Abstract
This paper establishes extension results for harmonic functions which vanish on a conical surface. These are based on a detailed analysis of expansions for the Green function of an infinite cone.
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1 Introduction
The Schwarz reflection principle gives a simple formula for extending a harmonic function h on a domain \(\omega \subset \mathbb {R}^{N}\) through a relatively open subset E of \(\partial \omega \) on which h vanishes, provided E lies in a hyperplane. A corresponding reflection formula holds when E lies in a sphere. When \(N \ge 3\) and N is odd, Ebenfelt and Khavinson [6] (cf. Chapter 12 of [16]) have shown that a point to point reflection law can only hold when the containing real analytic surface is either a hyperplane or a sphere. Thus more sophisticated methods are needed for extending a harmonic function which vanishes on any other type of set E.
This is the background to the following problem, which was posed by Dima Khavinson at various international conferences: if h is harmonic on an infinite cylinder and vanishes on the boundary, does it extend harmonically to all of \(\mathbb {R}^{N}\)? Of course, in the planar case, where h is harmonic on an infinite strip, the answer is readily seen to be positive by repeated application of the Schwarz reflection principle. In higher dimensions the problem was eventually also shown to have an affirmative answer [7] by analysis of the Green function of the cylinder. Subsequently, the authors investigated extension properties of harmonic functions on an annular cylinder \(\{x^{\prime }\in \mathbb {R} ^{N-1}:a<\left\| x^{\prime }\right\| <b\}\times \mathbb {R}\) that vanish on either one or both of the cylindrical boundary components (see [8, 10, 11]). The domain reflection results that emerged were noteworthy, given that reflection formulae for the harmonic functions themselves fail to exist. This raises the following general question.
Problem 1
For a domain \(\omega \) in \(\mathbb {R}^{N}\) and a subset E of \(\partial \omega \) identify a larger domain \(\omega _{E}\) such that each harmonic function on \(\omega \) which vanishes continuously on E has a harmonic extension to \(\omega _{E}.\)
Naturally we should assume that E is contained in a real-analytic surface, but the question is interesting even in the particular case where E is contained in the zero set of a polynomial. The cylindrical case corresponds to the polynomial \((x^{\prime },x_{N})\mapsto \left\| x^{\prime }\right\| ^{2}-1\). The next most natural case to consider is a cone. The analogue of Khavinson’s question above would then be: if h is harmonic on an infinite cone and vanishes on the boundary, does h extend harmonically to all of \(\mathbb {R}^{N}\), except for the negative axis of the cone? Again, in the planar case, such extension follows by repeated application of the Schwarz reflection principle.
A typical point of \(\mathbb {R}^{N}\) (\(N\ge 3\)) will be denoted by \( x=(x^{\prime },x_{N})\), where \(x^{\prime }\in \mathbb {R}^{N-1}\) and \( x_{N}\in \mathbb {R}\), and we will write \(\theta _{x}=\cos ^{-1}(x_{N}/\left\| x\right\| )\) when \(x\ne 0\). Let \(0<\theta _{*}<\pi \). We will show that harmonic functions h on the infinite cone
that vanish on \(\partial \Omega \) have an extension to the set
In fact, it is unnecessary to require that h vanishes at 0.
Theorem 1
Let \(0<\theta _{*}<\pi \). If h is a harmonic function on \( \Omega (\theta _{*})\) that continuously vanishes on \(\partial \Omega (\theta _{*})\backslash \{0\}\), then h has a harmonic extension to \( \Omega (\pi )\).
The proof of Theorem 1 is technically more challenging than the corresponding result for the cylinder. However, it also yields tools applicable to reflection results for functions that are harmonic on a domain of the form
and vanish on \(\partial \Omega (\theta _{*})\). Strikingly, a dichotomy emerges between the cases where \(\theta _{*}\le \pi /2\) and \(\theta _{*}>\pi /2\) , as we will now see.
Theorem 2
Let \(0\le \theta _{0}<\theta _{*}\le \pi /2\). If h is a harmonic function on the domain \(\Omega (\theta _{0},\theta _{*})\) that continuously vanishes on \(\partial \Omega (\theta _{*})\backslash \{0\}\) , then h has a harmonic extension to the domain \(\{x\in \mathbb {R} ^{N}\backslash \{0\}:\theta _{0}<\theta _{x}<2\theta _{*}-\theta _{0}\}.\)
Theorem 3
Let \(0\le \theta _{0}<\theta _{*}<\pi \), where \(\theta _{*}>\pi /2\). If h is a harmonic function on the domain \(\Omega (\theta _{0},\theta _{*})\) that continuously vanishes on \(\partial \Omega (\theta _{*})\backslash \{0\}\), then h has a harmonic extension to the domain
The conditions arising in Theorems 2 and 3, and their sharpness, will be discussed in Sect. 2. Theorems 1–3 answer particular cases of Questions 4 and 5 in [9]. Theorem 3 also has the following immediate corollary.
Corollary 4
Let \(\pi /2\le \theta _{*}<\pi \), and suppose that h is a harmonic function on the domain \(\Omega (0,\theta _{*})\) that continuously vanishes on \(\partial \Omega (\theta _{*})\backslash \{0\}\). Then h has a harmonic extension to the domain \(\left( \mathbb {R}^{N-1}\backslash \{0^{\prime }\}\right) \times \mathbb {R}\).
We now have a reasonably complete set of harmonic extension results for conical surfaces to complement those known for cylinders. Our hope is that these will suggest further steps towards addressing the broader question in Problem 1.
The extension of harmonic functions through conical surfaces is obviously related to extension properties of the Green function for a cone, and harmonic functions on conical domains are naturally related to Legendre functions. The plan of the paper is thus as follows. In Sect. 3 we assemble and develop some relevant material concerning Legendre functions. This is subsequently used, in conjunction with contour integration, to establish an expansion of the fundamental function that is adapted to the geometry of cones, and then two different expansions for the Green function of the cone \(\Omega (\theta _{*})\). The first of these latter expansions is used to establish the second and also has later application. The second yields extension properties of the Green function that are used in proving Theorem 1. Theorems 2 and 3 rely on both Theorem 1 and further extension properties of the Green function. These latter properties are established using bounds for ratios of conical functions that may be of independent interest.
2 Sharpness of results
The domain of extension in Theorem 2 is formed by angular reflection. This is natural, since in the planar analogue of the result the function h is harmonic in an angle and would extend to an angle of twice the aperture by Schwarz reflection. The sharpness of this result in higher dimensions is demonstrated by the following example.
Example 1
Let \(N=4\) and \(0<\theta _{0}<\theta _{*}<\pi \), where \(2\theta _{*}-\theta _{0}<\pi \), and define the planar angle
Further, let u be the Green potential in \(\omega (\theta _{*})\) of a dense sum of point masses on the half line \(\partial \omega (\theta _{0})\cap \omega (\theta _{*})\), and extend u to \(\omega (2\theta _{*}-\theta _{0})\) by the Schwarz reflection principle. The function
is, by computation of the Laplacian, harmonic on \(\Omega (2\theta _{*}-\theta _{0})\backslash \overline{\Omega (\theta _{0})}\), and it vanishes on \(\partial \Omega (\theta _{*})\backslash \{0\}\). It cannot be extended as a harmonic function because it is unbounded near every point of \( \partial \Omega (\theta _{0})\cup \partial \Omega (2\theta _{*}-\theta _{0})\).
Surprisingly, however, when \(\theta _{*}>\pi /2\) and \(2\theta _{*}-\theta _{0}<\pi \) the above example no longer gives a sharp bound for how far h can be extended. To compare the domains of extension in Theorems 2 and 3 we note that, if \(0\le \theta _{0}<\theta _{*}<\theta _{x}<\pi \), where \(\theta _{*}>\pi /2\), then
where the latter inequality follows from the observation that
(By \(\csc \) we mean \(1/\sin \).)
The sharpness of Theorem 3 is shown by the next example.
Example 2
Let \(N=3\) and \(0<\theta _{0}<\theta _{*}<\pi \), let \(y=(\sin \theta _{0},0,\cos \theta _{0})\) and \(w_{\theta }=(\sin \theta ,0,\cos \theta )\), and let S denote the unit sphere in \( \mathbb {R}^{3}\). The Green function \(\mathbf {G}_{\theta _{*}}\) for the Laplace-Beltrami operator on \(S\cap \Omega (\theta _{*})\) satisfies
(See, for example, formula (13) in [12].) Hence the function defined by \(h(x)=\mathbf {G}_{\theta _{*}}(x/\left\| x\right\| ,y)\), which satisfies the hypotheses of Theorem 3, has a singularity at \( w_{\theta }\) if \(\tan (\theta /2)\tan (\theta _{0}/2)=\left( \tan (\theta _{*}/2)\right) ^{2}\).
Let T denote the stereographic projection that maps a typical point x of \(S\backslash \{(0^{\prime },-1)\}\) to the point where the line through \( (0^{\prime },-1)\) and x meets the plane \(\mathbb {R}^{2}\times \{1\}\). Then any point of \(S\cap \partial \Omega (\theta )\) is mapped by T to a point of the form \((y^{\prime },1)\), where \(\left\| y^{\prime }\right\| =2\tan (\theta /2)\). Hence, in Theorem 3, the intersection of the enlarged domain with S is mapped by T to an annulus, of which the outer boundary circle is the image of the inner boundary circle under inversion in \(T(S\cap \partial \Omega (\theta _{*}))\).
3 Preparatory material
The ultraspherical (or Gegenbauer) polynomials \(C_{n}^{(\lambda )}\), where \( \lambda >0\) and \(n=0,1,2,...\), are defined by the equation
(see (4.7.23) in Szegö [21], where the notation \(P_{n}^{(\lambda )} \) is used instead). They satisfy the differential equation
and clearly
(see (4.7.4) and (4.7.5) in [21]). We will also need the fact that
(see Lemma 6(i) of [7]).
The Legendre (or Ferrers) functions of the first and second kinds, \(P_{\nu }^{\mu }\) and \(Q_{\nu }^{\mu }\), respectively, are defined on the interval \( (-1,1)\) by equations (14.3.1) and (14.3.2) of [19]. (That source uses Roman type, \(\mathrm {P}_{\nu }^{\mu }\) and \(\mathrm {Q}_{\nu }^{\mu }\), to distinguish functions defined on \((-1,1)\) from functions on \((1,\infty )\).) They satisfy the equation
(see (14.2.2) in [19]). We collect below some properties of these functions.
Lemma 5
-
(i)
The ultraspherical polynomials are connected to the Legendre functions by the formula
$$\begin{aligned} C_{n}^{\left( \frac{N-2}{2}\right) }(t)=\frac{2^{\frac{N-3}{2}}\Gamma \left( \frac{N-1}{2}\right) \Gamma (n+N-2)}{(1-t^{2})^{\frac{N-3}{4}}\Gamma (N-2)\Gamma (n+1)}P_{n+\frac{N-3}{2}}^{\frac{3-N}{2}}(t){ \ \ \ } (\left| t\right| <1,n=0,1,\ldots ). \end{aligned}$$ -
(ii)
If \(\mu \in \mathbb {R}\) and \(p\in \mathbb {Z}\), then \(P_{\mu +p}^{-\mu }(t)=(-1)^{p}P_{\mu +p}^{-\mu }(-t)\).
-
(iii)
If \(\mu \in \mathbb {R}\), then
$$\begin{aligned} (1-t^{2})\frac{dP_{\nu }^{-\mu }}{dt}(t)=(\nu +1)tP_{\nu }^{-\mu }(t)-(\mu +\nu +1)P_{\nu +1}^{-\mu }(t). \end{aligned}$$ -
(iv)
If \(-1<t<1\), then
$$\begin{aligned} (1-t^{2})\left( P_{\nu }^{-\mu }(t)\frac{dQ_{\nu }^{-\mu }}{dt}(t)-Q_{\nu }^{-\mu }(t)\frac{dP_{\nu }^{-\mu }}{dt}(t)\right)= & {} \frac{\Gamma (\nu -\mu +1)}{\Gamma (\nu +\mu +1)}, \end{aligned}$$(7)$$\begin{aligned} (1-t^{2})\left( P_{\nu }^{-\mu }(t)\frac{d}{dt}P_{\nu }^{-\mu }(-t)-P_{\nu }^{-\mu }(-t)\frac{d}{dt}P_{\nu }^{-\mu }(t)\right)= & {} \frac{2}{\Gamma (\nu +\mu +1)\Gamma (\mu -\nu )}. \nonumber \\ \end{aligned}$$(8) -
(v)
[Mehler–Dirichlet formula] If \(\mu \ge 0\), \(0<\theta <\pi \) and \(\nu \in \mathbb {C}\), then
$$\begin{aligned} P_{\nu }^{-\mu }(\cos \theta )=\frac{\sqrt{2}}{\sqrt{\pi }(\sin \theta )^{\mu }\Gamma (\mu + {\frac{1}{2}} )}\int _{0}^{\theta }\cos \left( \left( \nu +\frac{1}{2}\right) t\right) \left( \cos t-\cos \theta \right) ^{\mu - {\frac{1}{2}} }dt. \nonumber \\ \end{aligned}$$(9)In particular, \(P_{-\nu -1}^{-\mu }=P_{\nu }^{-\mu }\).
-
(vi)
If \(\mu \ge 0\) and \(0<\theta <\pi \), then the function \(\nu \mapsto P_{\nu }^{-\mu }(\cos \theta )\) has infinitely many zeros, all of which are real and simple. The positive zeros form an increasing sequence \((\nu _{m})\) which satisfies \(\nu _{m}>\mu +m-1\). All the remaining zeros are of the form \(\{-\nu -1:\nu \ \)is a positive zero\(\}\).
-
(vii)
If \(\nu \in \mathbb {C}\), then
$$\begin{aligned} \left| P_{\nu }^{-\mu }(\cos \theta )\right|\le & {} 2^{3/2}\sqrt{\pi } \left( \frac{\sin \theta }{1+\cos \theta }\right) ^{\mu }\frac{e^{\left| \mathrm{Im}\nu \right| \theta }}{\Gamma (\mu + {\frac{1}{2}} )}{ \ \ \ }\left( 0\le \theta \le \frac{\pi }{2}\right) , \\ \left| P_{\nu }^{-\mu }(\cos \theta )\right|\le & {} 2^{3/2}\sqrt{\pi } \left( \frac{1-\cos \theta }{\sin \theta }\right) ^{\max \{\mu , {\frac{1}{2}} \}}\frac{e^{\left| \mathrm{Im}\nu \right| \theta }}{\Gamma (\mu + {\frac{1}{2}} )}{ \ \ \ }\left( \frac{\pi }{2}<\theta <\pi \right) . \end{aligned}$$ -
(viii)
If \(\nu \ge \mu \ge 0\) and \(-1<t<1\), then
$$\begin{aligned} \left\{ P_{\nu }^{-\mu }(t)\right\} ^{2}+\left\{ \frac{2}{\pi }Q_{\nu }^{-\mu }(t)\right\} ^{2}\le \frac{4^{\mu }}{\pi \left( 1-t^{2}\right) ^{\max \{\mu , {\frac{1}{2}} \}}}\left\{ \frac{\Gamma \left( \frac{\nu +\mu +1}{2}\right) \Gamma (\nu -\mu +1)}{\Gamma \left( \frac{\nu -\mu }{2}+1\right) \Gamma (\nu +\mu +1)} \right\} ^{2}. \end{aligned}$$ -
(ix)
If \(\nu \ge \mu \ge 0\) and \(-1<t<1\), then
$$\begin{aligned} \left| \frac{d}{dt}\frac{P_{\nu }^{-\mu }(t)}{(1-t^{2})^{\mu /2}} \right| \le \frac{2^{\mu +1}}{\sqrt{\pi }\left( 1-t^{2}\right) ^{\max \{\mu +1,\mu /2+5/4\}}}\frac{\Gamma (\nu -\mu +1)\Gamma \left( \frac{\nu +\mu }{2}+1\right) }{\Gamma (\nu +\mu +1)\Gamma \left( \frac{\nu -\mu +1}{2} \right) }. \end{aligned}$$ -
(x)
If \(-1<t<1\) and \(\mu \ge 0\), then
$$\begin{aligned}&2\nu (\nu +1)\int _{t}^{1}\tau \left\{ P_{\nu }^{-\mu }(\tau )\right\} ^{2}d\tau =\left( (1-t^{2})\frac{dP_{\nu }^{-\mu }}{dt}\right) ^{2}\nonumber \\&\qquad +\left\{ P_{\nu }^{-\mu }(t)\right\} ^{2}\left\{ \nu (\nu +1)(1-t^{2})-\mu ^{2}\right\} . \end{aligned}$$(10)
Proof
(i)–(v). See (14.3.21), (14.9.10), (14.10.4), (14.2.4), (14.2.3) and (14.12.1) of [19].
(vi) It is shown in [17] (cf. Section 238 of [14]) that the function \(\nu \mapsto P_{\nu }^{-\mu }(\cos \theta )\) has infinitely many zeros, all of which are real. The argument in [18] shows that these zeros are simple and \(\nu _{m}>\mu +m-1\). (These results are given for the case where \(\mu >0\), but the arguments extend easily to cover also the case where \(\mu =0\).) The final assertion of (vi) is a consequence of (v), since \( P_{-\nu -1}^{-\mu }=P_{\nu }^{-\mu }\) and it follows from (9) that \( P_{\nu }^{-\mu }\ne 0\) when \(-1\le \nu \le 0\).
(vii) To see that this follows from (v) we note that
and
since \(\sin \phi \ge 2\phi /\pi \) on \((0,\pi /2)\), and \(\sin \) is concave and satisfies \(\sin (\phi /2)\ge (\sin \phi )/2\) on \((0,\pi )\). If \( 0\le \theta \le \pi /2\), the desired estimate now follows on noting that \( 1-\cos \theta =\left( \sin ^{2}\theta \right) /\left( 1+\cos \theta \right) \) . If \(\pi /2<\theta <\pi \), we instead note that \(\min \{\sin (\theta /2),\sin \theta \}\ge (\sin \theta )/\sqrt{2}\).
(viii) When \(\mu \ge {\frac{1}{2}} \) this follows on combining equations (5) and (19) in Durand [4], and when \(0\le \mu < {\frac{1}{2}} \) we instead use (5) and (23) there.
(ix) This follows on combining the first two lines of (29) with (5) in [4].
(x) This is equivalent to formula (5.3) in [15]. We recall the short proof here for completeness. Let F(t) denote the right hand side of (10). Then, by (6),
We see from (iii) and (vii) that \(F(t)\rightarrow 0\) as \(t\rightarrow 1-\), so the result follows. \(\square \)
If \(x^{\prime },y^{\prime }\in \mathbb {R}^{N-1}\), then we define \(\phi _{x^{\prime },y^{\prime }}\in [0,\pi ]\) by the equation
whenever the denominator is non-zero. We also recall that \(\cos \theta _{x}=x_{N}/\left\| x\right\| \). The following result shows how some of the above functions relate to harmonicity.
Proposition 6
Let \(w\in \mathbb {C}\), \(y^{\prime }\in \mathbb {R}^{N-1}\backslash \{0^{\prime }\}\) and \(k\in \mathbb {N}\cup \{0\}\). Then the function
is harmonic on \(\Omega (\pi )\) when suitably interpreted on the positive \( x_{N}\)-axis.
Proof
We will give the details when \(N\ge 4\) and leave the adjustments required when \(N=3\) to the reader. Let \(r=\left\| x\right\| \), \(\theta =\theta _{x}\) and \(\phi =\phi _{x^{\prime },y^{\prime }}\). Then
where
Since
it is enough to show that
where \(f(t)=P_{w- {\frac{1}{2}} }^{\frac{3-N}{2}-k}(t)\).
Now
so (3) yields
Thus it remains to check that
Next,
Thus
The above calculation is not valid when \(\theta _{x}=0\), or when \(\phi _{x^{\prime },y^{\prime }}\in \{0,\pi \}\). In the latter case we can use the continuity of \(C_{k}^{\left( \frac{N-3}{2}\right) }\) to see that the set
is a removable singularity for the harmonic function h, by Corollary 5.2.3 of [1]. A similar argument, combined with Lemma 5(vii), shows that the positive \(x_{N}\)-axis is also removable for h. \(\square \)
Corollary 7
Let \(\nu >0\), \(y^{\prime }\in \mathbb {R}^{N-1}\backslash \{0\}\) and \(k\in \mathbb {N}\cup \{0\}\). Then any function of the form
where \(A,B\in \mathbb {R}\), is harmonic on \(\Omega (\pi )\) when suitably interpreted on the positive \(x_{N}\)-axis.
Proof
We put \(w=\pm \left( \nu +\frac{N-2}{2}\right) \) in the proposition and use the fact that \(P_{\lambda }^{-\mu }=P_{-\lambda -1}^{-\mu }\), by Lemma 5(v). \(\square \)
Corollary 8
Let \(\lambda ,c\in \mathbb {R}\), \(y\in \mathbb {R}^{N}\backslash \{0\}\) and \(k\in \mathbb {N}\cup \{0\}\). Then the function
is harmonic on \(\Omega (\pi )\) when suitably interpreted on the positive \( x_{N}\)-axis.
Proof
We put \(w=i\lambda \) in the proposition, take real and imaginary parts of h, and expand \(\cos (\lambda \log \left\| x\right\| +c)\) using the addition formula. \(\square \)
Functions of the form \(P_{- {\frac{1}{2}} +i\lambda }^{-\mu }\) are known as conical (or Mehler) functions. We record below some of their further properties for future reference.
Lemma 9
(i) \(P_{- {\frac{1}{2}} +i\lambda }^{-\mu }>0\) and \(P_{- {\frac{1}{2}} +i\lambda }^{-\mu }=P_{- {\frac{1}{2}} -i\lambda }^{-\mu }\) on \((-1,1)\).(ii) If \(\mu \ge 0\), then the function \(\theta \mapsto P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(-\cos \theta )/P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(\cos \theta )\) is decreasing on \((0,\pi )\).(iii) If \(\mu \ge 0\), then the function \(\theta \mapsto P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(\cos \theta )\) is increasing on \((0,\pi )\).
Proof
(i) This is clear from the Mehler–Dirichlet formula (9).
(ii) Since \(\Gamma (\overline{z})=\overline{\Gamma (z)}\), it follows from (8) that the function \(t\mapsto P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(-t)/P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(t)\) is increasing on \((-1,1)\).
(iii) By definition,
so
Since the coefficients in the expansion
are all positive, we now see that the function \(\theta \mapsto P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(\cos \theta )\) is the product of two positive increasing functions on \((0,\pi )\). \(\square \)
In the next three sections we will adapt an argument outlined on pp.69–72 of Dougall [3] for \(\mathbb {R}^{3}\) to establish expansions of the Green function for \(\Omega (\theta _{*})\) in all dimensions.
4 An expansion of the fundamental function
If \(x,y\in \mathbb {R}^{N}\), then we define \(\gamma _{x,y}\in [0,\pi ]\) by the equation
whenever the denominator is non-zero. Since \(\left\langle x,y\right\rangle =\left\langle x^{\prime },y^{\prime }\right\rangle +x_{N}y_{N}\) and \( \left\| x^{\prime }\right\| =\left\| x\right\| \sin \theta _{x}\), it follows that
It will be convenient to define
We recall from p. 1938 of [5] (cf. equation (80) in [13]) an addition formula for \(P_{\nu }^{-\mu }\), namely
when \(\theta _{x}+\theta _{y}<\pi \). (The restriction in [5] that \( \phi _{x^{\prime },y^{\prime }}<\pi \) may be removed by dominated convergence, in the light of (5) and the asymptotic behaviour of \( P_{\nu }^{-\mu }\) for large \(\mu \), as described in (14.15.1) of [19].) Since
and analogous formulae hold for \(\theta _{-x}\) and \(\phi _{-x^{\prime },y^{\prime }}\), we can replace x by \(-x\) in (13), and use (4) and (12) to obtain
when \(\theta _{-x}+\theta _{y}<\pi \), that is, when \(\theta _{y}<\theta _{x}\) . When \(\mu =0\) the appropriate analogue of (13) may be found by combining equations (14.18.1) and (14.9.3) of [19]. This leads to the formula
where
Equations (14) and (15) are valid when \(\gamma _{x,y},\theta _{x},\theta _{y}\in (0,\pi )\) and \(\theta _{y}<\theta _{x}\).
Let
and suppose that \(\left\| y\right\| <\left\| x\right\| \) and \( 0<\gamma _{x,y}<\pi \). Then (2) and parts (i), (ii) of Lemma 5 show that
Hence
where
For any \(\kappa \in \mathbb {N}\) let \(c(\kappa )\) denote the contour around the boundary of the rectangle
oriented anticlockwise. The function \(z\mapsto P_{z+\frac{N-3}{2}}^{\frac{3-N }{2}}(-\cos \gamma _{x,y})\) is entire (see (14.3.1) and §15.2(ii) of [19]). Thus the residue theorem yields
since the singularities of the integrand in \(\mathbb {Z}\cap \left[ \frac{2-N }{2},0\right) \) are removable. By Lemma 5(vii) the above integrand is bounded in modulus by
on the top and bottom sides of the contour, and by
on the right hand side. Since we can parametrize the reverse path \(-c(\kappa )\) on the left hand side of the rectangle as \((2-N)/2+i\lambda \) \((-\kappa \le \lambda \le \kappa )\), we can let \(\kappa \rightarrow \infty \) in (18) to see that
(The convergence of this integral will become clear below.) Since
we see that
Hence
because \(P_{- {\frac{1}{2}} +i\lambda }^{\frac{3-N}{2}}(-\cos \gamma _{x,y})\) is real and symmetric in \( \lambda \), by Lemma 9(i). Combining this with (16), we see that
Noting that
by (5.11.9) in [19], we see from Lemma 5(vii) that the integral in (21) converges absolutely even when \(\left\| y\right\| =\left\| x\right\| \). It follows from dominated convergence and symmetry that (21) is valid for any non-zero choices of \(\left\| y\right\| \) and \(\left\| x\right\| \), provided \(\gamma _{x,y}\in (0,\pi )\). Since
by (12) and (20), we see from (21) that
We now make the additional assumption that \(0<\theta _{y}<\theta _{x}<\pi \), and deal first with the case where \(N\ge 4\). We can combine (24) with (14) to see that
In view of the positivity of \(P_{- {\frac{1}{2}} +i\lambda }^{\frac{3-N}{2}-k}\) (see Lemma 9(i)) and (5) the summand in (25) is bounded in absolute value by
In addition,
Thus the integral in (25) still converges when we replace the summand by its absolute value. In particular, we can thus allow \(\gamma _{x,y}\) to range over \((0,\pi ]\), by dominated convergence.
When \(N=3\) we instead combine (15) with (24) to see that
The analogue of (26) again holds, so the expansion in (27) has the same absolute convergence property.
We have established (25) and (27) for any \(x,y\in \mathbb {R }^{N}\backslash \{0\}\) satisfying \(0<\theta _{y}<\theta _{x}<\pi \). The integrals and summations are interchangeable, by Fubini’s theorem.
5 An expansion for the Green function
We assume in this section that \(x,y\in \mathbb {R}^{N}\backslash \{0\}\) and \( \theta _{x},\theta _{y}\in (0,\pi )\).
When \(N\ge 4\), \(y\in \Omega \) and \(x\in \overline{\Omega }\) we define
Since the function \(\theta \rightarrow P_{- {\frac{1}{2}} +i\lambda }^{\frac{3-N}{2}-k}(\cos \theta )\) is positive and increasing on \( (0,\pi )\), by Lemma 9, we see that
when \(\theta _{x}\le \theta _{*}\). It now follows from (26), with \(\theta _{x}=\theta _{*}\), and (5), that the right hand side of (28) is absolutely convergent, and from dominated convergence that \(h_{y}\) is continuous on \(\overline{\Omega }\), when suitably interpreted at points where \(\theta _{x}=0\). Further, by Fubini’s theorem and Corollary 8, the function \(h_{y}\) satisfies the volume mean value property in \(\Omega \), and so is harmonic there. It tends to 0 at infinity, by (26) again with \(\theta _{x}=\theta _{*}\). Since \( h_{y}(x)=\left\| x-y\right\| ^{2-N}\) on \(\partial \Omega \), by (25), it follows from the minimum principle that \(h_{y}\) is the greatest harmonic minorant of \(\left\| \cdot -y\right\| ^{2-N}\) on \(\Omega \). Hence, when \(0<\theta _{y}<\theta _{x}<\theta _{*}\), it follows from (25) and (28) that the Green function of \(\Omega \) is given by
where
The integration and summation can be interchanged in (29), by the absolute convergence of the expansions in (25) and (28). If \( \theta _{x}<\theta _{y}\), then we replace \(g_{k}(\lambda ,\theta _{x},\theta _{y})\) by \(g_{k}(\lambda ,\theta _{y},\theta _{x})\) in (29), by the symmetry of the Green function.
When \(N=3\) analogous reasoning shows that
when \(\theta _{y}<\theta _{x}\). This was asserted long ago in p.71(1) of [3], though full details of the proof were not provided. (That paper used \(P_{\nu }^{\mu }\) to denote what today is called \(P_{\nu }^{-\mu }\), as can seen from the definition on p.48.)
6 A second expansion for the Green function
We denote by \(n_{k,m}\) the mth positive zero of the entire function \(\nu \mapsto P_{\nu +\frac{N-3}{2}}^{\frac{3-N}{2}-k}(\cos \theta _{*})\) and note from Lemma 5(vi) that
Suppose that \(\left\| y\right\| <\left\| x\right\| \) and \(\theta _{x},\theta _{y}\in (0,\pi )\), and let
We recall that \(\Gamma (z)\) is holomorphic except for simple poles at the nonpositive integers, and that
Hence, by (12), the singularities of the function
lie at the integers j satisfying \(j\ge k\), and the residue at j is then
The singularities of f at such points are thus removable, in view of Lemma 5(ii). The remaining singularities of f in \(\left\{ \mathrm{Re} z>2-N\right\} \) are simple poles at the points \((n_{k,m})_{m\ge 1}\).
We will apply the residue theorem to the contour integral of f around the boundary \(d(\kappa )\) of the rectangle
oriented anticlockwise, where \(\kappa \in \mathbb {N}\). We recall from p.291 of [22] that, for fixed \(\mu \ge 0\) and \(\gamma ,\delta \in (0,\pi )\) ,
as \(\left| \nu \right| \rightarrow \infty \) in the set \(\left\{ \left| \mathrm {Arg}(\nu )\right| \le \pi -\delta \right\} \), whence
as \(\left| \nu \right| \rightarrow \infty \) in the set \(\left\{ \left| \mathrm {Arg}(\nu )\right| \le \pi -\delta ,\mathrm {dist}(\nu -\mu ,\mathbb {N})>\varepsilon \right\} \) for any \(\varepsilon >0\), by (12), (19) and Stirling’s formula. It follows that, for large \( \kappa \),
on the top and bottom sides of \(d(\kappa )\), and that
on the right hand side of \(d(\kappa )\). If we temporarily assume that \( \theta _{y}<\theta _{x}\), then we can apply the residue theorem and let \( \kappa \rightarrow \infty \) to see that
For any \(\mu \ge 0\), \(\nu >0\) and \(\tau _{0}\in (-1,1)\) satisfying \(P_{\nu }^{-\mu }(\tau _{0})=0\), we know from §11(I) of [2] (cf. §7 of [17]; the result is stated for the case where \(\mu >0\), but remains valid also when \(\mu =0\)) that
and from (8) that
Hence, by (12),
When \(N\ge 4\) we then see from (29), an interchange of summation and integration, and (32), that
and when \(N=3\) we use (30) in place of (29) to see that
We temporarily assumed above that \(\theta _{y}<\theta _{x}\). If \(\theta _{x}<\theta _{y}\), then we define \(x^{*}=(\left\| x\right\| /\left\| y\right\| )y\) and \(y^{*}=(\left\| y\right\| /\left\| x\right\| )x\). We then observe that \(G_{\Omega }(x^{*},y^{*})=G_{\Omega }(y,x)=G_{\Omega }(x,y)\), by (29) (or (30)) and the symmetry of the Green function, to arrive at (33) (or (34)) again. Our earlier assumption that \(\theta _{x},\theta _{y}\) are non-zero can be dropped provided the formulae are suitably interpreted. Thus these formulae hold when \(\theta _{x}\ne \theta _{y}\) and \(\left\| y\right\| <\left\| x\right\| \). The corresponding formulae when \( \left\| x\right\| <\left\| y\right\| \) are obtained by interchanging x and y in (33) and (34).
7 Extending the Green function of the cone
In preparation for the main result of this section we note the following lemma.
Lemma 10
If \(\nu \ge \mu \ge 0\), \(-1<t_{0}<1\ \)and \(P_{\nu }^{-\mu }(t_{0})=0\), then
Proof
It follows from parts (x), (iv) and then (viii) of Lemma 5 that
\(\square \)
Theorem 11
Let \(y\in \Omega \) and \(a>1\), and define
Then the formulae in (33) and (34) converge absolutely and uniformly to a harmonic function on \(\omega _{y,a}^{(1)}\), and when x and y are interchanged they converge absolutely and uniformly to a harmonic function on \(\omega _{y,a}^{(2)}\). In particular, \(G_{\Omega }(\cdot ,y)\) has a harmonic extension \(\widetilde{G}_{\Omega }(\cdot ,y)\) to the set \( \left( \Omega \backslash \{y\}\right) \cup \omega _{y,a}^{(1)}\cup \omega _{y,a}^{(2)}\). Further,
Proof
Suppose first that \(N\ge 4\) and \(\left\| x\right\| >a\left\| y\right\| \). We assume, without loss of generality, that \(1<a\le 2\), and define
By (31) we see that
which will allow us to apply Lemma 10 and some results from Lemma .
By Lemma 5(viii),
and, by Lemma 10,
where
Using the Legendre duplication formula,
we see that
When \(\theta _{*}/2<\theta _{y}<\theta _{*}\) we combine Lemma (ix) with the mean value theorem and use the concavity of \(\sin \theta \) on \((0,\pi )\) to see that
Using (38) again we see that
so
We next consider the case where \(0\le \theta _{y}\le \theta _{*}/2\). Let
whence
If h is a harmonic function on \(\Omega \), then \(h^{2}\) is subharmonic there, and so we can use the volume mean value inequality to see that
By Corollary 7 we can apply this inequality to the harmonic function given by
(interpreted, as usual, in the limiting sense on \(\{0\}^{N-1}\times (0,\infty )\)) to see that
whence
Since the sets
are bounded above by a constant C(a, N), we can use (39)–(41), (5) and (31), to see that
It follows that the expression for \(G_{\Omega }(x,y)\) in (33) converges absolutely to a harmonic function in \(\omega _{y,a}^{(1)}\) and satisfies the estimate (35) there.
For the set \(\omega _{y,a}^{(2)}\) we interchange x and y in (33) and argue similarly.
Analogous reasoning applies when \(N=3\). \(\square \)
8 Proof of Theorem 1
We will adapt the approach taken in Theorem 19 of [10]. Theorem 1 follows from the result below on letting \(c\rightarrow \infty \). We define
Theorem 12
Let \(c>1\) and let h be a harmonic function on the set \(\Omega \cap A(c)\) which continuously vanishes on \(\partial \Omega \cap A(c)\). Then h has a harmonic extension to the intersection of the sets
and
Proof
Let \(1<c^{\prime \prime }<c^{\prime }<c\). On \(\Omega \cap A(c^{\prime })\) we can write h as the difference, \(h_{1}-h_{2}\), of two positive harmonic functions that vanish on \(\partial \Omega \cap A(c^{\prime })\). (Each of these is a Dirichlet solution with non-negative boundary data.) Next, let \( h_{i}^{*}\) (\(i=1,2\)) be defined as \(h_{i}\) on \(\Omega \cap \overline{ A(c^{\prime \prime })}\), as 0 on \(\partial \Omega \) and also on \(\Omega \backslash A(c^{\prime })\), and extended to \(\Omega \) by solving the Dirichlet problem in \(\Omega \cap \left[ A(c^{\prime })\backslash \overline{ A(c^{\prime \prime })}\right] \). Then \(h_{i}^{*}\) is subharmonic on \( \Omega \backslash \overline{A(c^{\prime \prime })}\) and superharmonic on \( \Omega \cap A(c^{\prime })\), and continuously vanishes on \(\partial \Omega \) . By the Riesz decomposition theorem (Theorem 4.4.1 of [1]) and standard estimates of the Green function (cf. Theorems 4.2.4 and 4.2.5 of [1]) we can represent \(h_{i}^{*}\) as a Green potential \(G_{\Omega }\Lambda _{i}\), where \(\Lambda _{i}\) is a signed measure on \(\Omega \cap \left[ \partial A(c^{\prime })\cup \partial A(c^{\prime \prime })\right] \) satisfying
(More precisely, the Riesz decomposition theorem shows that \(h_{i}^{*}-G_{\Omega }\Lambda _{i}\) is harmonic on \(\Omega \), and the representation then follows from the fact that \(h_{i}^{*}\) and \(G_{\Omega }\Lambda _{i}\) both vanish at the boundary.)
Let \(a>1\). It follows from Theorem 11 that the formula
defines a harmonic extension of h from \(\Omega \cap \overline{A(c^{\prime \prime })}\) to the intersection of the sets
and
Since \(c^{\prime \prime }\) may be arbitrarily close to c, and a may be arbitrarily close to 1, the result follows. \(\square \)
9 Bounds for ratios of conical functions
Several authors have considered bounds on ratios of modified Bessel functions: see, for example, [20] and the references provided there. In this section we establish corresponding bounds on ratios of conical functions in preparation for the proofs of Theorems 2 and 3. We begin with two elementary lemmas concerning Riccati equations.
Lemma 13
Let h, \(\alpha \) \(\beta \) and \(\gamma \) be differentiable functions on an interval (a, b) such that
If \(\beta ^{\prime }h>0\), \(\alpha ^{\prime }\ge 0\), \(\gamma ^{\prime }\ge 0 \) and \(\lim \inf _{t\rightarrow a+}h^{\prime }(t)>0\), then \(h^{\prime }>0\) on (a, b).
Proof
Let
Then \(t_{0}>a\), by hypothesis. If \(t_{0}<b\), then \(h^{\prime }(t_{0})=0\) and so
This yields a contradiction, since \(h^{\prime }>h^{\prime }(t_{0})\) on \( (a,t_{0})\). Thus \(t_{0}=b\) as claimed. \(\square \)
Lemma 14
Suppose that
where h, A, B and C are all positive.
-
(i)
If \(h^{\prime }>0\) on (a, b), then \(0<h<B\).
-
(ii)
If \(h^{\prime }<0\) on (a, b), then \(0<B<h\).
Proof
Since \(h+C>0\) and \(A>0\), we see that \(h^{\prime }\) and \(h-B\) have opposite signs. \(\square \)
Proposition 15
Let \(0<\theta _{1}<\theta _{2}<\pi \) and \(\mu ,\lambda \in \mathbb {R}\). Then
where
Proof
We note from (14.10.2) in [19] that
and combine this with Lemma 5(iii) to see that
We also know from (14.10.1) in [19] that
and combine this with (46) to see that
Hence
and so
Equation (45) follows on substituting \(t=\cos \theta \). \(\square \)
Theorem 16
If \(\lambda \in \mathbb {R}\) and \(\mu >- {\frac{1}{2}} \), then
where \(h_{\mu }\) is as in Proposition 15,
and
Proof
Let \(F_{\mu }(\theta )=P_{- {\frac{1}{2}} +i\lambda }^{-\mu }(\cos \theta )\). We note from (46) and (48) that
and
Since \(h_{\mu }=F_{\mu +1}/F_{\mu }\) we now see that
Further,
by (14.8.1) of [19], so it follows from (49) that
The derivative of the function \(\theta \mapsto -(2\mu +1)\cot \theta \) is positive, because \(\mu >- {\frac{1}{2}} \). Since also \(h_{\mu }>0\), we can apply Lemma 13 to Eq. (49) to conclude that \(h_{\mu }^{\prime }>0\) on \((0,\pi )\).
It follows from Lemma 14 that \(h_{\mu }(\theta )\) is bounded above by the positive root of the equation
namely,
which equals \(f_{2}(\theta )\). Further, from (47),
whence \(h_{\mu }(\theta )\ge f_{1}(\theta )\). \(\square \)
10 Proofs of Theorems 2 and 3
Proposition 17
Let \(y\in \Omega \) and \(\delta >0\). If \(\theta _{*}\le \pi /2\), then \(G_{\Omega }(\cdot ,y)\) has a harmonic extension \(\overline{G} _{\Omega }(y,\cdot )\) to the set
and \(\overline{G}_{\Omega }(\cdot ,\cdot )\) is bounded on the set
Proof
We will give the argument when \(N\ge 4\). Only slight adjustments are required when \(N=3\). It is enough, by Corollary 8, to show that the expansion (29) (or, indeed, the expansion (28)) converges absolutely and uniformly when \(x,y\in \omega _{1,\delta }\). Let \(\mu =(N-3)/2+k\) and \(\Lambda =\lambda ^{2}+\left( \mu + {\frac{1}{2}} \right) ^{2}\).
By Lemma 9(ii),
when \((x,y)\in \omega _{1,\delta }\), where
By Theorem 16 and the formula (51) for \(f_{2}(\theta )\),
where
so
Hence, by Proposition 15,
We claim that
If \(\theta _{x}\le \pi /2\), this is clear from the monotonicity of H on \( (0,\pi /2]\) and the fact that \(\theta _{y}<\theta _{*}\). If \(\theta _{x}>\pi /2\), we use the symmetry of H about \(\pi /2\) as well as the above monotonicity to see that
because \(\theta _{y}+\theta _{x}-\theta _{*}\le \theta _{*}\le \pi /2\).
Since also \(\theta _{x}<2\theta _{*}-\theta _{y}\le \pi -\theta _{y}\), and so \(\left| \cot \right| \le \cot \theta _{y}\) on \((\theta _{*},\theta _{x})\), we now see from (55) that
by the equality in (53). Proposition 15 and Theorem also show that
Hence
Now
and
so
Since \(\theta _{y}+\theta _{x}-\theta _{*}\le \pi /2\), we now see from (56) that
when \((x,y)\in \omega _{1,\delta }\). It follows from (52) that
Since, by (26),
the proof is complete. \(\square \)
Proposition 18
Let \(\theta _{*}>\pi /2\), \(y\in \Omega \) and \(0<\delta <\min \{\theta _{*}-\theta _{y},\theta _{*}-\pi /2\}\). Then \( G_{\Omega }(\cdot ,y)\) has a harmonic extension \(\overline{G}_{\Omega }(y,\cdot )\) to the set
and \(\overline{G}_{\Omega }(y,\cdot )\) is bounded on the set
Proof
We modify the previous proof. Again we will assume, for simplicity, that \( N\ge 4\). This time we note that
where
It follows from our choice of \(\delta \) that
and from (1) that
Also, if \(0\le a<b\), then
Let \(H(\theta )\) be as in (54). Then
and
by (61). Also,
in view of (60). It follows from (63), (64), (62) and then (65) that
Hence, by (55), (53) and the fact that \(\log \sin \) is a primitive for \(\cot \),
Since, by Proposition 15 and Theorem 16,
we now see from (59) that
Now
so from (57) we have
Hence
when \((x,y)\in \omega _{2,\delta }\), since
It follows from (58) that
The argument is completed by observing from (26) that
when \((x,y)\in \omega _{2,\delta }\). \(\square \)
Proof of Theorem 2
Let \(\theta _{0}<\theta _{-}<\theta _{+}<\theta _{*}\) and \(1<c^{\prime \prime }<c^{\prime }\). As in the proof of Theorem 12, we can represent h in \(\left[ \Omega (\theta _{*})\backslash \overline{\Omega (\theta _{+})}\right] \cap A(c^{\prime \prime })\) as the potential \( G_{\Omega }\Lambda \) of a signed measure \(\Lambda \) on the union of the sets
Then \(h=h_{a}+h_{b}\), where
and
It follows from Theorem 12 that \(h_{a}\) has a harmonic extension to the intersection of the sets (42) and (43), and from Proposition 17 that \(h_{b}\) has a harmonic extension to the set \(\Omega (2\theta _{*}-\theta _{+})\backslash \overline{\Omega (\theta _{+})}\). The result now follows on letting \(c^{\prime \prime }\rightarrow \infty \) and \(\theta _{+}\rightarrow \theta _{0}+\). \(\square \)
Proof of Theorem 3
We follow the above argument except that we use Proposition 18 to see that \(h_{b}\) has a harmonic extension to the set
\(\square \)
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Gardiner, S.J., Render, H. Harmonic extension through conical surfaces. Math. Ann. 384, 1593–1627 (2022). https://doi.org/10.1007/s00208-021-02310-7
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DOI: https://doi.org/10.1007/s00208-021-02310-7