Abstract
We consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.
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1 Introduction
Let \({\mathbb {D}}\) be the open unit disc in the complex plane \({\mathbb {C}}\) and denote by \(\partial _z=\partial /\partial z\) and \(\bar{\partial }_z=\partial /\partial {\bar{z}}\) the usual complex partial derivatives. This work is concerned with second order partial differential operators of the form
where \(p,q\in {\mathbb {C}}\) are complex parameters. Of particular interest are solutions of the associated homogeneous equation
We say that a function u is (p, q)-harmonic if u is twice continuously differentiable in \({\mathbb {D}}\) (in symbols \(u\in C^2({\mathbb {D}})\)) and \(L_{p,q}u=0\) in \({\mathbb {D}}\), where \(L_{p,q}\) is as in (0.1). Notice that a function u is (p, q)-harmonic if and only if its complex conjugate \({\bar{u}}\) is \(({\bar{q}},{\bar{p}})\)-harmonic. Observe also that a (0, 0)-harmonic function is a harmonic function in \({\mathbb {D}}\) in the usual sense.
An interesting example of a (p, q)-harmonic function is the function
(see Theorem 1.4). Here powers are defined in the usual way using the principal branch of the logarithm, that is, we require that \(\log (1)=0\). Notice that the above functions \(u_{p,q}\) have the hermitian symmetry property that \({\bar{u}}_{p,q}=u_{{\bar{q}},{\bar{p}}}\) for \(p,q\in {\mathbb {C}}\).
Recall that elements of the unit circle \({\mathbb {T}}=\partial {\mathbb {D}}\) act on the unit disc \({\mathbb {D}}\) as rotations. On a function level we consider rotation operators
for \(e^{i\theta }\in {\mathbb {T}}\) acting on functions u in \({\mathbb {D}}\). A basic observation concerning the differential operator \(L_{p,q}\) is the commutativity relation
for \(e^{i\theta }\in {\mathbb {T}}\). This latter commutativity relation suggests an analysis of (p, q)-harmonic functions using concepts natural to classical Fourier analysis on the unit circle.
Let \({\mathbb {Z}}\) be the set of integers. For a suitably smooth function u in \({\mathbb {D}}\) we define its m-th homogeneous part by the formula
for \(m\in {\mathbb {Z}}\). Notice that the m-th homogeneous part \(u_m\) of u is the m-th Fourier coefficient of the vector-valued function
where \(R_{e^{i\theta }}\) is as in (0.3).
We set \(C^\infty ({\mathbb {D}})=\bigcap _{n=0}^\infty C^n({\mathbb {D}})\), where \(C^n({\mathbb {D}})\) is the space of n-times continuously differentiable functions in \({\mathbb {D}}\) for \(n\in {\mathbb {N}}=\{0,1,2,\dots \}\). We topologize these spaces in the usual way using the semi-norms
where \(j,k\in {\mathbb {N}}\) are non-negative integers and \(K\subset {\mathbb {D}}\) is a compact subset of \({\mathbb {D}}\).
Let us recall the classical hypergeometric function defined by
for \(a,b,c\in {\mathbb {C}}\) such that \(c\ne -1,-2,\dots \). Here \((a)_0=1\) and
for \(n=1,2,\dots \) are Pochhammer symbols.
Let us return to a (p, q)-harmonic function u. We show that the m-th homogeneous part of u has the form
for some \(c_m\in {\mathbb {C}}\) when \(m\in {\mathbb {N}}\) is a non-negative integer (see Theorem 4.3). Here F is the hypergeometric function (0.5). Hermitian symmetry leads to a similar formula for \(u_m\) when \(m\in {\mathbb {Z}}^-={\mathbb {Z}}{\setminus }{\mathbb {N}}\) is a negative integer (see Corollary 4.4). Notice that \(u_m\in C^\infty ({\mathbb {D}})\) for \(m\in {\mathbb {Z}}\).
A principal result of the present paper concerns the asymptotic behavior of the m-th homogeneous part \(u_m\) of a (p, q)-harmonic function u. We show that
for all \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact (see Theorem 4.6). This result enables us to use the classical root test to establish absolute convergence of the function series \(\sum _{m=-\infty }^\infty u_m\) in \(C^\infty ({\mathbb {D}})\) (see Corollary 4.7).
A further analysis leads to a function series characterization of (p, q)-harmonic functions. A function u in \({\mathbb {D}}\) is (p, q)-harmonic if and only if it has the form
for some sequence \(\{c_m\}_{m=-\infty }^\infty \) of complex numbers such that
(see Theorem 5.1). As mentioned above, the sums in (0.6) are absolutely convergent in the space \(C^\infty ({\mathbb {D}})\). As a consequence we have that \(u\in C^\infty ({\mathbb {D}})\) if u is (p, q)-harmonic (see Corollary 4.9). This characterization of (p, q)-harmonic functions improves on a result by Ahern et al. [1, Theorem 2.1] when specialized to our setting.
We then turn to coefficient formulas for (p, q)-harmonic functions. We show that
for \(m\in {\mathbb {N}}\), where the \(c_m\)’s are as in (0.6) (see Theorem 5.3). As a consequence, we have that
whenever u is a (p, q)-harmonic function (see Theorem 5.4) as well as a corresponding uniqueness result for such functions (see Corollary 5.5).
Let us return to the function \(u_{p,q}\) in (0.2). A calculation of partial derivatives at the origin of the function \(u_{p,q}\) leads to the series expansion
(see Theorem 6.3). This latter series expansion generalizes a well-known partial fraction decomposition formula for the classical Poisson kernel for \({\mathbb {D}}\) which is obtained for \(p=q=0\).
A main contribution of this paper concerns series expansion of (p, q)-harmonic functions. Of particular mention is a limit theorem for associated hypergeometric functions:
(see Theorem 2.6). Apart from its intrinsic interest, this limit theorem provides an efficient tool for the study of limit properties of homogeneous parts of (p, q)-harmonic functions.
The results of this paper have applications to Poisson integral representations of (p, q)-harmonic functions which is possible when \(p,q\in {\mathbb {C}}{\setminus }{\mathbb {Z}}^-\) are such that \({\text {Re}}(p)+{\text {Re}}(q)>-1\). In the final section of the paper we comment briefly on the connection to such theory.
We have traced the study of (p, q)-harmonic functions back to Daryl Geller [13]. Other significant contributions are those of Ahern and collaborators [1, 2]. Our interest in those topics [18,19,20,21] arose in connection to standard weights and earlier work of Garabedian [12]. Borichev and Hedenmalm [6] have established a connection to polyharmonic theory. Other recent related papers are [5, 8,9,10, 14, 16, 17, 22].
The authors thank the referee for a careful reading of the manuscript.
2 The function \(u_{p,q}\) is (p, q)-harmonic
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Observe that the functions \(u_{p,q}\) have the hermitian symmetry property \({\bar{u}}_{p,q}=u_{{\bar{q}}, {\bar{p}}}\). Indeed,
for \(z\in {\mathbb {D}}\). We shall discuss in this section some differentiation formulas for the functions \(u_{p,q}\). In particular, we shall show that the function \(u_{p,q}\) is (p, q)-harmonic (see Theorem 1.4).
Lemma 1.1
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Then
for \(z\in {\mathbb {D}}\).
Proof
Differentiating using the product rule for differentiation we have that
for \(z\in {\mathbb {D}}\), where the last equality is straightforward to check. This yields the conclusion of the lemma. \(\square \)
We next turn to the \(\bar{\partial }\)-derivative of \(u_{p,q}\).
Lemma 1.2
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Then
for \(z\in {\mathbb {D}}\).
Proof
We shall use the hermitian symmetry property \({\bar{u}}_{p,q}=u_{{\bar{q}}, {\bar{p}}}\). From Lemma 1.1 we have that
for \(z\in {\mathbb {D}}\). A complex conjugation now yields the conclusion of the lemma. \(\square \)
Following earlier practice from [20] we denote by \(A=z\partial -{\bar{z}}\bar{\partial }\) the angular derivative. We next calculate the angular derivative of \(u_{p,q}\).
Corollary 1.3
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Then
for \(z\in {\mathbb {D}}\).
Proof
The result is evident from Lemmas 1.1 and 1.2. \(\square \)
We mention that Corollary 1.3 generalizes [20, Theorem 1.11].
Recall the partial fraction formula
for the classical Poisson kernel for the unit disc. Formula (1.1) is straightforward to check.
Theorem 1.4
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Then \(L_{p, q}u_{p, q}=0\) in \({\mathbb {D}}\), where \(L_{p,q}\) is as in (0.1).
Proof
Recall Lemma 1.2. Notice that the differential operator \(z\partial \) satisfies the product rule for differentiation. It is straightforward to check that
whenever the formula makes sense. Differentiating using the product rule we have that
for \(z\in {\mathbb {D}}\). From Lemma 1.1 we now have that
for \(z\in {\mathbb {D}}\). Expanding the above product we see that
for \(z\in {\mathbb {D}}\). Multiplying by a factor \((1-|z|^2)/|z|^2\) we see that
for \(z\in {\mathbb {D}}\). Notice an appearance of the classical Poisson kernel in the rightmost term above. Using the partial fraction formula (1.1) we have that
for \(z\in {\mathbb {D}}\). A simplification of terms now leads to the formula
for \(z\in {\mathbb {D}}\). Recall Lemmas 1.1 and 1.2. In view of these two lemmas our latter formula (1.2) says that \(L_{p,q}u_{p,q}=0\) in \({\mathbb {D}}\). \(\square \)
3 A sequence of hypergeometric functions
Let us first consider a second order partial differential operator of the form
where \(p,q,r \in {\mathbb {C}}\) are complex parameters. A principal case is when \(r=pq\). Notice that \(L_{p,q;pq}=L_{p,q}\), where \(L_{p,q}\) is as in (0.1). The introduction of an additional parameter \(r\in {\mathbb {C}}\) allows for more general operators appearing in the study of conductivity problems, see for instance Calderón [7] or Astala and Päivärinta [4].
We shall evaluate the operator \(L_{p,q;r}\) on a complex-valued function u in the punctured disc \({\mathbb {D}}{\setminus } \{ 0 \}\) of the form
for some \(f\in C^2(0,1)\) and \(m\in {\mathbb {Z}}\).
We introduce also the ordinary differential operator
where \(a,b,c\in {\mathbb {C}}\) are complex parameters. Notice that the famous hypergeometric ordinary differential equation
takes the form \(H_{a,b;c}y=0\) using the operator \(H_{a,b;c}\).
Theorem 2.1
Let \(L_{p,q;r}\) be as in (2.1) for some \(p,q,r \in {\mathbb {C}}\). Let u be a function of the form (2.2) for some \(f\in C^2(0,1)\) and \(m \in {\mathbb {Z}}\). Then
where \(c=m+1\) and
Proof
A differentiation shows that
for \(z \in {\mathbb {D}} {\setminus } \{ 0 \}\), and similarly that
for \(z \in {\mathbb {D}} {\setminus } \{ 0 \}\). Another differentiation gives that
for \(z \in {\mathbb {D}} {\setminus } \{ 0 \}\). A calculation using these formulas gives that
for \(z \in {\mathbb {D}} {\setminus } \{ 0 \}\), where \(c=m+1\), the numbers a and b are as in (2.4) and \(H_{a,b;c}\) is as in (2.3). This yields the conclusion of the theorem. \(\square \)
Equations (2.4) say that a and b are the zeros of the quadratic polynomial
Notice that
In particular, the zeros of \(P_{p,q;r;m}\) are \(-p\) and \(m-q\) in the principal case when \(r=pq\).
Theorem 2.1 suggests a natural construction of (p, q)-harmonic functions.
Proposition 2.2
Let \(p,q\in {\mathbb {C}}\). Consider the function
where \(m\in {\mathbb {N}}\) and F is the hypergeometric function (0.5). Then \(u_m\) is a (p, q)-harmonic function.
Proof
Clearly \(u_m\in C^\infty ({\mathbb {D}})\). It is well-known that the hypergeometric function \(y=F(a,b;c;\cdot )\) satisfies the hypergeometric equation \(H_{a,b;c}y=0\) (see [3, Section 2.3]). The result now follows by Theorem 2.1. \(\square \)
Corollary 2.3
Let \(p,q\in {\mathbb {C}}\). Consider the function
where \(m\in {\mathbb {Z}}^-\) and F is the hypergeometric function (0.5). Then \(u_m\) is a (p, q)-harmonic function.
Proof
We consider the complex conjugate
By Proposition 2.2 we have that \({\bar{u}}_m\) is a \(({\bar{q}},{\bar{p}})\)-harmonic function. From hermitian symmetry we conclude that \(u_m\) is a (p, q)-harmonic function. \(\square \)
Following earlier practice, a function u in \({\mathbb {D}}{\setminus }\{0\}\) is said to be homogeneous of order \(m\in {\mathbb {Z}}\) with respect to rotations if it has the property that
for \(e^{i\theta }\in {\mathbb {T}}\). Notice that every function u of the form (2.2) is homogeneous of order m with respect to rotations.
Theorem 2.4
Let \(p,q\in {\mathbb {C}}\) and \(m\in {\mathbb {N}}\). Let \(u\in C^2({\mathbb {D}})\) be homogeneous of order m with respect to rotations. Then u is (p, q)-harmonic if and only if it has the form
for some \(c\in {\mathbb {C}}\), where F is the hypergeometric function (0.5).
Proof
From Proposition 2.2 we know that every function u of the form (2.5) is (p, q)-harmonic.
Assume next that u is (p, q)-harmonic. Since u is homogeneous of order m, we can put u on the form (2.2) for some \(f\in C^2(0,1)\). By Theorem 2.1 we have that
where \(H_{a,b;c}\) is as in (2.3). Below we shall check that
as \(x\rightarrow 0\). Condition (2.6) allows us to apply [18, Proposition 1.3] to conclude that f is a constant multiple of the hypergeometric function \(F(-p,m-q;m+1;\cdot )\). This will then complete the proof of the theorem.
We proceed to check (2.6). Recall formula (2.2). Since u is bounded near the origin we have that \(f(x)={\mathcal {O}}(1/x^{m/2})\) as \(x \rightarrow 0\). A differentiation of (2.2) gives that
Since \(\bar{\partial }u\) is bounded near the origin we have that \(f'(x)={\mathcal {O}}(1/x^{(m+1)/2})\) as \(x \rightarrow 0\). We have now checked (2.6). \(\square \)
Theorem 2.4 and its proof are modeled on [18, Theorem 2.1]. We have merely supplied some details.
We shall use the fact that the hypergeometric functions are analytic in \({\mathbb {D}}\). Let \(H({\mathbb {D}})\) be the space of analytic functions in \({\mathbb {D}}\). The space \(H({\mathbb {D}})\) is topologized in the usual manner using the semi-norms
for \(K\subset {\mathbb {D}}\) compact. Convergence in the space \(H({\mathbb {D}})\) is usually referred to as normal convergence in \({\mathbb {D}}\).
Recall the terminology that a subset \({\mathcal {F}}\) of \(H({\mathbb {D}})\) is called a normal family if every sequence of functions of \({\mathcal {F}}\) has a subsequence which converges in \(H({\mathbb {D}})\). The limit function is not required to belong to \({\mathcal {F}}\). We refer to Conway [11, Chapter VII] for background.
Recall also the binomial series:
for \(z\in {\mathbb {D}}\).
Lemma 2.5
Let \(p,q\in {\mathbb {C}}\) and consider the functions
for \(m\in {\mathbb {N}}\), where F is the hypergeometric function (0.5). Then \({\mathcal {F}}=\{f_m:\ m\in {\mathbb {N}}\}\) is a normal family of analytic functions in \({\mathbb {D}}\).
Proof
We shall prove that the functions in \({\mathcal {F}}\) are uniformly bounded on compact subsets of \({\mathbb {D}}\). The conclusion of the lemma then follows by a classical result of Montel (see Conway [11, Theorem VII.2.9]).
Let \(K\subset {\mathbb {D}}\) be compact. Since \(K\subset {\mathbb {D}}\) is compact there exists \(0<r<1\) such that \(\max _{z\in K}|z|<r\). Choose N such that
for \(m>N\). Observe that
Therefore \(|(m-q)_n/(m+1)_n |\le 1/r^n\) for \(n\in {\mathbb {N}}\) provided \(m>N\).
We now estimate the \(f_m\)’s with \(m>N\). From (0.5) we have that
Notice in this sum an appearance of the quotient \((m-q)_n/(m+1)_n\) considered in the previous paragraph. From the triangle inequality and the result of the previous paragraph we have that
for \(|z|<r\) and \(m>N\), where the last equality follows by the binomial series (2.8). This proves that the functions in \({\mathcal {F}}\) are uniformly bounded on K. \(\square \)
The following limit theorem will be much useful.
Theorem 2.6
Let \(p,q\in {\mathbb {C}}\). Then
with normal convergence, where F is the hypergeometric function (0.5).
Proof
Let the \(f_m\)’s be as in Lemma 2.5. From Lemma 2.5 we have that the set \({\mathcal {F}}=\{f_m:\ m\in {\mathbb {N}}\}\) is a normal family. From (0.5) we have that
for \(m,n\in {\mathbb {N}}\). Recall that the Pochhammer symbol \((\cdot )_n\) is a monic polynomial of degree n. Passing to the limit we have that \(\lim _{m\rightarrow \infty }f_m^{(n)}(0)=(-p)_n\) for \(n\in {\mathbb {N}}\).
From (2.8) we have that \(f^{(n)}(0)=(-p)_n\) for \(n\in {\mathbb {N}}\), where
A standard argument now yields that \(f_m\rightarrow f\) in \(H({\mathbb {D}})\) as \(m\rightarrow \infty \). Assume to reach a contradiction that there exists a compact set \(K\subset {\mathbb {D}}\) such that \(\{f_m\}_{m=0}^\infty \) does not converge uniformly to f on K. Passing to a subsequence we can assume that
for \(k=1,2,\dots \). Since the set \({\mathcal {F}}\) is a normal family, we can after passage to another subsequence if necessary, assume that \(f_{m_k}\rightarrow g\) in \(H({\mathbb {D}})\) as \(k\rightarrow \infty \) for some \(g\in H({\mathbb {D}})\). From the first paragraph of the proof we have \(g^{(n)}(0)=(-p)_n\) for \(n\in {\mathbb {N}}\), and a uniqueness argument gives that \(g=f\) in \({\mathbb {D}}\). Thus \(f_{m_k}\rightarrow f\) in \(H({\mathbb {D}})\) as \(k\rightarrow \infty \), which contradicts (2.9). \(\square \)
We emphasize that Theorem 2.6 appears much natural in view of the binomial series (2.8).
4 A generalized power series
From the product rule for differentiation we have that
for, say, \(f,g\in C^n[0,1)\).
Lemma 3.1
Let \(j,k,m\in {\mathbb {N}}\) be such that \(m \ge j\). Let u be a function of the form (2.2) for some \(f \in C^{\infty }[0,1)\). Then
for \(z \in {\mathbb {D}}\).
Proof
Recall (2.2). Differentiating with respect to \(\bar{z}\) we have that
for \(z\in {\mathbb {D}}\). Another differentiation using the product rule for differentiation gives that
for \(z\in {\mathbb {D}}\). This completes the proof of the lemma. \(\square \)
Let \(C^{\infty }({\mathbb {D}})\) be the set of smooth complex-valued functions in the unit disc \({\mathbb {D}}\). The space \(C^{\infty }({\mathbb {D}})\) is topologized by means of the family of semi-norms
where \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) is compact. Recall that \(u_m\rightarrow u\) in \(C^\infty ({\mathbb {D}})\) as \(m\rightarrow \infty \) means that \(\lim _{m\rightarrow \infty }\Vert u_m-u\Vert _{j,k;K}=0\) for all \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact.
A (formal) series \(\sum _{m=0}^\infty u_m \) of functions \(u_m\in C^\infty ({\mathbb {D}})\) for \(m\in {\mathbb {N}}\) is said to be absolutely convergent in \(C^\infty ({\mathbb {D}})\) if
whenever \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) is compact. By completeness of \(C^\infty ({\mathbb {D}})\) we have that every series absolutely convergent in \(C^\infty ({\mathbb {D}})\) is convergent in \(C^\infty ({\mathbb {D}})\).
Theorem 3.2
Let \(\lbrace f_m \rbrace _{m=0}^\infty \) be a sequence in \(C^{\infty }[0,1)\) such that
for \(n\in {\mathbb {N}}\) and \(0< r < 1\). Set
for \(m=0,1,2,\dots \). Then \(\limsup _{m \rightarrow \infty } \Vert u_m\Vert ^{1/m}_{j,k;K}<1\) for all \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact, where \(\Vert \cdot \Vert _{j,k;K}\) is as in (3.1).
Proof
Fix \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact. Set \(r=\max _{z\in K}|z|<1\). From Lemma 3.1 we have that
for \(z \in {\mathbb {D}}\) provided \(m\ge j\). We next apply the triangle inequality to see that
for \(m\ge j\), where we have used the notation (2.7). We shall next apply the m-th root to (3.2) and pass to the limit as \(m\rightarrow \infty \). In view of the assumption on the sequence \(\{f_m\}_{m=0}^\infty \) we conclude from (3.2) that
Since \(0<r<1\), this yields the conclusion of the theorem. \(\square \)
From the conclusion of Theorem 3.2 we have that \(\sum _{m=0}^\infty \Vert u_m\Vert _{j,k;K}<+\infty \) whenever \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) is compact. Thus the series \(\sum _{m=0}^\infty u_m\) is absolutely convergent in \(C^{\infty }({\mathbb {D}})\).
We can think of a function series
as a vector-valued power series. The sequence of coefficients \(\{f_m\}_{m=0}^\infty \) in (3.3) is now a sequence of functions in \(C^{\infty }[0,1)\). From this point of view, Theorem 3.2 generalizes a well-known root criteria for power series (see for instance Conway [11, Theorem III.1.3]).
Of particular concern to us are sequences \(\{f_m\}_{m=0}^\infty \) of the form
for \(m\in {\mathbb {N}}\), where \(p,q\in {\mathbb {C}}\) and F is the hypergeometric function (0.5). We next observe that Theorem 2.6 guarantees that every such sequence \(\{f_m\}_{k=0}^\infty \) satisfies the assumption of Theorem 3.2.
Proposition 3.3
Let \(p,q\in {\mathbb {C}}\). Then
for \(n\in {\mathbb {N}}\) and \(0< r < 1\), where F is the hypergeometric function (0.5).
Proof
Recall the fact that the complex derivative \(f\mapsto f'\) is continuous in the topology of normal convergence of analytic functions (see Conway [11, Theorem VII.2.1]). Recall Montel’s theorem characterizing normal families of analytic functions (see Conway [11, Theorem VII.2.9]). In view of these two results, the proposition follows from Theorem 2.6. \(\square \)
5 Analysis of homogeneous parts
Notice that the rotation operators \({\mathbb {T}}\ni e^{i\theta }\mapsto R_{e^{i\theta }}\) from (0.3) have the group properties that \(R_1=I\) is the identity and
for \(e^{i\theta },e^{i\tau }\in {\mathbb {T}}\).
Recall the notion of homogeneity with respect to rotations made precise in the paragraph before Theorem 2.4. Observe that a function u in \({\mathbb {D}}\) is homogeneous of order \(m\in {\mathbb {Z}}\) with respect to rotations if and only if \(R_{e^{i\theta }}u=e^{im\theta }u\) for \(e^{i\theta }\in {\mathbb {T}}\).
Proposition 4.1
Let \(u_m\) be the m-th homogeneous part of \(u\in C^n({\mathbb {D}})\) for some \(m\in {\mathbb {Z}}\), where \(n\in {\mathbb {N}}\cup \{\infty \}\). Then \(u_m\in C^n({\mathbb {D}})\) and \(u_m\) is homogeneous of order m with respect to rotations.
Proof
Differentiations under the integral in (0.4) show that \(u_m\in C^n({\mathbb {D}})\). From formula (0.4) we have that
in a vector-valued sense. Let \(e^{i\theta }\in {\mathbb {T}}\). From the group property of rotation operators we have that
in a vector-valued sense. A change of variables now gives that
Since \(e^{i\theta }\in {\mathbb {T}}\) is arbitrary, this yields that \(u_m\) is homogeneous of order m with respect to rotations. \(\square \)
Below we shall make use of the commutativity relation
for \(e^{i\theta }\in {\mathbb {T}}\). In order to prove (4.2) it suffices to check that the differential operators \(z\partial \) and \({\bar{z}}\bar{\partial }\) commute with the rotations \(R_{e^{i\theta }}\) which is evident.
We now return to (p, q)-harmonic functions.
Proposition 4.2
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function and denote by \(u_m\) its m-th homogeneous part for some \(m\in {\mathbb {Z}}\). Then \(u_m\) is (p, q)-harmonic.
Proof
From Proposition 4.1 we have that \(u_m\in C^2({\mathbb {D}})\) since \(u\in C^2({\mathbb {D}})\). Recall formula (4.1). Applying the operator \(L_{p,q}\) we have that
in a vector-valued sense. We now use the commutativity relation (4.2) to conclude that
where the last equality is evident since u is (p, q)-harmonic. \(\square \)
Let us next calculate the homogeneous parts of a (p, q)-harmonic function.
Theorem 4.3
Let \(p,q\in {\mathbb {C}}\). Let u be (p, q)-harmonic function and denote by \(u_m\) its m-th homogeneous part for some \(m\in {\mathbb {N}}\). Then
for some \(c_m\in {\mathbb {C}}\), where F is the hypergeometric function (0.5).
Proof
By Proposition 4.1, the function \(u_m\) is homogeneous of order m with respect to rotations. By Proposition 4.2, the function \(u_m\) is (p, q)-harmonic. The result now follows from Theorem 2.4. \(\square \)
Corollary 4.4
Let \(p,q\in {\mathbb {C}}\). Let u be (p, q)-harmonic function and denote by \(u_m\) its m-th homogeneous part for some \(m\in {\mathbb {Z}}{\setminus } {\mathbb {Z}}^+\). Then
for some \(c_m\in {\mathbb {C}}\), where F is the hypergeometric function (0.5).
Proof
The complex conjugate \({\bar{u}}_m\) is the \(-m=|m|\)-th homogeneous part of the function \({\bar{u}}\). By hermitian symmetry, the function \({\bar{u}}\) is \(({\bar{q}},{\bar{p}})\)-harmonic. From Theorem 4.3 we have that
for some \(a_m\in {\mathbb {C}}\). A complex conjugation now yields (4.4) with \(c_m={\bar{a}}_m\). \(\square \)
As a by-product from Theorem 4.3 and Corollary 4.4 we have that \(u_m\in C^\infty ({\mathbb {D}})\) for \(m\in {\mathbb {Z}}\) if u is (p, q)-harmonic.
Recall formula (0.4) for the homogeneous parts of a function u in \({\mathbb {D}}\). From the triangle inequality we have that
for \(0<r<1\) and \(m\in {\mathbb {Z}}\). Notice that (4.5) generalizes a well-known bound for Fourier coefficients.
We next estimate the constants \(c_m\) appearing in Theorem 4.3 or Corollary 4.4.
Lemma 4.5
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function and consider its m-th homogeneous part \(u_m\) for \(m\in {\mathbb {Z}}\). Let \(c_m\) be as in (4.4) or (4.4) depending on whether \(m\in {\mathbb {N}}\) or \(m\in {\mathbb {Z}}^-\). Then \(\limsup _{|m|\rightarrow \infty } |c_m|^{1/|m|}\le 1\).
Proof
For simplicity we consider the case \(m\in {\mathbb {N}}\). The case \(m\in {\mathbb {Z}}^-\) is analogous or follows by hermitian symmetry. Let \(0<r<1\). From (4.4) and (4.5) we have that
for \(m\in {\mathbb {N}}\). From Theorem 2.6 we have that
Therefore, a passage to the limit in (4.6) gives that
Since \(0<r<1\) is arbitrary we conclude that \(\limsup _{m\rightarrow \infty } |c_m|^{1/m}\le 1\). This yields the conclusion of the lemma. \(\square \)
The following theorem is our main result about the asymptotic behavior of the homogeneous parts of a (p, q)-harmonic function.
Theorem 4.6
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function and denote by \(u_m\) its m-th homogeneous part for \(m\in {\mathbb {Z}}\). Then \(u_m\in C^\infty ({\mathbb {D}})\) for \(m\in {\mathbb {Z}}\) and
for all \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact.
Proof
From Theorem 4.3 and Corollary 4.4 we have that \(u_m\in C^\infty ({\mathbb {D}})\) for \(m\in {\mathbb {Z}}\).
We consider first the case \(m\in {\mathbb {N}}\). From Theorem 4.3 we have (4.4) for \(m\in {\mathbb {N}}\), where \(c_m\in {\mathbb {C}}\). Notice that the function \(u_m\) in (4.4) is constructed from the function
as in Theorem 3.2. From Proposition 3.3 and Lemma 4.5 we have that the assumption of Theorem 3.2 is satisfied. An application of Theorem 3.2 yields the conclusion that \(\limsup _{m\rightarrow \infty }\Vert u_m\Vert _{j,k;K}^{1/m}<1\) for \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact.
We next consider the case \(m\in {\mathbb {Z}}^-\). The complex conjugate \({\bar{u}}_{m}\) of \(u_m\) is the \((-m)\)-th homogeneous part of the \(({\bar{q}},{\bar{p}})\)-harmonic function \({\bar{u}}\). From the result of the previous paragraph we thus have that
for \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact. This completes the proof of the theorem. \(\square \)
We point out that Theorem 2.6 forms an integral part in the proof of Theorem 4.6.
Recall the notion of absolute convergence in \(C^\infty ({\mathbb {D}})\), see the paragraph just before Theorem 3.2.
Corollary 4.7
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function with m-th homogeneous part \(u_m\) for \(m\in {\mathbb {Z}}\). Then the function series \(\sum _{m=-\infty }^\infty u_m\) is absolutely convergent in \(C^\infty ({\mathbb {D}})\).
Proof
Theorem 4.6 allows us to apply the root test to conclude that
for \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact. \(\square \)
The following lemma is well-known but included for the sake of completeness.
Lemma 4.8
Let \(u\in C^n({\mathbb {D}})\) for some \(n\in {\mathbb {N}}\) and denote by \(u_m\) its m-th homogeneous part for \(m\in {\mathbb {Z}}\). Then
in \(C^n({\mathbb {D}})\).
Proof
It is straightforward to check that
in a vector-valued sense, where
is the Fejér kernel. It is well-known that the \(K_N\)’s are non-negative and form an approximate identity as \(N\rightarrow \infty \). Since \(u\in C^n({\mathbb {D}})\), it is well-known that the function
is continuous from \({\mathbb {T}}\) into \(C^n({\mathbb {D}})\). The proof is now completed by a standard argument. We refer to Katznelson [15, Section I.2] for details. \(\square \)
We think of Lemma 4.8 as a version of Fejér’s theorem adapted to our context.
We next deduce that (p, q)-harmonic functions belong to the space \(C^\infty ({\mathbb {D}})\).
Corollary 4.9
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function. Then \(u\in C^\infty ({\mathbb {D}})\) and \(u=\sum _{m=-\infty }^\infty u_m\) in \(C^\infty ({\mathbb {D}})\), where \(u_m\) is the m-th homogeneous part of u for \(m\in {\mathbb {Z}}\).
Proof
From Lemma 4.8 we have that
in \(C^2({\mathbb {D}})\) since u has such regularity. From Corollary 4.7 we know that the function series \(\sum _{m=-\infty }^\infty u_m\) is absolutely convergent in \(C^\infty ({\mathbb {D}})\). We can thus omit the convergence factors in (4.7) and deduce that \(u=\sum _{m=-\infty }^\infty u_m\) in \(C^\infty ({\mathbb {D}})\). By completeness we have that \(u\in C^\infty ({\mathbb {D}})\). \(\square \)
6 A series expansion of harmonic functions
The analysis from Sect. 4 leads to a natural function series description of (p, q)-harmonic functions.
Theorem 5.1
Let \(p,q\in {\mathbb {C}}\). Then u is a (p, q)-harmonic function if and only if it has the form
for some sequence \(\{c_m\}_{m=-\infty }^\infty \) of complex numbers such that
where F is the hypergeometric function (0.5). Moreover, the sums in (5.1) are absolutely convergent in the space \(C^\infty ({\mathbb {D}})\) when (5.2) holds.
Proof
Consider first a (formal) function series of the form (5.1) with coefficient sequence \(\{c_m\}_{m=-\infty }^\infty \) satisfying (5.2). From Theorem 3.2 and Proposition 3.3 we have that
for \(j,k\in {\mathbb {N}}\) and \(K\subset {\mathbb {D}}\) compact, where
for \(m\in {\mathbb {N}}\) and
for \(m\in {\mathbb {Z}}^-\). The root test now applies to show that the series \(\sum _{m=-\infty }^\infty u_m\) in (5.1) is absolutely convergent in \(C^\infty ({\mathbb {D}})\). This defines a function \(u=\sum _{m=-\infty }^\infty u_m\) by (5.1). From Proposition 2.2 and Corollary 2.3 we have that each term \(u_m\) in (5.1) is (p, q)-harmonic. Since the series expansion (5.1) converges in the space \(C^\infty ({\mathbb {D}})\), we have that \(u\in C^\infty ({\mathbb {D}})\). Applying the operator \(L_{p,q}\) we have \(L_{p,q}u=\sum _{m=-\infty }^\infty L_{p,q}u_m =0\) in \(C^\infty ({\mathbb {D}})\), which shows that u is (p, q)-harmonic.
Consider next a (p, q)-harmonic function u. We proceed to derive (5.1). Denote by \(u_m\) the m-th homogeneous part of u for \(m\in {\mathbb {Z}}\). From Theorem 4.3 we have (4.4) for \(m\in {\mathbb {N}}\) and from Corollary 4.4 we have (4.4) for \(m\in {\mathbb {Z}}^-\), where \(c_m\in {\mathbb {C}}\) for \(m\in {\mathbb {Z}}\). From Corollary 4.9 we have (5.1) in the form that \(u= \sum _{m=-\infty }^\infty u_m\) in \(C^\infty ({\mathbb {D}})\). By Corollary 4.7 we have that the function series \(\sum _{m=-\infty }^\infty u_m\) in (5.1) is absolutely convergent in \(C^\infty ({\mathbb {D}})\). From Lemma 4.5 we have that (5.2) holds. \(\square \)
We emphasize that the function series (5.1) has an interpretation of homogeneous expansion of the (p, q)-harmonic function u.
We mention that Theorem 5.1 improves on a result by Ahern et al. [1, Theorem 2.1] when specialized to the present setting. Theorem 5.1 contains also some earlier results of Olofsson and Wittsten. Theorem 5.1 with \((p,q)=(0,\alpha )\), \(\alpha \in {\mathbb {R}}\), yields [21, Theorem 1.2]. Theorem 5.1 with \((p,q)=(\alpha /2,\alpha /2)\), \(\alpha \in {\mathbb {R}}\), yields [18, Theorem 2.2].
We shall next turn to formulas for the coefficients \(c_m\) appearing in (5.1).
Proposition 5.2
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function of the form (5.1) for some sequence \(\{c_m\}_{m=-\infty }^\infty \) of complex numbers satisfying (5.2). Then
for \(m\in {\mathbb {Z}}\).
Proof
We consider the case \(m\in {\mathbb {N}}\). Using the expansion (5.1) it is straightforward to check that
for \(0<r<1\). We next divide by \(r^m\) and pass to the limit to see that
This yields the conclusion of the proposition when \(m\in {\mathbb {N}}\).
The remaining case \(m\in {\mathbb {Z}}{\setminus }{\mathbb {N}}\) is proved similarly or follows by hermitian symmetry. We omit the details. \(\square \)
The integral quantity
in Proposition 5.2 is naturally thought of as a line integral. In fact,
for \(m\in {\mathbb {N}}\) and \(0<r<1\), where the circle of integration is traversed once in positive direction. These formulas are straightforward to check.
Theorem 5.3
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function of the form (5.1) for some sequence \(\{c_m\}_{m=-\infty }^\infty \) of complex numbers satisfying (5.2). Then
for \(m\in {\mathbb {N}}\).
Proof
We shall calculate the integral limit in Proposition 5.2. Let \(I_m(r)\) be as in (5.3). Recall from Corollary 4.9 that \(u\in C^\infty ({\mathbb {D}})\). Consider the Taylor expansion of u at the origin of degree \(m\ge 0\):
as \(z\rightarrow 0\). We have that
as \(r\rightarrow 0\), where the last equality follows by cancellation. A passage to the limit as \(r\rightarrow 0\) now yields that \(c_m= \partial ^m u(0)/m!\).
We now turn to the formula for \(c_{-m}\). A similar analysis as in the previous paragraph shows that
as \(r\rightarrow 0\). A passage to the limit as \(r\rightarrow 0\) yields that \(c_{-m}= \bar{\partial }^m u(0)/m!\). This completes the proof of the theorem. \(\square \)
The coefficient formulas in Theorem 5.3 leads to an addendum to Theorem 5.1.
Theorem 5.4
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function. Then
for \(z\in {\mathbb {D}}\), where F is the hypergeometric function (0.5).
Proof
Recall the series expansion (5.1) and (5.2) established in Theorem 5.1. By Theorem 5.3 we have that \(c_m= \partial ^m u(0)/m!\) and \(c_{-m}= \bar{\partial }^m u(0)/m!\) for \(m\in {\mathbb {N}}\). This yields the conclusion of the theorem. \(\square \)
As a by-product of Theorem 5.4 we have a uniqueness result of classical type.
Corollary 5.5
Let \(p,q\in {\mathbb {C}}\). Let u be a (p, q)-harmonic function. Assume that \(u(0)=0\) and
for \(m=1,2,\dots \). Then \(u(z)=0\) for all \(z\in {\mathbb {D}}\).
Proof
The result is evident by Theorem 5.4. \(\square \)
7 Further properties of the function \(u_{p,q}\)
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). In this section we derive the series expansion of \(u_{p,q}\) (see Theorem 6.3). Of interest are also properties of integral means of \(u_{p,q}\) (see Theorems 6.4 and 6.6).
Lemma 6.1
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Let \(m\in {\mathbb {N}}\). Then
where \(g_m\in C^\infty ({\mathbb {D}})\).
Proof
We prove the lemma by induction on \(m\in {\mathbb {N}}\). Clearly (6.1) holds for \(m=0\) with \(g_0=0\). By Lemma 1.1 we have that
with
This proves (6.1) for \(m=1\).
Assume next that (6.1) holds for some \(m=n\ge 1\), that is,
where \(g_n\in C^\infty ({\mathbb {D}})\). Differentiating we have that
for \(z\in {\mathbb {D}}\). We next use (6.2) to see that
for \(z\in {\mathbb {D}}\), where
This proves (6.1) for \(m=n+1\). The conclusion of the lemma now follows by the principle of induction. \(\square \)
Notice that formula (6.1) interprets naturally as a congruence in \(C^\infty ({\mathbb {D}})\) modulo \({\bar{z}} C^\infty ({\mathbb {D}})\).
Theorem 6.2
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\). Then
for \(m\in {\mathbb {N}}\).
Proof
We shall use Lemma 6.1. A point evaluation at the origin in (6.1) shows that \(\partial ^m u_{p,q}(0)=(p+1)_m\) for \(m\in {\mathbb {N}}\). Moreover, by hermitian symmetry \({\bar{u}}_{p,q}=u_{{\bar{q}},{\bar{p}}}\) we have that
for \(m\in {\mathbb {N}}\). \(\square \)
We can now derive the series expansion of the function \(u_{p,q}\).
Theorem 6.3
Let \(p,q\in {\mathbb {C}}\). Then
for \(z\in {\mathbb {D}}\), where F is the hypergeometric function (0.5).
Proof
Let \(u_{p,q}\) be as in (0.2). From Theorem 1.4 we know that the function \(u_{p,q}\) is (p, q)-harmonic. By Theorem 5.4 it has the function series expansion
for \(z\in {\mathbb {D}}\). The result now follows from Theorem 6.2. \(\square \)
We point out that Theorem 6.3 generalizes a well-known partial fraction decomposition formula for the classical Poisson kernel for \({\mathbb {D}}\), see formula (1.1). Suitably specialized Theorem 6.3 yields [21, Theorem 2.5] and [18, Theorem 3.2].
Notice that the series expansion in Theorem 6.3 is absolutely convergent in \(C^\infty ({\mathbb {D}})\).
We denote by \({\text {Re}}(z)\) and \({\text {Im}}(z)\) the real and imaginary parts of a complex number z, respectively. The standard Gamma function is defined by
for \({\text {Re}}(z)>0\). It is well-known that \(\Gamma \) continues to a meromorphic function in \({\mathbb {C}}\) with simple poles at the points \(z=0,-1,-2,\dots \).
We next turn to \(L^1\) means of \(u_{p,q}\).
Theorem 6.4
Let \(u_{p,q}\) be as in (0.2) for some \(p,q\in {\mathbb {C}}\) such that \({\text {Re}}(p)+{\text {Re}}(q)>-1\). Then
for \(0\le r<1\), where \(\Gamma \) is the Gamma function.
Proof
From definition of powers we have that
for \(z\in {\mathbb {D}}\). Notice that the complex number \(1-z\) lies in the open right half-plane when \(z\in {\mathbb {D}}\). Passing to absolute values we have that
for \(z\in {\mathbb {D}}\). We next apply [18, Theorem 3.1] with \(\alpha ={\text {Re}}(p)+{\text {Re}}(q)\) to conclude that
for \(0\le r<1\). This yields the conclusion of the theorem. \(\square \)
Remark 6.5
A similar analysis as in the proof of Theorem 6.4 shows that
if \({\text {Re}}(p)+{\text {Re}}(q)\le -1\). We omit the details.
A classical result known as Gauss summation formula says that
when \(a,b\in {\mathbb {C}}\) and \(c\in {\mathbb {C}}{\setminus }\{0,-1,-2,\dots \}\) are such that \({\text {Re}}(c)>{\text {Re}}(a)+{\text {Re}}(b)\) (see [3, Theorem 2.2.2]).
Theorem 6.6
Let \(p,q\in {\mathbb {C}}\) be such that \({\text {Re}}(p)+{\text {Re}}(q)>-1\). Then
where \(\Gamma \) is the Gamma function.
Proof
Recall the series expansion of the function \(u_{p,q}\) established in Theorem 6.3. By cancellation of terms we have that
for \(0\le r<1\). Since \({\text {Re}}(p)+{\text {Re}}(q)>-1\), we can apply Gauss summation formula (6.3) to calculate the limit of the above quantity as \(r\rightarrow 1\). This yields the conclusion of the theorem. \(\square \)
8 The Dirichlet problem: concluding remarks
In a restricted range of parameters, the class of (p, q)-harmonic functions can be analyzed in terms of their boundary values. We shall discuss in this section some rudiments of such theory.
Let \(p,q\in {\mathbb {C}}{\setminus }{\mathbb {Z}}^-\) be such that \({\text {Re}}(p)+{\text {Re}}(q)>-1\). The (p, q)-harmonic Poisson kernel is defined by
where
Notice that the constant \(c_{p,q}\) is a non-zero complex number in this range of parameters. By Theorem 1.4 the function \(K_{p,q}\) is (p, q)-harmonic. Theorems 6.4 and 6.6 ensure that the function \(K_{p,q}\) satisfies some standard properties for approximate identities.
The (p, q)-harmonic Poisson integral is defined by
for integrable functions \(f\in L^1({\mathbb {T}})\) on \({\mathbb {T}}\).
Let \(C({\mathbb {T}})\) be the space of continuous functions on \({\mathbb {T}}\) and fix \(\varphi \in C({\mathbb {T}})\). By the (p, q)-harmonic Dirichlet problem for \(\varphi \) we understand the problem of finding a (p, q)-harmonic function u such that \(\lim _{r\rightarrow 1}u_r=\varphi \) in \(C({\mathbb {T}})\), where
for \(0<r<1\). Following usual practice, we formulate this latter Dirichlet problem as
where \(L_{p,q}\) is as in (0.1).
Theorem 7.1
Let \(p,q\in {\mathbb {C}}{\setminus }{\mathbb {Z}}^-\) be such that \({\text {Re}}(p)+{\text {Re}}(q)>-1\). Let \(\varphi \in C({\mathbb {T}})\). Then a function u in \({\mathbb {D}}\) satisfies (7.1) if and only if it has the form
The proof of Theorem 7.1 follows a standard scheme for such results and is therefore omitted, see [18, 21] for details.
The present paper suggests a finer study of (p, q)-harmonic functions. Earlier results of such type concern Poisson integral representations, pointwise boundary limits, Green functions and Lipschitz continuity of generalized harmonic functions, see [18,19,20,21]. We mention here also work of Ahern and collaborators [1, 2].
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Klintborg, M., Olofsson, A. A series expansion for generalized harmonic functions. Anal.Math.Phys. 11, 122 (2021). https://doi.org/10.1007/s13324-021-00561-w
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DOI: https://doi.org/10.1007/s13324-021-00561-w