Reproducing kernels and minimal solutions of elliptic equations

Tomasz Łukasz Żynda; Jacek Józef Sadowski; Paweł Marian Wójcicki; Steven George Krantz

doi:10.1515/gmj-2022-2202

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Reproducing kernels and minimal solutions of elliptic equations

Tomasz Łukasz Żynda , Jacek Józef Sadowski , Paweł Marian Wójcicki and Steven George Krantz

Published/Copyright: November 11, 2022

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From the journal Georgian Mathematical Journal Volume 30 Issue 2

Abstract

Suppose that the set of square-integrable solutions of an elliptic equation which have a value at some given point equal to c is not empty. Then there is exactly one element with minimal L 2 -norm. Moreover, it is shown that this minimal element depends continuously on a domain of integration, i.e., on the set on which our solutions are defined, and on a weight of integration, i.e., on the deformation of an inner product. The theorems are proved using the theory of reproducing kernels and Hilbert spaces of square-integrable solutions of elliptic equations. We prove the existence of such a reproducing kernel using theory of Sobolev spaces. We generalize the well-known Ramadanov theorem. This is done in three different ways. Two of them are similar to the techniques used by I. Ramadanov and M. Skwarczyński(see [11, 14, 13]) , while the third method using weak convergence is new. Moreover, we show that our reproducing kernel depends continuously on a weight of integration. The idea of using the minimal norm property in such a proof is novel and, which is important, it needs the convergence of weights only almost everywhere.

Keywords: Reproducing kernel Hilbert space; elliptic operator; solutions of elliptic equations; Ramadanov theorem; continuous dependence of reproducing kernel

MSC 2020: 46E22; 35J15

1 Historic background

A version of the continuous dependence of general reproducing kernels on increasing or decreasing sequences of domains can be found in a seventy-year-old paper of N. Aronszajn (see [1]). In 1967, I. Ramadanov published his famous research paper [11] in which he considered the continuous dependence of the classical Bergman kernel on an increasing sequence of domains. His results were obtained later in a different way by M. Skwarczyński (see [14, 13]). As we will see in this paper, the method used by M. Skwarczyński can be applied to prove the continuous dependence of the reproducing kernels of the Hilbert space of square-integrable functions which are the algebraic kernel of some elliptic operator.

The dependence of two canonical kernels – of Bergman and of Szegö type – on a weight of integration was widely investigated. Z. Nehari (see [7]) considered weighted Bergman and weighted Szegö kernels. Z. Pasternak-Winiarski (see [9]) proved the analytic dependence of the weighted Bergman kernel, which plays an important role in physics (see [8]), without a hypothesis that the weight of integration must be continuous. There are also papers which give estimates for the weighted Szegö kernel (see, e.g., [17]). None of them, however, uses the minimal norm property of a reproducing kernel, which is done in this paper. Using this method we do not need the continuity of weights, which was an important requirements in [7]. We also assume that weights converge only almost everywhere, while in [9] a stronger topology was needed.

V. M. Malyshev (see [6]) considered the reproducing kernels of Hilbert spaces of functions connected with the kernel of a hypoelliptic operator, but his construction was a different idea.

Note also that usually “minimal solution of a differential equation” is understood in a different sense (see, e.g., [16, 4]) from that used in this paper.

2 Preliminaries

To begin with, let us recall some classical results which will be used later.

2.1 Reproducing kernel Hilbert spaces

Definition 2.1.

Let ℋ be a Hilbert space of functions defined on U. A function K defined on U × U such that for any z ∈ U and any f ∈ ℋ we have K ⁢ ( z , ⋅ ) ¯ ∈ ℋ and (reproducing property)

〈 f ⁢ ( ⋅ ) ∣ K ⁢ ( z , ⋅ ) ¯ 〉 = f ⁢ ( z ) ,

is called a reproducing kernel of the space ℋ .

It is well known that:

Theorem 2.2.

The following conditions are equivalent:

There exists a reproducing kernel of ℋ .
Functionals of point evaluation ℋ ∋ f ↦ f ⁢ ( z ) ∈ ℂ are continuous for any z ∈ U , i.e., for any z ∈ U there exists C z > 0 such that for any f ∈ ℋ ,

| f ⁢ ( z ) | ≤ C z ⁢ ∥ f ∥ .

Proof.

(i) ⇒ (ii) By the reproducing property and Cauchy–Schwarz inequality,

| f ⁢ ( z ) | = | 〈 f ∣ K ⁢ ( z , ⋅ ) ¯ 〉 | ≤ ∥ K ⁢ ( z , ⋅ ) ¯ ∥ ⋅ ∥ f ∥ = K ⁢ ( z , z ) ⁢ ∥ f ∥ .

(ii) ⇒ (i) If the functionals of point evaluation are continuous, then by the Riesz representation theorem, for each z ∈ U there exists e z ¯ ∈ ℋ such that

f ⁢ ( z ) = 〈 f ∣ e z ¯ 〉 .

A function K defined in the following way,

K ⁢ ( z , w ) := e z ⁢ ( w ) ,

is the reproducing kernel of the Hilbert space ℋ . ∎

It is also well known that, for any reproducing kernel K, we have

K ⁢ ( z , z ) ≥ 0 .

Indeed, by the reproducing property and the fact that the norm of any function is non-negative, we have

K ⁢ ( z , z ) = 〈 K ⁢ ( z , ⋅ ) ∣ K ⁢ ( z , ⋅ ) ¯ 〉 = ∥ K ⁢ ( z , ⋅ ) ¯ ∥ 2 ≥ 0 .

This fact will be used frequently in what follows.

Note that not every Hilbert space of functions is equipped with a reproducing kernel. Examples of Hilbert spaces with no corresponding reproducing kernel can be found in [10] or [18].

2.2 Partial differential equation theory

Definition 2.3.

If there exists a constant Θ > 0 such that

∑ i , j = 1 n a i ⁢ j ⁢ ( x ) ⁢ v i ⁢ v j ≥ Θ ⁢ | v | 2

for any x ∈ U and any v ∈ ℝ n in the definition above, then we say that D is an elliptic operator.

Often such a term is used to denote a uniformly elliptic operator. See, e.g., [3] for more details.

Theorem 2.4.

Let U ⊂ R n be a domain with a boundary of class C 1 . Let f be an element of the Sobolev space W k , p ⁢ ( U ) . If k > n p , then f is also an element of the Hölder space C k - [ n p ] - 1 , γ ⁢ ( U ¯ ) , where

γ = { [ n p ] + 1 - n p , n p ∉ ℤ , any positive number less than ⁢ 1 , n p ∈ ℤ .

Moreover, there exists a constant C 1 > 0 such that

] ] f [ [ U k - [ n p ] - 1 , γ ≤ C 1 [ [ f ] ] U k , p .

For more details, see [3, Theorem 6 in Section 5.6.3].

Theorem 2.5.

Let U be a domain in R n with a boundary of class C 1 . Let D be an elliptic operator such that (in divergence form)

D ⁢ u = - ∑ i , j = 1 n ( a i ⁢ j ⁢ ( u x i ) ) x j + ∑ i = 1 n b i ⁢ ( x ) ⁢ u x i , + c ⁢ ( x ) ⁢ u ,

where a i ⁢ j ∈ C 1 ⁢ ( U ) , b i , c ∈ L ∞ ⁢ ( U ) . Let f ∈ L 2 ⁢ ( U ) and u ∈ W 1 , 2 ⁢ ( U ) be a weak solution of the elliptic equation

D ⁢ u = f

in U. Then u ∈ W 2 , 2 ⁢ ( V ) for any compact set V ⊂ U and there exists a constant C 2 > 0 such that

[ [ u ] ] V 2 , 2 ≤ C 2 ⁢ ( ∥ f ∥ U + ∥ u ∥ U ) .

For more details, see [3, Theorem 1 in Section 6.3.1].

Moreover:

Theorem 2.6.

Let a i ⁢ j , b i , c ∈ C ∞ ⁢ ( U ) and f ∈ C ∞ ⁢ ( U ) . Let u ∈ W 1 , 2 ⁢ ( U ) be a weak solution of the elliptic equation

D ⁢ u = f

in U. Then u ∈ C ∞ ⁢ ( U ) , which means that it is in fact a strong solution.

For more details see [3, Theorem 3 in Section 6.3.1].

3 Reproducing kernel Hilbert space of square-integrable solutions of an elliptic equation

Let U be a domain in ℝ N . Let L 2 ⁢ ( U ) denote the class of measurable functions defined on U such that

∥ f ∥ U 2 := ∫ U | f ⁢ ( w ) | 2 ⁢ d w < ∞ .

(If U does not change in our considerations or is unspecified, we will simplify our notation to ∥ ⋅ ∥ .)

Let D be a linear differential operator defined on L 2 ⁢ ( U )

D = ∑ | a | ≤ n C a ⁢ ( x ) ⁢ ∂ a ,

where a = ( a 1 , a 2 , … , a n ) , | a | = a 1 + a 2 + … + a n and

∂ a = ∂ a 1 ∂ x 1 a 1 ⁢ ∂ a 2 ∂ x 2 a 2 ⁢ ⋯ ⁢ ∂ a n ∂ x n a n .

By L 2 ⁢ D ⁢ ( U ) we understand

{ f ∈ L 2 ⁢ ( U ) : D ⁢ f = 0 } ,

where the equality is understood in the strong sense, i.e., we consider f of class C k , where k is the rank of the operator D.

Note that if the coefficients of operator D are of class C ∞ , then by Theorem 2.6 a weak solution of the equation D ⁢ f = 0 is in fact a strong solution, which means that we can identify the elements of L 2 ⁢ D ⁢ ( U ) with their continuous representants, so that in particular the values of these functions at any point z ∈ U will be well defined.

Proposition 3.1.

Let f n be a sequence of functions such that D ⁢ f n = 0 for any n convergent to function f in L 2 ⁢ ( U ) topology. Then D ⁢ f = 0 . Moreover, if D is an elliptic operator with smooth coefficients, then the space L 2 ⁢ D ⁢ ( U ) is a closed subspace of L 2 ⁢ ( U ) .

Proof.

Let f n ∈ L 2 ⁢ D ⁢ ( U ) and suppose that f n → f in the L 2 ⁢ ( U ) topology. Let h be an element of some dense subspace of L 2 ⁢ ( U ) contained in the domain of D * . Then

0 = 〈 h ∣ D ⁢ f n 〉 = 〈 D * ⁢ h ∣ f n 〉 and 0 = 〈 D * ⁢ h ∣ f 〉 = 〈 h ∣ D ⁢ f 〉 ,

which implies

0 = 〈 h ∣ D ⁢ f 〉 .

Since h was chosen arbitrarily from a dense subspace of L 2 ⁢ ( U ) , we have D ⁢ f = 0 . Moreover, if D is an elliptic operator with smooth coefficients, then f is also a strong solution and L 2 ⁢ D ⁢ ( U ) is closed in L 2 ⁢ ( U ) . ∎

So we know that if only D has coefficients of C ∞ class, then L 2 ⁢ D ⁢ ( U ) is a Hilbert space. Such a Hilbert space is a generalization of the well-known Bergman space. Indeed, for example, for U ⊂ ℝ 2 and

D = ∂ ∂ ⁡ z ¯ = 1 2 ⁢ ( ∂ ∂ ⁡ x 1 + i ⁢ ∂ ∂ ⁡ x 2 ) ,

L 2 ⁢ D ⁢ ( U ) is a space of holomorphic and square-integrable functions on U.

Theorem 3.2.

Let U be a domain in R 2 with the boundary of class C 1 . Let D be an elliptic operator such that (in divergence form)

D ⁢ f = - ∑ i , j = 1 2 ( a i ⁢ j ⁢ ( f x i ) ) x j + ∑ i = 1 2 b i ⁢ ( x ) ⁢ f x i + c ⁢ ( x ) ⁢ f ,

where a i ⁢ j ∈ C 1 ⁢ ( U ) , b i , c ∈ L ∞ ⁢ ( U ) . Then there exists the reproducing kernel of L 2 ⁢ D ⁢ ( U ) .

If the coefficients of operator D are not of class C ∞ , then it is possible that the space of square integrable solutions in the strong sense of the equation D ⁢ f = 0 is not closed. In such a case, we can take the closure and define the reproducing kernel on it, using standard techniques.

A particular case of such a reproducing kernel is the reproducing kernel of the Hilbert space of square-integrable and harmonic functions. It was well investigated, including a direct formula for the unit ball (see, e.g., [2, 5, 12]).

In the remainder of this paper, if we say “elliptic operator”, we will mean that U ⊂ ℝ 2 is a domain with C 1 -boundary and coefficients in its divergence form that satisfy the hypotheses of the theorem above. We will also write “elliptic equation D ⁢ f = 0 ” in the same manner.

Proof of Theorem 3.2.

Let f ∈ L 2 ⁢ D ⁢ ( U ) and w ∈ U . Let r be sufficiently small for the ball

B ⁢ ( w , r ) = { z ∈ ℝ 2 : | w - z | < r }

to lie in U. Then for any z ∈ B ⁢ ( w , r ) , by Theorem 2.4, we have

| f ⁢ ( z ) | ≤ C B ⁢ ( w , r ) ⁢ [ [ f ] ] B ⁢ ( w , r ) 2 , 2 ,

where C B ⁢ ( w , r ) does not depend on f ∈ L 2 ⁢ D ⁢ ( U ) .

On the other hand, for f ∈ L 2 ⁢ D ⁢ ( U ) , by Theorem 2.5,

[ [ f ] ] B ⁢ ( w , r ) 2 , 2 ≤ C 2 ⁢ ∥ f ∥ U .

So we have shown that, for each compact set X ⊂ U , there exists C X such that for any z ∈ X and any function f ∈ L 2 ⁢ D ⁢ ( U ) ,

(3.1) | f ⁢ ( z ) | ≤ C X ⁢ ∥ f ∥ U .

This implies the continuity of the functionals of point evaluation, i.e., the functionals

E z : L 2 ⁢ D ⁢ ( U ) ∋ f ↦ f ⁢ ( z ) .

By Theorem 2.2, there exists the reproducing kernel of the considered space. ∎

Note that the ellipticity of the operator D cannot be skipped. For example, let U be the unit circle in ℝ 2 and

D = ∂ 2 ∂ ⁡ x ⁢ ∂ ⁡ y .

It is easy to show that f is a strong solution of the equation D ⁢ f = 0 if and only if

f ⁢ ( x , y ) = g ⁢ ( x ) + h ⁢ ( y )

for some C 2 -functions g and h. In particular,

h n ⁢ ( x , y ) := exp ⁡ ( - x 2 ⁢ n ) + exp ⁡ ( - y 2 ⁢ n ) ∈ L 2 ⁢ D ⁢ ( U )

for any n ∈ ℕ . We have

∫ - 1 1 exp ( - x 2 n ) d x = ≀ x n = t ≀ = 1 n ∫ - n n exp ( - t 2 ) d t < 1 n ∫ - ∞ ∞ exp ( - t 2 ) d t → 0 .

Similarly,

∫ - 1 1 exp ⁡ ( - 2 ⁢ x 2 ⁢ n ) ⁢ d x → 0 .

Let

S = { z = ( x , y ) ∈ ℝ 2 : - 1 < x < 1 , - 1 < y < 1 } .

Then

∫ U | exp ⁡ ( - x 2 ⁢ n ) + exp ⁡ ( - y 2 ⁢ n ) | 2 ⁢ d x ⁢ d y = ∫ U ( exp ⁡ ( - 2 ⁢ x 2 ⁢ n ) + exp ⁡ ( - 2 ⁢ y 2 ⁢ n ) + 2 ⁢ exp ⁡ ( - x 2 ⁢ n ) ⁢ exp ⁡ ( - y 2 ⁢ n ) ) ⁢ d x ⁢ d y

< ∫ S ( exp ⁡ ( - 2 ⁢ x 2 ⁢ n ) + exp ⁡ ( - 2 ⁢ y 2 ⁢ n ) + 2 ⁢ exp ⁡ ( - x 2 ⁢ n ) ⁢ exp ⁡ ( - y 2 ⁢ n ) ) ⁢ d x ⁢ d y

= ∫ - 1 1 d y ⁢ ∫ - 1 1 exp ⁡ ( - 2 ⁢ x 2 ⁢ n ) ⁢ d x + ∫ - 1 1 d x ⁢ ∫ - 1 1 exp ⁡ ( - 2 ⁢ y 2 ⁢ n ) ⁢ d y

+ 2 ⁢ ∫ - 1 1 exp ⁡ ( - x 2 ⁢ n ) ⁢ d x ⁢ ∫ - 1 1 exp ⁡ ( - y 2 ⁢ n ) ⁢ d y

= 2 ⁢ ∫ - 1 1 exp ⁡ ( - 2 ⁢ x 2 ⁢ n ) ⁢ d x + 2 ⁢ ∫ - 1 1 exp ⁡ ( - 2 ⁢ y 2 ⁢ n ) ⁢ d y + 2 ⁢ ( ∫ - 1 1 exp ⁡ ( - x 2 ⁢ n ) ⁢ d x ) 2

= 4 ⁢ ∫ - 1 1 exp ⁡ ( - x 2 ⁢ n ) ⁢ d x + 2 ⁢ ( ∫ - 1 1 exp ⁡ ( - x 2 ⁢ n ) ⁢ d x ) 2 → 0 .

We have shown that ∥ h n ∥ U → 0 . On the other hand, h n ⁢ ( 0 , 0 ) = 2 for any n ∈ ℕ . It means that the functional of point evaluation for z = ( 0 , 0 ) is not continuous and therefore, by Theorem 2.2, L 2 ⁢ D ⁢ ( U ) is not equipped with a reproducing kernel.

Note that V. M. Malyshev in [6] considered the reproducing kernels connected with Hilbert spaces of kernels of hypoelliptic operators. He stated that if there exists an embedding from L 2 ⁢ D ⁢ ( U ) into C ⁢ ( U ) , then functionals of point evaluation are continuous and therefore the reproducing kernel exists. Clearly, that embedding cannot be natural – i.e., he did not identify the element of L 2 ⁢ D ⁢ ( U ) with its continuous representant. Therefore, the space and reproducing kernel considered by him is a different idea.

The well-known Ramadanov theorem was proved by Ramadanov in [11] and it states that for an increasing sequence of domains, the limit of a sequence of classical Bergman kernels of domains is the Bergman kernel of the limit domain. The aim of this paper is to show that our reproducing kernels have the same property. In the proof we will need the following theorem by Stone.

Theorem 3.3.

Let F 1 ⋑ F 2 ⋑ F 3 ⋑ ⋯ be a sequence of closed subspaces of the Hilbert space H and F = ⋂ n = 0 ∞ F n . Let P i : H → F i and P : H → F be orthogonal projections. Then, for any f ∈ H , we have

P n ⁢ f → P ⁢ f .

See [15] for more details.

From now on, if we write L 2 ⁢ D ⁢ ( U ) , we will mean that D is a differential operator for which there exists a reproducing kernel of L 2 ⁢ D ⁢ ( U ) (e.g., D is an elliptic operator). Moreover, if the domain changes in our considerations, we will write K U to indicate the reproducing kernel of L 2 ⁢ D ⁢ ( U ) .

Now let us consider a measurable and almost everywhere positive function μ : U → ℝ . Such a function will be called a weight. By L 2 ⁢ ( U , μ ) we mean a Hilbert space of (classes of) functions for which

∥ f ∥ μ 2 := ∫ U | f ⁢ ( w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w < ∞

with the weighted inner product

〈 f ∣ g 〉 μ := ∫ U f ⁢ ( w ) ¯ ⁢ g ⁢ ( w ) ⁢ μ ⁢ ( w ) ⁢ d w .

Let D be a differential operator defined on L 2 ⁢ ( U , μ ) . Now we may define L 2 ⁢ D ⁢ ( U , μ ) as a space of elements from L 2 ⁢ ( U , μ ) which have a continuous representant with the same inner product for which D ⁢ f = 0 in the strong sense.

First, let us recall that if D does not have smooth coefficients, then L 2 ⁢ D ⁢ ( U ) may not be closed. If L 2 ⁢ D ⁢ ( U ) is not closed, we can take the closure of it. We will use the same symbol for the closure, which should not be misleading. In what follows, we will assume that L 2 ⁢ D ⁢ ( U ) is already closed.

It is easy to see that if there exist the constants C 1 , C 2 such that 0 < C 1 < μ < C 2 , then L 2 ⁢ D ⁢ ( U , μ ) is equal as a set with L 2 ⁢ D ⁢ ( U ) and the weighted inner product generates the same topology as the classical one. Indeed, to show that we need to write a simple inequality

C 1 ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ d w ≤ ∫ U | f ⁢ ( w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w ≤ C 2 ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ d w .

Consequently, L 2 ⁢ D ⁢ ( U , μ ) for μ bounded from above and below by non-zero constants is closed, i.e., is a Hilbert space.

If a weight is not bounded from below or from above by a non-zero constant, the topology of L 2 ⁢ D ⁢ ( U , μ ) may be different from the topology of L 2 ⁢ D ⁢ ( U ) and the spaces may be different as sets.

Nevertheless, state the following:

Proposition 3.4.

Let D be an elliptic operator with coefficients of class C ∞ . Suppose that for any compact set X ⊂ U , there exists C X such that for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) and any z ∈ X , we have

(3.2) | f ⁢ ( z ) | ≤ C X ⁢ ∥ f ∥ μ .

Then the space L 2 ⁢ D ⁢ ( U , μ ) is a closed subspace of L 2 ⁢ ( U , μ ) .

We will need the following lemma.

Lemma 3.5.

Let D be an elliptic operator. (Here we do not assume that its coefficients are of class C ∞ .) Let { f n } be a sequence of functions such that D ⁢ f n = 0 for any n which converges locally uniformly on U to some function f. Then D ⁢ f = 0 and for any compact set X ⊂ U , f ∈ L 2 ⁢ D ⁢ ( X ) .

Proof.

Since f n → f locally uniformly, we have

∫ X | f n ⁢ ( w ) - f ⁢ ( w ) | 2 ⁢ d w ≤ L ⁢ ( X ) ⁢ sup w ∈ X ⁡ | f n ⁢ ( w ) - f ⁢ ( w ) | 2 → 0 ,

where X is any compact subset of U and L ⁢ ( X ) is the Lebesgue measure of X. By Proposition 3.1, D ⁢ f = 0 . Using the fact that a compact set X ⊂ U can be chosen arbitrarily and the fact that D is a local operator ends the proof. ∎

Roughly speaking, this lemma states that the locally uniform limit of weak solutions of an elliptic equation is a weak solution of the same elliptic equation. If the coefficients of D are smooth, then in fact this proves that the locally uniform limit of strong solutions of an elliptic equation is also a strong solution of the same equation.

Proof of Proposition 3.4.

Let { f n } ⊂ L 2 ⁢ D ⁢ ( U , μ ) and suppose that f n → f in the L 2 ⁢ ( U , μ ) topology. By (3.2), f n converges to f locally uniformly. Using Lemma 3.5 and Proposition 3.1 completes the proof. ∎

From now on, D will mean an operator for which the reproducing kernel of L 2 ⁢ D ⁢ ( U ) exists, e.g., an elliptic operator. Moreover, by K μ we will denote the reproducing kernel of L 2 ⁢ D ⁢ ( U , μ ) without further reminding.

Definition 3.6.

Let μ be a weight on U. We say that it is admissible for D if for any compact set X ⊂ U , there exists C X such that for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) and any z ∈ C X , we have

| f ⁢ ( z ) | ≤ C X ⁢ ∥ f ∥ μ .

Note that by Theorem 2.2 this condition implies the existence of a reproducing kernel and if the coefficients of D are of class C ∞ , by Proposition 3.4 this condition implies that L 2 ⁢ D ⁢ ( U , μ ) is closed in L 2 ⁢ ( U , μ ) .

If L 2 ⁢ D ⁢ ( U , μ ) is not closed in L 2 ⁢ ( U , μ ) , we can take the closure and use the same symbol to denote it. As in the case of L 2 ⁢ D ⁢ ( U ) , it should not be misleading.

Proposition 3.7.

Let μ > C > 0 a.e. Then μ is admissible.

Proof.

By (3.1) for any compact set X ⊂ U there exists C X > 0 such that for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) ⊂ L 2 ⁢ D ⁢ ( U ) , we have

| f ⁢ ( z ) | 2 ≤ C X ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ d w .

Obviously,

∫ U | f ⁢ ( w ) | 2 ⁢ d w = 1 C ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ C ⁢ d w ≤ 1 C ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w . ∎

Theorem 3.8.

Let μ 1 , μ 2 be weights on U such that μ 1 is admissible and μ 2 ≥ μ 1 a.e. Then μ 2 is also admissible.

Proof.

If μ 1 is admissible, then for any compact set X ⊂ U there exists C X such that for any z ∈ X and any f ∈ L 2 ⁢ D ⁢ ( U , μ 1 ) ,

| f ⁢ ( z ) | ≤ C X ⁢ ∥ f ∥ μ 1 .

Since

∫ U | f ⁢ ( w ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ d w ≤ ∫ U | f ⁢ ( w ) | 2 ⁢ μ 2 ⁢ ( w ) ⁢ d w ,

we have that L 2 ⁢ D ⁢ ( U , μ 2 ) ⊂ L 2 ⁢ D ⁢ ( U , μ 1 ) and that for any f ∈ L 2 ⁢ D ⁢ ( U , μ 2 ) ,

| f ⁢ ( z ) | ≤ C X ⁢ ∥ f ∥ μ 2 . ∎

In particular, if μ is an admissible weight, then e μ and μ μ are also admissible weights, because e x > x and x x > x almost everywhere on the interval [ 0 , + ∞ [ .

Corollary 3.9.

Let μ 1 , μ 2 be weights on U and let μ 1 be admissible. Then μ 1 + μ 2 is also an admissible weight. In particular, the sum of admissible weights on the same domain is an admissible weight.

Theorem 3.10.

Let μ 1 , μ 2 be admissible weights on U such that μ 2 ≥ C > 0 a.e. Then μ 1 ⋅ μ 2 is an admissible weight.

Proof.

If μ 1 is admissible, then for any compact set X ⊂ U there exists C X such that for any z ∈ X and any f ∈ L 2 ⁢ D ⁢ ( U , μ 1 ) ,

| f ⁢ ( z ) | ≤ C X ⁢ ∥ f ∥ μ 1 .

Since

∫ U | f ⁢ ( w ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ d w = 1 C ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ C ⁢ d w ≤ 1 C ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ μ 2 ⁢ ( w ) ⁢ d w ,

we have that L 2 ⁢ D ⁢ ( U , μ 1 ⁢ μ 2 ) ⊂ L 2 ⁢ D ⁢ ( U , μ 1 ) and for any f ∈ L 2 ⁢ D ⁢ ( U , μ 1 ⁢ μ 2 ) ,

| f ⁢ ( z ) | ≤ C X ⁢ 1 C ⁢ ∥ f ∥ μ 1 ⁢ μ 2 . ∎

Corollary 3.11.

Let μ be an admissible weight on U and let α > 0 . Then α ⁢ μ is also an admissible weight.

From now on, by K μ we will denote the reproducing kernel of L 2 ⁢ D ⁢ ( U , μ ) without further reminding.

4 Minimal norm property

The content of this section holds true for general reproducing kernel Hilbert spaces and is well known. We have decided, however, to give details for completeness.

Theorem 4.1.

If K μ ⁢ ( z , z ) ≠ 0 , then

k z ⁢ ( ⋅ ) := K μ ⁢ ( z , ⋅ ) ¯ K μ ⁢ ( z , z )

is the only element of L 2 ⁢ D ⁢ ( U , μ ) with the following properties:

k z ⁢ ( z ) = 1 .
If m z ∈ L 2 ⁢ D ⁢ ( U , μ ) , m z ⁢ ( z ) = 1 and ∥ m z ∥ μ ≤ ∥ k z ∥ μ , then m z = k z . Moreover,

∥ k z ∥ μ = 1 K μ ⁢ ( z , z ) .

Proof.

By the reproducing property and Cauchy–Schwarz inequality, for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) and z ∈ U we have

| f ⁢ ( z ) | = | ∫ U f ⁢ ( w ) ⁢ K μ ⁢ ( z , w ) ⁢ μ ⁢ ( w ) ⁢ d w | ≤ ∫ U | f ⁢ ( w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w ⁢ ∫ U | K μ ⁢ ( z , w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w ,

i.e.,

(4.1) | f ⁢ ( z ) | ≤ K μ ⁢ ( z , z ) ⁢ ∥ f ∥ μ .

Moreover, K μ ⁢ ( z , z ) is the smallest possible constant for which inequality (4.1) holds. Indeed, let

E z : L 2 ⁢ D ⁢ ( U , μ ) ∋ f ↦ f ⁢ ( z ) ∈ ℂ

be a functional of point evaluation. By the Riesz correspondence theorem,

∥ E z ∥ μ * = ∥ K μ ⁢ ( z , ⋅ ) ¯ ∥ μ ,

but

∥ K μ ⁢ ( z , ⋅ ) ¯ ∥ μ = ∫ U K μ ⁢ ( z , w ) ⁢ K μ ⁢ ( z , w ) ¯ ⁢ d w = K μ ⁢ ( z , z ) .

At once ∥ K μ ⁢ ( z , ⋅ ) ¯ ∥ μ is by definition the smallest constant for which inequality (4.1) holds.

Now we have

1 K μ ⁢ ( z , z ) ≤ ∥ f ∥ μ | f ⁢ ( z ) | = ∥ f f ⁢ ( z ) ∥ μ .

But

∫ U | K μ ⁢ ( z , w ) ¯ K μ ⁢ ( z , z ) | 2 ⁢ d w = 1 K μ ⁢ ( z , z )

by the reproducing property. To end the proof, we need only to show that if ∥ m z ∥ μ = ∥ k z ∥ μ , then m z = k z . Note that for f z := 1 2 ⁢ ( m z + k z ) we have f ⁢ ( z ) = 1 and

∥ f z ∥ μ = ∥ 1 2 ⁢ ( m z + k z ) ∥ μ ≤ 1 2 ⁢ ( ∥ m z ∥ μ + ∥ k z ∥ μ ) = ∥ k z ∥ μ .

On the other hand, we have shown above that

∥ f z ∥ μ ≥ ∥ k z ∥ μ ,

so ∥ f z ∥ μ = ∥ k z ∥ μ . Since in our case the triangle inequality is in fact an equality and each Hilbert space is strictly convex, there exists α > 0 such that m z = α ⁢ k z . Thus

∥ 1 2 ⁢ ( m z + k z ) ∥ + μ = 1 2 ⁢ ( α + 1 ) ⁢ ∥ k z ∥ μ .

Since

∥ f z ∥ μ = ∥ k z ∥ μ ,

we see that α = 1 and thus m z = k z . ∎

Now we will investigate the case in which f ⁢ ( z ) = 0 for each f ∈ L 2 ⁢ D ⁢ ( U , μ ) and some z ∈ U .

Theorem 4.2.

The following conditions are equivalent for a point z ∈ U :

f ⁢ ( z ) = 0 for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) .
K μ ⁢ ( z , z ) = 0 .
K μ ⁢ ( z , ⋅ ) ≡ 0 .

Proof.

(i) ⇒ (ii) If for some z ∈ U we have f ⁢ ( z ) = 0 for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) , then in particular for g ⁢ ( ⋅ ) = K μ ⁢ ( z , ⋅ ) ¯ we have g ⁢ ( z ) = 0 .

(ii) ⇒ (iii) Because

∫ U | K μ ⁢ ( z , w ) | 2 ⁢ d w = K μ ⁢ ( z , z ) = 0

and the integrated function is continuous and non-negative, K μ ⁢ ( z , ⋅ ) ≡ 0 on U.

(iii) ⇒ (i) By the reproducing property, for any f ∈ L 2 ⁢ D ⁢ ( U , μ ) , we have

f ⁢ ( z ) = ∫ U f ⁢ ( w ) ⁢ K μ ⁢ ( z , w ) ⁢ d w = ∫ U f ⁢ ( w ) ⋅ 0 ⁢ d w = 0 . ∎

Corollary 4.3.

Let U ⊂ R 2 , z ∈ U and let the weight μ be bounded from below by a non-zero constant. In the set V z μ := { f ∈ L 2 ⁢ ( U , μ ) : D ⁢ f = 0 ⁢ and ⁢ f ⁢ ( z ) = 1 } of weighted square-integrable solutions of the elliptic equation D ⁢ f = 0 , for which f ⁢ ( z ) = 1 , if it is not empty, there exists exactly one element f 0 such that

∥ f 0 ∥ μ = min f ∈ V z μ ⁢ ∥ f ∥ μ .

Such an element in what follows will be called a minimal z-solution in weight μ of the elliptic equation D ⁢ f = 0 on U.

In particular, the above corollary and the whole section are true for the weight μ ≡ 1 . For such a weight we will write V z instead of V z μ and say “minimal z-solution” instead of “minimal z-solution in weight 1”.

Note that if the elliptic operator D has c ⁢ ( x ) = 0 in divergence form and the weight μ is bounded from below by a non-zero constant and integrable on U, then all constant functions are elements of L 2 ⁢ D ⁢ ( U , μ ) and therefore the set V z μ is not empty. In particular, the weight μ ≡ 1 is integrable on a bounded domain.

5 Dependence of the reproducing kernel of L 2 ⁢ D ⁢ ( U , μ ) on a weight of integration

Theorem 5.1.

Let { μ n } be a sequence of weights on U convergent almost everywhere to μ such that there exist C 1 , C 2 > 0 for which C 1 < μ < C 2 and C 1 < μ n < C 2 a.e. for each n ∈ N . Then for any z ∈ U ,

lim n → ∞ ⁡ K μ n ⁢ ( z , ⋅ ) ,

where the limit above is locally uniform on U, exists and is equal to K μ ⁢ ( z , ⋅ ) .

By K ⁢ ( z , w ) we will mean the pointwise limit of K μ n ⁢ ( z , w ) .

Lemma 5.2.

Let { μ n } be a sequence of weights on U convergent almost everywhere to μ such that μ > c > 0 a.e. Let z ∈ U be arbitrary but fixed and let the locally uniform limit of K μ n ⁢ ( z , ⋅ ) exist on U. Then the following conditions are equivalent:

K μ n ⁢ ( z , z ) → K μ ⁢ ( z , z ) ;
K μ n ⁢ ( z , ⋅ ) → K μ ⁢ ( z , ⋅ ) locally uniformly on U.

Proof.

We need only to show the implication (i) ⇒ (ii).

By Lemma 3.5, the locally uniform limit K is an element of the kernel of the operator D. By Fatou’s lemma and our assumptions,

∫ U | K ⁢ ( z , w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w ≤ lim inf n → ∞ ⁡ ∫ U | K μ n ⁢ ( z , w ) | 2 ⁢ μ n ⁢ ( w ) ⁢ d w = lim inf n → ∞ ⁡ K μ n ⁢ ( z , z ) = K μ ⁢ ( z , z ) .

We have shown that K ⁢ ( z , ⋅ ) ∈ L 2 ⁢ D ⁢ ( U , μ ) . Therefore,

∥ K ⁢ ( z , ⋅ ) ¯ ∥ μ ≤ K μ ⁢ ( z , z )

and if only K μ ⁢ ( z , z ) > 0 ,

∥ K ⁢ ( z , ⋅ ) ¯ K μ ⁢ ( z , z ) ∥ μ ≤ 1 K μ ⁢ ( z , z ) ,

so by Theorem 4.1,

K ⁢ ( z , w ) K μ ⁢ ( z , z ) = K μ ⁢ ( z , w ) K μ ⁢ ( z , z ) ,

which means that

K ⁢ ( z , w ) = K μ ⁢ ( z , w ) .

Because K μ n ⁢ ( z , z ) → K μ ⁢ ( z , z ) for almost every z ∈ U , in fact K ⁢ ( z , w ) = K μ ⁢ ( z , w ) for any z , w ∈ U , as K μ is continuous on U × U .

If K μ ⁢ ( z , z ) = 0 , then by Theorem 4.2, we also have K ⁢ ( z , ⋅ ) ≡ 0 and K μ ⁢ ( z , ⋅ ) ≡ 0 , so K ⁢ ( z , ⋅ ) = K μ ⁢ ( z , ⋅ ) . ∎

Lemma 5.3.

Let μ 1 , μ 2 be the weights on U such that 0 < c < μ 1 ≤ μ 2 a.e. Then for any z ∈ U we have

K μ 2 ⁢ ( z , z ) ≤ K μ 1 ⁢ ( z , z ) .

Proof.

First, let us assume that K μ 1 ⁢ ( z , z ) and K μ 2 ⁢ ( z , z ) are greater than 0. By Theorem 4.1 it is true that

1 K μ 1 ⁢ ( z , z ) = ∫ U | K μ 1 ⁢ ( z , w ) K μ 1 ⁢ ( z , z ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ d w ≤ ∫ U | K μ 2 ⁢ ( z , w ) K μ 2 ⁢ ( z , z ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ d w .

Since μ 1 ≤ μ 2 , it follows that

∫ U | K μ 2 ⁢ ( z , w ) K μ 2 ⁢ ( z , z ) | 2 ⁢ μ 1 ⁢ ( w ) ⁢ d w ≤ ∫ U | K μ 2 ⁢ ( z , w ) K μ 2 ⁢ ( z , z ) | 2 ⁢ μ 2 ⁢ ( w ) ⁢ d w .

Because

∫ U | K μ 2 ⁢ ( z , w ) K μ 2 ⁢ ( z , z ) | 2 ⁢ μ 2 ⁢ ( w ) ⁢ d w = 1 K μ 2 ⁢ ( z , z ) ,

finally we have that

1 K μ 2 ⁢ ( z , z ) ≤ 1 K μ 1 ⁢ ( z , z ) ,

which ends the proof.

Now let K μ 1 ⁢ ( z , z ) = 0 . Then by Theorem 4.2, K μ 1 ⁢ ( z , ⋅ ) ≡ 0 . Since μ 1 ≤ μ 2 , we have K μ 2 ⁢ ( z , ⋅ ) ¯ ∈ L 2 ⁢ D ⁢ ( U , μ 1 ) . Then again by Theorem 4.2 we have K μ 2 ⁢ ( z , ⋅ ) ≡ 0 , so K μ 2 ⁢ ( z , z ) ≤ K μ 1 ⁢ ( z , z ) .

If K μ 2 ⁢ ( z , z ) = 0 , then obviously K μ 2 ⁢ ( z , z ) ≤ K μ 1 ⁢ ( z , z ) . ∎

Proof of Main Theorem 5.1.

Let X ⊂ U be any compact set. We have

∫ X | K μ n ⁢ ( z , w ) | 2 ⁢ d w ≤ ∫ U | K μ n ⁢ ( z , w ) | 2 ⁢ d w .

Moreover,

1 C 1 ⁢ ∫ U | K μ n ⁢ ( z , w ) | 2 ⁢ C 1 ⁢ d w ≤ 1 C 1 ⁢ ∫ U | K μ n ⁢ ( z , w ) | 2 ⁢ μ n ⁢ ( w ) ⁢ d w = K μ n ⁢ ( z , z ) .

By Lemma 5.3,

K μ n ⁢ ( z , z ) ≤ K C 1 2 ⁢ ( z , z ) ,

where K C 1 / 2 denotes the reproducing kernel of L 2 ⁢ D ⁢ ( U , C 1 2 ) . This means that the sequence { K μ n ⁢ ( z , ⋅ ) } is bounded in L 2 ⁢ D ⁢ ( U ) . By Theorem 2.5 we claim that { K μ n ⁢ ( z , ⋅ ) } is bounded also in the Sobolev space W 2 , 2 ⁢ ( X ) . Now, by Theorem 2.4, we see that { K μ n ⁢ ( z , ⋅ ) } is also bounded in the Hölder space C 0 , γ ⁢ ( X ¯ ) for any γ > 0 . This means that the hypotheses of the Arzelá–Ascoli theorem are satisfied and in our sequence { K μ n ⁢ ( z , ⋅ ) } there exists a subsequence which is locally uniformly convergent to some function K. Without losing generality, we may identify such a convergent sequence with the whole sequence.

We need only to show that K is the reproducing kernel of L 2 ⁢ D ⁢ ( U , μ ) . We divide our proof into two cases.

Case 1: Let K μ ⁢ ( z , z ) = 0 . Then by Theorem 4.2 also K μ ⁢ ( z , ⋅ ) ≡ 0 . In addition, since L 2 ⁢ D ⁢ ( U , μ ) is equal as a set to L 2 ⁢ D ⁢ ( U , 1 ) , we have K ⁢ ( z , ⋅ ) ¯ ∈ L 2 ⁢ D ⁢ ( U , 1 ) and again by Theorem 4.2, K ⁢ ( z , ⋅ ) ≡ 0 . So we have shown that K μ ⁢ ( z , ⋅ ) = K ⁢ ( z , ⋅ ) .

Case 2: Let K μ ⁢ ( z , z ) > 0 . Since all μ n and μ are uniformly bounded from below and above by non-zero constants, all spaces L 2 ⁢ D ⁢ ( U , μ n ) and L 2 ⁢ D ⁢ ( U , μ ) are pairwise equal as sets. By (4.1) and Theorem 4.1 for any f ∈ L 2 ⁢ D ⁢ ( U , μ n ) ,

| f ⁢ ( z ) | ≤ K μ n ⁢ ( z , z ) ⁢ ∥ f ∥ μ n .

Taking the limit, we get

| f ⁢ ( z ) | ≤ K ⁢ ( z , z ) ⁢ ∥ f ∥ μ ,

where Lebesgue’s dominated convergence theorem can be used to show that ∥ f ∥ μ n → ∥ f ∥ μ . For f ⁢ ( ⋅ ) := K μ ⁢ ( z , ⋅ ) ¯ , we obtain

K μ ⁢ ( z , z ) ≤ K ⁢ ( z , z ) ⁢ K μ ⁢ ( z , z )

and therefore

K μ ⁢ ( z , z ) ≤ K ⁢ ( z , z ) .

So is K ⁢ ( z , z ) > 0 .

By Fatou’s lemma and our assumptions,

∫ U | K ⁢ ( z , w ) | 2 ⁢ μ ⁢ ( w ) ⁢ d w ≤ lim inf n → ∞ ⁡ ∫ U | K μ n ⁢ ( z , w ) | 2 ⁢ μ n ⁢ ( w ) ⁢ d w

= lim inf n → ∞ ⁡ K μ n ⁢ ( z , z ) = K ⁢ ( z , z ) .

Therefore, by Lemma 3.5, K ⁢ ( z , ⋅ ) ¯ ∈ L 2 ⁢ D ⁢ ( U , μ ) and

∥ K ⁢ ( z , ⋅ ) ¯ K ⁢ ( z , z ) ∥ μ ≤ 1 K ⁢ ( z , z ) .

By Theorem 4.1, K ⁢ ( z , z ) ≤ K μ ⁢ ( z , z ) .

So we have shown that K μ ⁢ ( z , z ) = K ⁢ ( z , z ) . By Lemma 5.2, the sequence K μ n ⁢ ( z , ⋅ ) converges locally uniformly to K μ ⁢ ( z , ⋅ ) . ∎

Corollary 5.4.

Let μ n be a sequence of weights convergent to μ a.e. on U. Let f n denote a minimal z-solution in weight μ n of the elliptic equation D ⁢ u = 0 . Let f be a minimal z-solution in weight μ of the elliptic equation D ⁢ u = 0 . Then

lim n → ∞ ⁡ f n = f ,

where the limit above is locally uniform on U.

6 The Ramadanov theorem for an increasing sequence of domains

The aim of this section is to prove the following:

Theorem 6.1.

Let D be an elliptic operator. Let { U n } be an increasing sequence of domains and U = ⋃ n = 1 + ∞ U n . Then for each z ∈ U ,

lim n → ∞ ⁡ K U n ⁢ ( z , ⋅ )

exists and is equal to K U ⁢ ( z , ⋅ ) , where the limit above is locally uniform on U.

In the remainder of this section, K ⁢ ( z , w ) will mean the pointwise limit of K U n ⁢ ( z , w ) .

Before giving the proof of the main theorem, we will prove some lemmas.

Lemma 6.2.

With hypotheses as in the theorem above, we have

K U n + 1 ⁢ ( z , z ) ≤ K U n ⁢ ( z , z )

for any z ∈ U n . Moreover,

K U ⁢ ( z , z ) ≤ K U n ⁢ ( z , z )

for any n ∈ N , which in the limit becomes

K U ⁢ ( z , z ) ≤ K ⁢ ( z , z ) .

Proof.

First let us assume that K U n ⁢ ( z , z ) > 0 and K U n + 1 ⁢ ( z , z ) > 0 . By the reproducing property and Theorem 4.1,

1 K U n ⁢ ( z , z ) = ∫ U n | K U n ⁢ ( z , w ) ¯ K U n ⁢ ( z , z ) | 2 ⁢ d w ≤ ∫ U n | K U n + 1 ⁢ ( z , w ) ¯ K U n + 1 ⁢ ( z , z ) | 2 ⁢ d w .

By the properties of the integral and again by the reproducing property,

∫ U n | K U n + 1 ⁢ ( z , w ) ¯ K U n + 1 ⁢ ( z , z ) | 2 ⁢ d w ≤ ∫ U n + 1 | K U n + 1 ⁢ ( z , w ) ¯ K U n + 1 ⁢ ( z , z ) | 2 ⁢ d w = 1 K U n + 1 ⁢ ( z , z ) .

In order to prove the second part of the lemma for the case of K U ⁢ ( z , z ) > 0 and K U n ⁢ ( z , z ) > 0 for any n ∈ ℕ , we just need to swap K U n + 1 ⁢ ( z , z ) with K U ⁢ ( z , z ) in the considerations above.

Now let us assume that K U n ⁢ ( z , z ) = 0 for some n ∈ ℕ . Then, for m > n , also K U m ⁢ ( z , z ) = 0 and K U ⁢ ( z , z ) = 0 . Indeed, if K U n ⁢ ( z , z ) = 0 , then by Theorem 4.2 for any f ∈ L 2 ⁢ D ⁢ ( U n ) we have f ⁢ ( z ) = 0 . But if g ∈ L 2 ⁢ D ⁢ ( V ) for V ⊃ U n , then g is also an element of L 2 ⁢ D ⁢ ( U n ) , so that for any g ∈ L 2 ⁢ D ⁢ ( V ) , we have g ⁢ ( z ) = 0 . Again by Theorem 4.2, K U m ⁢ ( z , z ) = 0 and K U ⁢ ( z , z ) = 0 .

If K U ⁢ ( z , z ) = 0 , then obviously K U ⁢ ( z , z ) ≤ K U n ⁢ ( z , z ) for any n ∈ ℕ and K U ⁢ ( z , z ) ≤ K ⁢ ( z , z ) . ∎

Lemma 6.3.

Let U 1 , U 2 , … be a sequence of domains and let U be a limit domain in the sense of pointwise limit of the sequence of indicator functions of sets U 1 , U 2 , … . Let f be an arbitrary positive-valued function defined and integrable on ⋃ n = 1 + ∞ U n . Then

lim n → ∞ ⁡ ∫ U n f ⁢ ( w ) ⁢ d w = ∫ U f ⁢ ( w ) ⁢ d w .

Proof.

Let χ X be the indicator function of set X. Then obviously f ⁢ ( w ) ⁢ χ U n ⁢ ( w ) ≤ f ⁢ ( w ) and, by the Lebesgue dominated convergence theorem,

lim n → ∞ ⁡ ∫ U n f ⁢ ( w ) ⁢ d w = lim n → ∞ ⁡ ∫ ⋃ n = 1 + ∞ U n f ⁢ ( w ) ⁢ χ U n ⁢ ( w ) ⁢ d w = ∫ ⋃ n = 1 + ∞ U n lim n → ∞ ⁡ f ⁢ ( w ) ⁢ χ U n ⁢ ( w ) ⁢ d ⁢ w = ∫ U f ⁢ ( w ) ⁢ d w . ∎

Lemma 6.4.

Let U 1 , U 2 , … be a sequence of domains convergent to domain U in the sense of pointwise convergence of the sequence of indicator functions of sets U n , which satisfies the following condition:

For each compact set X ⊂ U there exists m such that for n > m , X ⊂ U n .

Let z ∈ U be arbitrary but fixed and let the locally uniform limit of K U n ⁢ ( z , ⋅ ) exist on U. Then the following conditions are equivalent:

K U n ⁢ ( z , z ) → K U ⁢ ( z , z ) .
K U n ⁢ ( z , ⋅ ) → K U ⁢ ( z , ⋅ ) locally uniformly on U.

Note that if { U n } is an increasing sequence of domains or if U ⊂ U n for large enough n, then condition (A) is satisfied. Moreover, it is easy to see that, in the case of an increasing or decreasing sequence of domains, ⋃ n = 1 + ∞ U n and ⋂ n = 1 + ∞ U n , are equivalent to the limits introduced in the lemma.

Proof of Lemma 6.4.

We need only prove the implication (i) ⇒ (ii).

Let X ⊂ U be compact. Then there exists m ∈ ℕ such that for any n > m , X ⊂ U n . By Fatou’s lemma,

In conclusion, ∥ K ⁢ ( z , ⋅ ) ¯ ∥ U ≤ K U ⁢ ( z , z ) and if only K U ⁢ ( z , z ) > 0 , we have

∥ K ⁢ ( z , ⋅ ) ¯ K U ⁢ ( z , z ) ∥ U ≤ 1 K U ⁢ ( z , z ) .

By Lemma 3.5, K ⁢ ( z , ⋅ ) ¯ ∈ L 2 ⁢ D ⁢ ( U ) . By Theorem 4.1,

K ⁢ ( z , ⋅ ) = K U ⁢ ( z , ⋅ ) ,

which ends the proof.

On the other hand, if K U ⁢ ( z , z ) = 0 , then by Theorem 4.2 also K U ⁢ ( z , ⋅ ) ≡ 0 and K ⁢ ( z , ⋅ ) ≡ 0 , so

K U ⁢ ( z , ⋅ ) = K ⁢ ( z , ⋅ ) . ∎

Proof of Main Theorem 6.1.

Let X ⊂ U be any compact set. We have

∥ K U n ⁢ ( z , ⋅ ) ∥ X 2 ≤ ∥ K U n ⁢ ( z , ⋅ ) ∥ U 2 = K U n ⁢ ( z , z ) .

By Lemma 6.2,

0 ≤ K U n + 1 ⁢ ( z , z ) ≤ K U n ⁢ ( z , z ) ≤ K U 1 ⁢ ( z , z ) ,

which means that the sequence { K U n ⁢ ( z , ⋅ ) } is bounded in L 2 ⁢ D ⁢ ( U ) . By Theorem 2.5 we claim that { K U n ⁢ ( z , ⋅ ) } is bounded also in the Sobolev space W 2 , 2 ⁢ ( X ) . Now, by Theorem 2.4, we see that { K U n ⁢ ( z , ⋅ ) } is also bounded in the Hölder space C 0 , γ ⁢ ( X ¯ ) for any γ > 0 . This means that the hypotheses of the Arzelá–Ascoli theorem are satisfied and in our sequence { K U n ⁢ ( z , ⋅ ) } there exists a subsequence which is locally uniformly convergent to some function K. We need only to show that the limit of such a convergent subsequence is the reproducing kernel of the indicated space. Without loss of generality, we may identify such a convergent subsequence with the whole sequence.

Note that lim n → ∞ ⁡ K U n ⁢ ( z , z ) must exist, because the sequence { K U n ⁢ ( z , z ) } is bounded and monotonic.

By condition (A), there exists m such that X ⊂ U n for n > m . Then, by Fatou’s lemma and our assumptions,

∫ X | K ⁢ ( z , w ) | 2 ⁢ d w ≤ lim inf n → ∞ ⁡ ∫ X | K U n ⁢ ( z , w ) | 2 ⁢ d w ≤ lim inf n → ∞ ⁡ ∫ U n | K U n ⁢ ( z , w ) | 2 ⁢ d w .

But

(6.1) lim inf n → ∞ ⁡ ∫ U n | K U n ⁢ ( z , w ) | 2 ⁢ d w = lim inf n → ∞ ⁡ K U n ⁢ ( z , z ) = K ⁢ ( z , z ) .

By Lemma 3.5 and the arbitrariness of the choice of a compact set X, we have K ⁢ ( z , ⋅ ) ¯ ∈ L 2 ⁢ D ⁢ ( U ) . Now we need to show that K ⁢ ( z , ⋅ ) = K U ⁢ ( z , ⋅ ) . We consider two cases.

Case 1: Let K U ⁢ ( z , z ) = 0 for some z ∈ U . By Theorem 4.2, also K ⁢ ( z , z ) = 0 , as the value of K ⁢ ( z , ⋅ ) ¯ at the point z. By (6.1), K ⁢ ( z , ⋅ ) ≡ 0 on U, since | K ⁢ ( z , ⋅ ) | 2 is continuous and non-negative. Using again Theorem 4.2, we conclude that K ⁢ ( z , ⋅ ) = K U ⁢ ( z , ⋅ ) .

Case 2: Let K U ⁢ ( z , z ) > 0 . By Lemma 6.2, we also have K ⁢ ( z , z ) > 0 . Moreover, by (6.1),

∥ K ⁢ ( z , ⋅ ) ¯ ∥ U ≤ K ⁢ ( z , z )

and

∥ K ⁢ ( z , ⋅ ) ¯ K ⁢ ( z , z ) ∥ U ≤ 1 K ⁢ ( z , z ) .

By Theorem 4.1, K ⁢ ( z , z ) ≤ K U ⁢ ( z , z ) .

Obviously,

L 2 ⁢ D ⁢ ( U ) ⊂ L 2 ⁢ D ⁢ ( U n + 1 ) ⊂ L 2 ⁢ D ⁢ ( U n )

for any n ∈ ℕ . Therefore, for f ∈ L 2 ⁢ D ⁢ ( U ) , we can write inequality (4.1) in the following form:

(6.2) | f ⁢ ( z ) | ≤ K U n ⁢ ( z , z ) ⁢ ∫ U n | f ⁢ ( w ) | 2 ⁢ d w .

Taking the limit in (6.2) and using Lemma 6.3, we get

| f ⁢ ( z ) | ≤ K ⁢ ( z , z ) ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ d w .

In particular, for f ⁢ ( ⋅ ) := K U ⁢ ( z , ⋅ ) ¯ ∈ L 2 ⁢ D ⁢ ( U ) , we have

| K U ⁢ ( z , z ) | ≤ K ⁢ ( z , z ) ⁢ ∫ U | K U ⁢ ( z , w ) | 2 ⁢ d w = K ⁢ ( z , z ) ⁢ K U ⁢ ( z , z ) ,

so K U ⁢ ( z , z ) ≤ K ⁢ ( z , z ) .

Finally, K U ⁢ ( z , z ) = K ⁢ ( z , z ) and using Lemma 6.4 ends the proof. ∎

Corollary 6.5.

Let U n be an increasing sequence of bounded domains and U = ⋃ n = 1 + ∞ U n . Let f n z be a minimal z-solution of the elliptic equation D ⁢ f = 0 on U n and let f z be a minimal z-solution of the elliptic equation D ⁢ f = 0 on U. Then

f n z → f z

and the convergence above is locally uniform on U.

7 The Ramadanov theorem for a decreasing sequence of domains

First, we will define D ⁢ ( U ) as a set of the functions f from L 2 ⁢ D ⁢ ( U ) for which there exists a domain V ⊃ U ¯ such that f ∈ L 2 ⁢ D ⁢ ( V ) .

Theorem 7.1.

Let D be an elliptic operator. Let { U n } be a decreasing sequence of domains and U = int ⁢ ( ⋂ n = 1 + ∞ U n ) a bounded domain. If D ⁢ ( U ) is dense in L 2 ⁢ D ⁢ ( U ) , then

lim n → ∞ ⁡ K U n ⁢ ( z , ⋅ )

exists and is equal to K U ⁢ ( z , ⋅ ) , where the limit above is locally uniform on U.

Note that

∫ U ¯ f ⁢ ( w ) ⁢ d w = ∫ U f ⁢ ( w ) ⁢ d w

for any integrable function f. We can extend any function f defined on U by 0 to ∂ ⁡ U .

As in the preceding section, we denote the pointwise limit of K U n ⁢ ( z , w ) by K ⁢ ( z , w ) .

Proof.

Let z ∈ U and f ∈ L 2 ⁢ D ⁢ ( U ) . Let h ∈ D ⁢ ( U ) . Then, for n large enough, we have h ∈ L 2 ⁢ D ⁢ ( U n ) . By inequality (4.1),

(7.1) | h ⁢ ( z ) | ≤ K U n ⁢ ( z , z ) ⁢ ∫ U n | h ⁢ ( w ) | 2 ⁢ d w .

Taking the limit in (7.1) and using Lemma 6.3, we get

| h ⁢ ( z ) | ≤ K ⁢ ( z , z ) ⁢ ∫ U | h ⁢ ( w ) | 2 ⁢ d w .

We know that K ⁢ ( z , z ) exists because, in a fashion similar to Lemma 6.2, it can be shown that

K U ⁢ ( z , z ) ≥ K U n + 1 ⁢ ( z , z ) ≥ K U n ⁢ ( z , z ) .

(One notable difference here is that if K U n ⁢ ( z , z ) = 0 for some n, then also K U m ⁢ ( z , z ) = 0 for m < n . Also, if K U ⁢ ( z , z ) = 0 , then for each n ∈ ℕ we have K U n ⁢ ( z , z ) = 0 .)

As a conclusion,

K U ⁢ ( z , z ) ≥ K ⁢ ( z , z ) ,

i.e., K U n ⁢ ( z , z ) is increasing and bounded from above.

On the other hand, by our assumptions there exists a sequence { h n } ⊂ D ⁢ ( U ) such that h n → f in the L 2 ⁢ ( U ) sense. So we can write

| f ⁢ ( z ) | ≤ K ⁢ ( z , z ) ⁢ ∫ U ¯ | f ⁢ ( w ) | 2 ⁢ d w = K ⁢ ( z , z ) ⁢ ∫ U | f ⁢ ( w ) | 2 ⁢ d w .

Taking f ⁢ ( ⋅ ) = K U ⁢ ( z , ⋅ ) ¯ , we get

K U ⁢ ( z , z ) ≤ K ⁢ ( z , z ) ⁢ K U ⁢ ( z , z )

and if K U ⁢ ( z , z ) > 0 , then

K U ⁢ ( z , z ) ≤ K ⁢ ( z , z ) .

If K U ⁢ ( z , z ) = 0 , then also K ⁢ ( z , z ) = 0 . In conclusion, K U ⁢ ( z , z ) = K ⁢ ( z , z ) .

Now let X ⊂ U be any compact set. We have

∥ K U n ⁢ ( z , w ) ∥ X 2 ≤ ∥ K U n ∥ U 2 ≤ K U n ⁢ ( z , z ) ≤ K U ⁢ ( z , z ) ,

so the sequence { K U n ⁢ ( z , ⋅ ) } is bounded in L 2 ⁢ D ⁢ ( U ) . By Theorem 2.5 we claim that { K U n ⁢ ( z , ⋅ ) } is bounded also in the Sobolev space W 2 , 2 ⁢ ( X ) . Now by Theorem 2.4 we see that { K U n ⁢ ( z , ⋅ ) } is also bounded in the Hölder space C 0 , γ ⁢ ( X ¯ ) for any γ > 0 . This means that the hypotheses of the Arzelá–Ascoli theorem are satisfied and in our sequence { K U n ⁢ ( z , ⋅ ) } there exists a subsequence which is locally uniformly convergent to some function K. Using Lemma 6.4 ends the proof. ∎

8 The Ramadanov theorem and orthogonal projections

The main aim of this section is to prove the following theorem.

Theorem 8.1.

Let D be an elliptic operator. Let U = ⋃ n = 1 + ∞ U n , U 1 ⋐ U 2 ⋐ U 3 ⋐ ⋯ . Then

lim n → ∞ ⁡ K U n ⁢ ( z , ⋅ ) = K U ⁢ ( z , ⋅ ) ,

where the limit is the limit in norm, for any z ∈ U .

Note that this theorem is stronger than Theorem 6.1, because the convergence here is the convergence in norm and by (4.1) it implies the locally uniform convergence. (Recall that we may extend K U n ⁢ ( z , ⋅ ) to U by zero.)

In order to prove this theorem, we will need the following lemma.

Lemma 8.2.

Let U = ⋃ n = 1 + ∞ U n , U 1 ⋐ U 2 ⋐ U 3 ⋐ ⋯ . Let F n := { f ∈ L 2 ⁢ ( U ) : f | U n ∈ L 2 ⁢ D ⁢ ( U n ) } . Let P n be the orthonormal projection onto F n . Then

P ⁢ f = lim n → ∞ ⁡ P n ⁢ f ,

where f ∈ L 2 ⁢ ( U ) .

Proof.

By definition, F n is a closed subspace of L 2 ⁢ ( U ) for any natural number n. Moreover, f ∈ F n + 1 implies that f ∈ F n . Indeed,

(8.1) ∫ U n | f ⁢ ( w ) | 2 ⁢ d w ≤ ∫ U n + 1 | f ⁢ ( w ) | 2 ⁢ d w < ∞

and since D ⁢ f = 0 on U n + 1 , we have that also D ⁢ f = 0 on U n . So we know that F 1 ⋑ F 2 ⋑ F 3 ⋑ ⋯ . Of course, L 2 ⁢ D ⁢ ( U ) = ⋂ n = 1 + ∞ F n because of the fact that D ⁢ f = 0 on U is equivalent to D ⁢ f = 0 on each U n and because of inequality (8.1). Using Stone’s theorem ends the proof. ∎

Proof of Main Theorem 8.1.

Let P : L 2 ⁢ ( U ) → L 2 ⁢ D ⁢ ( U ) be orthogonal projection. Then, for any h ∈ L 2 ⁢ ( U ) , we have

h = h 1 + h 2 ∈ L 2 ⁢ D ⁢ ( U ) ⊕ L 2 ⁢ D ⁢ ( U ) ⊥ .

Obviously,

∫ U h ⁢ ( w ) ⁢ K U ⁢ ( z , w ) ⁢ d w = ∫ U h 1 ⁢ ( w ) ⁢ K U ⁢ ( z , w ) ⁢ d w = h 1 ⁢ ( z ) .

Thus we can write

( P ⁢ h ) ⁢ ( z ) = ∫ U h ⁢ ( w ) ⁢ K U ⁢ ( z , w ) ⁢ d w .

Now let X ⊂ U , z ∈ X , and let h be defined in the following way:

h ⁢ ( z ) ⁢ ( w ) := { K X ⁢ ( z , w ) for ⁢ w ∈ X , 0 for ⁢ w ∈ U ∖ X .

Obviously, such an h ⁢ ( z ) ⁢ ( ⋅ ) is an element of L 2 ⁢ ( U ) .

For any f ∈ L 2 ⁢ D ⁢ ( U ) we have

〈 f ∣ P ⁢ h 〉 = 〈 P ⁢ f ∣ h 〉 = 〈 f ∣ h 〉 = ∫ X f ⁢ ( w ) ⁢ K X ⁢ ( z , w ) ⁢ d w = f ⁢ ( z ) .

Since the only element in L 2 ⁢ D ⁢ ( U ) with reproducing property is its reproducing kernel, we have

K U ⁢ ( z , ⋅ ) = ( P ⁢ h ) ⁢ ( z ) , z ∈ U .

Using the lemma for K U n ⁢ ( z , ⋅ ) = ( P n ⁢ h ) ⁢ z ends the proof. ∎

Corollary 8.3.

Let U n be an increasing sequence of domains and let U = ⋃ n = 1 + ∞ U n . Let f n z be the minimal z-solution of the elliptic equation D ⁢ f = 0 on U n and let f z be the minimal z-solution of the elliptic equation D ⁢ f = 0 on U. Then

f n z → f z

and this limit is the limit in the norm.

9 One more proof of the Ramadanov theorem

Theorem 8.1 can be proved in another way. Let f ∈ L 2 ⁢ ( U ) . Then

∫ U χ U n ⁢ ( w ) ⁢ K U n ⁢ ( z , w ) ⁢ f ⁢ ( w ) ⁢ d w = [ P n ⁢ f ] ⁢ ( z ) ,

where χ X is the indicator function of the set X and P n is the orthogonal projection of L 2 ⁢ ( U ) onto L 2 ⁢ D ⁢ ( U n ) as in the proof from the previous Section. By Stone’s theorem, [ P n ⁢ f ] ⁢ ( z ) → [ P ⁢ f ] ⁢ ( z ) , where P is the orthogonal projection of L 2 ⁢ ( U ) onto L 2 ⁢ D ⁢ ( U ) . So we have

∫ U χ U n ⁢ ( w ) ⁢ K U n ⁢ ( z , w ) ⁢ f ⁢ ( w ) ⁢ d w → ∫ U χ U ⁢ ( w ) ⁢ K U ⁢ ( z , w ) ⁢ f ⁢ ( w ) ⁢ d w .

Since f was chosen arbitrarily from L 2 ⁢ ( U ) , we conclude that χ U n ⁢ K U n ⁢ ( z , ⋅ ) converges weakly to χ U ⁢ K U ⁢ ( z , ⋅ ) . Now we need only to show that

lim n → ∞ ⁡ ∫ U χ U n ⁢ | K U n ⁢ ( z , w ) | 2 ⁢ d w ≤ ∫ U χ U ⁢ | K U ⁢ ( z , w ) | 2 ⁢ d w

to prove that, in fact, χ U n ⁢ K U n ⁢ ( z , ⋅ ) converges to χ U ⁢ K U ⁢ ( z , ⋅ ) in the strong topology of L 2 , i.e., we need to show that

lim n → ∞ ⁡ K U n ⁢ ( z , z ) ≤ K U ⁢ ( z , z ) .

This can be done in the same way as in the proof of the main theorem from Section 4.

10 Concluding remarks

In the whole of this section D will be an elliptic operator with smooth coefficients.

It is easy to see that the results on minimal solutions of an elliptic equation can be generalized.

Theorem 10.1.

Let U ⊂ R 2 , z ∈ U and let the weight μ be bounded from below by a non-zero constant. In the set V z , c μ := { f ∈ L 2 ⁢ ( U , μ ) : D ⁢ f = 0 ⁢ and ⁢ f ⁢ ( z ) = c } of weighted square-integrable solutions of the elliptic equation D ⁢ f = 0 , for which f ⁢ ( z ) = c , if it is not empty, there exists exactly one element f 0 such that

∥ f 0 ∥ μ = min f ∈ V z , c μ ⁢ ∥ f ∥ μ .

Such an element in what follows will be called a minimal ( z , c ) -solution in weight μ of the elliptic equation D ⁢ f = 0 on U.

Obviously, if μ is integrable on U and c ⁢ ( x ) = 0 in the divergence form of an elliptic equation, then each constant function is an element of L 2 ⁢ D ⁢ ( U , μ ) and therefore V z , c is not empty and 0 is the unique minimal element for c = 0 .

In particular, it is true for the weight μ ≡ 1 and bounded domain U.

Theorem 10.2.

Let μ n be a sequence of weights convergent to μ a.e. on U such that there exist C 1 , C 2 > 0 for which C 1 < μ n < C 2 for any n ∈ N and C 1 < μ < C 2 a.e. Let f n denote a minimal ( z , c ) -solution in weight μ n of the elliptic equation D ⁢ u = 0 . Let f be a minimal ( z , c ) -solution in weight μ of the elliptic equation D ⁢ u = 0 . Then

lim n → ∞ ⁡ f n = f ,

where the limit above is locally uniform on U.

Theorem 10.3.

Let U n be an increasing sequence of bounded domains and U = ⋃ n = 1 + ∞ U n . Let f n z be a minimal ( z , c ) -solution of the elliptic equation D ⁢ f = 0 on U n and let f z be a minimal ( z , c ) -solution of the elliptic equation D ⁢ f = 0 on U. Then

f n z → f z

in the topology L 2 ⁢ ( U ) .

Proof of Theorems 10.2 and 10.3.

It is obvious that, for c ≠ 0 , the linear operator

A ⁢ f := c ⁢ f

is a bijection between V z μ and V z , c μ and

∥ A ⁢ f ∥ μ = | c | ⋅ ∥ f ∥ μ .

Therefore, in V z , c μ there is exactly one element f c with minimal weighted L 2 -norm and

f c = c ⁢ f 1 ,

where f 1 is the unique element of V z μ with minimal weighted L 2 -norm.

Now let us consider the case c = 0 . Clearly, zero is the only element of V z , 0 μ with minimal norm for any U n and U, and zero is locally the uniform limit of the sequence of zero functions. ∎

Dedicated to the memory of Lech Kaczyński, the last political romantic

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Received: 2022-02-18

Revised: 2022-06-08

Accepted: 2022-08-02

Published Online: 2022-11-11

Published in Print: 2023-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/gmj-2022-2202

Keywords for this article

Reproducing kernel Hilbert space; elliptic operator; solutions of elliptic equations; Ramadanov theorem; continuous dependence of reproducing kernel

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