SHARP ESTIMATES FOR HOLOMORPHIC FUNCTIONS ON THE UNIT BALL OF C n

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1 SHARP ESTIMATES FOR HOLOMORPHIC FUNCTIONS ON THE UNIT BALL OF C n ADAM OSȨKOWSKI Abstract. Let S n denote the unit sphere in C n. The purpose the paper is to establish sharp L log L estimates for H and P +, the Hilbert transform and the Riesz projection on S n, respectively. The proof rests on the existence of a certain superhamonic function on C satisfying appropriate majorization conditions. 1. Introduction Suppose that f(ζ) = ˆf(n)ζ n Z n is a complex-valued integrable function on the unit circle = {ζ C : ζ = 1}. Here, as usual, ˆf(n) = 1 f(eiθ )e inθ dθ denotes the n-th Fourier coefficient of f. For p 1, let H p (T, C) consist of all functions f satisfying ˆf(n) = 0 for n < 0. Then H p (T, C) is a closed subspace of L p (T, C) and can be identified with the space of analytic functions on the unit disc B 1. The Riesz projection (or analytic projection) P + : L p (, C) H p (, C), is the operator given by P + f(ζ) = f + (ζ) = n 0 ˆf(n)ζ n, ζ. One defines the complementary co-analytic projection on by P = I P +. These two projections are closely related to another classical operator, the Hilbert transform (conjugate function) on, which is given by Hf(ζ) = i n Z sgn(n) ˆf(n)ζ n, ζ. Here sgn(n) = n/ n for n 0 and sgn(0) = 0. A classical theorem of M. Riesz 16 asserts that the operator P + (equivalently, the Hilbert transform H) is bounded on L p (, C) for 1 < p <. The question about the precise value of the norms of these operators has gathered some interest in the literature. For p = k, k = 1,,..., the exact values of the norms of H were determined by Gohberg and Krupnik 4, who showed that H Lp (,C) L p (,C) = cot(/(p)). For the remaining values of 1 < p <, the norms of the operator H acting on real L p spaces were found by Pichorides 14 and, independently, by Cole (unpublished work, see Gamelin 3): H L p (,R) L p (,R) = cot(/(p )), 000 Mathematics Subject Classification. Primary: 4B35, 3A10. Secondary: 4A50, 3A50. Key words and phrases. harmonic function, holomorphic function, Riesz projection, Hilbert transform, best constants. Partially supported by the NCN grant DEC-01/05/B/ST1/

2 ADAM OSȨKOWSKI where p = max{p, p/(p 1)}. Consult also Essén and Verbitsky 0. These norms do not change while passing to the complex L p spaces (see e.g. Pe lczyński 13): H L p (,C) L p (,C) = cot(/(p )), 1 < p <. For the Riesz projection, Hollenbeck and Verbitsky 6 (see also 7) proved that P ± Lp (,C) L p (,C) = csc(/p), 1 < p <, solving a conjecture which was open for almost 40 years. For related weak-type bounds, consult e.g. the works of Davis 1, Janakiraman 8, Kolmogorov 9, Osȩkowski 11, 1 and Tomaszewski 18, 19. For a treatment of the subject from a wider perspective, see Gohberg and Krupnik 5, Krupnik 10 and the classical monograph of Zygmund 1. Our purpose is to study another class of sharp estimates of the above type. Actually, we will work in a slightly more general, n-dimensional setting. Let C n be the n-dimensional complex space, equipped with the usual norm z = ( z 1 + z z n ) 1/. Let B n be the unit ball of C n and let σ n denote the rotation-invariant normalized Borel measure on the unit sphere S n = B n. For a σ n -measurable function f : S n C, we define the associated L log L norm by the formula f L log L = f log + f dσ n. S n If a real-valued f satisfies f L log L < and its Poisson integral Pf (see e.g. Chapter 5 in 17) is pluriharmonic, then we shall write f H log H(S n ; R). In such a case, there is a unique function g : S n R (up to set of σ n -measure 0) satisfying gdσ S n n = 0 and such that the Poisson extension of f +ig is a holomorphic function on the unit ball. Such a g will be denoted by Hf and called the Hilbert transform of f; furthermore, we can define the Riesz projection of f by P + f = (f + ihf + fdσ S n n )/. One easily checks that for n = 1, these definitions of H and P + are consistent with those given at the beginning of the paper. Clearly, both H and P + can be extended to act on complex-valued functions f H log H(S n ; C) (i.e., those satisfying f L log L < and whose Poisson extension is pluriharmonic), by setting Hf = H(Rf) + ih(if) and P + f = P + (Rf) + ip + (If). Let us turn to the formulation of the main results of this paper. The first statement is a sharp bound for the Hilbert transform. Theorem 1.1. Assume that f H log H(S n ; C) satisfies fdσ S n n = 0. Then (1.1) Hf(z) log Hf(z) dσ n (z) c 1 f(z) log ( c f(z) ) dσ n (z), S n S n where c 1 = 1 and c = e /. Both c 1 and c are the best possible, even if f is assumed to be real-valued. The second theorem is a sharp L log L estimate for Riesz projection. interestingly, it holds true only for real-valued functions. Theorem 1.. Assume that f H log H(S n ; R) satisfies fdσ S n n = 0. Then (1.) P + f(z) log P + f(z) dσ n (z) c 3 f(z) log ( c 4 f(z) ) dσ n (z), S n S n Quite

3 RIESZ PROJECTION 3 where c 3 = 1/ and c 4 = e /4 /. Both constants c 3 and c 4 are the best possible. The inequality (1.) does not hold with any constants c 3 and c 4 when f is allowed to take complex values. The change in the constants for P + when passing from the real to the complex case occurs also for the L p bounds (see 6 and 0), so the above phenomenon is perhaps not that surprising. What may be a little unexpected is that actually no finite constants work in the complex case. This is strictly related to the fact that the function t t log t is neither convex nor concave, and has negative and positive numbers in its range. The proof will be based on the existence of a certain special superharmonic function satisfying appropriate majorization condition. Theorem 1., as well as the sharpness of (1.1), are shown in the next section. The final part of the paper is devoted to the proof of (1.1).. Proof of Theorem 1. and sharpness of (1.1) Consider the function U : R 0, ) R, given by U(x, y) = ( 4 R cos(θ) R log R cos(θ) + R sin(θ) θ ). Here x = R cos θ, y = R sin θ denote the usual polar coordinates (with R 0 and θ 0, ). We extend the function to the whole plane R by setting U(x, y) = U(x, y). Note that U is also symmetric with respect to the y-axis: U(x, y) = U( x, y) for all x, y. The key properties of the function U are gathered in a lemma below. Recall the constants c 3 and c 4, introduced in the statement of Theorem 1. above. Lemma.1. (i) We have the majorization (.1) U(x, y) (x + y ) log(x + y ) 1/ 4c 3 x log(c 4 x ), x, y R. (ii) The function U is superharmonic. Proof. (i) It is enough to establish the majorization in the first quadrant, i.e. for nonnegative x and y. Then it takes the equivalent form H(θ) = ( 4 cos(θ) + sin(θ) θ ) + cos θ log( e /4 cos θ), θ 0, /, and one computes that H (θ) = cos(θ) ( θ ) sin(θ) log( cos θ). Now, if θ < /4, then cos(θ) ( θ ) < 0 and sin(θ) log( cos θ) > 0, so H is negative. Similarly, if θ > /4, then cos(θ) ( θ ) > 0 and sin(θ) log( cos θ) < 0, which implies that H is positive. Consequently, H has a global minimum at θ = /4; since H(/4) = 0, the majorization (.1) holds true. (ii) First note that U is harmonic in the upper halfplane. This follows at once from the identity U(x, y) = R ( z log( iz) z /4 ), where z = x + iy and log is the principal branch of the complex logarithm. Therefore, U is also harmonic in the halplane {(x, y) : y < 0}. Consequently, the superharmonicity will follow if we show that lim y 0 U y (x, y) 0 for all x R. Actually,

4 4 ADAM OSȨKOWSKI by symmetry, it is enough to consider positive x only. We have lim y 0 R y = 0 and lim y 0 θ y = 1/x, so lim y 0 U y (x, y) = x < 0. This completes the proof. Proof of Theorem 1.. Let us start with the case n = 1. Pick any function f as in the statement. Since fdσ 1 = 0, we have P + f = (f + ihf)/ and hence P + f = f + Hf. Apply the majorization (.1) to obtain P + f(z) log P + f(z) 4c 3 f(z) log(c 4 f(z) ) dσ 1 (z) U(f(z), Hf(z))dσ 1 (z). However, the Poisson extension Pf + ihf is analytic on the unit disc and U is superharmonic on the plane. Consequently, the composition U(Pf + ihf) is superharmonic on B 1 and hence the latter integral does not exceed U(f(0), Hf(0)) = U(0, 0) = 0. Thus we have obtained P + f(z) log P + f(z) dσ 1 (z) 4c 3 f(z) log(c 4 f(z) )dσ 1 (z). It remains to apply this estimate to the function f/ to get (1.1) for n = 1. When n, we note that P + f(z) log P + f(z) c 3 f(z) log(c 4 f(z) ) dσ 1 (z) S n = P + f(wz) log P + f(wz) c 3 f(wz) log(c 4 f(wz) ) dσ 1 (z)dσ n (w). S n However, for any w S n we see that the inner integral is nonpositive. This follows at once from the one-dimensional case we have just established, applied to the function z f(wz). This implies that the integral over S n is also nonpositive, and this completes the proof of (1.). Sharpness of (1.1) and (1.). We will show that the estimates are sharp in the case n = 1. To do this, we will use a certain modification of an example exploited by Pichorides in 14. We will start with the estimate (1.). Pick κ < c 4, let γ (0, /4) be a given parameter and consider the holomorphic function F (z) = ( i 1+z 1 z ) γ/ i defined on the unit disc B 1. Since z 1+z 1 z maps B1 onto the left halfplane {z C : Rz 0}, we see that the function F + i maps B 1 onto the angle {z C : Rz tan γ Iz}. In particular, we see that the unit circle is mapped onto the boundary of the angle, and hence (.) F (e iθ ) + i = RF (eiθ ) sin γ = IF (eiθ ) + 1 cos γ for θ,. Let us take a look at the difference F (z) log F (z) dσ 1 (z) RF (z) log (κ RF (z) ) dσ 1 (z) = I + II,

5 RIESZ PROJECTION 5 where I = II = F (z) log F (z) F (z) + i log F (z) + i dσ 1 (z), F (z) + i log F (z) + i RF (z) log (κ RF (z) ) dσ 1 (z). We will show that I is bounded and II tends to infinity as γ /4; this will clearly imply that no constants c < c 3, κ < c 4 work in (1.), since F = P + (R(F )). Let us start with the term I. Note that I = F (z) log F (z) F (z) + i dσ 1 (z) + ( IF (z) 1 ) log F (z) + i dσ1 (z). Both integrals are bounded as γ /4: the identity 1+eiθ = i cot(θ/) implies 1 e iθ F (z) 1 + cot(θ/) γ/. Thus, substituting z = e iθ, we see that we must guarantee that both integrands do not explode too fast in the neighborhood of θ = 0. However, we have F (z) log F (z) F (z) + i = O(cot(θ/)4γ/ 1 ) and ( IF (z) 1 ) log F (z)+i = O ( cot(θ/) γ/ 1 log cot(θ/) ) as θ 0. Hence lim sup γ /4 I <. We turn our attention to II. By (.), we have II = (1 sin γ) F (z) + i log F (z) + i dσ 1 (z) sin γ log(κ sin γ) F (z) + i dσ 1 (z) (.3) = 1 sin γ sin γ = 1 sin γ cot(θ/) 4γ/ log cot(θ/) γ/ dθ cot(θ/) 4γ/ log ( κ sin γ ) dθ 8γ 0 r 4γ/ log rdr 1 + r 4 sin γ log ( κ sin γ ) r 4γ/ dr r, where the latter equality follows from the substitution r = cot(θ/). Now pick an arbitrary η (0, 1); then we have (1 + r ) 1 ηr for sufficiently large r; say, r > N = N(η). Thus, II 1 sin γ 4 sin γ = 1 sin γ 4 sin γ 8γ η r 4γ/ log rdr N log ( κ sin γ ) 1 + r 4γ/ dr 8γ η log ( κ sin γ ) N 4γ/ 1 4γ/ 1 N log N + 1 4γ/ (1 4γ/) γ/

6 6 ADAM OSȨKOWSKI Now, if we let γ /4, then both terms 1 sin γ 8γ η N 4γ/ 1 1 4γ/ log N and 4 sin γ log ( κ sin γ ) converge to certain finite constants; on the other hand, for the remaining two terms, we can write 1 sin γ 8γ η N 4γ/ 1 (1 4γ/) 4 sin γ log ( κ sin γ ) 1 4γ/ ( = 1 4γ ) 1 1 sin γ γ 4γ ηn 4γ/ 1 sin γ log ( κ sin γ ). Now if we let γ /4, the expression in the square brackets converges to η/4 log(κ/ )/. However, we have assumed at the beginning that κ < c 4 = e /4. Therefore, if η is a priori taken sufficiently close to 1, the expression in the square brackets converges to a positive constant and, consequently, II. This shows that the constants c 3, c 4 are indeed the best possible in (1.). The reasoning for the estimate (1.1) is similar, actually it exploits the same function F as above. We pick κ 0 < c and write where I = II = IF (z) log IF (z) c 1 RF (z) log (κ 0 RF (z) ) dσ 1 (z) = I + II, IF (z) log IF (z) IF (z) + i log IF (z) + i dσ 1 (z), IF (z) + 1 log IF (z) + 1 c 1 RF (z) log (κ 0 RF (z) ) dσ 1 (z). Now, as above, the term I remains bounded when we let γ /4. To handle II, note that (.) implies II = (1 sin γ) F (z) + i log F (z) + i dσ 1 (z) + (cos log cos γ sin γ log(κ 0 sin γ)) F (z) + i dσ 1 (z) S 1 (1 sin γ) F (z) + i log F (z) + i dσ 1 (z) sin γ log κ 0 F (z) + i dσ 1 (z), since cos γ log cos γ sin γ log cos γ for γ 0, /4. Now, compare this to the first expression on the right of (.3). We have κ 0 < c = e / and hence there is κ < c 4 = e /4 such that for γ sufficiently close to /4, sin γ log κ 0 < sin γ log(κ sin γ). But this implies that II as γ /4, as we have shown above. This proves the optimality of the constants c 1 and c, since IF = H(RF ). We conclude this section by indicating an example which shows that (1.) does not hold for complex-valued functions, no matter what c 3 and c 4 are. To see this, fix these two constants and consider the function f(z) = c 1 4 ζ 1 /, ζ, where a is a fixed constant. Then, from the very definition of P +, we have P + f = 0, so the

7 RIESZ PROJECTION 7 left-hand side of (1.) vanishes; on the other hand, we have f(ζ) log(c 4 f(ζ) ) < 0 for all ζ. Hence the right-hand side is negative and the inequality (1.) fails to hold. 3. Proof of (1.1) The reasoning will again be based on the properties of a certain special superharmonic function on the plane. Consider the holomorphic G : C \ 0, ) C, given by ( G(z) = z log iz 1/) z/4. Here log denotes the principal branch of the complex logarithm and the root is given by z 1/ = R 1/ e iθ/ for any z = Re iθ with θ 0, ). Then Φ = RG is a harmonic function on C\0, ). We easily check that lim y 0 Φ(x, y) = lim y 0 Φ(x, y) and hence Φ extends to a continuous function on C. Actually, this extension (still denoted by Φ) is superharmonic, which follows from the estimate lim y 0 Φ y (x, y) = lim y 0 Φ y (x, y) < 0 for any x > 0. Compare this to the analogous reasoning from the proof of Lemma.1. The next step in our analysis is to establish the following majorization. Theorem 3.1. For any w, z C we have the estimate ( ) (3.1) Φ(wz) w z w z ( ) log w + z c c 1 w + z log. Proof. Define a = (w z)/ and b = (w + z)/, and let us try to express the left-hand side in terms of these auxiliary variables. We easily compute that wz = (a + b)( b ā) = b a + ii(a b); furthermore, a little calculation reveals that i(wz) 1/ a = b + ( b a ) + (I(a b)) b sgn(i(a b)) a + ( b a ) + (I(a b)). Consequently, we compute that Φ(wz) = ( b a ) log I(a b) arctan ( b a ) + (Ia b) + 4 b a + ( b a ) + (I(a b)) a b + ( b a ) + (I(a b)). By continuity, it suffices to show (3.1) for nonzero a. Let us minimize the lefthand side over a, b, keeping a and b fixed. To do this, set u = b a R, v = I(a b) 0, and consider the function H(v) = Φ(wz) = u log u + v + u + u v arctan + v 4 u + u + v. By the direct differentiation, we see that H uv (v) = (u + v ) arctan u/ u + v + 1 u/ u + v + 1.

8 8 ADAM OSȨKOWSKI Observe that there is α 0, such that u u + v = cos α and v u + v = sin α (the restriction on α is due to the inequality v 0). Thus we have H (v) = (sin α α + ) / 0, where the latter elementary estimate follows from a straightforward analysis of a derivative. Summarizing, we have shown that the function H is nonincreasing and hence it is enough to check (3.1) for the largest possible value of v, i.e., under the assumption v = a b. However, then the majorization becomes ( b a ) log( b + a ) + a b arctan b 4 a a log( a ) b log( e / b ), or, after some equivalent rearranging, J(s) = s 1 log s + 1 where s = b / a 0. We derive that and J (s) = s log s + 1 s arctan s 4 (s 1) + s log(e / s) 0, arctan s s s log(e/ s) J (s) = log s e / s + 1. Therefore, J is concave on 0, s 0 and convex on s 0, ), for some positive number s 0 ; furthermore, we have J(0) = log + /4 > 0 and J(1) = J (1) = 0. Thus J has to be nonnegative on the whole halfline 0, ) and the claim is proved. Proof of (1.1). As previously, the main difficulty lies in showing the bound for n = 1. Let f : C be as in the statement and write f = P + f + P f. Then PP + f and P P f, the Poisson extensions of P + f and P f to the disc B 1, are holomorphic. The function Φ is superharmonic and hence, by the well-known fact from the theory of analytic functions (see e.g. Range 15), the composition (w, z) Φ(wz) is plurisuperharmonic on C. This, in turn, implies that the function Φ ( 4PP + fp P f ) is superharmonic. Hence, by the mean-value property and (3.1), 0 = Φ(0) Φ ( 4P + f(z)p f(z) ) dσ 1 (z) S 1 Hf(z) log( Hf(z) ) c 1 f(z) log( c f(z) ) dσ 1 (z), where we have used the identities Hf = i(p f P + f) and f = P + f + P f. It remains to apply the above estimate to the function f/, and multiply both sides by to get the claim in the case n = 1. The passage to the higher-dimensional setting is done exactly in the same manner as in the preceding section. We omit the straightforward details.

9 RIESZ PROJECTION 9 References 1 B. Davis, On the weak (1, 1) inequality for conjugate functions, Proc. Amer. Math. Soc. 4 (1974), pp E. M. Essén, A superharmonic proof of the M. Riesz conjugate function theorem, Ark. Mat. (1984), pp T. W. Gamelin, Uniform algebras and Jensen measures, Cambridge University Press, London, I. Gohberg and N. Krupnik, Norm of the Hilbert transformation in the L p space, Funct. Anal. Pril. (1968), pp in Russian; English transl. in Funct. Anal. Appl. (1968), pp I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vol. I. II. Operator Theory: Advances and Appl. Vols. 53, 54. Birkhäuser, B. Hollenbeck and I. E. Verbitsky, Best Constants for the Riesz Projection, J. Funct. Anal. 175 (000), pp B. Hollenbeck and I. E. Verbitsky, Best constant inequalities involving the analytic and coanalytic projections, Operator Theory: Adv. Appl. vol. 0. Birkhäuser (010), pp P. Janakiraman, Best weak-type (p, p) constants, 1 p for orthogonal harmonic functions and martingales, Illinois J. Math. 48 No. 3 (004), pp A. N. Kolmogorov, Sur les fonctions harmoniques conjugées et les séries de Fourier, Fund. Math. 7 (195), pp N. Krupnik, Survey on the Best Constants in the Theory of One-dimensional Singular Integral Operators, Operator Theory: Advances and Applications 0 (010), pp A. Osȩkowski, Sharp inequalities for differentially subordinate harmonic functions and martingales, Canadian Math. Bull. 55 (01), pp A. Osȩkowski, Sharp weak type inequalities for Hilbert transform and Riesz projection, Israel J. Math. 19 (01), pp A. Pe lczyński, Norms of classical operators in function spaces, in Colloque en l honneur de Laurent Schwartz, Vol. 1, Asterisque 131 (1985), pp S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (197), pp R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics Vol. 108, Springer-Verlag, New York, M. Riesz, Sur les fonctions conjugées, Math. Z. 7 (197), pp W. Rudin, Function theory in the unit ball of C n, Springer-Verlag, New York, B. Tomaszewski, The best constant in a weak-type H 1 -inequality, complex analysis and its applications, Complex Variables 4 (1984), pp B. Tomaszewski, Some sharp weak-type inequalities for holomorphic functions on the unit ball of C n, Proc. Amer. Math. Soc. 95 (1985), pp I. E. Verbitsky, Estimate of the norm of a function in a Hardy space in terms of the norms of its real and imaginary part, Mat. Issled. 54 (1980), pp in Russian; English transl. in Amer. Math. Soc. Transl. () 14 (1984), pp A. Zygmund, Trigonometric series Vol., Cambride University Press, London, Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha, Warsaw, Poland address: ados@mimuw.edu.pl