Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane

Abstract

We consider the Weinstein equation, also known as the equation governing generalized axisymmetric potentials (GASP), with complex coefficients $L_mu={\Delta } u+(m/x)\partial _x u =0$, $m\in {\mathbb {C}}$. We generalize results known for $m\in {\mathbb {R}}$ to the case $m\in {\mathbb {C}}$. In particular, explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities are presented, a Green’s formula for GASP in the right half-plane ${\mathbb {H}}^+$ for $\mathrm{Re}\,m<1$ is established. We prove a new decomposition theorem for the GASP in annular domains for $m\in {\mathbb {C}}$, which is in fact a generalization of the Bôcher’s decomposition theorem. In particular, using bipolar coordinates, it is proved for annuli that a family of solutions for the GASP equation in terms of associated Legendre functions of first and second kind is complete. This family is shown to be a Riesz basis in some non-concentric circular annuli.

1 Introduction

In this article, we study the Weinstein differential operator

well-defined on the right half-plane $\mathbb {H}^+\!=\{(x,y)\in {{\mathbb {R}}}^2: x>0\}=\{z\in {\mathbb {C}}: \mathrm{Re}\,z>0\}$ with the convention . This class of operators is also called operators governing axisymmetric potentials. They have been studied quite extensively in cases $m\in \mathbb {N}$ and $m\in {{\mathbb {R}}}$ in [10–13, 16–18, 22–24, 29–33, 36–38, 41, 51–68]. We will focus exclusively on the case $m\in {\mathbb {C}}$, recalling in the course of the paper some results for integer values of m. The Weinstein equation reads

$$\begin{aligned} L_mu=0. \end{aligned}$$

(1)

The main motivation for which we consider the case $m\in {\mathbb {C}}$ is that if we complexify the coordinates by writing $z=x+iy$, (1) takes the form

which is a particular case of the equation

considered with $\alpha ,\beta \in {\mathbb {C}}$ in [49, Equation (5.7), p. 20].

Equation (1) also appears in physics in the study of the behavior of plasma in a tokamak. The role of tokamak, which has a toroidal geometry, is to control location of the plasma in its chamber by applying magnetic fields on its boundary. It is possible to assume that plasma is axially symmetric what reduces this problem to a plane section in $\mathbb {H}^+$, where the magnetic flux in the vacuum between the plasma and the circular boundary of the chamber satisfies a second-order elliptic nonlinear partial differential equation, the so-called Grad–Shafranov equation, which reduces to the homogeneous equation (1) with $m=-1$ (Fig. 1).

Fig. 1

Section of a Tokamak

Note that in this instance, (1) takes place in an annular domain rather than in a simply connected domain, see [8, 9, 46]. This fact motivates our decomposition theorem, Theorem 5.9.

In the sequel, the sense in which the solutions are studied will be specified. We will also look at solutions to the equation in the sense of distributions

$$\begin{aligned} L_mu=\delta _{(x,y)}, \end{aligned}$$

where $\delta _{(x,y)}$ denotes the Dirac mass at $(x,y)\in \mathbb {H}^+$.

The mentioned above class of operators was first considered by Weinstein in [54], where he studied the case $m\in \mathbb {N}^*$. He also established a relation between the axisymmetric potentials for $m\in \mathbb {N}^*$ and harmonic functions on ${{\mathbb {R}}}^{m+2}$, see Proposition 2.4.

In [20, 58, 59], Weinstein and Diaz–Weinstein established the correspondence principle between axisymmetric potentials $L_m$ and $L_{2-m}$, see Proposition 2.3. They deduced an expression of a fundamental solution (where the singular point is taken on the y-axis) for $m\in {{\mathbb {R}}}$ and established a link between the Weinstein equation and Tricomi equations and their fundamental solutions.

Let us return to the book [49]. Studying elliptic equations with analytic coefficients, Vekua provided means to express their fundamental solutions by using the Riemann functions, introduced earlier (see e.g. [28]) in the real hyperbolic context, he also investigated generalized elliptic equations with complex operators ${\partial }_z$ and ${\partial }_{\overline{z}}$. In heuristic words, in the same way as a harmonic function is the real part of a holomorphic function, or the sum of a holomorphic and an anti-holomorphic function, Vekua established the fact that solutions to elliptic equations, and therefore GASP, can be written as a sum of two functionals, one applied to an arbitrary holomorphic function and the other applied to an arbitrary anti-holomorphic function. These functionals can be written explicitly in terms of the Riemann function, by using the hypergeometric functions [49] or fractional derivations [16]. In [34], Henrici gave a very interesting introduction to the work of Vekua.

More recently, basing on the work of Vekua, Savina [44] gave a series representation of fundamental solutions for the operator $\widehat{L}u={\Delta } u+a{\partial }_x u+b{\partial }_y u+cu$ and studied the convergence of this series. She also provided an application of her results to the Helmholtz equation.

In [31], Gilbert studied the non-homogeneous Weinstein equation, i.e. the case $m\ge 0$, and gave an integral formula for this class of equations. In particular, an explicit solution was given when the second member depends only on one variable.

Some Dirichlet problems are considered in [40, 41] in a special geometry, the so-called “geometry with separable variable”.

Even if some results presented in this paper are known for real values of m, we make a totally self-contained presentation involving elementary technics not necesseraly used in the papers mentioned above. For instance, usual arguments involving estimates of hypergeometric integrals are replaced by arguments using the Lebesgue dominated convergence theorem. Our main result is a decomposition theorem for axisymmetric potentials which is new also for real values of m. We obtain a Liouville-type result for the solutions of Weinstein equation on $\mathbb {H}^+$, with an interesting observation that there is a loss of strict ellipticity of the Weinstein operator on the boundary of $\mathbb {H}^+$. An application of the decomposition theorem is given by showing that an explicit family of axisymmetric potentials constructed by introduction of bipolar coordinates is a Riesz basis in some annuli.

The plan of the paper is the following. In Sect. 2, we recall preliminary information about fundamental solutions for linear partial differential operators with non-constant coefficients. Proposition 2.1 provides with a connection between fundamental solutions for $L_m$ and fundamental solutions for $L_m^\star$, where $L_m^\star$ denotes the formal adjoint of $L_m$. The Weinstein principle [59], valid for m real and complex, establishes a connection between $L_m$ and $L_{2-m}$. We state it without proof as Proposition 2.3. Proposition 2.4, valid only for $m\in \mathbb {N}$, is fundamental in the sense that one can compute a fundamental solution for $L_m$ just knowing the usual fundamental solution for the Laplacian in ${{\mathbb {R}}}^{m+2}$. The corresponding computations are done for $m\in \mathbb {N}$ and for $m\in \mathbb {Z}$ in Sect. 3.

The extension of formulas for fundamental solutions to the case $m\in {\mathbb {C}}$ is the core of Sect. 4. First we describe in Proposition 4.2 in an elementary way the behavior of fundamental solutions near their singularities. Next we use the corresponding estimates to establish the main result of the section, Theorem 4.4.

Section 5 is dedicated to the decomposition theorem. First we modify the fundamental solutions built earlier in order to get fundamental solutions which vanish on the boundary of $\mathbb {H}^+$. Next, in Proposition 5.3, we show that if u is a solution for $L_mu=0$ which vanishes on the boundary of $\mathbb {H}^+$, then $u\equiv 0$ on $\mathbb {H}^+$. Let us emphasize here that, although this statement looks obvious, this is not the case due to the loss of ellipticity of $L_m$ on the boundary of $\mathbb {H}^+$. Let us mention that Proposition 5.3 is a consequence of the maximum principle for pseudo-analytic functions given in a recent paper by Chalendar–Partington [14] for more general function $\sigma$ than $x^m$, but in [14] there is an additional assumption on $\sigma$, which in our case corresponds to the assumption $|m|\ge 1$. The proof of Proposition 5.3 is quite long, but not difficult, it follows careful estimates of fundamental solutions in some parts of $\mathbb {H}^+$. Finally the decomposition theorem, Theorem 5.9, is proved. Its proof is similar to the proof of the Bôcher’s decomposition theorem presented in [4]. We end Sect. 5 with a Poisson formula for axisymmetric potentials in $\mathbb {H}^+$, Proposition 5.10.

In Sect. 6, we consider the case where the annular domain is a kind of annulus. We introduce very classical (in physics) bipolar coordinates, cf. [39], in which the GASP equation has a form presented in Theorem 6.2. Next, applying the method of separation of variables we obtain a basis of solutions in disks and complements of disks in $\mathbb {H}^+$, see Theorem 6.3. In Sect. 7 it is shown that this basis forms a Riesz basis.

2 Notations and preliminaries

Throughout $\mathbb {H}^+\!=\{(x,y)\in {{\mathbb {R}}}^2: x>0\}$ stands for the right half-plane, all scalar functions are assumed to be complex valued. If ${\Omega }$ is an open set in ${{\mathbb {R}}}^n$, $n\in \mathbb {N}^*$, let ${\mathscr {D}}({\Omega })$ designate the space of $C^\infty$-functions compactly supported on ${\Omega }$, where $\mathrm{supp }\,f=\overline{\{x\in {\Omega }: f(x)\ne 0\}}$. If K is a compact set in ${\Omega }$, let ${\mathscr {D}}_K({\Omega })$ be the set of functions $\varphi \in {\mathscr {D}}({\Omega })$ such that $\mathrm{supp}\,\varphi \subset K$.

The partial derivatives of a differentiable function u on an open set ${\Omega }\subset {{\mathbb {R}}}^n$ will be denoted by ${\partial }u/ {\partial }x_i$, ${\partial }_{x_i}u$ or $u_{x_i}$, $i\in \{1,\dots ,n\}$. If $\alpha =(\alpha _1,\ldots ,\alpha _n)\in \mathbb {N}^n$ is a multi-index, we denote

$$\begin{aligned} {\partial }^\alpha ={\partial }_{x_1}^{\alpha _1}\ldots {\partial }_{x_n}^{\alpha _n} ={{\partial }^{|\alpha |}\over {\partial }x_1^{\alpha _1}\ldots {\partial }x_n^{\alpha _n}} \end{aligned}$$

with $|\alpha |=\alpha _1+\cdots +\alpha _n$.

It is assumed that the reader is familiar with the terminology of distributions and we refer to [35].

Let L be a linear differential operator on ${\Omega }$,

$$\begin{aligned} L=\!\sum _{|\alpha |\le N}\!a_{\alpha }{\partial }^\alpha ,\qquad N\in \mathbb {N}, \end{aligned}$$

where the summation runs over the multi-indices $\alpha$ of length $|\alpha |\le N$, $a_\alpha$ are $C^\infty ({\Omega })$-functions. If T is a distribution, then $LT=\sum _{|\alpha |\le N}a_\alpha {\partial }^\alpha T$. Denote by $L^\star$ the adjoint operator of L in the sense of distributions, namely,

$$\begin{aligned} L^\star T=\!\sum _{|\alpha |\le N}\!(-1)^{|\alpha |}{\partial }^\alpha (a_\alpha T). \end{aligned}$$

One can easily check, if $f,g\in {\mathscr {D}}({\Omega })$, we have

$$\begin{aligned} \langle Lf,g\rangle =\langle f,L^\star g\rangle . \end{aligned}$$

Let $a\in {\Omega }$ and L be a differential operator on ${\Omega }$. A fundamental solution for L on ${\Omega }$ at $a\in {\Omega }$ is a distribution $T_a$ such that

$$\begin{aligned} LT_a=\delta _a, \end{aligned}$$

where the equality is understood in the sense of distributions on ${\Omega }$. This equality can be rewritten as

$$\begin{aligned} \varphi (a)=\langle LT_a,\varphi \rangle =\langle T_a,L^\star \varphi \rangle , \qquad \varphi \in {\mathscr {D}}({\Omega }). \end{aligned}$$

In particular, if $a\in {\Omega }$ and $T_a$ is a fundamental solution to $L^\star$ at a on ${\Omega }$ and if $g\in {\mathscr {D}}({\Omega })$ is such that $g=L(\varphi )$ with $\varphi \in {\mathscr {D}}({\Omega })$, then

$$\begin{aligned} \varphi (a)=\langle T_a,g\rangle , \qquad a\in {\Omega }. \end{aligned}$$

Indeed, we have

$$\begin{aligned} \varphi (a)=\langle \delta _a,\varphi \rangle =\langle L^\star T_a,\varphi \rangle =\langle T_a,L\varphi \rangle =\langle T_a,g\rangle , \qquad a\in {\Omega }. \end{aligned}$$

These fundamental solutions are therefore a good tool for solving equations $L\varphi =g$ on ${\mathscr {D}}({\Omega })$ if $g\in {\mathscr {D}}({\Omega })$.

If $m\in \mathbb {N}^*$, the Laplacian in ${{\mathbb {R}}}^m$ will be denoted by ${\Delta }_m$, or ${\Delta }$ when $m=2$. For $m\in {\mathbb {C}}$, $L_m$ denotes the Weinstein operator:

If $f(x,y)=(f_1(x,y),f_2(x,y))$ is a ${C}^1$-vector ${\mathbb {C}}^2$-valued function on an open set in ${{\mathbb {R}}}^2$,

Similarly, if $f:{{\mathbb {R}}}^2\rightarrow {\mathbb {C}}$ is a ${C^1}$-scalar ${\mathbb {C}}^2$-valued function on an open set in ${{\mathbb {R}}}^2$,

With these notations, the operator $L_m$ can be written as follows:

By the Schwarz rule, if u is a function defined on a connected open set in $\mathbb {H}^+$ such that , where $\sigma :\mathbb {H}^+\rightarrow {\mathbb {C}}^*$ is a ${C}^1$-function, then there is a function v which satisfies the well-known generalized Cauchy–Riemann system of equations

and v satisfies the conjugate equation , see for example [7]. This observation justifies the fact that we call $L_{-m}$, $m\in {\mathbb {C}}$, the conjugate operator of $L_m$.

The adjoint operator of $L_m$ is

where $u\in {C}^2(\mathbb {H}^+)$, $(x,y)\in \mathbb {H}^+$. This definition is given on $\mathbb {H}^+$ but it is easily transposed to the case of an open set ${\Omega }$ of $\mathbb {H}^+$.

In the case where the functions involved do not depend only on x and y, we will write $L_{m,x, y}$ instead of $L_{m}$, which means that the partial derivatives are related to the variables x and y, and all other variables are considered to be fixed.

If $u\in {\mathscr {D}}(\mathbb {H}^+)$, we define $S_m u,Du\in {\mathscr {D}}(\mathbb {H}^+)$ as

These operators satisfy the following property.

Proposition 2.1

The operator $S_m$ conjugates $L_m^\star$ and $L_m$, D conjugates $L_{-m}^\star$ and $L_m$, which means that

$$\begin{aligned} S_mL_m^\star =L_mS_m,\qquad L_{-m}^\star D=DL_m. \end{aligned}$$

Remark 2.2

• 1. Let $m\in {\mathbb {C}}$, $S_m$ and $L_m S_m$ are self-adjoint operators, i.e. $S_m=S_m^\star$ and $L_m S_m=(L_m S_m)^\star$.

• 2. Let $\sigma :{\Omega }\rightarrow {\mathbb {C}}$ be a ${C}^1$-function which does not vanish, consider the operator defined on ${C}^2({\Omega })$ as follows:

  

  where ${\Omega }$ is an open set in ${{\mathbb {R}}}^2$. Then

  

  Indeed, if $u,v\in {\mathscr {D}}({\Omega })$, then, by using the derivation in the sense of distributions, we have

  $$\begin{aligned} \langle P_{\sigma }u,v\rangle&= \int _{{\Omega }}{1\over \sigma (x,y)}\,\mathrm{div}\bigl (\sigma (x,y)\nabla u(x,y)\bigr )v(x,y)\,dxdy\\&= -\int _{{\Omega }}\sigma \nabla u{\cdot }\nabla \biggl ({v\over \sigma }\biggr )\,dxdy= \int _{{\Omega }}u\mathrm{div}\biggl (\sigma \nabla \biggl ({v\over \sigma }\biggr )\biggr )dxdy\\&= \langle u,P_{\sigma }^\star v\rangle . \end{aligned}$$

  Define $S_\sigma$ as

  $$\begin{aligned} (S_\sigma u)(x,y)={1\over \sigma (x,y)}\,u(x,y), \qquad u\in C^2({\Omega }). \end{aligned}$$

  Then, one can easily check that $S_\sigma P_\sigma ^\star =P_\sigma S_\sigma$, hence $S_\sigma$ conjugates $P_\sigma$ and $P_\sigma ^\star$. The operators $P_\sigma$ and $S_\sigma$ are a generalization of $L_m$ and $S_m$ with the conjugation relation preserved.

If m is a positive integer, introduce an operator $T_m:u\mapsto v$ defined as follows: $T_m$ maps a function u defined on an open set ${\Omega }$ of $\mathbb {H}^+$ to the function

$$\begin{aligned} v(x_1,\ldots ,x_{m+2})=u\Bigl (\sqrt{x_1^2 +\cdots +x_{m+1}^2},x_{m+2}\Bigr ). \end{aligned}$$

The following two propositions can be found in the Weinstein paper [59] in the case $m\in {{\mathbb {R}}}$. They can be checked by a direct computation for m real and complex, so we omit the proofs.

Proposition 2.3

(Weinstein principle [59]) Let ${\Omega }$ be a relatively compact open set in $\mathbb {H}^+$, if $u:{\Omega }\rightarrow {\mathbb {C}}$ is ${C}^2$, then for all $m\in {\mathbb {C}}$,

$$\begin{aligned} L_mu=x^{1-m}L_{2-m}\bigl [x^{m-1}u\bigr ]. \end{aligned}$$

Proposition 2.4

([54]) Let ${\Omega }$ be a relatively compact open set in $\mathbb {H}^+$. If $u\in {C}^2({\Omega })$ and $m\in \mathbb {N}$, then ${\Delta }_{m+2}(T_mu)=T_m(L_mu)$.

These properties will allow us to calculate fundamental solutions for $L_m$ and $L_m^\star$ for $m\in \mathbb {N}$, and, thereafter, for $m\in \mathbb {Z}$. Finally, estimates of formulas for $L_m,L_m^\star$, $m\in \mathbb {Z}$, will show that these formulas actually provide fundamental solutions for $L_m$ and $L_m^\star$ in the case $m\in {\mathbb {C}}$.

3 Integral expressions of fundamental solutions for integer values of m

Let us recall the definition of the Dirac mass in a point $(x,y)\in {{\mathbb {R}}}^2$:

$$\begin{aligned} \langle \delta _{(x,y)},\varphi \rangle =\varphi (x,y),\qquad \varphi \in {\mathscr {D}}({{\mathbb {R}}}^2). \end{aligned}$$

Proposition 3.1

(partially in [20, 53, 54]) Let $m\in \mathbb {N}^\star$. For $(x,y)\in \mathbb {H}^+$ and $(\xi ,\eta )\in \mathbb {H}^+$,

is a fundamental solution on $\mathbb {H}^+$ for the operator $L^\star _{m,\xi ,\eta }$ at the fixed point $(x,y)\in \mathbb {H}^+$, which means

in the sense of distributions. Moreover, if $(\xi ,\eta )\in \mathbb {H}^+$ is fixed, then

in the sense of distributions, which means that $E_m$ is a fundamental solution on $\mathbb {H}^+$ for the operator $L_{m,x,y}$ at the fixed point $(\xi ,\eta )\in \mathbb {H}^+$.

Proof

Let $m\in \mathbb {N}^*$. Recall that

is a fundamental solution for the Laplacian on ${{\mathbb {R}}}^{m+2}$, i.e. in the sense of distributions ${\Delta }_{m+2} E=\delta _0$, where $\omega _{m+2}$ is the area of the unit sphere in ${{\mathbb {R}}}^{m+2}$. Thus, for all $v\in {\mathscr {D}}({{\mathbb {R}}}^{m+2})$,

where $\tau =(\tau _1,\dots ,\tau _{m+2})$.

Applying this relation to the function $v=T_mu$, where $u\in {\mathscr {D}}(\mathbb {H}^+)$, and, due to Proposition 2.4, for all $(x,y)\in \mathbb {H}^+$ we have

We will simplify this integral expression. For this, we will consider the following hyper-spherical coordinates:

where $\xi =\sqrt{\xi _1^2+\cdots +\xi _{m+1}^2}\ge 0$, $\theta _m\in (-\pi ,\pi )$ and $\theta _1,\ldots ,\theta _{m-1}\in (0,\pi )$. The absolute value of the determinant of the Jacobian matrix defined by this system of coordinates is

Then, for all $(x,y)\in \mathbb {H}^+$,

(2)

with

Let

then $E_m$ can be written as

Also, due to (2) we have

Moreover, since for all $(x,y),(\xi ,\eta )\in \mathbb {H}^+$,

$$\begin{aligned} E_m(x,y,\xi ,\eta )=\biggl ({ x\over \xi }\biggr )^{\!-m}\! E_m(\xi ,\eta ,x,y), \end{aligned}$$

and by Proposition 2.1, $S_m$ conjugates $L_m^\star$ and $L_m$, we have

in the sense of distributions. Hence

and the proof is complete. $\square$

This proposition and the Weinstein principle imply the following result.

Proposition 3.2

(partially in [20, 53, 54]) Let . For $(x,y),(\xi ,\eta )\in \mathbb {H}^+$,

is a fundamental solution on $\mathbb {H}^+$ for the operator $L^\star _{m,\xi ,\eta }$ at the fixed point $(x,y)\in \mathbb {H}^+$ and it is also a fundamental solution on $\mathbb {H}^+$ for the operator $L_{m,x,y}$ at the fixed point $(\xi ,\eta )\in \mathbb {H}^+$.

Proof

For all $m\in \mathbb {N}^*$, $u\in {\mathscr {D}}(\mathbb {H}^+)$ and $(x,y)\in \mathbb {H}^+$ we have

and by the Weinstein principle, Proposition 2.3, we have

Denoting $v(x,y)=x^{m-1}u(x,y)$, we obtain

then, for all , $v\in {\mathscr {D}}(\mathbb {H}^+)$ and $(x,y)\in \mathbb {H}^+$, putting $m=2-m'$, we have

The proof of the second statement is similar. $\square$

4 Fundamental solutions for the Weinstein equation with complex coefficients

In this section, we will generalize the result obtained in the previous section for $m\in \mathbb {Z}$ to the case $m\in {\mathbb {C}}$.

Let $m\in {\mathbb {C}}$. If $\mathrm{Re}\,m\ge 1$ put

(3)

and if $\mathrm{Re}\,m<1$ put

(4)

here, if $\alpha >0$ is a real number and $\mu$ is a complex number, . Both values are well defined as the integrals on the right-hand side converge in the Lebesgue sense.

Proposition 4.1

For $m\in {\mathbb {C}}$ and $(\xi ,\eta )\in \mathbb {H}^+$ fixed, we have

and for $(x,y)\in \mathbb {H}^+$ fixed, we have

Proof

For convenience, denote

To prove the first equality of the proposition, it suffices to show that

Let us compute the partial derivatives of the function $f_m$:

We then have

Note that

hence

Noting that

we have

Integrating by parts, we have

and the result is deduced in the case $\mathrm{Re} \,m\ge 1$. The same argument is valid for $\mathrm{Re}\,m<1$. The second equality of the proposition can be deduced immediately from the fact that $S_m$ conjugates $L_m^\star$ and $L_m$, see Proposition 2.1. $\square$

In the sequel, we will denote

The following proposition describes the behavior of $E_m$ defined by (3) and (4) near its singularity. In particular, we show that the behavior of $E_m$ is close to the behavior of fundamental solutions for the Laplacian. This fact is well known for elliptic operators. But we emphasize here that in our proof the estimates of elliptic integrals are elementary (obtained using the dominated convergence theorem) and we do not use estimates arising from classical estimates of hypergeometric functions.

Proposition 4.2

Let $m\in {\mathbb {C}}$. For $(x,y)\in \mathbb {H}^+$ fixed,

Proof

We start with $\mathrm{Re}\,m\ge 1$. In this case, we have

Note that when $d\rightarrow 0$, k tends to $+\infty$.

Claim 4.3

Proof

Putting $u=\sin \theta /2$, we have

Note that

where, due to monotone convergence, the right-hand side tends to

as $k\rightarrow +\infty$. The change of variable $u={\text{ sh }}\,t/\sqrt{k}$ gives us

Since $\mathrm{th}^{m-1}t$ tends to 1 as $t\rightarrow +\infty$, we deduce that as $k\rightarrow +\infty$,

The proof is complete. $\blacksquare$

Due to Claim 4.3, we have

as $d\rightarrow 0+$. The case $\mathrm{Re }\,m<1$ is analogous. $\square$

Now, we can prove the main result of this section which shows that $E_m$ are fundamental solutions not only for $m\in \mathbb {N}$ but for all $m\in {\mathbb {C}}$.

Theorem 4.4

Let $m\in {\mathbb {C}}$. For $(x,y),(\xi ,\eta )\in \mathbb {H}^+$, $E_m$ defined by (3) and (4) is a fundamental solution on $\mathbb {H}^+$ for the operator $L^\star _{m,\xi ,\eta }$ at the fixed point $(x,y)\in \mathbb {H}^+$, which means that on $\mathbb {H}^+$

in the sense of distributions. Moreover, if $(\xi ,\eta )\in \mathbb {H}^+$ is fixed, then on $\mathbb {H}^+$

in the sense of distributions, which means that $E_m$ is a fundamental solution on $\mathbb {H}^+$ for the operator $L_{m,x,y}$ at the fixed point $(\xi ,\eta )\in \mathbb {H}^+$.

Proof

Let $m\in {\mathbb {C}}$ and $u\in \mathscr { D}(\mathbb {H}^+)$. Let $(x,y)\in \mathbb {H}^+$ and ${\varepsilon }>0$ be such that $D((x,y),{\varepsilon })\subset \mathbb {H}^+$, where $D((x,y),{\varepsilon })$ is a disk with center (x, y) and radius ${\varepsilon }$. Put

We use the fact that $L_m^\star (E_m)=0$ on . An elementary computation gives

Let us recall the Green formula: If ${\Omega }$ is an open set in ${{\mathbb {R}}}^2$ with a piecewise ${C}^1$-differentiable boundary, then

where $\mathbf {n}$ is the outer unit normal vector to ${\partial }{\Omega }$ and ds the arc length element on ${\partial }{\Omega }$ (positively oriented), $X=(X_1,X_2):\overline{\Omega }\rightarrow {\mathbb {C}}^2$ is a ${C}^1$-vector field.

Applying this formula to the open set , where U is a regular open set in $\mathbb {H}^+$ containing the support of u, we have

Proposition 4.2 implies

as ${\varepsilon }\rightarrow 0+$ because $\lim _{{\varepsilon }\rightarrow 0}{\varepsilon }\ln {\varepsilon }=0$. Then, if we want to prove that $\lim _{{\varepsilon }\rightarrow 0} \mathrm{I}_{\varepsilon }$ exists, we have to prove the existence of

and this limit will be equal to the limit of $\mathrm{I}_{\varepsilon }$.

Now, assume that $\mathrm{Re}\,m\ge 1$. Denote by $\mathrm{J}_{\varepsilon }$ the integral in the previous expression. A computation gives

where $k=4x\xi /{\varepsilon }^2$.

Claim 4.5

Proof

We put $u=\sin \theta /2$, then

Note that

as $k\rightarrow +\infty$. The change of variable $u={\text{ sh }}\, t/\sqrt{k}$ gives us

Since $\mathrm{th}^{m+1}t$ tends to 1 as $t\rightarrow +\infty$, it follows that as $k\rightarrow +\infty$

Claim 4.6

Proof

Putting as previously $u=\sin \theta /2$, we have

Note that

Let us estimate the right-hand side of this equality:

as $k\rightarrow +\infty$. As seen in the proof of Claim 4.5, we have

as $k\rightarrow +\infty$. Due to ($\star$) and ($\star \star$), we have

as $k\rightarrow +\infty$. The change of variable $u={\text{ sh }}\,t/\sqrt{k}$ gives

It follows that as $k\rightarrow +\infty$,

Thus

and

as $k\rightarrow +\infty$. This completes the proof. $\blacksquare$

Let us return to the proof of Theorem 4.4. Claim 4.3 implies

which tends to 0 as ${\varepsilon }\rightarrow 0+$.

Claim 4.5 implies

which tends to 0 as ${\varepsilon }\rightarrow 0+$.

Finally, Claim 4.6 implies

which tends to u(x, y) as ${\varepsilon }\rightarrow 0+$.

So we have proved that for all $m\in {\mathbb {C}}$ such that $\mathrm{Re}\,m>0$,

therefore $E_m$ indeed is a fundamental solution for the operator $L_m^\star$ for all $m\in {\mathbb {C}}$ with $\mathrm{Re}\,m>0$. The case $m\in {\mathbb {C}}$ with $\mathrm{Re}\,m\le 1$ is similar.

Due to Proposition 2.1, we also have dual assertions for fundamental solutions for the operator $L_m$. $\square$

The following proposition is roughly a consequence of the previous theorem. Of course, it is a classical statement, but we would like to present its short proof.

Proposition 4.7

Let $m\in {\mathbb {C}}$ and let ${\Omega }$ be a relatively compact open set in $\mathbb {H}^+$ whose boundary is piecewise $C^1$-differentiable. Then, for $(x,y)\in {\Omega }$ and $u\in {C}^2(\overline{\Omega })$, we have

where $u=u(\xi ,\eta )$, $E_m=E_m(x,y,\xi ,\eta )$, $\mathbf {n}$ is the outer unit normal vector to ${\partial }{\Omega }$ and ds is the arc length element on ${\partial }{\Omega }$ (positively oriented).

Proof

Indeed, if $u\in {C}^2(\overline{\Omega })$, for $(x,y)\in {\Omega }$ and ${\varepsilon }>0$ such that $\overline{D((x,y),{\varepsilon })}\subset {\Omega }$, we have

By the Green formula, the latter integral is equal to

and, as we saw in the previous proof, it tends to

as ${\varepsilon }\rightarrow 0$. Due to integrability of $E_m$ near (x, y) we have

and the proof is complete. $\square$

5 Liouville-type result and decomposition theorem for axisymmetric potentials

In the previous section, we have seen that fundamental solutions $E_m$ in the complex case have different expressions depending on whether $\mathrm{Re}\,m<1$ or $\mathrm{Re}\,m\ge 1$. Hence the behavior of $E_m$ will be different in each case.

We will modify fundamental solutions so that they vanish at the boundary of $\mathbb {H}^+$, which means that they tend to zero on the y-axis and at infinity. Expression (4) satisfies this property: $E_m(x,y,{\cdot },{\cdot })$ tends to 0 as $x\rightarrow 0+$ and $\Vert (x,y)\Vert \rightarrow +\infty$; whereas (3) does not. Consider

It is also a fundamental solution on $\mathbb {H}^+$ and it satisfies the required property. Let us put

• for $\mathrm{Re}\,m<1$,

  $$\begin{aligned} F_m(x,y,\xi ,\eta )=E_m(x,y,\xi ,\eta ), \end{aligned}$$

• for $\mathrm{Re}\,m\ge 1$,

  

We will need the following definition of convergence on the boundary of $\mathbb {H}^+$.

Definition 5.1

Let $u:\mathbb {H}^+\!\rightarrow {{\mathbb {R}}}$ be a function defined on $\mathbb {H}^+$. We write

$$\begin{aligned} \lim _{{\partial }\mathbb {H}^+}u=0 \end{aligned}$$

if and only if for all ${\varepsilon }>0$ there exists $N\in \mathbb {N}$ such that for all $n>N$ and all $(x,y)\in H^+$, $x\le {1/n}$ or $\Vert (x,y)\Vert \ge n$ implies $|u(x,y)|\le {\varepsilon }$.

Proposition 5.2

Let $u:\mathbb {H}^+\!\rightarrow {\mathbb {C}}$. We have $\lim _{{\partial }\mathbb {H}^+}u=0$ if and only if

$$\begin{aligned} \lim _{\Vert (x,y)\Vert \rightarrow +\infty }\!\!u(x,y)=0\qquad \hbox {and}\qquad \lim _{(0,y)}u=0,\quad y\in {{\mathbb {R}}}. \end{aligned}$$

Proof

The direct implication is easy. Conversely, assume $\lim _{\Vert (x,y)\Vert \rightarrow +\infty }u(x,y)=0$ and $\lim _{(0,y)}u=0$, $y\in {{\mathbb {R}}}$. Let ${\varepsilon }>0$, then there is $A>0$ such that for all $(\xi ,\eta )\in \mathbb {H}^+$, $\sqrt{\xi ^2+\eta ^2}\ge A$ implies $|u(\xi ,\eta )|\le {\varepsilon }$. Similarly, for all $y\in {{\mathbb {R}}}$, there is $\alpha _y\in (0,1)$ such that for all $(\xi ,\eta )\in \mathbb {H}^+$, $\sqrt{\xi ^2+(\eta -y)^2}<\alpha _y$ implies $|u(\xi ,\eta )|\le {\varepsilon }$.

The interval $[-A,A]$ is compact. By the Lebesgue covering lemma, there is $\alpha >0$ such that for all $y'\in [-A,A]$, the ball $B(y',\alpha )$ is included in one of the balls $B(y,\alpha _{y})$ with $y\in [-A,A]$. In particular, if $(\xi ,\eta )\in \mathbb {H}^+$ is such that $0<\xi <\alpha$, then $|u(\xi ,\eta )|\le {\varepsilon }$. This completes the proof. $\square$

The following proposition is a Liouville-type result for axisymmetric potentials in the right half-plane. As we mentioned in the introduction, this result is not trivial due to the loss of strict ellipticity of the Weinstein operator on the y-axis. Let us mention that in [5, Theorem 7.1] one can find an interesting result on the description of a class of non-strictly elliptic equations with unbounded coefficients.

Proposition 5.3

Let $u\in C^2(\mathbb {H}^+)$ be such that $L_mu=0$ and $\lim _{{\partial }\mathbb {H}^+}u=0$. Then $u\equiv 0$ on $\mathbb {H}^+$.

Proof

For $(\xi ,\eta )\in \mathbb {H}^+$ and $N\in \mathbb {N}^*$, define

where $\theta _1$ and $\theta _2$ are smooth functions on ${{\mathbb {R}}}$, valued on [0, 1] and such that $\theta _1(t)=1$ for $t\ge 1$, $\theta _1(t)= 0$ for $t\le {1/2}$, $\theta _2(t)=1$ for $t\in [-{1/2},{1/2}]$, and $\theta _2(t)=0$ for $t\in {{\mathbb {R}}}{\setminus }(-1,1)$. Assume also that all derivatives of $\theta _1$ and $\theta _2$ vanish at $\{-1,-{1/2},$ ${1/2},1\}$ (Fig. 2).

Fig. 2

The functions $\theta _1$ and $\theta _2$

If $u\in {C}^2(\mathbb {H}^+)$ satisfies $L_mu=0$, then $u\phi _N\in {C}^2(\mathbb {H}^+)$ and it is compactly supported on $\mathbb {H}^+$. Throughout the following, we fix $(x,y)\in \mathbb {H}^+$. For N sufficiently large, due to Proposition 4.7 (true if $E_m$ is replaced by $F_m$), we have

(because the function $L_m(u\phi _N)$ is identically zero in a neighborhood of the singularity of $F_m$), thus

where $D_1,\ldots , D_8$ are the following domains (which depend on N) (Fig. 3):

Fig. 3

Domains $D_{i}$

Since $\lim _{{\partial }\mathbb {H}^+}\!u=0$,

We will estimate integrals over sets $D_1,\ldots ,D_8$ separately, see auxiliary lemmas below. Recall that, if $(u_N)_N$ and $(v_N)_N$ are complex sequences, $u_N=\mathrm{O}(v_N)$ means that there exists a constant M such that, for every N sufficiently large, $|u_N|\le M|v_N|$; $u_N=\mathrm{o}(v_N)$ means that for every ${\varepsilon }>0$, for every N sufficiently large, $|u_N|\le {\varepsilon }|v_N|$.

Lemma 5.4

On $D_1$, we have

$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N)\qquad \hbox {and} \qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=0. \end{aligned}$$

On $D_2\cup D_4$, we have

$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=0\qquad \hbox {and} \qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ). \end{aligned}$$

On $D_3$, we have

$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr )\qquad \hbox {and}\qquad \sup {\biggl |{{\partial }\phi _N \over {\partial }\eta }\biggr |}=0. \end{aligned}$$

On $D_5\cup D_8$, we have

$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N) \qquad \hbox {and}\qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |} =\mathrm{O}\biggl ({1\over N}\biggr ). \end{aligned}$$

On $D_6\cup D_7$, we have

$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr )\qquad \hbox {and}\qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ). \end{aligned}$$

On $D_1\cup D_5\cup D_8$, we have

$$\begin{aligned} \sup |L_{-m}(\phi _N)|=\mathrm{O}(N^2). \end{aligned}$$

On $D_2\cup D_3\cup D_4\cup D_6\cup D_7$, we have

$$\begin{aligned} \sup |L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$

Proof

For $(\xi ,\eta )\in D_1$, $\phi _N(\xi ,\eta )=\theta _1(N\xi )$ and thus

which give us

$$\begin{aligned} \sup _{D_1}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N),\qquad \sup _{D_1}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=0,\qquad \sup _{D_1}|L_{-m}(\phi _N)|=\mathrm{O}(N^2) \end{aligned}$$

since the derivatives of $\theta _1$ are bounded and for $(\xi ,\eta )\in D_1$ one gets $\xi \ge {1/(2N)}$.

For $(\xi ,\eta )\in D_2$, $\phi _N(\xi ,\eta )=\theta _2({\eta / N})$ and thus

$$\begin{aligned}&\displaystyle {{\partial }\phi _N\over {\partial }\xi }(\xi ,\eta )=0,\qquad {{\partial }\phi _N\over {\partial }\eta }(\xi ,\eta )={1\over N}\,\theta _2'\biggl ({\eta \over N}\biggr ),&\\&\displaystyle L_{-m}\phi _N(\xi ,\eta )={1\over N^2}\,\theta _2''\biggl ({\eta \over N}\biggr ),&\end{aligned}$$

which give us

$$\begin{aligned} \sup _{D_2}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=0,\qquad \sup _{D_2}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\qquad \sup _{D_2}|L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$

The same works for $D_4$.

For $(\xi ,\eta )\in D_3$, $\phi _N(\xi ,\eta )=\theta _2({\xi / N})$ and thus

which give us

$$\begin{aligned} \sup _{D_3}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\qquad \sup _{D_3}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=0, \qquad \sup _{D_3}|L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$

For $(\xi ,\eta )\in D_5, \phi _N(\xi ,\eta )=\theta _1(N\xi )\theta _2({\eta / N})$ and thus

which give us

$$\begin{aligned} \sup _{D_5}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N),\qquad \sup _{D_5}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\qquad \sup _{D_5}|L_{-m}(\phi _N)|=\mathrm{O}(N^2). \end{aligned}$$

The same works for $D_8$.

For $(\xi ,\eta )\in D_6$, $\phi _N(\xi ,\eta )=\theta _2({\xi / N})\theta _2({\eta / N})$ and thus

which give us

$$\begin{aligned} \sup _{D_6}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\quad \; \sup _{D_6}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\quad \;\sup _{D_6}|L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$

The same works for $D_7$. $\blacksquare$

We now estimate the following quantities for $i\in \{1,\,\ldots ,\,8\}$:

$$\begin{aligned} \int _{D_i}\!\! |F_m| \,d\xi d\eta , \qquad \int _{D_i} \!\!|{\partial }_\xi F_m| \,d\xi d\eta \qquad \int _{D_i}\!\! |{\partial }_\eta F_m| \,d\xi d\eta . \end{aligned}$$

Lemma 5.5

For $\mathrm{Re}\, m<1$, we have:

• for $i=1$,

  $$\begin{aligned} \int _{D_i}\!\!|F_m|\,d\xi d\eta =\mathrm{O}\biggl ({1\over N^2}\biggr ),\qquad \int _{D_i}\biggl |{{\partial }F_m\over {\partial }\xi }\biggr |\,d\xi d\eta =\mathrm{O}\biggl ({1\over N}\biggr ); \end{aligned}$$

• for $i=2, 4$,

  $$\begin{aligned} \int _{D_i}\!\!|F_m|\,d\xi d\eta =\mathrm{O}({N^2}),\qquad \int _{D_i}\biggl |{{\partial }F_m\over {\partial }\eta }\biggr |\,d\xi d\eta =\mathrm{O}({N}); \end{aligned}$$

• for $i=3$,

  $$\begin{aligned} \int _{D_i}\!\!|F_m|\,d\xi d\eta =\mathrm{O}({N^2}),\qquad \int _{D_i}\biggl |{{\partial }F_m\over {\partial }\xi }\biggr |\,d\xi d\eta =\mathrm{O}({N}); \end{aligned}$$

• for $i=5,8$,

  

• for $i=6,7$,

  

Proof

By definition, for $\mathrm{Re}\,m<1$,

Therefore there is a constant $C_1$ such that for all $(\xi ,\eta )\in \mathbb {H}^+$, we have

(5)

Similarly, we have

and as before, as for all $\theta \in [0,\pi ]$,

there exists a constant $C_2$ such that for all N large enough and all $(\xi ,\eta )\in \mathbb {H}^+$, we have

(6)

Finally, as

there exists a constant $C_3$ such that for all N large enough and all $(\xi ,\eta )\in \mathbb {H}^+$, we have

(7)

Using these inequalities, we estimate integrals on domains $D_i$.

$\underline{\hbox {On }D_1}$: Inequality (5) implies

Then, thanks to (6), we have

$\underline{\hbox {On }D_2}$: Due to inequality (5), we have

Then, thanks to (7), we have

$\underline{\hbox {On }D_3}$: Due to inequality (5), we have

Then, thanks to (6), we have

$\underline{\hbox {On }D_4}$: This case is analogous to the case $D_2$.

$\underline{\hbox {On }D_5}$: Due to inequality (5), we have

$$\begin{aligned} \int _{D_5}|F_m|\,d\xi d\eta&=\mathrm{O}(1)\int _{{1/ 2N}}^{{1/ N}}d\xi \int _{{N/2}}^N {\xi \, d\eta \over \bigl [(x-\xi )^2+(\eta -y)^2\bigr ]^{1-{\mathrm{Re\,} m/ 2}}} \\&=\mathrm{O}\biggl (\frac{1}{N^2}\biggr )\int _{{N/2}}^N { d\eta \over (\eta -y)^{2-\mathrm{Re\,} m}} \\&=\mathrm{O}\biggl (\frac{1}{N^2}\biggr )\biggl [{1\over (N-y)^{1-\mathrm{Re\,}m}}-{1\over ({N/2}-y)^{1-\mathrm{Re\,}m}}\biggr ]=\mathrm{O}\biggl (\frac{1}{N^{3-\mathrm{Re\,}m}}\biggr ). \end{aligned}$$

Then, thanks to (6), we have

Estimate (7) gives

$$\begin{aligned} \int _{D_5}\biggl |{{\partial }F_m\over {\partial }\eta }\biggr |\,d\xi d\eta =\mathrm{O}(1)\int _{{1/ 2N}}^{{1/ N}}d\xi \int _{{N/2}}^{{N}} {\xi \,d\eta \over |\eta -y|^{3-\mathrm{Re\,}m}}=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$

$\underline{\hbox {On }D_6}$: Due to (5), we have

$$\begin{aligned} \int _{D_6}\!\!|F_m|\,d\xi d\eta&=\mathrm{O}(1)\int _{{N/2}}^{{N}}d\xi \int _{{N/2}}^N {\xi \, d\eta \over \bigl [(x-\xi )^2+(\eta -y)^2\bigr ]^{1-{\mathrm{Re\,} m/ 2}}}\\&=\mathrm{O}(N^2)\int _{{N/2}}^N { d\eta \over (\eta -y)^{2-\mathrm{Re\,} m}}\\&=\mathrm{O}(N^2)\biggl [{1\over (N-y)^{1-\mathrm{Re\,}m}}-{1\over ({N/2}-y)^{1-\mathrm{Re\,}m}}\biggr ]=\mathrm{O}\bigl (N^{1+\mathrm{Re\,}m}\bigr ). \end{aligned}$$

Then, thanks to (6), we have

Estimate (7) gives

$$\begin{aligned} \int _{D_6}\biggl |{{\partial }F_m\over {\partial }\eta }\biggr |\,d\xi d\eta&=\mathrm{O}(1)\int _{{N/ 2}}^{{N}}d\xi \int _{{N/2}}^{{N}} {\xi \, d\eta \over |\eta -y|^{3-\mathrm{Re\,}m}}\\&=O(N^2)\int _{{N/2}}^{{N}} {d\eta \over |\eta -y|^{3-\mathrm{Re\,}m}} =\mathrm{O}\bigl (N^{\mathrm{Re\,}m}\bigr ). \end{aligned}$$

$\underline{\hbox {On }D_7, D_8}$: These cases are analogous to the cases $D_6$ and $D_5$, respectively.$\blacksquare$

Lemma 5.6

Lemma 5.5 remains true for $\mathrm{Re }\,m\ge 1$.

Proof

For $\mathrm{Re}\,m\ge 1$, we have

Since for all $(\xi ,\eta )\in \mathbb {H}^+$, we have

then

(8)

and there is a constant $C'_1$ such that for all $(\xi ,\eta )\in \mathbb {H}^+$, we have

(9)

This inequality does not suffice to estimate integrals over $D_1$. We shall improve it as follows. Rewrite $F_m$ as

where

For $(x,y)\in \mathbb {H}^+$, $\theta \in [0,\pi ]$ and $\eta \in {{\mathbb {R}}}$ fixed, define a function $g_m$ on $[-1/N,1/N]$, with $1/N<x$, by

This function is well defined because

and the last term is greater than $(x-1/N)^2>0$.

We have

$$\begin{aligned} K_m(x,y,\xi ,\eta ,\theta )=g_m(\xi )-g_m(-\xi ) \end{aligned}$$

thus

$$\begin{aligned} \bigl |K_m(x,y,\xi ,\eta ,\theta )\bigr |\le 2\xi \!\!\sup _{[-\xi ,\xi ]}\!|g_m'|\le 2|m|\xi \, {|\xi -x|+2x\over \bigl [(x-\xi )^2+(y-\eta )^2\bigr ]^{1+\mathrm{Re\,}m/2}}, \end{aligned}$$

which implies that there exists a constant $c'_1$ such that for all $(\xi ,\eta )\in D_1$,

$$\begin{aligned} |F_m|\le c'_1\,{\xi ^{\mathrm{Re\,}m+1}\over \bigl [(x-\xi )^2+(\eta -y)^2\bigr ]^{1+\mathrm{Re\,}m/2}}. \end{aligned}$$

(10)

Similarly, we have

(11)

and as before, for all $\theta \in [0,\pi ]$,

and thanks to (8), for all $\theta \in [0,\pi ]$,

These estimates, (11) and (9) show that there is a constant $C'_2$ such that for large enough N and all $(\xi ,\eta )\in \mathbb {H}^+$, we have

(12)

We can improve this inequality on $D_1$, by using inequality (10) instead of (9), then there are two constants $C''_2$ and $C'''_2$ (which do not depend on N) such that for all $(\xi ,\eta )\in D_1$,

(13)

Finally,

Similarly, there is a constant $C'_3$ such that for all N large enough and all $(\xi ,\eta )\in \mathbb {H}^+$, we have

(14)

Thanks to these inequalities, we can now estimate the corresponding integrals over domains $D_i$.

$\underline{\hbox {On }D_1}$: Due to (10), we have

Then thanks to (13),

$\underline{\hbox {On }D_2}$: Due to (9), we have

because we integrate a bounded function (independent of N) on a domain with measure controlled by $\mathrm{O}(N^2)$.

Then, inequality (14) implies

$\underline{\hbox {On }D_3}$: Due to (9), we have

Then, thanks to (12), we have

$\underline{\hbox {On }D_4}$: This case is analogous to the case $D_2$.

$\underline{\hbox {On }D_5}$: Due to (9), we have

Then, thanks to (12), we have

Applying inequality (14), we have

$\underline{\hbox {On }D_6}$: Due to (9), we have

Then, thanks to (12), we obtain

Finally, inequality (14) implies

$\underline{\hbox {On }D_7, D_8}$: These cases are analogous to the cases $D_6$ and $D_8$, respectively. $\blacksquare$

In the following table, we summarize results obtained on the previous lemmas:

i		$\int _{D_i}\!|F_m|$	$(|{\partial }_{\xi }\phi _N|,|{\partial }_{\eta }\phi _N|)$	$\int _{D_i}\!|{\partial }_{\xi }F_m|$	$\int _{D_i}\!|{\partial }_{\eta }F_m|$
1	$\mathrm{O}(N^2)$	$\mathrm{O}(1/N^2)$	$(\mathrm{O}(N),0)$	$\mathrm{O}({1/ N})$	$\times$
2	$\mathrm{O}(1/N^2)$	$\mathrm{O}(N^2)$	$(0,\mathrm{O}({1/ N}))$	$\times$	$\mathrm{O}(N)$
3	$\mathrm{O}(1/N^2)$	$\mathrm{O}(N^2)$	$(\mathrm{O}({1/ N}),0)$	$\mathrm{O}(N)$	$\times$
4	$\mathrm{O}(1/N^2)$	$\mathrm{O}(N^2)$	$(0,\mathrm{O}({1/ N}))$	$\times$	$\mathrm{O}(N)$
5	$\mathrm{O}(N^2)$	$\mathrm{O}(1/N^2)$	$(\mathrm{O}(N),\mathrm{O}({1/ N}))$	$\mathrm{O}({1/ N})$	$\mathrm{O}({1/ N^2})$
6	$\mathrm{O}(1/N^2)$	$\mathrm{O}(N^2)$	$(\mathrm{O}({1/ N}),\mathrm{O}({1/ N}))$	$\mathrm{O}(N)$	$\mathrm{O}(N)$
7	$\mathrm{O}(1/N^2)$	$\mathrm{O}(N^2)$	$(\mathrm{O}({1/ N}),\mathrm{O}({1/ N}))$	$\mathrm{O}(N)$	$\mathrm{O}(N)$
8	$\mathrm{O}(N^2)$	$\mathrm{O}(1/N^2)$	$(\mathrm{O}(N),\mathrm{O}({1/ N}))$	$\mathrm{O}({1/ N})$	$\mathrm{O}({1/ N^2})$

We can easily check that for each $i\in \{1,\, \ldots ,\, 8\}$, the quantities

are bounded. Therefore,

$$\begin{aligned} u(x,y)=\mathrm{o}(1)\qquad \text {as}\quad N\rightarrow +\infty . \end{aligned}$$

Thus $u\equiv 0$ and this completes the proof of Proposition 5.3.$\square$

Proposition 5.7

Let $u\in {\mathscr {D}}(\mathbb {H}^+)$ and let $(x,y)\in \mathbb {H}^+$, define

then $\lim _{\Vert (x,y)\Vert \rightarrow +\infty } U=0$, and for all $y\in {{\mathbb {R}}}$, $\lim _{(0,y)} U=0$. Moreover, and for all $(x,y)\not \in { \mathrm supp }\,u$ we have $L_{m,x,y}U(x,y)=0$.

Proof

Fix $(\xi ,\eta )$. For ${\mathrm{Re\,}}\,m<1$,

hence $F_m(x,y,\xi ,\eta )\rightarrow 0$ as $\Vert (x,y)\Vert \rightarrow +\infty$. For $\mathrm{Re}\,m\ge 1$,

hence $F_m(x,y,\xi ,\eta )\rightarrow 0$ as $\Vert (x,y)\Vert \rightarrow +\infty$. So the first statement of the proposition is shown.

For the second statement, for $\mathrm{Re}\,m<1$, we have

as $(x,y)\rightarrow (0,y')$, which implies the desired result.

Now, assume that $\mathrm{Re}\,m\ge 1$. Let $(\xi ,\eta )$ be fixed in the support of u, which is a compact set in $\mathbb {H}^+$. In particular, there exist $M>0$ and $\alpha >0$ which do not depend on u such that $\Vert (\xi ,\eta )\Vert \le M$ and $\xi \ge 2\alpha$. Let y be in ${{\mathbb {R}}}$. Denote

By the mean value inequality, for $x>0$ near 0, we have

and

then

In particular,

$$\begin{aligned} \sup _{\mathop {(\xi ,\eta )\in \mathrm{supp\,} u}\limits _{ y\in {{\mathbb {R}}}}} \!\!\!|F_m(x,y)|=\mathrm{O}(x) \end{aligned}$$

as $x\rightarrow 0+$. The second statement is proved.

The last statement can be deduced from the fact that if $(x,y)\not =(\xi ,\eta )$ are both in $\mathbb {H}^+$, then

Remark 5.8

If $U\in {\mathscr {D}}(\mathbb {H}^+)$, then $L_{m,x,y}U=u$, but this identity is not necessarily true if $U\not \in {\mathscr {D}}(\mathbb {H}^+)$. In particular, we cannot say that in Proposition 5.7 we have $L_mU=u$.

Now, we will prove a decomposition theorem for axisymmetric potentials, it is interesting to compare it with the known result in [6, Section 4, Theorem 2]. The fundamental difference is that in this paper, the conductivity is not extended by reflection through the boundary ${\partial }{\Omega }$ to the whole domain.

Note that, due to our construction of fundamental solutions, the proof of this theorem is more or less the same as the proof of the decomposition theorem in [4, Chapter 9]. Note also that in our situation, the domain of our functions is $\mathbb {H}^+$ not ${\mathbb {C}}$.

Theorem 5.9

Let $m\in {\mathbb {C}}$. Let ${\Omega }$ be an open set in $\mathbb {H}^+$ and let K be a compact set in ${\Omega }$. If satisfies $L_mu=0$ in , then u has a unique decomposition as

$$\begin{aligned} u=v+w, \end{aligned}$$

where $v\in {C}^2({\Omega })$ satisfies $L_mv=0$ in ${\Omega }$ and satisfies $L_mw=0$ in with $\lim _{{\partial }\mathbb {H}^+} w=0$.

Proof

For $E\subset {\mathbb {C}}$ and $\rho >0$, define $E_\rho =\{x\in {\mathbb {C}}: d(x,E)<\rho \}$, i.e. $E_\rho$ is a neighborhood of E.

First, assume that ${\Omega }$ is a relatively compact open set in $\mathbb {H}^+$. Choose $\rho$ small enough so that $K_\rho$ and $({\partial }{\Omega })_\rho$ are disjoint. There is a function $\varphi _\rho \in {\mathscr {D}}(\mathbb {H}^+)$ compactly supported on such that $\varphi _\rho \equiv 1$ in a neighborhood of (Fig. 4).

Fig. 4

$\varphi _{\rho }\equiv 1$ on the grey domain

For , denote

$$\begin{aligned} F_z(\zeta )=F_m(x,y,\xi ,\eta ),\qquad L_\zeta =L_{m,\xi ,\eta }\quad \qquad \mathrm{for}\quad \zeta =\xi +i\eta . \end{aligned}$$

Thanks to Proposition 4.7, we have

Then, the last result of Proposition 5.7 shows us that $v_\rho$ satisfies $L_mv_\rho =0$ on and $w_\rho$ satisfies $L_mw_\rho =0$ on . We also have $\lim _{{\partial }\mathbb {H}^+}w_\rho =0$.

Now, assume that $\sigma <\rho$. As previously, we obtain the decomposition $u=v_\sigma + w_\sigma$ on . We claim that $v_\rho =v_\sigma$ on and $w_\rho =w_\sigma$ on . To see this, note that if , then $v_\rho (z)+w_\rho (z)=v_\sigma (z)+w_\sigma (z)$.

The function $w_\rho -w_\sigma$ satisfies (1) on $\mathbb {H}^+{\setminus } K_\rho$, which is equal to $v_\sigma -v_\rho$ on ${\Omega }{\setminus }(K_\rho \cup ({\partial }{\Omega })_\rho )$, therefore $v_\sigma -v_\rho$ extends to a solution of (1) on ${\Omega }{\setminus }({\partial }{\Omega })_\rho$. Finally, $w_\rho -w_\sigma$ extends to a solution of (1) on $\mathbb {H}^+$, and $\lim _{{\partial }\mathbb {H}^+}(w_\rho -w_\sigma )=0$. Due to Proposition 5.3, we have $w_\rho =w_\sigma$, and hence $v_\rho =v_\sigma$.

For $z\in {\Omega }$, we can define $v(z)=v_\rho (z)$ for $\rho$ small enough so that . Similarly, for , we put $w(z)=w_\rho (z)$ for small $\rho$. Thus we have established the desired decomposition $u=v+w$.

Now, assume that ${\Omega }$ is an arbitrary domain of $\mathbb {H}^+$ and let u be a solution of $L_mu=0$ on . Choose $a\in \mathbb {H}^+$ and R large enough so that $K\subset D(a,R)$ and D(a, R) is a relatively compact set in $\mathbb {H}^+$. Let $\omega ={\Omega }\cap D(a,R)$. Note that K is a compact set in $\omega$ which is a relatively compact open set in $\mathbb {H}^+$ and u satisfies (1) on . Applying the results demonstrated for relatively compact open sets, we obtain

$$\begin{aligned} u(z)=\widetilde{v}(z)+\widetilde{w}(z) \end{aligned}$$

for , where $\widetilde{v}$ satisfies (1) on the set $\omega$ and $\widetilde{w}$ satisfies (1) on with $\lim _{{\partial }\mathbb {H}^+}\widetilde{w}=0$. Note that $V=u-\widetilde{w}$ satisfies (1) on and V can be extended to a solution of (1) in a neighborhood of K because $V=\widetilde{v}$ on $\omega$. The sum $u=V+\widetilde{w}$ provides with the desired decomposition of u.

If we have another decomposition $u=v+w$ with $v\in {C}^2({\Omega })$, $L_mv=0$ and with , $L_mw=0$ and $\lim _{{\partial }\mathbb {H}^+}w=0$, then we have $V-v=w-\widetilde{w}$ on . The function $w-\widetilde{w}$ can be extended on $\mathbb {H}^+$ to a solution of $L_m(w-\widetilde{w})=0$ on $\mathbb {H}^+$ with $\lim _{{\partial }\mathbb {H}^+}(w-\widetilde{w})=0$. Thanks to Proposition 5.3, we obtain $w=\widetilde{w}$, then $V=v$, which completes the proof of the decomposition theorem. $\square$

The following proposition is a Poisson formula for axisymmetric potentials in $\mathbb {H}^+$.

Proposition 5.10

Let $m\in {\mathbb {C}}$ be such that $\mathrm{Re}\,m<1$ and $u:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ be a continuous and bounded function. Then there is a unique axisymmetric potential $U\in {C}^2(\mathbb {H}^+)$ such that $\lim _{\Vert (x,y)\Vert \rightarrow +\infty } U(x,y)=0$ and for all $y\in {{\mathbb {R}}}$,

$$\begin{aligned} \lim _{(0,y)}U=u(y). \end{aligned}$$

Moreover, we have for all $(x,y)\in \mathbb {H}^+$,

(15)

where

Proof

Let us first show that (15) is a solution of $L_m U=0$. Let $f(x,y)=x^{1-m}/ (x^2+(y-\eta )^2)^{1-{m/2}}$. Interchanging differentiation and integration, it suffices to prove that $L_m f=0$. We have

Then,

and we deduce that $L_m f(x,y)=0$.

We have

By a change of variable $t=(y-\eta )/ x$, we obtain

Thanks to the dominated convergence theorem, it suffices to show that

To see this, according [1, p. 258], note that

where $\mathrm{B}$ is the Euler beta function and

By the duplication formula for the $\mathrm{\Gamma }$ function,

and by the recurrence formula $\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)$, we obtain the desired result:

The uniqueness follows from Proposition 5.3. The proof is complete. $\square$

Remark 5.11

One may ask if there is a reproducing formula for the case $\mathrm{Re\,}m\ge 1$. Let $m\in \mathbb {N}^*$ and let $u\in C^2(\overline{\mathbb {H}^+})$ be such that $L_mu=0$ on $\mathbb {H}^+$, then the function v defined on ${{\mathbb {R}}}^{m+2}$ by

$$\begin{aligned} v(x_1,\ldots ,x_{m+2})=u\Bigl (\sqrt{x_1^2+\cdots +x_{m+1}^2},x_{m+2}\Bigr ) \end{aligned}$$

is harmonic on . In particular, if $m\ge 2$, by [19, Proposition 18, p. 310], v can be extended to a harmonic function on ${{\mathbb {R}}}^{m+2}$, which tends to 0 at infinity. We then deduce that $v\equiv 0$ hence $u\equiv 0$. This shows that solving $L_mu=0$ with u tending to 0 at infinity and with prescribed values of u on the y-axis is a problem which does not make sense. In this case, the fact that there is no solution to this Dirichlet problem is a consequence of the loss of ellipticity of $L_m$ on the boundary of $\mathbb {H}^+$. Therefore, we do not deal with the case $\mathrm{Re\,}m\ge 1$.

6 Fourier–Legendre decomposition

First, we will introduce a specific system of coordinates $(\tau ,\theta )$ called bipolar coordinates, see [39]. The numerical applications on extremal bounded problems using this system of coordinates can be found in [25–27].

Let $\alpha >0$. Suppose that there is a positive charge at and a negative charge at $B=(\alpha ,0)$ (the absolute values of the two charges are identical). The potential generated by these charges at a point M is (modulo a multiplicative constant) (Fig. 5).

Fig. 5

Bipolar coordinates

Definition 6.1

The coordinates

are called bipolar coordinates.

The bipolar coordinates are related to the Cartesian coordinates by the following formulas:

Let $R>0$ and $a=\sqrt{R^2+\alpha ^2}$, the disk with center (a, 0) and radius R is defined in terms of bipolar coordinates as

The right half-plane is defined as

$$\begin{aligned} \mathbb {H}^+=\bigl \{(\tau ,\theta ):\tau \in (0+\infty ], \theta \in [0,2\pi )\bigr \}. \end{aligned}$$

The level lines $\tau =\tau _0$ are circles with center $(\alpha \coth \tau _0,0)$ and radii $\alpha /{\text{ sh }}\,\tau _0$. This implies that for all $\tau _0,\tau _1$ such that $0<\tau _0<\tau _1$, the set $\{(\tau ,\theta ): \tau \ge \tau _0\}$ is a closed disk and the set $\{(\tau ,\theta ):0<\tau <\tau _1\}$ is the complement in $\mathbb {H}^+$ of the closed disk $\{\tau \ge \tau _1\}$ (Fig. 6).

Fig. 6

Level lines (with $\alpha =1$)

The following theorem is well known in physics for $m=-1$ [3, 15, 42, 45, 47, 48]. We extend it to $m\in {\mathbb {C}}$.

Theorem 6.2

Let u be a solution to $L_mu=0$ in an open set in $\mathbb {H}^+$. Let

where, by definition,

Then

$$\begin{aligned} {{\partial }^2v_m\over {\partial }\tau ^2}+{{\partial }^2v_m\over {\partial }\theta ^2}+\coth \tau \,{{\partial }v_m\over {\partial }\tau }+\biggl ({1\over 4}-{(m-1)^2\over 4\,{\text{ sh }}^2\tau }\biggr ) v_m=0. \end{aligned}$$

(16)

Proof

We have

Thus, we obtain

and

and

In particular, we have

Therefore, we obtain

According to definition of $v_m$,

Denote

then

and

Hence the equation can be rewritten as

with

and

This completes the proof. $\square$

Let us apply the method of separation of variables, i.e. assume $v_m$ has the form $v_m(\tau ,\theta )=A_m(\tau )B_m(\theta )$. Then (16) becomes

The term on the right depends only on $\theta$ and the left-hand side depends only on $\tau$, thus we deduce that both are constant. Let $n\in {\mathbb {C}}$ be such that this constant is equal to $n^2$. Then we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {A_m''+\coth \tau \, A_m'+\biggl (\frac{1}{4} -\frac{(m-1)^2}{4\,{\text{ sh }}^2\tau }-n^2\biggr )\,A_m=0,}\\ \displaystyle {B_m''+n^2B_m=0.} \end{array}\right. } \end{aligned}$$

The function $B_m$ is a $2\pi$-periodic function as $\theta$ represents an angle, therefore n should necessarily be an integer.

To examine the equation satisfied by $A_m$, we carry out the following change of function

$$\begin{aligned} A_m(\tau )=C_m({\text{ ch }}\,\tau ). \end{aligned}$$

Then, $C_m$ satisfies

which can be rewritten as

This equation is called the hyperbolic associated Legendre equation. Note that if we put $z={\text{ ch }}\,\tau$ and $u(z)=C_m({\text{ ch }}\,\tau )$, then

where

This equation is called the associated Legendre equation, and it can be reduced to the Legendre equation if $\mu =0$:

Two independent solutions of (LA) are given in Appendix, where they are denoted by $P_\nu ^\mu ({\text{ ch }}\,\tau )$ and $Q_\nu ^\mu ({\text{ ch }}\,\tau )$.

Theorem 6.3

Let $m\in {\mathbb {C}}$ and $0<\tau _0$. Let u be a smooth solution to $L_mu=0$ on the disk $\tau \ge \tau _0$ and let v be a smooth solution to $L_mv=0$ on $\mathbb {H}^+{\setminus }\{\tau >\tau _0\}$, which is the complement in $\mathbb {H}^+$ of the disk $\{\tau >\tau _0\}$, and assume that $\lim _{{\partial }\mathbb {H}^+}v=0$. Then there are two sequences $(a_n)_{n\in \mathbb {Z}}$ and $(b_n)_{n\in \mathbb {Z}}$ of $\ell ^2(\mathbb {Z})$ (rapidly decreasing) such that

The sequence $(a_n)$ is unique. In addition, the convergence of the first series is uniform on every compact set $[\tau _1,\tau _2]$ of the disk $\tau >\tau _0$ with $\tau _0\le \tau _1<\tau _2$. And the convergence of the second one is uniform on every compact set $[\tau _3,\tau _4]$ of the complement of the disk $\tau >\tau _0$ in $\mathbb {H}^+$ with $0<\tau _3<\tau _4\le \tau _0$.

If $\mathrm{Re}\,m<1$, then the sequence $(b_n)$ is unique.

Proof

Indeed, decomposing the function

as Fourier series with respect to the variable $\theta$, yields the following Fourier expansion for $u(\tau _0,{\cdot })$:

where $a_n\in \ell ^2(\mathbb {Z})$ satisfies

The function is a smooth function of the variable $\theta$, therefore the sequence $(a_n)_n$ is rapidly decreasing as $|n|\rightarrow +\infty$.

The function

coincides with u on the circle $\tau =\tau _0$. Let us see that $\widetilde{u}$ is well defined on the disk $\tau \ge \tau _0$. Indeed, thanks to Proposition 8.1, as $|n|\rightarrow +\infty$,

and this equivalence is uniform on all compact sets $[\tau _1,\tau _2]$ with $0<\tau _0\le \tau _1<\tau _2$. It follows that the series which defines $\widetilde{u}$ is norm convergent on any compact set $[\tau _1,\tau _2]$ of the disk $\tau \ge \tau _0$. The same is true for derivatives with respect to $\tau$ and $\theta$ (which are also expressed through the associated Legendre functions, see Appendix).

Due to the fact that the solution of an elliptic equation is uniquely determined by its boundary values (this follows from the maximum principle), we deduce that $\widetilde{u}$ is the unique axisymmetric potential on the disk $\tau \ge \tau _0$ which coincides with u on the circle $\tau =\tau _0$.

For v, the proof is similar. Indeed, decomposing the function

as Fourier series with respect to the variable $\theta$, yields the following Fourier expansion for :

where $b_n\in \ell ^2(\mathbb {Z})$ satisfies

The function is a smooth function of the variable $\theta$, therefore the sequence $(b_n)_n$ is rapidly decreasing as $|n|\rightarrow +\infty$.

The function

coincides with v on the circle $\tau =\tau _0$. Let us see that $\widetilde{v}$ is well defined on the complement of the disk $\tau >\tau _0$. Indeed, thanks to Proposition 8.1, as $|n|\rightarrow +\infty$,

and this equivalence is uniform on all compact sets $[\tau _1,\tau _2]$ with $0<\tau _1<\tau _2\le \tau _0$. It follows that the series which defines $\widetilde{v}$ is norm convergent on any compact set $[\tau _1,\tau _2]$ of the complement of the disc $\tau >\tau _0$. The same is true for derivatives with respect to $\tau$ and $\theta$.

We will show that

$$\begin{aligned} \lim _{\tau \rightarrow 0+}\!\widetilde{v}=0. \end{aligned}$$

If $\mathrm{Re}\,m<1$, by formula (18), for $n\in \mathbb {N}$ we have

hence

$$\begin{aligned} \lim _{\tau \rightarrow 0+}\!P_{n-{1/2}}^{(m-1)/2}({\text{ ch }}\,\tau )=0. \end{aligned}$$

In addition, for $n>1-{\mathrm{Re\,}m/2}$, we have

thus

By Proposition 8.1, we obtain

So, $\lim _{\tau \rightarrow 0+}\widetilde{v}=0$.

The uniqueness of $(b_n)$ for $\mathrm{Re}\,m<1$ follows from the following fact established in the next section: the families

form a Riesz basis. $\square$

Corollary 6.4

The solution of the Dirichlet problem for $L_mu=0$ on D((a, 0), R), with $u=\varphi$ on ${\partial }D((a,0),R)$, is given by

where $\{\tau =\tau _0\}$ corresponds to the circle with center (a, 0) and radius R and

Similarly, the function

(17)

is a solution to $L_mv=0$ on $\mathbb {H}^+{\setminus } D((a,0),R)$, which is equal to $\varphi$ on ${\partial }D((a,0),R)$.

Moreover, if $\mathrm{Re}\,m<1$, then v satisfies $\lim _{{\partial }\mathbb {H}^+}\!v\!=\!0$, and (17) is the unique solution of the Dirichlet problem $L_mv=0$ on $\mathbb {H}^+{\setminus } D((a,0),R)$ which vanishes on ${\partial }\mathbb {H}^+$.

7 Riesz basis

We will prove that the first group of functions of the following family:

is a basis of solutions on the disk $\tau \ge \tau _1$ and the second one is a basis of solutions on $\tau \le \tau _0$, which is the complement on $\mathbb {H}^+$ of some disk, with $0<\tau _0<\tau _1$. This fact is known for $m=-1$, namely, for $\mu =1$. We extend this result for complex m.

Let us recall the definition of a Riesz basis. The sequence $(x_n)_{n\in \mathbb {N}}$ is called a quasi-orthogonal or Riesz sequence of a Hilbert space X if there are two constants $c,C>0$ such that for all sequences $(a_n)_{n\in \mathbb {Z}}$ in $\ell ^2$, we have

$$\begin{aligned} c^2\sum _n|a_n|^2\le \biggl \Vert \sum _n a_n x_n\biggr \Vert ^2 \le C^2\sum _n|a_n|^2. \end{aligned}$$

If the family $(x_n)_{n\in \mathbb {Z}}$ is complete, it is called a Riesz basis. The matrix of scalar products $\{\langle x_i,x_j\rangle \}_{i,j}$ is called the Gram matrix associated to $\{x_i\}_i$. Let us recall the following characterization of a Riesz basis by the Gram matrix.

Property

([43, p. 170]) A family $\{x_i\}_i$ is a Riesz basis for some Hilbert space X if $\{x_i\}_i$ is complete in X and the Gram matrix associated to $\{x_i\}_i$ defines a bounded and invertible operator on $\ell ^2(\mathbb {N})$.

Let ${\mathscr {A}}$ and ${\mathscr {B}}$ be the families of solutions to $L_mu=0$, respectively, inside the disk $\tau >\tau _0$ and outside the disk $\tau >\tau _1$, with $0<\tau _0<\tau _1$:

Let ${\mathscr {C}}$ be the union of these two families,

Denote the annulus defined in terms of bipolar coordinates $\{0<\tau _0<\tau <\tau _1\}$ by ${\mathbb {A}}$. The space $L^2({\partial }{\mathbb {A}})$ is equipped with the following inner product: for $f,g\in L^2({\partial }{\mathbb {A}})$,

Proposition 7.1

${\mathscr {C}}$ is a Riesz basis in the Hilbert space $L^2({\partial }{\mathbb {A}})$.

Proof

To find the Gram matrix for ${\mathscr {C}}$, first calculate all its elements. For $n\in \mathbb {Z}$,

In all other cases, the inner product is zero, hence the Gram matrix is diagonal by blocks and each block is the following matrix:

The Gram matrix G has the form

$$\begin{aligned} \displaystyle G= \left( \begin{matrix} M_0 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ 0 &{}\quad M_{-1} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad 0 &{}\quad M_1 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad 0 &{}\quad M_{-2}&{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad 0 &{}\quad \ddots &{}\quad \ddots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad M_{-n}&{}\quad \ddots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad M_n &{}\quad \ddots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \end{matrix}\right) \!. \end{aligned}$$

The determinant of $M_n$ is

Let us show that $M_n$ is invertible. Suppose the contrary, then $\det M_n=0$, which is equivalent to

It can be rewritten as

with $P_{n-1/2}^{(m-1)/2}({\text{ ch }}\,\tau _0), Q_{n-1/2}^{(m-1)/2}({\text{ ch }}\,\tau _1)\ne 0$. Therefore, there is $\lambda \in {\mathbb {C}}{\setminus }\{0\}$ (which depends on $m,n, \tau _0$ and $\tau _1$) such that

Then, by the asymptotic behavior of associated Legendre functions (see Proposition 8.1 in Appendix), on the one hand, we have

and on the other hand, we have

what implies that $\tau _0=\tau _1$, whereas it is not possible. Hence the matrix $M_n$ is invertible and this completes the proof. $\square$

8 Appendix: Associated Legendre functions of first and second kind

In this section, we provide formulas of integral representation for the associated Legendre functions of the first and the second kind with $z={\text{ ch }}\,\tau >1$, see [2, 39, 50].

with $\mathrm{Re}\,\nu >-1$ and $\mathrm{Re}\,(\mu +\nu )<0$.

(18)

with $\mathrm{Re}\,\mu <{1/2}$.

with $\mathrm{Re}\,\mu <{1/2}$.

with $\mathrm{Re}\,\mu >-{1/2}$, $\mathrm{Re}\,(\nu -\mu +1)<0$ and $\mathrm{Re}\,(\nu +\mu +1)>0$.

with $\mathrm{Re}\,\mu <{1/2}$ and $\mathrm{Re}\,(\mu +\nu +1)>0$.

with $\mathrm{Re}\,\nu >-1$ and $\mathrm{Re}\,(\mu +\nu +1)>0$, see [50, pp. 4–6].

There are the following relations between the Legendre functions, see [50, p. 6] and [2, Formula 8.2.2]:

for $\nu -\mu \not \in \mathbb {Z}$. In particular, for $\nu =n-{1/2}$ and $n\in \mathbb {Z}$, we have

There hold the following Whipple formulas relating the associated Legendre functions of first and second kind, see [50, p. 6]:

There hold the following recursion formulas, see [50, pp. 6–7]:

All of these formulas are used to calculate the values $P_\nu ^\mu ({\text{ ch }}\,\tau )$ and $Q_\nu ^\mu ({\text{ ch }}\,\tau )$ for all $\tau >0$ and $(\mu ,\nu )\in {\mathbb {C}}^2$.

If $\mu$ and $\tau$ are fixed, the following proposition describes the behavior of associated Legendre functions of the first and second kind when $\nu =n-{1/2}$, $n\in \mathbb {Z}$, as $|n|\rightarrow +\infty$.

Proposition 8.1

Fix $\tau >0$ and $\mu \in {\mathbb {C}}$. Then if $\nu =n-{1/2}$ and $n\in \mathbb {Z}$, we have:

These equivalences are locally uniform with respect to the variable $\tau$, i.e. uniform on the whole interval $[\tau _0,\tau _1]$ with $0<\tau _0<\tau _1$.

Proof

If $\nu =n-{1/2}$ and $n\in \mathbb {N}$, see [50, p. 48], we have

By the Stirling formula as $\nu \rightarrow +\infty$

consequently,

which gives us the first estimate.

The second estimate is obtained thanks to the relation $P_\nu ^\mu =P_{-\nu -1}^\mu$. The third estimate follows from [50, Formula (8.3)]:

and the last estimate follows from the fact that for $\nu =n-{1/2}$ and $n\in \mathbb {Z}$, we have

$$\begin{aligned} Q_{-\nu -1}^\mu =Q_\nu ^\mu . \end{aligned}$$

The local uniformity of these equivalences implies from explicit expressions of $P_\nu ^\mu$ and $Q_\nu ^\mu$ in terms of hypergeometric functions available in [21, pp. 124–138] and estimates of these hypergeometric functions which are locally uniform [50, pp. 178–182]. $\square$