OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS IN THE MULTIDIMENSIONAL HALF-SPACE. Gershon Kresin. Vladimir Maz ya

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1 Manuscript submitted to AIMS Journals Volume X Number X XX X Website: pp. X XX OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS IN THE MULTIDIMENSIONAL HALF-SPACE Gershon Kresin Department of Computer Science and Mathematics Ariel University Center of Samaria Ariel Israel Vladimir Maz ya Department of Mathematical Sciences University of Liverpool M&O Building Liverpool L69 3BX UK Department of Mathematics Linköping University SE Linköping Sweden Dedicated to Louis Nirenberg on the occasion of his eighty fifth birthday Abstract. A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a multidimensional half-space is obtained under the assumption that function s boundary values belong to L p. This representation is concretized for the cases p 1 and. 1. Introduction. There is a series of sharp estimates for the first derivative of a function f analytic in the upper half-plane C {z C : Iz > } with different characteristics of the real part of f in the majorant part see e.g. [6]). We mention in particular the Lindelöf inequality in the half-plane f z) 1 Iz and two equivalent inequalities f z) πiz {Rfζ) Rfz)} 1.1) ζ C Rfζ) 1.) ζ C and f z) 1 πiz osc C Rf) 1.3) osc C Rf) is the oscillation of Rf on C and z is an arbitrary point in C. Inequalities for analytic functions with certain characteristics of its real part as majorants are called real-part theorems in reference to the first assertion of such a kind the celebrated Hadamard real-part theorem fz) C z 1 z max Rfζ). ζ 1 Mathematics Subject Classification. Primary: 35B3; Secondary: 35J5. Key words and phrases. Real-part theorems multidimensional harmonic functions estimates of the gradient Khavinson s problem. 1

2 G. KRESIN AND V. MAZ YA Here z < 1 and f is an analytic function on the closure D of the unit disk D {z : z < 1} vanishing at z. This inequality was first obtained by Hadamard with C 4 in 189 [4]. The following refinement of Hadamard real-part theorem due to Borel [1 ] Carathéodory [8 9] and Lindelöf [1] fz) f) z 1 z R {fζ) f)} 1.4) ζ <1 and corollaries of the last sharp estimate are often called the Borel-Carathéodory inequalities. Sometimes 1.4) is called Hadamard-Borel-Carathéodory inequality see e.g. Burckel [3]). The collection of real-part theorems and related assertions is rather broad. It involves assertions of various form see e.g. [6] and the bibliography collected there). Obviously the inequalities for the first derivative of an analytic function 1.1)- 1.3) can be restated as estimates for the gradient of a harmonic function. For example inequality 1.) can be written in the form uz) πy uζ) 1.5) ζ R u is a harmonic function in the half-plane R {z x y) R : y > }. In the present work we find a representation for the sharp coefficient C p x) in the inequality ux) C p x) u p 1.6) u is harmonic function in the half-space R n { x x x n ) : x R x n > } represented by the Poisson integral with boundary values in L p R ) p is the norm in L p R ) 1 p x R n. It is shown that C p x) C p x 1 n p)/p n and explicit formulas for C p in 1.6) for p 1 are given. Note that a direct consequence of 1.6) is the following sharp limit relation for the gradient of a harmonic function in the n-dimensional domain Ω with smooth boundary: lim x Ox np 1)/p { ux) : u Ω p 1 } C p x O x O x is a point at Ω nearest to x Ω compare with Theorem in [7] a relation of the same nature for the values of solutions to elliptic systems was obtained). In Section we characterize C p in terms of an extremal problem on the unit hemisphere in R n. In Section 3 we reduce this problem to that of finding of the remum of a certain double integral depending on a scalar parameter and show that C 1 n 1) n 1)Γ n/) nn 1) C π n/ n n 1)Γ ) n n π n/ is the area of the unit sphere in R n. In Section 5 we treat the more difficult case of p. We anticipate the proof of the main result by deriving in Section 4 an algebraic inequality to be used for finding

3 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 3 an explicit formula for C. Solving the variational problem stated in Section 3 we find 4n 1))/ Γ n ) ). In particular C 4n 1))/ 1 n n/ πn n/ Γ C C 3 3 π. for n 3 and n 4 respectively. As a trivial corollary of the inequality ux) 4n 1))/ 1 n n/ x n which is equivalent to 1.6) with p we find uy) 1.7) y R n ux) n 1))/ 1 n n/ x n osc R n u) 1.8) osc R n u) is the oscillation of u on R n. The sharp inequalities 1.7) and 1.8) are multidimensional generalizations of analogues of the real-part theorems 1.) and 1.3) respectively. The sharp constant in the inequality u Kx x ) uy) x y <1 u is a harmonic function in the three-dimensional unit ball B and x B was found by Khavinson [5] who suggested in a private conversation that the same constant Kx ) should appear in the stronger inequality ux ) Kx ) uy). y <1 When dealing in Section 4 with the analogue of Khavinson s problem for the multidimensional half-space we show that in fact the sharp constants in pointwise estimates for the absolute value of the normal derivative and of the modulus of the gradient of a harmonic function coincide. We also show that similar assertions hold for p 1 and p. {. Auxilliary assertion. We introduce some notation used henceforth. Let R n x x x n ) : x x 1... x ) R x n > } S {x R n : x 1} S {x R n : x 1 x n > } and S {x R n : x 1 x n < }. Let e σ stand for the n-dimensional unit vector joining the origin to a point σ on the sphere S. By p we denote the norm in the space L p R ) that is { } 1/p f p fx ) p dx R if 1 p < and f ess { fx ) : x R }. Next by h p R n ) we denote the Hardy space of harmonic functions on R n which can be represented as the Poisson integral ux) R x n y x n uy )dy.1)

4 4 G. KRESIN AND V. MAZ YA with boundary values in L p R ) 1 p y y ) y R. Now we find a representation for the best coefficient C p x; z) in the inequality for the absolute value of derivative of u at x R n in the arbitrary direction z S assuming that L p R ). In particular we obtain a formula for the constant in a similar inequality for the modulus of the gradient. Lemma 1. Let u h p R n ) and let x be an arbitrary point in R n. The sharp coefficient C p x; z) in the inequality ux) z) C p x; z) u p is given by C p x; z) C p z)x 1 n p)/p n.) C 1 z) e n ne σ e n )e σ z ) ) n e σ e n.3) σ S { C p z) e n ne σ e n )e σ z ) } p 1)/p p/p 1) ) n/p 1)dσ eσ e n.4) S for 1 < p < and C z) S e n ne σ e n )e σ z ) dσ..5) In particular the sharp coefficient C p x) in the inequality is given by ux) C p x) u p C p x) C p x 1 n p)/p n.6) C p C p z)..7) z 1 Proof. Let x x x n ) be a fixed point in R n. The representation.1) implies u [ δ ni x i y x n nx ] ny i x i ) y x n uy )dy that is R ux) [ e n R y x n nx ] ny x) y x n uy )dy R e n ne xy e n )e xy y x n uy )dy e xy y x) y x 1. For any z S ux) z) e n ne xy e n )e xy z) R y x n uy )dy..8) Hence C 1 x; z) y R e n ne xy e n )e xy z) y x n.9)

5 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 5 and { C p x; z) e n ne xy e n )e xy z ) q } 1/q R y x nq dy.1) for 1 < p p 1 q 1 1. Taking into account the equality x n y x e xy e n ).11) by.9) we obtain C 1 x; z) ) n e n ne xy e n )e xy z) xn y R x y x x n n e n ne σ e n )e σ z ) ) n. e σ e n σ S Replacing here e σ by e σ we arrive at.) for p 1 with the sharp constant.3). Let 1 < p. Using.11) and the equality 1 y x nq 1 x nq n1 n xn y x ) nq 1) x n y x n and replacing q by p/p 1) in.1) we conclude that.) holds with the sharp constant { C p z) e n ne σ e n )e σ z ) } p 1)/p p/p 1) ) n/p 1) eσ e n dσ S S {σ S : e σ e n ) < }. Replacing here e σ by e σ we arrive at.4) for 1 < p < and at.5) for p. By.8) we have ux) e n ne xy e n )e xy z) z 1 R y x n uy )dy. Hence by the permutation of rema.1).9) and.) { C p x) e n ne xy e n )e xy z ) q for 1 < p and z 1 R y x nq dy } 1/q C p x; z) C p z)x 1 n p)/p n.1) z 1 z 1 C 1 x) z 1 e n ne xy e n )e xy z) y R y x n C 1 x; z) C 1 z)x n n..13) z 1 z 1 Using the notation.7) in.1) and.13) we arrive at.6).

6 6 G. KRESIN AND V. MAZ YA Remark 1. Formula.4) for the coefficient C p z) 1 < p < can be written with the integral over the whole sphere S in R n { C p z) 1/p en ne σ e n )e σ z ) p/p 1) } ) p 1)/p eσ e n n/p 1) dσ. S A similar remark relates.5) as well as formula.3): C 1 z) en ne σ e n )e σ z ) ) n e σ e n. σ S 3. The case 1 p <. The next assertion is based on the representation for C p obtained in Lemma 1. Proposition 1. Let u h p R n ) and let x be an arbitrary point in R n. The sharp coefficient C p x) in the inequality is given by and C p ) p 1)/p if 1 < p <. Here γ ux) C p x) u p 3.1) C p x) C p x 1 n p)/p n 3.) C 1 { 1 π dϕ 1 γ n 1) 3.3) / F np ϕ ϑ; γ) dϑ} p 1)/p 3.4) F np ϕ ϑ; γ) Gn ϕ ϑ; γ) p/p 1) cos n/p 1) ϑ sin n ϑ sin n 3 ϕ 3.5) with G n ϕ ϑ; γ) n cos ϑ 1) nγ cos ϑ sin ϑ cos ϕ. 3.6) In particular nn 1) C n. 3.7) For p 1 and p the coefficient C p x) is optimal also in the weaker inequality obtained from 3.1) by replacing u by u/ x n. Proof. The equality 3.) was proved in Lemma 1. i) Let p 1. Using.3).7) and the permutability of two rema we find e n ne σ e n )e σ z ) ) n e σ e n C 1 z 1 σ S ) en ne σ e n )e σ n eσ e n. 3.8) σ S Taking into account the equality ) 1/ e n ne σ e n )e σ e n ne σ e n )e σ e n ne σ e n )e σ 1 n n)e σ e n ) ) 1/

7 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 7 and using 3.8) we arrive at the sharp constant 3.3). Furthermore by.3) C 1 e n ) 1 neσ e n ) ) n n 1) e σ e n. σ S 1 n Hence by C 1 C 1 e n ) and by 3.3) we obtain C 1 C 1 e n ) which completes the proof in the case p 1. ii) Let 1 < p <. Since the integrand in.4) does not change when z S is replaced by z we may assume that z n e n z) > in.7). Let z z z n e n. Then z e n ) and hence zn z 1. Analogously with σ σ 1... σ σ n ) S we associate the vector σ e σ σ n e n. Using the equalities σ e n ) σ n 1 σ and z e n ) we find an expression for e n ne σ e n )e σ z ) as a function of σ : e n ne σ e n )e σ z ) z n nσ n eσ z ) z n nσ n σ σ n e n z ) z n e n [ z n nσ n σ z ) ] z n σ n [ n1 σ ) 1 ] z n n 1 σ σ z ). 3.9) Let B {x x 1... x ) R : x < 1}. By.4) and 3.9) taking into account that dσ dσ / 1 σ we may write.7) as C p z S { B H np σ σ z ) ) 1 σ ) } n/p 1) p 1)/p dσ 1 σ { H np σ σ z ) ) 1 σ ) } p 1)/p n1 p)/p 1) dσ 3.1) z S B H np σ σ z ) ) [ n1 σ ) 1 ] z n n 1 σ σ z ) p/p 1). 3.11) Let B n {x R n : x < 1}. Using the well known formula see e.g. [11] 3.3.3)) g x a x) ) 1 dx 1 r dr g r a r cos ϕ ) sin n ϕ dϕ B n we obtain H np σ σ z ) ) 1 σ ) n1 p)/p 1) dσ B 1 r n 1 r ) n1 p)/p 1) dr H np r r z ) cos ϕ z n sin n 3 ϕdϕ. 3.1) Making the change of variable r sin ϑ in 3.1) we find B H np σ σ z ) ) 1 σ ) n1 p)/p 1) dσ sin n 3 ϕdϕ / 3.13) H np sin ϑ z sin ϑ cos ϕ ) sin n ϑ cos n/p 1) ϑdϑ

8 8 G. KRESIN AND V. MAZ YA by 3.11) H np sin ϑ z sin ϑ cos ϕ ) n cos ϑ 1)z n n z cos ϑ sin ϑ cos ϕ p/p 1). Introducing here the parameter γ z /z n and using the equality z z n 1 we obtain H np sin ϑ z sin ϑ cos ϕ ) 1 γ ) p/p 1) G n ϕ ϑ; γ) p/p 1) 3.14) G n ϕ ϑ; γ) is given by 3.6). By 3.1) taking into account 3.13) and 3.14) we arrive at 3.4). iii) Let p. By 3.4) 3.5) and 3.6) C γ { 1 π dϕ 1 γ / F n ϕ ϑ; γ) dϑ} 1/ 3.15) F n ϕ ϑ; γ) [ n cos ϑ 1)nγ cos ϑ sin ϑ cos ϕ ] cos n ϑ sin n ϑ sin n 3 ϕ. 3.16) The equalities 3.15) and 3.16) imply C γ 1 { I1 γ } 1/ I 3.17) 1 γ I 1 sin n 3 ϕ dϕ / n cos ϑ 1) sin n ϑ cos n ϑ dϑ π nn 1)Γ n ) Γ n1 ) 3.18) 8n 1)! I n sin n 3 ϕ cos ϕ dϕ By 3.17) we have π nγ n 8n 1)! ) Γ n1 ) / sin n ϑ cos n ϑ dϑ. 3.19) C max { I 1/ 1 I 1/ } which together with 3.18) and 3.19) gives C I 1/ 1 n 1)Γ ) n. n π n/ Hence 3.7) follows. Since z S and the remum in γ z /z n in 3.15) is attained for γ we have C C e n ).

9 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 9 4. An auxilliary algebraic inequality. Here we prove an algebraic inequality to be used later for deriving an explicit formula for the sharp constant in the estimate for the modulus of gradient in the case p. Lemma. For all x and any µ 1 the inequality holds ) µ 1 ) µ 1 µ 1 µ 1 x µ1 µ3µ 1) x µ x 1 µx µ 1) 1 x). 4.1) The equality sign takes place only for µ 1 or x 1. Proof. Clearly the inequality 4.1) becomes equality for µ 1 or x 1. Suppose that µ 1 ). i) The case x < 1. Let us write 4.1) in the form µ 1 µ x ) µ 1 Introducing the notation F x) µx x 1 µx ) µ 1 x x ) µ 1 µ 1 µ x µ3µ 1) µ 1) 1 x). ) µ 1 µx x x 4.) 1 µx we find F x) µ 1 ) µ µ 1 x µ 1 ) ) µ 1 µx x µ 1 µ x 1 µx 1 µx ) µ1 [ F µµ 1) µ 1 x 1) x) µ 1) µ µ x 1 µx µ 1 ] ) µ 1 µx x 1 µx) 4.3) 1 µx ) µ ) µ 1 F µµ 1) µ 1 x) µ 1) µµ 1) µx x µ x x1 µx) ) 1 µx By Taylor s formula with Lagrange s remainder term F x) F 1)F 1)x 1) 1 F t) x 1) x 1) 1 F t) x 1) 4.5) x [ 1) and t x 1). Note that F ) < F 1). In fact by 4.3) F ) µ ) µ F µ3µ 1) 1) µ 1 µ µ 1) which together with the obvious inequality implies F ) < F 1). Next we show that µ 1 µ3µ 1) e < µ 1 µ 1) F t) < max { F ) F 1) } µ3µ 1) µ 1) 4.6) for any t 1). Suppose the opposite assertion holds i.e. there exists a point t 1) at which 4.6) fails. Hence F t) attains its maximum value on [ 1] at an inner point τ i.e. F τ) max t [;1] F t) max { F ) F 1) } µ3µ 1) µ 1). 4.7)

10 1 G. KRESIN AND V. MAZ YA Taking into account 4.4) we have F µµ 1) µ 1 τ) µ 1) µ τ which is equivalent to ) µ 1 µτ τ 1 µτ Combined with 4.3) this implies F µµ 1) µ 1 τ) µ 1) µ τ µµ 1) µ 1 < µ 1) µ τ Therefore Setting F τ) < we rewrite 4.8) as µ µ 1) µ µ 1) ) µ µµ 1) τ1 µτ) 3 τ1 µτ)3 µ 1) 3 ) µ1 { 1 τ1 µτ) ) µ1 { ) µ 1 µτ τ 1 µτ ) µ µ 1. µ τ 1 τ)1 µτ) µ 1 } 1 τ)1 µτ) µ 1 µ 1)µ τ) }. ) µ1 µ 1 { µ 1) 1 τ)1 µτ)} µ τ µ 1 µ τ ) µ1 { µτ µ 1)τ µ}. 4.8) ητ) µτ µ 1)τ µ µ τ) µ1 4.9) F τ) < µµ 1) µ 1 ητ). 4.1) Noting that η µτµ 1)1 τ) τ) µ τ) µ > for < τ < 1 by 4.9) and 4.1) we find max F t) F τ) < µµ 1) µ 1 µ3µ 1) η1) t [;1] µ 1). The latter contradicts 4.7) which proves 4.6) for all t 1). Thus it follows from 4.) 4.5) and 4.6) that ) µ 1 ) µ 1 µ 1 µx x x µ3µ 1) < x x 1) µ x 1 µx µ 1) for all x < 1. This means that for µ > 1 and x < 1 the strict inequality 4.1) holds. ii) The case x > 1. Since the function Gx) µ 1 µ x ) µ 1 µx x 1 µx ) µ 1 x x µ3µ 1) x 1) µ 1) satisfies the equality ) 1 G 1 x x Gx) we have by part i) that Gx) < for x < 1 and hence Gx) < for x > 1. Thus the strict inequality 4.1) holds for µ > 1 and x > 1.

11 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 11 Corollary 1. For all y and any natural n the inequality holds P ny) P n y) n 4n 1)3n )y n n 4.11) P n y) 1 y y) ) ) n )/. 4.1) 1 n 1) 1 y y Proof. Suppose that < x 1. We introduce the new variable y 1 x [ ). 4.13) x Solving the equation y x 1/ x 1/ )/ in x we find x 1 y y i.e. ) 1 x 1 y y 1 y y) Putting µ n 1 we write 4.1) as 1 x n ) n ) n n x n 1)3n )1 x) n 1 x 1 n 1)x n n. 4.14) By 4.13) we have 1 x) 4y x hence 4.14) can be rewritten in the form 1 x n 1 x ) n x ) n n 4n 1)3n )y 1 n 1)x n n. 4.15) ) Setting x 1 y y in the first term on the left-hand side of 4.15) we obtain n 1 y y) 1 x n 1 x ) n ) ) n Pny). 4.16) 1 n 1) 1 y y Similarly putting x 1 y y) in the second term on the left-hand side of 4.15) we find x 1 n 1)x ) n 1 y y) n ) ) n Pn y). 4.17) 1 n 1) 1 y y Using 4.16) and 4.17) we can rewrite 4.15) as 4.11). We give one more corollary of Lemma containing an alternative form of 4.1) with natural µ. However we are not going to use it henceforth. Corollary. For all x and any natural n the inequality holds n1 n 1 k k3 ) { } 1 1) k ) k ) k 1 x) k. 4.18) n x 1 nx The equality sign takes place only for x 1.

12 1 G. KRESIN AND V. MAZ YA Proof. We set µ n n in 4.1): ) ) n 1 n 1 x n1 n3n 1) x n x 1 nx n 1) 1 x). Multiplying the last inequality by n 1) we write it as ) n 1) n1 n1 n 1)x ) ) n 1) x n3n 1)1 x). 4.19) n x 1 nx We rewrite the first term in 4.19): n 1) n1 n x ) [n x) 1 x)]n1 n x ) n x) n 1)n x)1 x) nn 1) n1 1 x) k3 n 1 k ) 1 x) k. 4.) n x) k Similarly the second term in 4.19) can be written as ) n1 n 1)x [1 nx) 1 x)]n1 ) ) 1 nx ) ) n 1) 1 nx 1 x) 1 nx 1 nx nn 1) n1 1 x) k3 1) k n 1 k ) 1 x) k 1 nx ) k. 4.1) Using 4.) and 4.1) in 4.19) we arrive at 4.18) with the left-hand side as the sum of rational functions. 5. The case p. The next assertion is the main theorem of this paper. It is based on the representation for the sharp constant C p 1 < p < ) obtained in Proposition 1. To find the explicit formula for C we solve an extremal problem with a scalar parameter entering the integrand in a double integral. Theorem 1. Let u h R n ) and let x be an arbitrary point in R n. The sharp coefficient C p x) in the inequality is given by ux) C x) u 5.1) C x) 4n 1))/ 1 n n/ x n. 5.) For p the absolute value of the derivative of a harmonic function u with respect to the normal to the boundary of the half-space at any x R n has the same remum as ux). Proof. We pass to the limit as p in 3.1) 3.) 3.4) and 3.5). This results in C x) C x 1 n 5.3) C γ Here by 3.6) 1 γ sin n 3 ϕ dϕ / Gn ϕ ϑ; γ) sin n ϑ dϑ. 5.4) G n ϕ ϑ; γ) n cos ϑ 1) nγ cos ϑ sin ϑ cos ϕ.

13 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 13 We are looking for a solution of the equation n cos ϑ 1) nγ cos ϑ sin ϑ cos ϕ 5.5) as a function ϑ of ϕ. We can rewrite 5.5) as the second order equation in tan ϑ: tan ϑ nγ cos ϕ tan ϑ 1 n. Since ϑ π/ we find that the nonnegative root of this equation is ϑ γ ϕ) arctan h γϕ) 5.6) 1/ h γ ϕ) nγ cos ϕ 4n 1) n γ cos ϕ). 5.7) Taking into account that the function G n ϕ ϑ; γ) is nonnegative for ϑ ϑ γ ϕ) ϕ π and using the equalities ϑ sin n 3 ϕ dϕ we write 5.4) as C γ γ By 5.6) G n ϕ ϑ; γ) sin n ϑdϑ [ cos ϑ γ cos ϕ sin ϑ ] sin ϑ 4 / 1 γ 4 1 γ By 5.9) and 5.1) we find G n ϕ ϑ; γ) sin n ϑ dϑ γ sin n 3 ϕ dϕ ϑγϕ) sin n 3 ϕ cos ϕ dϕ G n ϕ ϑ; γ) sin n ϑdϑ [ cos ϑγ ϕ)γ cos ϕ sin ϑ γ ϕ) ] sin ϑ γ ϕ) sin n 3 ϕdϕ. 5.8) h γ ϕ) sin ϑ γ ϕ) 5.9) 4 h γϕ) cos ϑ γ ϕ). 5.1) 4 h γϕ) cos ϑ γ ϕ) γ cos ϕ sin ϑ γ ϕ) γh γϕ) cos ϕ 4 h γϕ) Using 5.9) 5.11) and the identity γh γ ϕ) cos ϕ 4 h γϕ) n h γ ϕ) is defined by 5.7) we can write 5.8) as C γ n 1 γ Introducing the parameter α hγ ϕ) 4 h γ ϕ) ) n )/ sinn 3 ϕ dϕ. nγ n )

14 14 G. KRESIN AND V. MAZ YA and taking into account 5.7) we obtain C α 4 n 1) )/ n 4n 1)α P n α cos ϕ ) sin n 3 ϕ dϕ 5.1) with P n defined by 4.1). The change of variable t cos ϕ in 5.1) implies C α 4 n 1) )/ n 4n 1)α 1 Integrating in 5.13) over 1 ) and 1) we have C α 1 4 n 1) )/ n 4n 1)α P n αt)1 t ) n 4)/ dt. 5.13) 1 P n αt) P n αt) 1 t ) 4 n)/ dt. Applying the Schwarz inequality we see that { A 1 n Pn αt)p n αt) ) } 1/ C dt 5.14) α n 4n 1)α 1 t ) 4 n)/ By 4.1) which implies A n 4 ) )/ P n y)p n y) Combining 5.16) and 4.11) we obtain { 1 } 1/ dt. 5.15) 1 t ) 4 n)/ 4n 1)y n ) n)/ P n y)p n y) n n. 5.16) Pn y)p n y) ) P n y)pn y) n n n 4)3n )y n n n n. Therefore Pn αt) P n αt) ) ) 4 n 1)3n ) n n 1 n α t. 5.17) By 5.14) 5.15) 5.17) and by t ) n 4)/ dt t 1 t ) n 4)/ dt π Γ n ) Γ ) π Γ n ) n 1)Γ we find C 4ω n π n 1) )/ Γ ) n n n n/ Γ ) 3n )α ) 1/ α n 4n 1)α. 5.18) Note that d dα n 3n )α ) αn )n n 4n 1)α n 4n 1)α ) < for α > )

15 OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS 15 therefore the remum in α on the right-hand side of 5.18) is attained for α. Thus C 4ω n π n 1) )/ Γ ) n n n/ Γ ) 4n 1))/ ) n n/ Besides in view of 5.1) and 4.1) C 4 n 1) )/ n P n ) sin n 3 ϕ dϕ 4 n 1) )/ sin n 3 ϕ dϕ n n n )/ 4ω n π n 1) )/ Γ ) n n n/ Γ ) 4n 1))/ 1 n n/ which together with 5.3) and 5.19) proves the equality 5.). Since z S and the remum with respect to the parameter α nγ n 1 n z z n n 1 in 5.1) is attained for α it follows that the absolute value of the directional derivative of a harmonic function u with respect to the normal e n to R n taken at an arbitrary point x R n has the same remum as ux). Remark. Inequality 5.1) can be written in the form ux) C x) uy). 5.) y R n Using here 5.) we arrive at the explicit sharp inequality ux) 4n 1))/ 1 n n/ x n uy) y R n which generalizes the real value analog 1.5) of 1.) to harmonic functions in the n-dimensional half-space. From 5.) it follows that ux) C x) y R n with an arbitrary constant ω. Minimizing 5.1) in ω we obtain ux) C x) osc R n u) is the oscillation of u on R n. Inequalities 5.) 5.) imply the sharp estimate ux) n 1))/ 1 n n/ x n uy) ω 5.1) osc R n u) 5.) osc R n u) which is an analogue of 1.3) for harmonic functions in R n.

16 16 G. KRESIN AND V. MAZ YA Acknowledgments. The research of the first author was ported by the KAMEA program of the Ministry of Absorption State of Israel and by the Ariel University Center of Samaria. The second author was partially ported by the UK Engineering and Physical Sciences Research Council grant EP/F5563/1. REFERENCES [1] E. Borel Démonstration élémentaire d un théorème de M. Picard sur les fonctions entières C.R. Acad. Sci ) [] E. Borel Méthodes et Problèmes de Théorie des Fonctions Gauthier-Villars Paris 19. [3] R.B. Burckel An Introduction to Classical Complex Analysis V. 1 Academic Press New York-San Francisco [4] J. Hadamard Sur les fonctions entières de la forme e GX) C.R. Acad. Sci ) [5] D. Khavinson An extremal problem for harmonic functions in the ball Canad. Math. Bull. 35) 199) 18. [6] G. Kresin and V. Maz ya Sharp Real-Part Theorems. A Unified Approach Lect. Notes in Math. 193 Springer-Verlag Berlin-Heidelberg-New York 7. [7] G. Kresin and V. Maz ya Sharp pointwise estimates for solutions of strongly elliptic second order systems with boundary data from L p Appl. Anal. 86 N. 7 7) [8] E. Landau Über den Picardschen Satz Vierteljahrschr. Naturforsch. Gesell. Zürich ) [9] E. Landau Beiträge zur analytischen Zahlentheorie Rend. Circ. Mat. Palermo 6 198) [1] E. Lindelöf Mémoire sur certaines inégalités dans la théorie des fonctions monogènes et sur quelques propriétés nouvelles de ces fonctions dans le voisinage d un point singulier essentiel Acta Soc. Sci. Fennicae 35 N ) [11] A. P. Prudnikov Yu. A. Brychkov and O. I. Marichev Integrals and Series Vol. 1 Elementary Functions Gordon and Breach Sci. Publ. New York address: kresin@ariel.ac.il address: vlmaz@mai.liu.se address: vlmaz@liv.ac.uk