SOME PROPERTIES OF CONJUGATE HARMONIC FUNCTIONS IN A HALF-SPACE ANATOLY RYABOGIN AND DMITRY RYABOGIN Abstract. We prove a multi-dimensional analog of the Theorem of Hard and Littlewood about the logarithmic bound of the L p - average of the conjugate harmonic functions, < p. We also give sufficient conditions for a harmonic vector to belong to H p + + ), < p.. Introduction and statements of main results The following result of Hard and Littlewood [6] is classical. Theorem. Let < p, and let fz) = uz) + ivz) be an analtic function in the unit disc D := {z C : z < }, such that 2π /p ) ) v) =, 2) M p r, u) := ure iθ ) dθ) p C, r <. 2π Then 2) M p r, v) AC + AC ) /p. log r In this paper we prove an analog of Theorem for conjugate harmonic functions in + + =, ). The case p < n )/n leads to additional difficulties, since F p is subharmonic, provided p n )/n, [5]. We refer the reader to the classical works [3], [5], [7], [5], [2], [4], [8] for the histor and different results related to the classes S p + + ), h p + + ), H p + + ), all definitions are given in Section 2). We have Theorem 2. Let < p, and let F, F x, ) = Ux, ), V x, ), V 2 x, ),..., V n x, )), x, ) + +, be the harmonic vector such that 3) ) V i x, i =,..., n, 2) M p, U) C, 3) M p, F ) C. Then 4) M p, V ) AC + AC log /p. 99 Mathematics Subject Classification. Primar 3E25, secondar 42B25. Ke words and phrases. Hard spaces, subharmonic functions.
2 ANATOLY RYABOGIN AND DMITRY RYABOGIN The third condition in 3) appears after the application of the Main Theorem of calculus, see 8), 9). The logarithmic bound comes from the estimate ) p Ux, ξ) dx ACt p in the integral, t p dt Ux, ξ) ) see Lemmata 3, 4, 5. To control the logarithmic blow up, we use the Littlewood-Pale - tpe condition: Ip) := t p dt Ux, ξ) ) <. Our second result is Theorem 3. Let < p, and let F = U, V,..., V n ) be the harmonic vector such that 5) ) F x, 2) M p, F ) C, 3) Ip) <. Then F H p. It is unlikel that F H p implies Ip) <. The paper is organized as follows. In section 2 we give all necessar definitions and auxiliar results used in the sequel. In Section 3 and 4 we prove Theorems 2 and 3. For convenience of the reader we split our proofs into elementar Lemmata. 2. Auxiliar results Let Ux, ) be a harmonic function in + +, ). We sa that the vector-function V x, ) = V x, ),..., V n x, )) is the conjugate of Ux, ) in the sense of M. Riesz [4], [6], if V k x, ), k =,..., n are harmonic functions, satisfing the generalized Cauch-Riemann conditions: U n + V k =, x k k= V i x k = V k x i, U = V i, i k, k =,..., n. x i If Ux, ) and V x, ) are conjugate in + + in the above sense, then the vectorfunction F x, ) = Ux, ), V x, )) = Ux, ), V x, ),..., V n x, )) is called a harmonic vector. Define M p, F ) = U 2 x, ) + n i= V 2 i x, )) p/2dx ) /p, p >. Now we define the space H p + + ). We follow the work of Fefferman and Stein[4]. Let Ux, ) be a harmonic function in + +, and let U j j 2 j 3...j k denote a component
H p SPACES OF CONJUGATE HARMONIC FUNCTIONS 3 of a smmetric tensor of rank k, j i n, i =,..., n. Suppose also that the trace of our tensor is zero, meaning n U jjj3...j k x, ) =, j 3,..., j k. j= The tensor of rank k + can be obtained from the above tensor of rank k b passing to its gradient: U j j 2...j k j k+ x, ) = U j j x 2 j 3...j k x, )), x =, j k+ n. jk+ Definition [4]).We sa that U H p + + ), p >, if there exists a tensor of rank k of the above tpe with the properties: p/2dx U... x, ) = Ux, ), Uj)x, )) 2 <, j) = j,...j k ). It is well-known that the function j) ) p/2 Uj) 2 x, ) is subharmonic for p pk = j) n k)/n + k ), see [3],[4],[6]. We remind that the radial and the non-tangential maximal functions are defined as follows: F + x) = F x, ), N α F )x ) = F x, ). x,) Γ α x ) Here Γ α x ) = {x, ) + + : x x < α}, α >, is an infinite cone with the vertex at x. It is well-known [4] that F H p + + ) N α F ) L p F + L p, p >. We also define the weak maximal function W F x, ) = F x, ζ), >. ζ The above expression is understood as folows: we fix x, and for fixed we find the remum over all ζ. We will use the following results. Lemma. [4], p.73). Suppose w is harmonic in + +, and M p, u) C for some p, < p <. Then 6) x ux, ) A n/p, < <. Theorem 4. [5], p.267). Let < p, a >, let w : + + [, ) be a function such that w p is subharmonic and satisfies J a,p := t ap wx, t) p dxdt < +, + +
4 ANATOLY RYABOGIN AND DMITRY RYABOGIN and for each x, t) [, + ) let w a x, t) := Γa) + s a wx, s + t) ds. Then w a is subharmonic on + + and is finite a.e. on, and for all t, wx, t) p dx ACa, n, p) J a,p. Theorem 5. [5], p.269). Let m N, p n )/m + n ) if n = we pose p > ), and let u : + + R be harmonic. Then, for all t >, m ux, t) p dx Am, n, p)t mp 3t/2 ds ux, s) p dx. t/2 Lemma 2. [3], p.2464). Let p > and let F = U, V,..., V n ) be such that V i x, i =,..., n, M p, U) C. Then M p, k F ) AC k, k N. Notation. We denote b Di k fx, ) the partial derivative of the function f of the order k with respect to x i, i =, 2,..., n +. The notation fx, ) x means that fx, ) converges to uniforml with respect to x, provided, k fx) = k fx),..., k fx) ). Everwhere below the constants Ak, n), C, K depend x k x k n onl on the parameters pointed in parentheses, and ma be different from time to time. 3. Proof of Theorem 2. Lemma 3. Let p >, and let F = U, V,..., V n ) satisf M p, F ) C. Then, 7) M p, V i ) AC + Ux, ξ) dt) i =,,..., n, V = U. Proof. B the Main Theorem of Calculus, and the Cauch-Riemann equations, we have V i x, t) Ux, t) 8) V i x, ) V i x, ) = dt = dt, t x i i =, 2,..., n, 9) Ux, ) Ux, ) = Then, and the result follows. M p, V i ) M p, V i ) + Ux, t) t dt., Ux, ξ) dt)
H p SPACES OF CONJUGATE HARMONIC FUNCTIONS 5 Lemma 4. Let p > and let < <. Then Ux, ξ) dt) ) p Proof. Denote Ux, ξ) ξ Ψx, ) := Following [6] consider Φx, ) := Ψx, ) p p B definition of Ω), ) \Ω) ) + 2 p. t p dt Ux, ξ) ) Ux, ξ) dt. Ψx, ) ) p, Ω) := {x : Φx, ) > }. Ψx, ) p dx p \Ω) Ψx, ) ). Next, the reasons which are similar to those in [6], impl a Φx, ξ) 2) Φx, ) dx Φx, a) dx = dξ dx, < < a. Ω) Ωa) Moreover, using 2 Ψ/ 2 for almost ever < ξ <, we have Φ Ψ = p Ψp + p ξp Ψ ) p p ξ p Ψ ) p 2 Ψ 2 p Ψ )Ψ p + ξ p Ψ ) p ) 2p ξ p Ψ ) p. Here the last inequalit follows from the definition of Ωξ) and < p <. Since Ψx, ) Ψx, ) ), the function Φx, ) is negative, provided is sufficientl close to, and we can take a such that Ωa) =. Hence, 2) ields a Φx, ) dx 2p ξ p Ψx, ξ) p dξ x, ξ)) dx Ω) 3) 2p Adding Ω) Ψx, ) p dx ξ p dξ Ω) with ), and using 3), we obtain ). Ωξ) Ωξ) Φx, ) dx + p Ψx, ξ) p x, ξ)) dx. Ω) Ψx, ) )
6 ANATOLY RYABOGIN AND DMITRY RYABOGIN The next result is crucial. Lemma 5. Let p > and let F = U, V,..., V n ) be such that 4) ) V i x, i =,..., n, 2) M p, U) C. Then 5) ) /p φ ij x, ξ) ) AC, ξ where φ ij x, ) is a coordinate of V i x, ), j =,..., n +, x n+ =, i =,..., n, V = U. Proof. Fix p > and let l = inf{j N : p > p j := n )/j + n )}. Since V i x, ) x, we ma use the following relation see [5] or [4]) We have where φ ij x, ) = Rx, ) := 2l 2)! 2l 2)! 2l 2)! s ) 2l 2 D 2l n+ φ ij x, s)ds = s 2l 2 D 2l n+ φ ij x, s + )ds. φ ij x, ξ) Rx, ), ξ To prove 5) it is enough to show that 6) M p, R) AC. Since ) p l Dn+φ l ij x, ξ) ) s 2l 2 l Dn+φ l ij x, s + ξ) ds. ξ is subharmonic [3], the function ) p w p x, s + ) := l Dn+φ l ij x, s + ξ) ξ is also subharmonic, and we ma appl Theorem 4 take a = 2l, A = Al, n, p)) to obtain Rx, ) p dx A A s 2l )p ds s 2l )p ds l Dn+φ l ij x, s + ξ) ) = ξ p/pjdx. l Dn+φ l ij x, s + ξ) j) p ξ
H p SPACES OF CONJUGATE HARMONIC FUNCTIONS 7 B the choice of p j, we have p/p j > and we ma use the well-known [4] L p/p j - boundedness of the maximal operator: Rx, ) p dx A s 2l )p ds l Dn+φ l ij x, s + ) p dx. Since Dn+φ l ij x, ) is the l-th derivative of V i, we use Lemma 2 to get 7) l Dn+φ l ij x, ) p dx 2l F x, ) p dx C 2lp. This gives Rx, ) p dx Al, n, p)c s 2l )p s + ) 2lp ds = Al, n, p)c p, and 6) is proved. Proof of Theorem 2. The proof follows from Lemmata 3, 4, 5. 4. Proof of Theorem 3. Lemma 6. Let p >. Then F = U, V,..., V n ) H p iff ) F x, 2) Ux, η) ) < C. η Proof. Let F H p, then both ) and 2) are well-known, [4]. We prove the converse in two steps. At first we show that 8) Ux, ) ) C. Then we prove that 8) implies 9) V i, ) ) p L ), i =,..., n. To prove 8), we observe that Ux, ) = Ux, η) = lim η η Ux, η). Hence, using 2) and Fatou s Lemma, we obtain Ux, ) ) lim Ux, η) ) C. η It remains to show 9). Using Cauch-Riemann equations, we have V i x, ) = [V i ]... x, ) = )k k )! s k D k n+v i x, s + )ds =
8 ANATOLY RYABOGIN AND DMITRY RYABOGIN 2) ) k k )! s k D k n+d i Ux, s + )ds, ) p/2 where k is chosen such that the function Uj) 2 x, ) is subharmonic, p pk = j) n k)/n + k ), see [3],[4],[6]). Since the expression in 2) is one of the tensor coordinates of U j), see [4], page 69), we have ) p/2dx V i x, ) Uj)x, )) 2 = ) p/2dx Uj)x, 2 ) C The last estimate is proved in [4], page 7. j) j). Ux, ) ) Lemma 7. Let p > and let F = U, V,..., V n ) be such that V i x, i =,..., n. Then F H p, provided 2) ) M p, F ) C, 2) M p, U) C, 3) Ip) <. Proof. At first we prove that ) 22) 2 p V i x, ) AC + G G, Ux, ) ) G. This follows from 8), 9) the inequalit ) p ) p p, 2 p V i x, ) V i x, ) + Ux, ) ) j =,, 2,..., n, V = U, and the first condition in 2). Now we finish the proof. Assume that the lemma is not true. Then N > there exists a set E, < me) <, such that V i x, ) ) 4N, E for some i =,,..., n, V = U. Hence, taking G = E in 22) we have Ux, ) ) 2N. Then E Ux, ) = and Lemma Fatou impl ) lim Ux, ξ) ξ E Ux, ξ) = lim ξ E ξ Ux, ξ) Ux, ) ) 2N.
H p SPACES OF CONJUGATE HARMONIC FUNCTIONS 9 But this contradicts the third condition of the lemma, since ) Ip) d Ux, ξ) d Ux, ξ) ) N. ξ ξ Lemma 8. Let p > and let F = U, V,..., V n ) be the harmonic vector satisfing conditions of Theorem 3. Then 8) holds. Proof. We argue as in the previous Lemma. We have 9), and ) 2 p Ux, ) Ux, ) p dx + leads to a contradiction. E Ux, ) ) Proof of Theorem 3. The proof follows from Lemmata 8, 7 and 6. References [] A. A. Bonami, Integral inequalities for conjugate harmonic functions of several variables, Math. Sbornik., Vol. 87 29, 972), No. 2, p. 88-23. [2] D. L. Burkholder, R. V. Gand and M. L. Silverstein, A maximal functions characterizations of the class H p, Trans. AMS., 57, 97), 37-53. [3] A. P. Calderón and A. Zgmund, On higher gradients of harmonic functions, Studia Math. 24, 964), No. 2, 2-266. [4] C. Fefferman, E. M. Stein, H p spaces of several variables, Acta Math., 29, No 3, 972), 37-93. [5] T. M. Flett, Inequalities for the p-th mean values of harmonic and subharmonic functions with p, Proc. Lon. Math. Soc., Ser. 3, 2, No. 3, 97), 249-275. [6] G. H. Hard, J. E. Littlewood, Some properties of conjugate functions, J. Reine und Angew. Math., 67, 932), 45-432. [7] V. I. Krlov, On functions regular in the half-plane, Math., Sb., 939), 648), pp.95-38. [8] U. Kuran, Classes of subharmonic functions in, ), Proc. Lond. Math. Soc., Ser 3., 6, 966), No. 3, 473-492. [9] I. Privalov, Subharmonic functions, Moscow, 937. [] M. Riesz,, Proc. Lon. Math. Soc., Ser. 3, 2, No. 3, 97), 249-275. [] A. Rabogin, Conjugate harmonic functions of the Hard class. Russian), Izv. Vssh. Uchebn. Zaved. Mat., 99), no. 9, 47-53; translation in Soviet Math. Iz. VUZ) 35 99), no. 9, 46-5. [2] A. Rabogin, Boundar values of conjugate harmonic functions of several variables, Russian), Izv. Vssh. Uchebn. Zaved. Mat., 98), no. 2, 5-54. [3] A. Rabogin, D. Rabogin, Hard spaces and partial derivatives of conjugate harmonic functions, Proc. AMS, 27), 35, no. 8, 246-247. [4] E. M. Stein, Singular integrals and differentiabilit properties of functions, Princeton Universit Press, Princeton NJ, 97. [5] E. M. Stein and G. Weiss, On the theor of harmonic functions of several variables, Acta Math., 3, 96), pp. 26-62. [6] E. M. Stein and G. Weiss, Generalization of the Caush-Riemann equations and representation of the rotation group, Amer. J. Math.,9, 968), 63-96. [7] E. M. Stein and G. Weiss, An introduction to harmonic analsis on Euclidean spaces, Princeton Universit Press, Princeton NJ, 969. [8] T. Wolff, Counterexamples with harmonic gradient in R 3, Essas in honor of E. M. Stein, Princeton Mathematical Series, 42, 995), 32-384.
ANATOLY RYABOGIN AND DMITRY RYABOGIN Anatol Rabogin, Department of Phsics and Mathematics, Stavropol State Universit, Pushkin s Street, Stavropol, Russia Dmitr Rabogin, Department of Mathematical Sciences, Kent State Universit, Kent, OH 44242, USA E-mail address: rabogin@math.kent.edu