BOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1. (k > 1)

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1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, BOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1 (k > 1) S. TOPURIA Abstract. Boundary properties of first-order partial derivatives of the Poisson integral are studied in the half-space R + (k > 1). k+1 The boundary properties of the Poisson integral for a circle were thoroughly studied by Fatou [1]. In particular, he showed that the following theorems are valid: Theorem A. If there exists a finite f (x ), then u(f; r, x) f (x ), re ix e x ix where u(f; r, x) is the Poisson integral for a circle, and the symbol re ix e ix means that the point re ix tends to e ix along the paths which are nontangential to the circumference (see [], p. 1, and [3], p. 156). Theorem B. If there exists a finite or infinite D 1 f(x ) which is a first symmetric derivative of f at the point x (see [], p. 99 1), i.e., then f(x + h) f(x h) D 1 f(x ), h h u(f; r, x ) D 1 f(x ). r 1 x 1991 Mathematics Subject Classification. 31B5. Key words and phrases. The Poisson integral for the half-space, angular it, derivatives of the integral, differential properties of the density X/97/11-585$1.5/ c 1998 Plenum Publishing Corporation

2 586 S. TOPURIA In [4] a continuous π-periodic function f(x) is constructed such that D 1 f(x ), but u(f; r, x) re ix e x ix does not exist. Thus it is shown that Theorem B cannot be strengthened in the sense of the existence of an angular it. An analogue of Theorem A for a half-plane R + is proved in [5, Theorem 4], while an analogue of Theorem B given in [6, Theorem 1] shows that this theorem cannot be strengthened in the sense of the existence of an angular it. The question as to the validity of Fatou s theorem for a bicylinder was considered in [7], where it is proved that in the neighborhood of some point the density of the Poisson integral can have no smoothness that would ensure the existence of a boundary value of partial derivatives of the Poisson integral at the considered point. Furthermore, in this paper sufficient conditions are found for the convergence of first- and second- order partial derivatives of the Poisson integral for a bicylinder, and it is shown that the results obtained cannot be strengthened (in the definite sense). The boundary properties of the integral D k u(f; r, ϑ 1, ϑ,..., ϑ k, ϕ) were studied in [8] (see also [9], p. 118), where u(f; r, ϑ 1, ϑ,..., ϑ k ϕ) is the Poisson integral for the unit sphere in (k > ), and D k is the Laplace operator on the sphere, i.e., the angular part of the Laplace operator written in terms of spherical coordinates (see [9], p. 14). The boundary properties of first- and second- order partial derivatives of the Poisson integral for the unit sphere in R 3 are given a detailed consideration in [1, 11, 1], but for the half-space R 3 + in [13], [14], [15]. In [14] it is shown that there exists a continuous function of two variables f(x, y) L(R ) which, at the point (x, y ), has the partial derivatives f x(x, y ) and f y(x, y ), but the integrals u(f;x,y,z) x and u(f;x,y,z) y (u(f; x, y, z) is the Poisson integral for R 3 +) of this function have no values at the point (x, y ) even along the normal. Hence the question arises how to generalize the notion of derivatives of a function of many variables so that a Fatou type theorem would hold for the integral u(f; x, x k+1 ) (u(f; x, x k+1 ) is the Poisson integral for R+ k+1 (k > 1)). In this paper, the notion of a generalized partial derivative is introduced for a function of many variables and Fatou type theorems are proved on boundary properties of first-order partial derivatives of the Poisson integral for a half-space. These results complement and generalize the author s studies in [13], [14], [15]. In particular, in this paper it is shown that the boundary properties of derivatives of the Poisson integral for a half-space essentially depend on the sense in which the integral density is differentiable. Examples are constructed testifying to the fact that the results obtained are unimprovable (in the definite sense).

3 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL Notation, Definitions, and Auxiliary Propositions The following notation is used in this paper: is a k-dimensional Euclidean space (R R 1 ); x (x 1, x,..., x k ), t (, t,..., t k ), x (x 1, x,..., x k ) are the points (vectors) of the space ; (x, t) k x i t i is the scalar product; i1 x (x, x); x + t (x 1 +, x + t,..., x k + t k ); e i (i 1,,..., k) is the coordinate vector. Let (see [16], p. 174) M {1,,..., k} (k N, k ), B be an arbitrary subset from M and B M \ B. For any x and an arbitrary set B M, the symbol x B denotes a point from whose coordinates with indices from the set B coincide with the corresponding coordinates of the point x, while coordinates with indices from the set B are zeros (x M x, B \ i B \ {i}); if B {i 1, i,..., i s }, 1 s k (i l < i r for l < r), then x B (x i1, x i,..., x is ) R s ; m(b) is the number of elements of the set B; L( ) is the set of functions f(x) f(x 1, x,..., x k ) such that f(x) (1 + x ) k+1 L( ); +1 + {(x, x k+1 ) +1 ; x k+1 > }; u(f; x, x k+1 ) is the Poisson integral of the function f(x) for the half-space +1 +, i.e., u(f; x, x k+1 ) x k+1γ( k+1 ) π k+1 f(t) dt ( t x + x k+1 ) k+1 In investigating the boundary properties of the partial derivatives ϑ u f (r, ϑ, ϕ) and ϕ u f (r, ϑ, ϕ) of the spherical Poisson integral u f (r, ϑ, ϕ) for the summable function f(ϑ, ϕ) on the rectangle [, π] [, π], Dzagnidze introduced the notion of a dihedral-angular it [1] which is applicable to +1 + in the manner as follows: if the point N R+ k+1 converges to the point P(x, ) under the condition x k+1 ( (x i x i ) ) 1/ C >, 1 then we shall write N(x, x k+1 ) P(x, ). When B M, we have an angular xb convergence and thus we write N(x, x k+1 ) P(x, ). Finally, the notation N(x, x k+1 ) P(x, ) means that the point N(x, x k+1 ) remaining in +1 + converges to P(x, ) without any restrictions. 1 Here and further C denotes absolute positive constants which, generally speaking, may be different in different relations. i B.

4 588 S. TOPURIA It is known that ϑ u f (r, ϑ, ϕ) and ϕ u f (r, ϑ, ϕ) have dihedral-angular its if partial derivatives of the function f(ϑ, ϕ) exist in a strong sense [1], [1]. This notion admits various generalizations when the function depends on three and more variables and we shall also discuss them below. Let u R. We shall consider the following derivatives of the function f(x): 1. Denote the it f(x B + x B + ue i ) f(x B + x ) B (u,x B ) (,x ) u B by: (a) f x i (x ) for B ; (b) D xi (x B )f(x ) for i B ; (c) D xi(x B )f(x ) for i B.. Denote the it f(x B + x B + ue i ) f(x B + x ue B i) (u,x B ) (,x B ) u by: (a) D x i (x ) for B ; (b) D x i(x B ) f(x ) for i B ; (c) D xi(x B )f(x ) for i B. The following propositions are valid: (1) If B B 1, the existence of D xi (x B1 )f(x ) implies the existence of D xi(x B )f(x ) and D xi (x B1 )f(x ) f x i (x ). The converse does not hold. () The existence of D xi (x B1 )f(x ) implies the existence of D xi (x B )f(x ) and their equality. (3) The existence of D xi (x B )f(x ) implies the existence of D xi (x B1 )f(x ) and D xi(x B )f(x ) D xi(x B\i )f(x ) f x i (x ). (4) If f x i (x) is a continuous function at x, then for any B M all derivatives D xi (x B )f(x ) exist and D xi(x B )f(x ) f x i (x ). Indeed, by virtue of the Lagrange theorem f(x B + x + ue B i) f(x B + x ) B f x i [x B + x + θ(x)ue B i]u u u f x i [x B + x + θ(x)ue B i], < θ < 1. Hence we conclude that statement (4) is valid. (5) There exists a function f(x) for which D xi(x)f(x ) exist, but on an everywhere dense set in the neighborhood of the point x there are no f x i (x).

5 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL 589 (6) If the function f(x) has finite derivatives D x1 (x,...,x k )f(x ), D x(x 3,...,x k )f(x ),..., D xk 1 (x k )f(x ) at the point x, then its continuity at x with respect to the argument x k is a necessary and sufficient condition for f(x) to be continuous at x (see [1], p.15). (7) The existence of the derivatives D x1(x,...,x k )f(x ), D x (x 3,...,x k )f(x ),..., D x1 (x,...,x k )f(x ) and f x k (x ) implies the existence of the differential df(x ) (see [1], p. 16). In what follows it will be assumed that f L( ). Lemma. The equalities (t i x i )f(t t i e i + x i e i )dt I 1 ( t x + xk+1 ), k+3 hold for any (x, x k+1 ). Proof. We have I 1 I (k + 1)x k+1γ( k+1 ) π k+1 1 f(x + t t i e i ) ds(t M\i ) (t i x i ) dt ( t x + x k+1 ) k+3 t i dt i ( t + xk+1 ) k+3 1. Further, if we use the spherical coordinates, ρ, θ 1,..., θ k, ϕ, we shall have I (k + 1)x k+1γ( k+1 ) t i dt i π k+1 ( t + xk+1 ) k+3 ) (k+1)x k+1γ( k+1 π k+1 ρ sin ϑ 1 sin ϑ sin ϑ i 1 cos ϑ i (ρ + x k+1 ) k+3 ρ k 1 sin k ϑ 1 sin k i ϑ i 1 sin ϑ k dρdϑ 1 dϑ k dϕ k+1 (k+1)γ( π k 1 ) ρ k+1 dρ (1 + ρ ) k+3 k {}}{ π π sin k ϑ 1 sin k 1 ϑ sin k i+ ϑ i 1 cos ϑ i sin k i 1 ϑ i sin ϑ k dϑ 1 dϑ dϑ k k+1 (k + 1)Γ( ) π k 1 kγ( k ) π (k + 1)Γ( k+1 ) π k kγ( k ) 1.

6 59 S. TOPURIA. Boundary Properties of the Integral u(f;x,x k+1) The following theorem is valid. Theorem 1. (a) If a finite derivative D xi (x)f(x ) exists at the point x, then f(x ). (1) (x,x k+1 ) (x,) (b) There is a continuous function f L( ) such that for any B M, m(b) < k, at the point x (,,..., ) all derivatives D xi (x B )f(), i 1, k, but the its do not exist. Proof. (a) Let x, C k u(f;, x k+1 ), i 1, k, x k+1 + k+1 (k+1)γ( ) π k+1 C k x k+1 By virtue of the lemma we have. It is easy to check that (t i x i )f(t) dt ( t x + x k+1 ) k+3 D xi (x)f() t [ i f(x + t) f(x + t ti e i ) ] C k x k+1 ( t + x k+1 ) D k+3 xi (x)f() dt t i where I 1 C k x k+1 I 1 + I,, I C k x k+1 V δ V δ is the ball with center at and radius δ. Let ε >. Choose δ > such that f(x + t) f(x + t t ie i ) D xi (x)f() < ε for x < δ, t < dl. Hence I 1 < C k x k+1 ε t i V δ t i dt ( t + x k+1 ) k+3 CV δ, <.

7 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL 591 < C k x k+1 ε It is likewise easy to show that t i dt ( t + x k+1 ) k+3 ε. () I. (3) (x,x k+1 ) (x,) Equalities () and (3) imply the validity of (1). (b) Let D ( < ; t < ;..., t k < ). Define the function f as follows: f(t) { k+1 t t k if (, t,..., t k ) D, if (, t,..., t k ) CD. Clearly, f(t) is continuous in and D xi(x B )f(), i 1, k, for any B when m(b) < k. If in the integral C k x k+1 (t i x i )f(t) dt ( t x + x k+1 ) k+3 we use spherical coordinates, then for the considered function we shall have u(f;, x k+1 ) t i f(t) dt C k x k+1 ( t + x k+1 ) k+3 Hence Cx k+1 Cx k+1 ρ k+ k k+1 dρ (ρ + x k+1 ) k+3 > Cx k+1 ρ k 1 ρ k (ρ + x k+1 ) k+3 x k+1 > Cx k+1 x k+1 ρ k+ k k+1 dρ x k+3 k+1 u(f;, x k+1 ) +. x k+1 + x ρ k 1 dρ ρ k+ k k+1 dρ (ρ + x k+1 ) k+3 C k+1 x k+1. ρ k 1 dρ > Corollary 1. If finite derivatives D xi(x)f(x ), i 1, k, exist at the point x, then (x,x k+1 ) (x,) d xu(f; x, x k+1 ) df(x ).

8 59 S. TOPURIA Corollary. If f has a continuous partial derivative at the point x, then f (x,x k+1 ) (x x,) (x ). i Corollary 3. (a) If f is a continuously differentiable function at the point x, then (x,x k+1 ) (x,) d xu(f; x, x k+1 ) df(x ). (b) There exists a differentiable function f(, t ) at the point (, ) such that df(, ) but the its u(f; x 1, x, x 3 ) u(f; x 1, x, x 3 ), (x 1,x,x 3 ) (,,) x 1 (x 1,x,x 3 ) (,,) x do not exist. Proof of assertion (b) of Corollary 3. We set D [, 1;, 1]. Let { 5 t 3 f(, t ) 1 t 3 for (, t ) D, for (, t ) [, ;, [ ], ;, [, and continue f onto the set ], ;, [\D so that f L(R ). It is easy to check that f(, t ) is differentiable at the point (, ) and f (, ) f t (, ). Let (x 1, x, x 3 ) (,, ) for x 1, x 3 x, x >. Then for the considered function u(f;, x, x 3 ) 3x 3 x 1 π 3x π 3x π > 3x π x x x x x x f(, t ) d dt [t 1 + (t x ) + x 4 ]5/ f(, t + x ) d dt (t 1 + t + x4 )5/ 5 t 3 1 (t + x ) 3 d dt (t 1 + t + x4 )5/ > 5 t x 3 ddt (t 1 + t + x4 )5/ >

9 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL 593 > 3x x 3/5 π x x x x x x 6/5 d dt (4x) 4 + 4x 4 + x4 )5/ 1 16π 5 x for x +. Theorem. If a finite derivative D xi(x M\i )f(x ) exists at the point x, then f(x ). (x,) x i Proof. Let x. By virtue of the lemma we have the equality C k x k+1 D xi (x x M\i )f() i t i (t i x i ) [ f(t) f(t ti e i ) ( t x + x k+1 ) k+3 t i ] D xi(x M\i )f() dt I 1 + I, where I 1 C k x k+1, I C k x k+1. V δ CV δ Let ε > and choose δ > such that f(t) f(t t ie i ) D xi(x t M\i )f() < ε i for r < δ. Now, I 1 < C k x k+1 ε < C k x k+1 ε + C k x k+1 ε x i t i (t i x i ) dt ( t x + x k+1 ) k+3 t i dt ( t + x k+1 ) k+3 t i dt ε + Cx k+1 ε x i ε + Cxk+ k+1 x i ε x k+3 k+1 + ( t + xk+1 ) k+3 ρ k dρ (ρ + x k+1 ) k+3 ρ k dρ (1 + ρ ) k+3 < ( 1 + C x i x k+1 ) ε.

10 594 S. TOPURIA Hence we obtain I 1. (x,) x i In a similar manner we prove that I. (x,) x i Theorem 3. If at the point x there exist finite derivatives D xi(x M\i )f(x ) and D xj(x B )f(x ), i j, B M \ {i, j}, then f(x ), (x,) x i f(x ). (x,) x j x j x i Proof. Let x. By virtue of the lemma we have C k x k+1 x j (t j x j )f(t) dt ( t x + x k+1 ) k+3 (t j x j ){[f(t) f(t t i e i )]+[f(t t i e i ) f(t t i e i t j e j )]}dt C k x k+1 ( t x + x k+1 ) k+3 (t j x i )[f(t) f(t t i e i )] C k x k+1 ( t x + x k+1 ) dt + k+3 (t j x i )[f(t t i e i ) f(t t i e i t j e j )] +C k x k+1 ( t x + xk+1 ) dt I k I, where I 1 C k x k+1 C k x k+1 t i (t j x j ) ( t x + x k+1 ) k+3 t i (t j x j ) + C k x k+1 D xi (x M\i )f() ( t x +x k+1 ) k+3 t i (t j x j ) dt f(t) f(t t ie i ) t i dt [ f(t) f(t ti e i ) t i ( t x + x k+1 ) k+3 ] D xi (x M\i )f() dt+ I 1 + I 1.

11 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL 595 It is easy to see that I 1 and I 1 <C t i (t j x j ) k x k+1 Hence we obtain Now we shall show that Indeed, ( t x + xk+1 ) k+3 f(t) f(t t ie i ) I 1 I 1. x i x i I D xj (x M\{i,j} )f(). x j I C k x k+1 t j t i t j (t j x j ) ( t x + x k+1 ) k+3 D xi (x M\i )f() dt. [ f(t ti e i ) f(t t i e i t j e j ) ] D xj (x M\{i,j} )f() dt+d xj (x M\{i,j} )f(). This readily implies Finally, we obtain I D xj(x M\{i,j} )f() f(). x j x j f(). x j x j x i x j By a similar reasoning we prove Theorem 4. If at the point x there exist finite derivatives D x1 (x,x 3,...,x k )f(x ), D x(x 3,...,x k )f(x ),..., D xk 1 (x k )f(x ), f x k (x ), then f(x ), (x,) x 1 x 1 x 1 (x,) x 1 x x f(x ) x,...

12 596 S. TOPURIA f(x ). (x,) x k x k Corollary. If at the point x there exist finite derivatives D x1(x M\1 )f(x ), D x (x M\{1,} )f(x ),..., D xk 1 (x k )f(x ), f x k (x ), then d x u(f; x, x k+1 ) df(x ). (x,) Theorem 5. (a) If at the point x there exists a finite derivative D x i(x M\i ) f(x ), then u(f; x x i e i + x i e i, x k+1 ) D (x x ie i +x i ei,x k+1 ) (x x,) f(x ). i (b) There exists a continuous function f(x) such that Dx i(x M\i ) f(x ), but the it (x,) does not exist. Proof. (a) Let x. The validity of (a) follows from the equality u(f; x x i e i + x i e i, x k+1 ) Dx (x M\i ) f(x ) i t [ i f(t + x xi e i ) f(t + x x i e i t i e i ) C k x k+1 ( t + x k+1 ) k+3 t i ] D xi(x M\i )f(x ) dt. (b) We set D 1 [, 1;, 1], D p 1, ;, 1]. Let t1 t for (, t ) D 1, f(, t ) t1 t for (, t ) D, for t and continue f(, t ) onto the set R + \ (D 1 D ) so that f L(R ). It is easy to check that Dt 1 (t ) f(). Let x 1 x and (x 1, x, x 3 ) (,, ) so that x and x 3 x 1. Then for the constructed function we have u(f; x 1, x, x 3 ) 3x 3 x 1 π R ( x 1 )f(, t ) d dt [( x 1 ) + (t x ) + x 3 ]5/

13 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL 597 { Cx [ 1+x1 Cx 1 + x 1 1 x 1 1 x 1 ( x 1 ) t d dt [( x 1 ) + t + x 3 ]5/ + ( x 1 ) t d dt [( x 1 ) + t + x 3 ]5/ { x1 Cx } + o(1) (t1 x 1 ) t (t 1 + t + x 1 )5/ ddt + (t1 + x 1 ) t (t 1 + t + x 1 )5/ ddt 1 x 1 1 x 1 1 x 1 ] + o(1) (t1 + x 1 ) t (t 1 + t + x 1 )5/ ddt + [ ( + x 1 ) t ( x 1 ) t ] (t 1 + t + x 1 )5/ d dt Cx 1 (I 1 + I ) + o(1), } where I 1 I > > x x 1 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 [ ( + x 1 ) t (x 1 ) t ] (t 1 + t + x 1 )5/ d dt >, 4 t ( + x 1 x 1 ) (t 1 + t + x 1 )5/ d dt > 4 t ( + x x 1 ) (t 1 + t + x 1 )5/ d dt > x 4 1 x 1 ( x 1 x 1 ) 1 (9x d dt 1 )5/ 18 Thus, by the chosen path, we obtain u(f; x 1,, x 1 ) x 1 > C 4 x1, 1. x 5 1 4

14 598 S. TOPURIA which yields u(f;x1,,x1) x 1 + when (x 1, x, x 3 ) (,, ) by the chosen path. By a similar reasoning we prove Theorem 6. (a) If at the point x there exists a finite derivative D x i (x) f(x ), i 1, k, then D (x,x k+1 ) (x x,) f(x ). i (b) There exists a continuous function f(x) such that for any B M, m(b) < k, all derivatives D x i (x B ) f(), i 1, k, but the its do not exist. u(f;, x k+1 ) x k Statement (a) of Theorem 1 is a corollary of statement (a) of Theorem The validity of (b) follows from statement (b) of Theorem 1. Theorem 7. (a) If f has a total differential df(x ) at the point x, then d x u(f; x, x k+1 ) df(x ). (4) (x,) (b) there exists a continuous function f which has partial derivatives of any order, but the its do not exist. u(f; x, x k+1 ) x k+1 + Proof. (a) By virtue of the lemma we have (x ) C k x k+1 (t i x i ) k t ν ν1 ( t x + x k+1 ) k+3 f() f(t) f() k k t ν ν1 ν1 f() t i dt.

15 PARTIAL DERIVATIVES OF THE POISSON INTEGRAL 599 This equality implies f(), i 1, k. Thus equality (4) is valid. (b) Consider the function 4 ( t )(t 1 ) for (, t ) D {(, t ) : f(, t ) 1 < ; t }, for (, t ) CD. This function is continuous in R, has partial derivatives of any order at the point (, ) which are equal to zero, but u(f;,, x 3 ) 3x 4 t 3 1 ( t )(t 1 dt ) 1 x 1 π (t 1 + t + dt > x 3 )5/ > Cx 3 > Cx 3 > Cx 3 x 3 1 x 3 d x 3 x 3 d x 3 x 3 d ( t )(t 1 ) (t 1 + t + x 3 )5/ dt > ( 3 )( 1 ) ( x 3 )5/ dt > t 1 dt x 5 C 3 x 4 3 C x3 + for x 3 +. x 3 t 5/ 1 d x 3 References 1. P. Fatou, Séries trigonoméetriques et séries de Taylor. Acta Math. 3(196), A. Zygmund, Trigonometric series, I. Cambridge University Press, N. K. Bari, Trigonometric series. (Russian) Fizmatgiz, S. B. Topuria, Boundary properties of the differentiated Poisson integral in a circle. (Russian) Trudy Gruz. Politekhn. Inst. 3(176)(1975), 13.

16 6 S. TOPURIA 5. A. G. Jvarsheishvili, On analytic functions in a half-plane. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 31(1966), S. B. Topuria, Boundary properties of the differentiated Poisson integral in a half-plane. (Russian) Soobshch. Akad. Nauk Gruz. SSR 78(1975), No., S. B. Topuria, Summation of the differentiated Fourier series by the Abel method. (Russian) Dokl. Akad. Nauk SSSR 9(1973), No. 3, S. B. Topuria, Summation of the differentiated Fourier Laplace series by the Abel method. (Russian) Trudy Gruz. Polytekhn. Inst. 7(147) (1971), S. B. Topuria, Fourier Laplace series on the sphere. (Russian) Tbilisi University Press, Tbilisi, O. P. Dzagnidze, Boundary properties of derivatives of the Poisson integral for a ball and representation of a function of two variables. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 98(199), O. P. Dzagnidze, Boundary properties of second-order derivatives of the spherical Poisson integral. Proc. A. Razmadze Math. Inst. 1(1993), O. P. Dzagnidze, On the differentiability of functions of two variables and of indefinite double integrals. Proc. A. Razmadze Math. Inst. 16(1993), S. B. Topuria, Solution of the Dirichlet problem for a half-space. (Russian) Dokl. AN SSSR 195(197), No. 3, S. B. Topuria, Boundary properties of the differentiated Poisson integral and solution of the Dirichlet problem in a half-space. (Russian) Trudy Gruz. Polytekhn. Inst. 6(197)(1977), S. B. Topuria, Boundary properties of derivatives of the Poisson integral in a half-space and representation of a function of two variables. (Russian) Dokl. Akad. Nauk 353(1997), No L. V. Zhizhiashvili, Some problems of the theory of trigonometric Fourier series and their conjugates. (Russian) Tbilisi University Press, Tbilisi, (Received ; revised ) Author s address: Department of Mathematics (No. 63) Georgian Technical University 77, M. Kostava St., Tbilisi 3893 Georgia