GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 1, 1999, 19-32

A RADIAL DERIVATIVE WITH BOUNDARY VALUES OF THE SPHERICAL POISSON INTEGRAL *

O. DZAGNIDZE

Abstract. A formula of a radial derivative \frac{∂}{∂r} U_{f} (r, θ, ϕ) is obtained with the aid of derivatives with respect to θ and to ϕ of the functions closely connected with the spherical Poisson integral U_{f} (r, θ, ϕ) and the boundary values are determined for \frac{∂}{∂r} U_{f} (r, θ, ϕ). The boundary values are also found for partial derivatives with respect to the Cartesian coordinates \frac{∂}{∂x} U_{F}, \frac{∂}{∂y} U_{F} and \frac{∂}{∂z} U_{F}.

Introduction

0.1. Let the function F (X, Y, Z) be summable on the two-dimensional unit sphere σ with center at the origin. The Poisson integral corresponding to this function F will be denoted by U_{F} (x, y, z), where (x, y, z) is a point in the open unit ball 𝔹 bounded by σ. This integral written in terms of spherical ccordinates will be denoted by U_{f}(r, θ, ϕ).

We shall represent the spehrical kernel of the Poisson integral P_{r} by the series (see, for instance, [2], pp. 335 and 143)

P_{r}(θ, ϕ; θ′, ϕ′) = 1 + ∑_{n=1}^{∞}(2n + 1)r^{n}P_{n}(cos θ)P_{n}(cos θ′) +

+2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} \frac{(n − m)!}{(n + m)!} P_{nm}(cos θ)P_{nm}(cos θ′) cos m(ϕ − ϕ′). (0.1)

0.2. In the case of the unit circle, the radial derivative of the Poisson integral v_{λ}(r, ϕ) for a function λ ∈ L(0, 2π) and the derivative with respect

19 to ϕ of the conjugate Poisson integral ṽ_{λ}(r, ϕ) are connected through the equality

1991 Mathematics Subject Classification. 33A45, 33C55.

Key words and phrases. Spherical Poisson integral, allied functions (integrals, kernels), radial derivative, angular limit, limit.

* The results of this paper were announced in the author’s paper [1]. The terms used there are slightly different from those of this paper.

1072-947X/99/0100-0021$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

\frac{∂}{∂r} v_{λ}(r, ϕ) = \frac{1}{r} \frac{∂}{∂ϕ} ṽ_{λ}(r, ϕ).

As is well known, the above equality is based on the property of the function v_{λ}(r, ϕ) + iṽ_{r}(r, ϕ) to be analytic with respect to the complex variable re^{iϕ}, 0 ≤ r < 1, 0 ≤ ϕ ≤ 2π. The equality is immediately obtained from the representations of v_{λ}(r, ϕ) and ṽ_{λ}(r, ϕ) as the series

v_{λ}(r, ϕ) = \frac{a_{0}}{2} + ∑_{n=1}^{∞} r^{n}(a_{n} cos nϕ + b_{n} sin nϕ),

ṽ_{λ}(r, ϕ) = ∑_{n=1}^{∞} r^{n}(a_{n} sin nϕ − b_{n} cos nϕ)

since the exponent of r coincides with multiplicity of the polar angle ϕ. Moreover, if the conjugate function λ̃ is summable on (0, 2π), then the equality ṽ_{λ}(r, ϕ) = v_{λ}(r, ϕ) holds by virtue of Smirnov’s theorem ([3], p. 263; [4], p. 583). Therefore by the well-known Fatou’s theorem ([3], p. 100) the radial derivative \frac{∂}{∂r} v_{r}(r, ϕ) will have an angular limit at the point (1, ϕ_{0}) if the conjugate function λ̃ with λ̃(ϕ) ∈ L(0, 2π) has the finite derivative (λ̃)′(ϕ_{0}) at the point ϕ_{0}.

It should be noted in the first place that there exists no analogue of the theory of analytic functions of a complex variable in a three-dimensional real space. In the second place, the exponent of r in series (0.2) differs from the multiplicity of the polar angle ϕ. Hence to obtain a formula for \frac{∂}{∂r} U_{f}(r, θ, ϕ) we have to overcome some difficulties. That this is so is seen from the fact that in the final result the products cos mϕ · P_{nm}(cos θ) and sin mϕ · P_{nm}(cos θ) must remain unchanged after all transformations if we want to remain in the class of harmonic functions; such an intention is dictated by the problem itself. At the same time, the trigonometric system (cos mϕ, sin mϕ) has some properties which the system (P_{nm}(cos(θ)) does not possess. For instance, a derivative of the trigonometric system is the same system to within a constant multiplier with respect to ϕ. The system (P_{nm}(cos(θ)) does not possess the latter property! Furthermore, the trigonometric system is bounded, while the system (P_{nm}(cos(θ)) is unbounded! These and other differences (for instance, the 2π-periodicity with respect to ϕ and the non-π-periodicity with respect to θ) between these two systems make series (0.2) have different properties with respect to θ and to ϕ (a similar observation is obviously true for series (0.3)). The same reason accounts for different preconditions with respect to θ and to ϕ, which are imposed on the boundary function f(θ, ϕ) from equalities (0.2) and (0.3). Nevertheless the system of spherical functions (P_{n}(cos(θ), cos mϕ · P_{nm}(cos θ), sin mϕ · P_{nm}(cos θ)) is successfully used to solve many important spatial problems of theoretical and mathematical physics as well as of mechanics. Theoretical geodesics which studies the external gravitational field of the Earth and other planets makes active use of the method of Laplace series.

0.3. In what follows we shall use the representation of the spherical Poisson integral as a series (see, e.g., [5], p. 444)

U_{f}(r, θ, ϕ) = a_{00} + ∑_{n=1}^{∞} a_{n0}r^{n}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ),  (0.2)

coinciding with the Abel–Poisson mean values (A-mean values) of the following Fourier–Laplace series S[f] for the function f(θ, ϕ) ∈ L(R) (see, e.g., [5], p. 444):

S[f] = a_{00} + ∑_{n=1}^{∞} a_{n0}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ).  (0.3)

The Legendre polynomials P_{n}(x) and associated Legendre functions P_{nm}(x) figuring in these equalities are defined on [−1, 1] by the following equalities:

P_{n}(x) = \frac{1}{n!2^{n}} · \frac{d^{n}}{dx^{n}} (x^{2} − 1)^{n}, n = 0, 1, 2, …,  (0.4)
P_{nm}(x) = (1 − x^{2})^{\frac{1}{2} m} \frac{d^{m}}{dx^{m}} P_{n}(x) =  (0.5)
= \frac{(1 − x^{2})^{\frac{1}{2} m}}{n!2^{n}} \frac{d^{n+m}}{dx^{n+m}} (x^{2} − 1)^{n},  (0.6)
1 ≤ m ≤ n, n = 1, 2, …,

where (1 − x^{2})^{\frac{1}{2}} is a non-negative value of the square root. Note that we have P_{n}^{0}(x) = P_{n}(x) and P_{nm}(x) = 0 for m > n.

0.4. The formula for \frac{∂}{∂r} U_{f}(r, θ, ϕ) will contain the functions U_{f}^{∗} and Ũ_{f}^{∗} which are allied with U_{f} and defined as follows (see [6], §1). To the spherical Poisson integral U_{f}(r, θ, ϕ), f ∈ L(R), given in the form of series (0.2) there correspond the following functions allied with respect to θ and to (θ, ϕ):

U_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} a_{n0}λ_{n0}r^{n}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ)  (0.7)

and

Ũ_{f}^{∗}(r, θ, ϕ) = ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ), (0.8)

where the numbers λ_{nm} are given by the equalities

λ_{nm} = \frac{1}{n + m} + \frac{1}{n − m + 1} for 0 ≤ m ≤ n, n = 1, 2, …. (0.9)

Analogously, to the spherical Poisson kernel P_{r}(θ, ϕ; θ′, ϕ′) there correspond the following kernels allied with respect to θ and (θ, ϕ) (see [6], §2):

P_{r}^{∗}(θ, ϕ; θ′, ϕ′) = ∑_{n=1}^{∞}(2n + 1)λ_{n0}r^{n}P_{n}(cos θ)P_{n}(cos θ′) +

+2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} λ_{nm} \frac{(n − m)!}{(n + m)!} ×

×P_{nm}(cos θ)P_{nm}(cos θ′) cos m(ϕ − ϕ′),  (0.10)
P̃_{r}^{∗}(θ, ϕ; θ′, ϕ′) =

+2 ∑_{n=1}^{∞}(2n + 1)r^{n} ∑_{m=1}^{n} λ_{nm} \frac{(n − m)!}{(n + m)!} ×

×P_{nm}(cos θ)P_{nm}(cos θ′) sin m(ϕ − ϕ′).  (0.11)

Finally, for each function f ∈ L^{2}(R) there are functions f^{∗}(θ, ϕ) ∈ L^{2}(R) and f̃^{∗}(θ, ϕ) ∈ L^{2}(R) allied with respect to θ and to (θ, ϕ) respectively, such that (see [6], §4)

U_{f}^{∗}(r, θ, ϕ) = U_{f^{∗}}(r, θ, ϕ), Ũ_{f}^{∗}(r, θ, ϕ) = U_{f^{∗}}(r, θ, ϕ).  (0.12)

§ 1. A Radial Derivative of the Spherical Poisson Integral and Kernel

Our aim now is to derive a formula for the radial derivative \frac{∂}{∂r} U_{f}(r, θ, ϕ) of the spherical Poisson integral U_{f}(r, θ, ϕ) for f ∈ L(R), and also a formula for \frac{∂}{∂r} P_{r}. At the end of the paper a formula for \frac{∂^{2}}{∂r^{2}} U_{f}(r, θ, ϕ) will be given too.

1.1. For the derivative \frac{∂}{∂r} U_{f} we have

Theorem 1.1. For every function f(θ, ϕ) ∈ L(R) the equality

2r \frac{∂}{∂r} U_{f}(r, θ, ϕ) + U_{f}(r, θ, ϕ) − a_{00} =

= \frac{∂^{2}}{∂θ^{2}} U_{f}^{∗}(r, θ, ϕ) + cot θ \frac{∂}{∂θ} U_{f}^{∗}(r, θ, ϕ) + cot^{2} θ \frac{∂^{2}}{∂ϕ^{2}} U_{f}^{∗}(r, θ, ϕ) −

−\frac{∂}{∂ϕ} Ũ_{f}^{∗}(r, θ, ϕ)  (1.1)

holds, where the allied harmonic functions U_{f}^{∗} and Ũ_{f}^{∗} in the ball 𝔹 are defined by equalities (0.7) and (0.8).

By introducing the operator

T_{θ,ϕ} = \frac{1}{sin θ} \frac{∂}{∂θ} ( sin θ \frac{∂}{∂θ} ) + cot^{2} θ \frac{∂^{2}}{∂ϕ^{2}} =
= \frac{∂^{2}}{∂θ^{2}} + cot θ \frac{∂}{∂θ} + cot^{2} θ \frac{∂^{2}}{∂ϕ^{2}} ,  (1.2)

equality (1.1) can be rewritten as

2r \frac{∂}{∂r} U_{f}(r, θ, ϕ) + U_{f}(r, θ, ϕ) − a_{00} =

= T_{θ,ϕ}U_{f}^{∗}(r, θ, ϕ) − \frac{∂}{∂ϕ} Ũ_{f}^{∗}(r, θ, ϕ).  (1.3)

Proof. From equality (0.2) we obtain

r \frac{∂}{∂r} U_{f}(r, θ, ϕ) = ∑_{n=1}^{∞} nr^{n}a_{n0}P_{n}(cos θ) +

+ ∑_{n=1}^{∞} nr^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ).  (1.4)

Our next aim is to find convenient expressions for nP_{n}(cos θ) and nP_{nm}(cos θ). If in equality (0.5) we replace P_{n}(x) by its value from the equality

(2n + 1)P_{n}(x) = P′_{n+1}(x) − P′_{n−1}(x), P_{−1}(x) = 0

(see,for instance, [2], p. 33, equality (34), or [7], p. 228, equality (7.8.2)), then we obtain the relations

(2n + 1)P_{nm}(x) = (1 − x^{2})^{\frac{m}{2}} \frac{d^{m}}{dx^{m}} [P′_{n+1}(x) − P′_{n−1}(x)] =

= (1 − x^{2})^{− \frac{1}{2}} [(1 − x^{2})^{\frac{m+1}{2}} \frac{d^{m+1}}{dx^{m+1}} P_{n+1}(x) −

− (1 − x^{2})^{\frac{m+1}{2}} \frac{d^{m+1}}{dx^{m+1}} P_{n−1}(x)] =

= (1 − x^{2})^{− \frac{1}{2}} [P_{n+1,m+1}(x) − P_{n−1,m+1}(x)].

Hence

(2n + 1)P_{nm}(x) =
= (1 − x^{2})^{− \frac{1}{2}} [P_{n+1,m+1}(x) − P_{n−1,m+1}(x)], −1 < x < 1. (1.5)

Further, the differentiation of equality (0.5) gives

\frac{d}{dx} P_{nm}(x) = −\frac{mx}{1 − x^{2}} (1 − x^{2})^{\frac{m}{2}} \frac{d^{m}}{dx^{m}} P_{n}(x) +

+ \frac{(1 − x^{2})^{\frac{m+1}{2}}}{(1 − x^{2})^{\frac{1}{2}}} · \frac{d^{m+1}}{dx^{m+1}} P_{n}(x) =

= −\frac{mx}{1 − x^{2}} P_{nm}(x) + \frac{1}{(1 − x^{2})^{\frac{1}{2}}} P_{n,m+1}(x).

Thus

\frac{d}{dx} P_{nm}(x) = \frac{1}{(1 − x^{2})^{\frac{1}{2}}} P_{n,m+1}(x) − \frac{mx}{1 − x^{2}} P_{nm}(x)  (1.6)

from which we have

P_{n,m+1}(x) = \frac{mx}{(1 − x^{2})^{\frac{1}{2}}} P_{nm}(x) + (1 − x^{2})^{\frac{1}{2}} \frac{d}{dx} P_{nm}(x).  (1.7)

Using the latter equality to define successively P_{n+1,m+1}(x), P_{n−1,m+1} and substituting them into (1.5), we obtain the equality

(2n + 1)P_{nm}(x) = \frac{d}{dx} [P_{n+1,m}(x) − P_{n−1,m}(x)] +
+ \frac{mx}{1 − x^{2}} [P_{n+1,m}(x) − P_{n−1,m}(x)]  (1.8)

which for x = cos θ and 0 < θ < π can be rewritten as

(2n + 1)P_{nm}(cos θ) = \frac{1}{sin θ} \frac{d}{dθ} [P_{n−1,m}(cos θ) − P_{n+1,m}(cos θ)] +

+m \frac{cos θ}{sin^{2} θ} [P_{n+1,m}(cos θ) − P_{n−1,m}(cos θ)].  (1.9)

Hence we find

nP_{nm}(cos θ) = \frac{1}{2 sin θ} \frac{d}{dθ} [P_{n−1,m}(cos θ) − P_{n+1,m}(cos θ)] +

+ m \frac{cos θ}{2 sin^{2} θ} [P_{n+1,m}(cos θ) − P_{n−1,m}(cos θ)] −

− \frac{1}{2} P_{nm}(cos θ).  (1.10)

For m = 0 the latter equality implies the equality

nP_{n}(cos θ) = \frac{1}{2 sin θ} \frac{d}{dθ} [P_{n−1}(cos θ) − P_{n+1}(cos θ)] − \frac{1}{2} P_{n}(cos θ). (1.11)

By the substitution of the expressions we have found for nP_{n}(cos θ) and nP_{nm}(cos θ) into (1.4) we obtain the following equality, using in doing so the right of termwise differentiation of the sries (see the arguments from [6], §1):

2r \frac{∂}{∂r} U_{f}(r, θ, ϕ) =

= \frac{1}{sin θ} \frac{d}{dθ} ∑_{n=1}^{∞} a_{n0}r^{n}[P_{n−1}(cos θ) − P_{n+1}(cos θ)] − ∑_{n=1}^{∞} a_{n0}r^{n}P_{n}(cos θ) +

+\frac{1}{sin θ} \frac{∂}{∂θ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ) ×

×[P_{n−1,m}(cos θ) − P_{n+1,m}(cos θ)] +

+\frac{cos θ}{sin^{2} θ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} m(a_{nm} cos mϕ + b_{nm} sin mϕ) ×

×[P_{n+1,m}(cos θ) − P_{n−1,m}(cos θ)] −

− ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ).

Hence

2r \frac{∂}{∂r} U_{f}(r, θ, ϕ) + U_{f}(r, θ, ϕ) − a_{00} =

= \frac{1}{sin θ} \frac{d}{dθ} ∑_{n=1}^{∞} a_{n0}r^{n}[P_{n−1}(cos θ) − P_{n+1}(cos θ)] +

+\frac{1}{sin θ} \frac{∂}{∂θ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} cos mϕ + b_{nm} sin mϕ) ×

×[P_{n−1,m}(cos θ) − P_{n+1,m}(cos θ)] +

+\frac{cos θ}{sin^{2} θ} \frac{∂}{∂ϕ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} (a_{nm} sin mϕ − b_{nm} cos mϕ) ×

×[P_{n+1,m}(cos θ) − P_{n−1,m}(cos θ)] ≡
≡ \frac{1}{sin θ} A + \frac{1}{sin θ} B + \frac{cos θ}{sin^{2} θ} C.  (1.12)

Now we are to find expressions for the differences contained within the square brackets from equality (1.12).

The well-known equality ([2], p. 107, equality (41)^{1} with replaced cos θ = x, cot θ = x(1 − x^{2})^{− \frac{1}{2}} ; [7], p. 241, equality (7.12.9), or [8], p. 99, equality (46′) with the same replacement)

(−1)^{m+2}P_{n,m+2}(x) + (−1)^{m+1} \frac{2(m + 1)x}{(1 − x^{2})^{\frac{1}{2}}} P_{n,m+1}(x) +
+(−1)^{m}(n − m)(n + m + 1)P_{nm}(x) = 0  (1.13)

implies, after reducing by (−1)^{m} and replacing m by (m − 1), the equality

P_{n,m+1}(x) = \frac{2mx}{(1 − x^{2})^{\frac{1}{2}}} P_{nm}(x) + (n + m)(n − m + 1)P_{n,m−1}(x). (1.14)

By (1.7) and (1.14) we have the equality

(1 − x^{2}) \frac{d}{dx} P_{nm}(x) = mxP_{nm}(x) +

+(n + m)(n − m + 1)(1 − x^{2})^{\frac{1}{2}} P_{n,m−1}(x), −1 < x < 1. (1.15)

Next, if in equality (1.5) m is replaced by (m − 1), then we obtain the equality

P_{n+1,m}(x) − P_{n−1,m}(x) = (2n + 1)(1 − x^{2})^{\frac{1}{2}} P_{n,m−1}(x), (1.16)
1 ≤ m ≤ n, −1 < x < 1.

1Here the Legendre functions are defined together with multipliers (−1)^{k}. This fact is taken into account in equality (1.13), i.e., in (1.13) P_{nk}(x) are defined by equality (0.5).

Now, by equalities (1.15) and (1.16) we find the product (1−x^{2})^{\frac{1}{2}} P_{n,m−1}(x) and equate them to each other. This results in the equality

P_{n+1,m}(x) − P_{n−1,m}(x) = \frac{(2n + 1)(1 − x^{2})}{(n + m)(n − m + 1)} \frac{d}{dx} P_{nm}(x) −

−\frac{m(2n + 1)x}{(n + m)(n − m + 1)} P_{nm}(x), −1 < x < 1.  (1.17)

Thus

P_{n−1,m}(cos θ) − P_{n+1,m}(cos θ) = \frac{(2n + 1) sin θ}{(n + m)(n − m + 1)} \frac{d}{dθ} P_{nm}(cos θ) +

+\frac{m(2n + 1) cos θ}{(n + m)(n − m + 1)} P_{nm}(cos θ), 0 < θ < π.  (1.18)

For m = 0 equality (1.18) gives rise to the equality

P_{n−1}(cos θ) − P_{n+1}(cos θ) = \frac{(2n + 1) sin θ}{n(n + 1)} \frac{d}{dθ} P_{n}(cos θ), (1.19)
0 < θ < π.

Now, taking into account equalities (1.18) and (1.19) we define A, B and C appearing in (1.12). In doing so, we make use of the numbers λ_{nm} from equality (0.9).

A = \frac{d}{dθ} ( sin θ \frac{d}{dθ} ∑_{n=1}^{∞} λ_{n0}a_{n0}r^{n}P_{n}(cos θ)) =

= cos θ \frac{d}{dθ} ∑_{n=1}^{∞} λ_{n0}a_{n0}r^{n}P_{n}(cos θ) +

+ sin θ \frac{d^{2}}{dθ^{2}} ∑_{n=1}^{∞} λ_{n0}a_{n0}r^{n}P_{n}(cos θ).  (1.20)

Further,

B = \frac{∂}{∂θ} [ sin θ \frac{∂}{∂θ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ)] +

+ \frac{∂}{∂θ} [ cos θ ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}m(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ)].

After performing the above-indicated differentiations and taking into account the equality

m(a_{nm} cos mϕ + b_{nm} sin mϕ) = \frac{∂}{∂ϕ} (a_{nm} sin mϕ − b_{nm} cos mϕ),

we have

B =

= cos θ \frac{∂}{∂θ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ) +

+ sin θ \frac{∂^{2}}{∂θ^{2}} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ) −

− sin θ \frac{∂}{∂ϕ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ) +

+ cos θ \frac{∂^{2}}{∂θ∂ϕ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ). (1.21)

In a similar manner we obtain

C = − sin θ \frac{∂^{2}}{∂θ∂ϕ} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm} ×

×(a_{nm} sin mϕ − b_{nm} cos mϕ)P_{nm}(cos θ) +

+ cos θ \frac{∂^{2}}{∂ϕ^{2}} ∑_{n=1}^{∞} r^{n} ∑_{m=1}^{n} λ_{nm}(a_{nm} cos mϕ + b_{nm} sin mϕ)P_{nm}(cos θ). (1.22)

The substitution of the found values of A, B and C into equality (1.12) shows that the one-dimensional series with cot θ \frac{d}{dθ} and the double-series with cot θ \frac{∂}{∂θ} are summable. The situation for series with \frac{∂^{2}}{∂θ^{2}} is similar. In the obtained series we have 0 ≤ m ≤ n. The sum of the series with cot θ \frac{∂^{2}}{∂θ∂ϕ} and (− cot θ \frac{∂^{2}}{∂θ∂ϕ}) is equal to zero. Further, in the series with

cot^{2} θ \frac{∂^{2}}{∂ϕ^{2}} , where 1 ≤ m ≤ n, we add λ_{n0}a_{n0}r^{n}P_{n}(cos θ) which does not depend on ϕ. This will not affect the result, since the derivative has been taken with respect to ϕ. Now m will vary from zero to n inclusive.

Summarizing the above arguments, we obtain equality (1.1) with the functions U_{f}^{∗}(r, θ, ϕ) and Ũ_{f}^{∗}(r, θ, ϕ) from equalities (0.7) and (0.8). It should be noted that formula (1.1) contains no mixed derivative \frac{∂^{2}}{∂θ∂ϕ} whose presence would make the situation much more difficult on the whole.

1.2. By the method of proving Theorem 1.1 we can obtain a formula for the radial derivative \frac{∂}{∂r} P_{r} of the Poisson kernel P_{r}.

Theorem 1.2. The equalities

2r \frac{∂}{∂r} P_{r} + P_{r} − 1 =

\frac{∂^{2}}{∂θ^{2}} P_{r}^{∗} + cot θ \frac{∂}{∂θ} P_{r}^{∗} + cot^{2} θ \frac{∂^{2}}{∂ϕ^{2}} P_{r}^{∗} − \frac{∂}{∂ϕ} P̃_{r}^{∗},  (1.23)
2r \frac{∂}{∂r} P_{r} + P_{r} − 1 = T_{θ,ϕ}P_{r}^{∗} − \frac{∂}{∂ϕ} P̃_{r}^{∗}  (1.24)

hold, where the allied harmonic kernels P_{r}^{∗} and P̃_{r}^{∗} in the ball 𝔹 are defined by equalities (0.10) and (0.11), and the operator T_{θ,ϕ} by equality (1.2).

Proof. If we itroduce the values A_{00} = 1, A_{n0} = (2n + 1)P_{n}(cos θ′), A_{nm} = 2(2n+1)\frac{(n−m)!}{(n+m)!}P_{nm}(cos θ′) cos mϕ′, B_{nm} =2(2n+1)\frac{(n−m)!}{(n+m)!}P_{nm}(cos θ′) sin mϕ′, and use the equality cos m(ϕ − ϕ′) = cos mϕ cos mϕ′ + sin mϕ sin mϕ′, then series (0.1) will take the form of series (0.2) with the coefficients we have just introduced. From the resulting series we can obtain equality (1.23).

§ 2. Boundary values of the radial derivative \frac{∂}{∂r} U_{f}

The formula obtained in §1 for \frac{∂}{∂r} U_{f} makes it possible to find the boundary values of \frac{∂}{∂r} U_{f} with the aid of the author’s previous results. By equality (1.1) it is clear that the fact that \frac{∂}{∂r} U_{f} has an angular limit (limit) imposes restrictions of differential nature on the functions f^{∗} and f̃^{∗}.

2.1. The corresponding result for f ∈ L^{2}(R) consists in the following statement:

Theorem 2.1. Let the function f ∈ L^{2}(R). If the function f is continuous at the point (θ_{0}, ϕ_{0}) with 0 < θ_{0} < π, the function f̃^{∗} has a continuous partial derivative at (θ_{0}, ϕ_{0}), and the function f^{∗} is twice differentiable^{2} (twice continuously differentiable) at the point (θ_{0}, ϕ_{0}), then the radial derivative \frac{∂}{∂r} U_{f}(r, θ, ϕ) has an angular limit (has a limit) at the point (1, θ_{0}, ϕ_{0}) which is equal to the value of

\frac{1}{2} [a_{00} − f(θ_{0}, ϕ_{0}) + \frac{∂^{2}f^{∗}}{∂θ^{2}} (θ_{0}, ϕ_{0}) + cot θ_{0} \frac{∂f^{∗}}{∂θ} (θ_{0}, ϕ_{0}) +

cot^{2} θ_{0} \frac{∂^{2}f^{∗}}{∂ϕ^{2}} (θ_{0}, ϕ_{0}) − \frac{∂f̃^{∗}}{∂ϕ} (θ_{0}, ϕ_{0})]  (2.1)

or, which is the same,

\frac{1}{2} [a_{00} − f(θ_{0}, ϕ_{0}) + T_{θ,ϕ}f^{∗}(θ_{0}, ϕ_{0}) − \frac{∂f̃^{∗}}{∂ϕ} (θ_{0}, ϕ_{0})].  (2.2)

2The function λ(x, y) is called twice differentiable at the point (x_{0}, y_{0}) if in a neighborhood of (x_{0}, y_{0}) there exist partial derivatives λ′_{x}(x, y), λ′_{y}(x, y) and they are differentiable at (x_{0}, y_{0}). Moreover, if λ′_{x}(x, y) and λ′_{y}(x, y) are continuously differentiable at the point (x_{0}, y_{0}), then the function λ(x, y) is called twice continuously differentiable at (x_{0}, y_{0}).

Proof. The fact that the function f is continuous at the point (θ_{0}, ϕ_{0}) implies that by the well known theorem (see, for instance, [9]. p. 209; [10], p. 243) the Poisson integral U_{f}(r, θ, ϕ) has a limit at (θ_{0}, ϕ_{0}) (though instead of the continuity it is sufficient that f have a finite limit at (θ_{0}, ϕ_{0})). Since f ∈ L^{2}(R), the functions U_{f}^{∗} and Ũ_{f}^{∗} in equality (1.1) can be replaced by U_{f^{∗}} and U_{f^{∗}}, respectively, in accordance with equalities (0.12). The continuity

of the partial derivative \frac{∂}{∂ϕ} f̃^{∗} at the point (θ_{0}, ϕ_{0}) implies that \frac{∂}{∂ϕ} U_{f^{∗}} (see [11], Theorem 8.2) has a limit at the point (1, θ_{0}, ϕ_{0}) which is equal to the

value of \frac{∂_{f^{∗}}}{∂ϕ} (θ_{0}, ϕ_{0}). Further, since the function f^{∗} is twice differentiable (twice continuously differentiable) at the point (θ_{0}, ϕ_{0}), all second order derivatives of U_{f^{∗}} have angular limits at the point (1, θ_{0}, ϕ_{0}) (limits) which are equal to the values of the corresponding derivatives of f^{∗} at (θ_{0}, ϕ_{0}) (see [12], Theorems 3.6 and 5.5).

§ 3. boundary values of derivatives with respect to cartesian derivatives of the Poisson integral

Applying the well known rules, derivatives with respect to the Cartesian coordinates of U_{F} (x, , y, z) can be expressed by derivatives with respect to spherical coordinates (for −1 < z < 1, 0 < θ < π, 0 < r < 1) as follows (see, for instance, [13], p.107):

\frac{∂}{∂x} U_{F} = sin θ cos ϕ \frac{∂}{∂r} U_{f} + \frac{1}{r} cos θ cos ϕ \frac{∂}{∂θ} U_{f} −
− \frac{1}{r} \frac{sin ϕ}{sin θ} \frac{∂}{∂ϕ} U_{f},  (3.1)
\frac{∂}{∂y} U_{F} = sin θ sin ϕ \frac{∂}{∂r} U_{f} + \frac{1}{r} cos θ sin ϕ \frac{∂}{∂θ} U_{f} +
+ \frac{1}{r} \frac{cos ϕ}{sin θ} \frac{∂}{∂ϕ} U_{f},  (3.2)
\frac{∂}{∂z} U_{F} = cos θ \frac{∂}{∂r} U_{f} − \frac{1}{r} sin θ \frac{∂}{∂θ} U_{f}.  (3.3)

The derivatives figuring in the right-hand sides of equalities (3.1)–(3.3) have angular limits (limits) at the point (1, θ_{0}, ϕ_{0}) with 0 < θ_{0} < π, if f(θ, ϕ) is differentiable (continuously differentiable) at the point (θ_{0}, ϕ_{0}) (see [11], Theorems 6.1 and 8.3). On the other hand, for \frac{∂}{∂r} U_{f} to have an angular limit (limit), we had to impose certain restrictions on f^{∗} and f̃^{∗}. Hence the following statements are valid.

Theorem 3.1. Let the function f ∈ L^{2}(R) be differentiable (continuously differentiable) at the point (θ_{0}, ϕ_{0}) with 0 < θ_{0} < π. If the function f̃^{∗} has a partial derivative \frac{∂}{∂ϕ} f̃^{∗} continuous at (θ_{0}, ϕ_{0}) and the function f^{∗} is twice differentiable (twice continuously differentiable) at the point (θ_{0}, ϕ_{0}), then the derivatives of (3.1)–(3.3) have angular limits (have limits) at (1, θ_{0}, ϕ_{0}) which are calculated by equalities (3.1)–(3.3), in whose righthand sides (θ, ϕ) is replaced by (θ_{0}, ϕ_{0}), \frac{∂}{∂θ} U_{f} and \frac{∂}{∂ϕ} U_{f} by \frac{∂f}{∂θ} (θ_{0}, ϕ_{0}) and \frac{∂f}{∂ϕ} (θ_{0}, ϕ_{0}), respectively, and, finally \frac{∂}{∂r} U_{f} is replaced by value (2.1) or (2.2).

Theorem 3.2. Let for the function f ∈ L(R) there exist summable allied functions f^{∗} and f̃^{∗} on R such that U_{f}^{∗} = U_{f^{∗}} and Ũ_{f}^{∗} = U_{f^{∗}}. If the function f is differentiable (continuously differentiable) at the point (θ_{0}, ϕ_{0}) with 0 < θ_{0} < π, the function f̃^{∗} has a partial derivative \frac{∂}{∂ϕ} f̃^{∗} continuous at (θ_{0}, ϕ_{0}) and the function f^{∗} is twice differentiable (twice continuously differentiable) at the point (θ_{0}, ϕ_{0}), then the derivatives of (3.1)–(3.3) have angular limits (have limits) at the point (1, θ_{0}, ϕ_{0}) which are calculated by equalities (3.1)–(3.3) in whose right-hand sides (θ, ϕ) is replaced by (θ_{0}, ϕ_{0}), \frac{∂}{∂θ} U_{f} and \frac{∂}{∂ϕ} U_{f} by \frac{∂f}{∂θ} (θ_{0}, ϕ_{0}) and \frac{∂f}{∂ϕ} (θ_{0}, ϕ_{0}) respectively and, finally \frac{∂}{∂r} U_{f} is replaced by value (2.1) or (2.2).

Remark 3.1. Since the spherical Poisson integral is a harmonic function in the open unit ball, it satisfies the Laplace equation

2r \frac{∂U_{f}}{∂r} + r^{2} \frac{∂^{2}U_{f}}{∂r^{2}} + \frac{∂^{2}U_{f}}{∂θ^{2}} + cot θ \frac{∂U_{f}}{∂θ} + \frac{1}{sin^{2} θ} \frac{∂^{2}U_{f}}{∂ϕ^{2}} = 0. (3.4)

If the first term here is replaced by its value from formula (1.3), then for r^{2} \frac{∂^{2}U_{f}}{∂r^{2}} we shall have

r^{2} \frac{∂^{2}U_{f}}{∂r^{2}} = −a_{00} + U_{f} − T_{θ,ϕ}U_{f}^{∗} +
+ \frac{∂}{∂ϕ} Ũ_{f}^{∗} − \frac{∂^{2}U_{f}}{∂θ^{2}} − cot θ \frac{∂U_{f}}{∂θ} − \frac{1}{sin^{2} θ} \frac{∂^{2}U_{f}}{∂ϕ^{2}} .  (3.5)

To find the boundary values for \frac{∂^{2}U_{f}}{∂r^{2}} it is sufficient to use in an appropriate manner the same facts as we used in finding the boundary values for \frac{∂U_{f}}{∂r}.

In a similar manner we obtain the formula

r^{2} \frac{∂^{2}P_{r}}{∂r^{2}} = − 1 + P_{r} − T_{θ,ϕ}P_{r}^{∗} + \frac{∂P̃_{r}^{∗}}{∂ϕ} − \frac{∂^{2}P_{r}}{∂θ^{2}} −

− cot θ \frac{∂P_{r}}{∂θ} − \frac{1}{sin^{2} θ} \frac{∂^{2}P_{r}}{∂ϕ^{2}} .  (3.6)

References

1. O. Dzagnidze, Associated integrals, functions, series and a radial derivative of the spherical Poisson integral. Proc. A. Razmadze Math. Inst. 114(1997), 107–111.

2. E. W. Hobson, The theory of spherical and ellipsoidal harmonics. Cambridge University Press, 1931.

3. A. Zygmund, Trigonometric series, v. I. Cambridge Univ. Press, 1959.

4. N. K. Bari, Trigonometric series. (Russian) Fizmatgiz, Moscow, 1961.

5. R. Courant und D. Hilbert, Methoden der Mathematischen Physik, Bd. I. Verlag von Julius Springer, Berlin, 1931.

6. O. P. Dzagnidze, Allied integrals, functions and series for the unit sphere. Georgian Math. J. 5(1998), No. 3, 213–232.

7. N. N. Lebedev, Special functions and their applications. (Russian) Fizmatgiz, Moscow–Leningrad, 1963.

8. L. Robin, Fonctions sphériques de Legendre et fonctions sphéroïdales, t. II. Gauthier–Villars, Paris, 1958.

9. V. I. Levin and Yu. I. Grosberg, Differential equations of mathematical physics. (Russian) Gostekhizdat, Moscow–Leningrad, 1951.

10. R. Courant und D. Hilbert, Methoden der Mathematischen Physik, Bd. II. Verlag von Julius Springer, Berlin, 1937.

11. O. P. Dzagnidze, Boundary values of derivatives of the Poisson integral for a ball and the representation of functions of two variables. (Russian) Trudy Tbiliss. Mat. Inst. 98(1990), 52–98.

12. O. P. Dzagnidze, Boundary properties of second order derivatives of the spherical Poisson integral. (Russian) Proc. A. Razmadze Math. Inst. 102(1993), 9–27.

13. D. Jackson, Fourier series and orthogonal polynomials. The Carus Mathematical Monographs, No. 6, Published by the Mathematical Association of America, Oberlin, Ohio, 1941.

(Received 18.12.1995)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia