝑚(𝑙+1)}(Ω)⊂𝐷(𝐴_𝐹^𝑙+1). (3.27) Then using the results of [2] (see Theorem 4.3.2/1) we may conclude that 𝐵^{2𝑚(𝑙+1)}_2,2(Ω)⊂𝑊^∘₂^{2𝑚(𝑙+1)}(Ω)⊂𝐷(𝐴_𝐹^(𝑙+1)). (3.28) Consequently, by Theorem 4.3.2/2 of [2] and by Peetre's method of real interpolation (see, e.g., [2]), we have 𝐵^𝑛/2_2,1(Ω)=(𝐿²(Ω),𝐵^{2𝑚(𝑙+1)}_2,2(Ω))_{𝑛/4𝑚(𝑙+1),1}⊂(𝐿²(Ω),𝐷(𝐴_𝐹^(𝑙+1)))_{𝑛/4𝑚(𝑙+1),1}. (3.29) But the latter space is the interpolation space 𝐷^𝑛/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^∘𝑛/2_2,1(Ω) due to the equality (see Theorem 4.3.2/1 of [2]) 𝐵^𝑛/2_2,1(Ω)=𝐵^∘𝑛/2_2,1(Ω). (3.30) 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^∘𝑠₂(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).