1. 1-9}. 930:...) ‘ fl v!.'.x:JWl|ufi”v4. I....: i . a ... I z .. . ‘1: All? 7 . II «n . ‘ .912A 5‘Xil...‘$ ‘I A IV. .‘ f.a . lllvlt..r 4 5 ntll.2l-.|)\l a...’ ll 1:01.: i! .1..|Il ‘ .4; . . l. .v.. «4%.. » ‘ , ‘ .....:fl.l. ‘ P». . «1.x rice... .1 f .1. a .., . . 5.5.93.1: , V x . ‘ tn. . v E . . . :3 xx: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 4 Michigan State University ( LIBRARY ] K This is to certify that the dissertation entitled HARMONIC BLOCH FUNCTIONS ON THE UPPER HALF SPACE presented by Hedi Ajmi has been accepted towards fulfillment of the requirements for PhoDo degree in Mathematics WM C KM/ Major professor J 29 gg 92 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE ll JL 7 =3 W T1 i T—il J MSU Is An Affirmative ActiorVEqual Opportunity Institution omens-9.1 HARMONIC BLOCH FUNCTIONS ON THE UPPER HALF SPACE By Hedi Ajmi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1992 ABSTRACT HARMONIC BLOCH FUNCTIONS ON THE UPPER HALF SPACE Hedi Ajmi It is known that a holomorphic Bloch function f on the open unit disk of the complex plane need not have radial limit at any boundary point. Nevertheless, Ullrich showed that such an f does have “boundary values” in an average sense. Furthermore, these average boundary values reflect the behavior of f inside the disk in a manner analogous to the case of bounded holomorphic functions. These results transfer easily to the upper half plane. They are also valid if we merely assume that f is harmonic; here we need only remember that a real harmonic function is Bloch if and only if its harmonic conjugate is Bloch (by Cauchy-Riemann equations). In this paper we generalize these results to harmonic Bloch functions on upper half spaces of arbitrary dimension. In higher dimensions, a new problem arises: A harmonic Bloch function may well have harmonic conjugates that are not Bloch. However, we show it is always possible to choose harmonic conjugates that are Bloch. (This choice is unique up to an additive constant.) From this we obtain several results relating the “boundary values” of a harmonic Bloch function to its behavior inside the domain. We also obtain a number of “bounded mean oscillation” characterizations of the harmonic Bloch space as well as characterizations involving higher derivatives (one of which may be a little surprising). To my father (posthume). To my dear mother. To my beautiful wife Saeeda Rayssi. To my wonderful children Ameur, Allem, and Wah'd (I have not seen him yet). To my sisters Fatma, Zohra, and Aycha. To my brothers Sidi, Ayadi, Hachmi, Chedh', Ali, Moncef, and Faycal. ACKNOWLEDGMENTS I would like to thank Dr. Wade Ramey, my dissertation advisor, for all his help, encouragement and advice. His knowledge and enthusiasm were invaluable. Besides helping with mathematics, Dr. Ramey insisted (forcefully) that I type my disserta- tion on my own, using Latex—a task that at first seemed to me unsurmountable, considering that I had never used a computer before. I do thank Dr. Ramey for this prodding. Dr. Ramey made also a tremendous positive impact on me in dealing with personal or general matters. I also thank the following professors: Dr. Sheldon Axler, Dr. Joel Shapiro, Dr. Peter Lappan and Dr. Bill Brown for agreeing to be on my commitee. The course on harmonic funtions that I took with Dr. Axler was especially helpful to me in my studies. As well I thank Dr. Michael Frazier for helpful conversations concerning Poisson integrals of BMO functions. I thank the authorities at the Ministry of Defense and at the Ministry of Education of the goverment of Tunisia for giving me the opportunity to study in the US. In that regard, two people deserve my gratitude: MM. Youssef Baraket and Habib Bouzguenda. I am also very grateful to the mathematics department at M.S.U for its support. Here also I specifically thank Dr. Charles Seebeck and Dr. David Yen. Finally I thank MM. Mondher Essid and André Burago for their help with Latex and I wish good luck to my friend Radhouane. ii Chapter 1 Chapter 2 Chapter 3 Chapter 4 Bibliography TABLE OF CONTENTS Introduction ................................................. 1 Basic properties of harmonic Bloch functions ......................................... 18 Harmonic conjugate functions ........................... 34 Boundary behavior of harmonic Bloch functions ......................................... 42 56 iii Chapter 1 Introduction Let D = {z E C : Izl < 1} denote the open unit disk of the complex plane C. A holomorphic function f on D is said to be a Bloch function if llflls = 815p (1 - |z|’)|f’(z)| < 00. The Bloch space 80(D) is the set of all holomorphic Bloch functions on D. (The sub- script a stands for “analytic” ; later we will be looking at harmonic Bloch functions.) Even though N f “3 is not a norm, we will refer to N f “B as the Bloch norm of f. The quantity | f (0)] + II f ”B does define a norm on Ba(D), and equipped with this norm, 3.,(D) is a Banach space. Let us discuss some of the history of this subject. The basic idea of a Bloch function goes back to the following remarkable theorem of André Bloch. Bloch’s theorem: There exists a finite positive constant b such that if g is a holomorphic function on D, normalized so that g(0) = 0 and g’(0) = 1, then there is a disk A contained in D on which 9 is one-to-one and such that g(A) contains a disk of radius b. Although the exact value of b is unknown, it is known that .43 < b < .47. Before we discuss the connection between Bloch’s theorem and the Bloch norm, we introduce some notation. For a E C and r > 0, we denote the set {2 E C : Iz—al < r} by D(a,r). For a holomorphic function f on D and a point z in D, let d 1(2) be the supremum of all r > O for which there exists an open connected neighborhood (I of 2: such that f is one-to-one on Q and f(fl) = D(f(z),r). (If f’(z) = 0, in which case f cannot be one-to—one on any neighborhood of 2, then we set d ,(z) = 0.) Here is the connection between Bloch’s theorem and the Bloch norm: For any holomorphic function f on D, sng) 3 1|st 5 %sgp (11(2)- This follows (easily, but not obviously) from Bloch’s theorem and the invariant form of the Schwarz Lemma. Besides this very appealing connection between the geometrically defined d f(2') and the analytically defined II f “3, a number of other interesting properties of the holomorphic Bloch space have been obtained; we discuss some of these below. The main purpose of this paper is to explore some of these properties for the harmonic Bloch space, especially in higher dimensions. A For the harmonic Bloch space, we take as our principal setting the open upper halfospace H = Hn of R” defined by H = {(a:,y) 2:1: E 11”“, y > 0}. A harmonic function u on H is said to be a Bloch function if Hulls = sup y|V(U(x,y))l < 00, where the supremum is taken over all (:r, y) E H, and Vu denotes the gradient of u. In this paper the term “harmonic” means “real-valued and harmonic” unless otherwise stated. We denote the vector space of harmonic Bloch functions on H by 8 (H) As above, we call Hulls the Bloch norm of u, even though it is not a norm; we obtain a norm on 8(H) by adding |u(0,1)| to ”“llB- We call 8(H) the harmonic Bloch space of the upper half-space. For the convenience of the reader, let us sketch a proof that, equipped with this norm, 8(H) is a Banach space. The fact that 8(H) is a real vector space is clear. If (u) is a Cauchy sequence in 8(H), then it is easy to see that (Vuk) is uniformly Cauchy on compact subsets of H. Therefore, because (u) converges at the point (0,1), (u) must converge uniformly on compact subsets of H to some function u. The function u is harmonic, and it is easy to see that u E 8(H) and that (uh) converges to u in norm. Letting B = Bn = {2: E R" : le < 1} denote the unit ball of R", we could also discuss the Bloch space 8 (B ) (whose definition can be guessed easily). When n = 2, there is no great difference between 8(H) and 8(3). Indeed, the map iz+l ¢= 2+, is easily seen to induce an isomorphism between these two spaces. In higher dimen- sions, the natural replacement for the map 43 is an appropriate modification of the Kelvin transform (see 7.15 of [ABR]). Unfortunately, this map does not carry 8(B) into 8(H). Thus when n > 2, there is no obvious correspondence between 8 (B) and 8 (H). So why choose H over B? The main reason is that in H, we have a transitive group of self-mappings that preserves harmonic functions—namely the family of maps generated by horizontal translations and dilations. This is a luxury that we don’t find in B. We do have the rotations of B (which correspond to horizontal translations of H), but there is nothing in B resembling the dilation structure of H (which allows us to pull all of H in towards the point 0 in the boundary of H). On the other hand, the fact that H is unbounded creates a few problems of its own. In the remainder of this introduction, we discuss some analogies between 86(D) and 8 (H) The numbered theorems below are the ones whose proofs appear later in this paper. GROWTH ESTIMATES: The next theorem, which is well-known, shows that a holo- morphic Bloch function cannot grow faster than a logarithm near the boundary of D. Theorem: Iff E 8(D), then 1 |f(2)| S |f(0)| + llfllslosfil—zl for all z 6 D. This easily follows from integrating f’ from 0 to z. A similar type of growth estimate (requiring a little more work) is valid in 8(H). 2.4 Theorem: Ifu E 8(H), then IU($,y)l S lu(0a1)l+llull8ll +|103y|+ 2108(1+l$l)l for all (x,y) E H. CONFORMAL INVARIANCE: The invariant form of the Schwarz’s Lemma states that if (,0 : D ——+ D is a holomorphic function, then (1 - |2|2)| 0. For such a and r and any function u defined on H, we may define the horizontal translation Tau by “raider, 3/) = "(x + a, y), and the r-dilate u, by u,(z) = u(rz). Clearly, u, and Tau are harmonic on H whenever u is harmonic on H. Also, a straightforward computation shows ll‘raUIls = IIUIIB = Ilurlls for all a and r as above. Hence, 8(H) is invariant under horizontal translations and dilations. We’ll see later on that the dilation invariance of the Bloch norm is crucial. This is a property that the Bloch norm shares in common with the L°°-norm and the EMU- norm. We should thus expect (and will prove later) that 8(H) “behaves like” an L°°-space or a BMO-space in some respects. Note that most norms do not have this property. For example, the LP-norm is not dilation invariant if 1 S p < 00. When n = 2, we have more than horizontal translations and dilations. For ex- ample, the map 2 -> 1 /E = z/Izl2 preserves H and harmonicity, and allows us to interchange the boundary points 0 and 00. It also preserves the Bloch space (this is straightforward computation). When n > 2, we could hope that the Kelvin trans- form, which preservee H and harmonicity (see 4.4 of [ABR]), has analogous properties. Recall that the Kelvin transform of a function u defined on H is Klu](z) = |2|""u(-|zi|,-) for all z E H. Unfortunately, the Kelvin transform does not preserve 8(H) when n > 2. Indeed, letting u E l we have K[l](z) = |z|"“ ¢ 8(H), since yIVK[1](0,y)| = (n —2)y2'" is unbounded. Thus, when n > 2, there seem to be no other self-mappings of H, other than the ones generated by the dilations and the horizontal translations, that preserve 8 (H) HIGHER DERIVATIVES: In dealing with higher derivatives the following notation is useful. We denote the k-th derivative of a holomorphic function f by fl“). In the several variable case, we define a multi—index a to be an n-tuple of nonnegative integers (011, . . . ,an). We use |a| to mean 011 + . . . + an. We denote the of,“ partial derivative with respect to the j‘“ coordinate variable by Df", and when j = n we often write D3“ instead of 03*. The partial differentiation operator D“ is defined to be Di" . . . Dz" (D? denotes the identity operator). The following theorem characterizes the holomorphic Bloch functions in terms of their higher derivatives. Here (and in the rest of this paper) the expression A( f) z B( f) means that there are two positive constants c and C such that the nonnegative quantities A( f), B( f) satisfy the inequalities 0A(f) S B(f) S CA(f) for all f under consideration. Theorem: Let f be holomorphic on D, and let m be an integer greater than 1. Then llflls % 8gp (1 - IZI)”lf(’"’(Z)I + Z |f("’(0)l- k=1 A straightforward induction argument, using Cauchy’s estimates and the funda- mental theorem of calculus, gives the theorem. One might think that a natural analogue in H for the above theorem would be: If u is harmonic on H and m is an integer greater than 1, then 1-2 IIUIIIW 821p 31'" Z |D°U($,y)l+ Z |D°'U(0.1)l- Ia|=m Ial 1. In fact, if m > 1, then 1.2 fails for any nonconstant harmonic polynomial of degree less than m. The reason is that, unlike a holomorphic polynomial (which is in 8.,(D)), a nonconstant harmonic polynomial is never in 8(H). Thus, the harmonic polynomials should somehow be taken into consideration in the statement of the analogue of the above theorem in H. We do this in the following way: Let ’Pm denote the set of all harmonic polynomials of degree less or equal to m and define P". = ' f llu + H pg,” ||u + PllB for any harmonic function u on H. Note that although the above quantity could be infinite for some u, it behaves like a quotient norm in the sense that it vanishes on ’Pm. We are now able to state the analogue we are after in 8(H). 2.11 Theorem: If u is a harmonic function on H and m is a positive integer, then ||u+7’ -1||~8;1{p y'" 2: |D°UI- Ial=m If we know ahead of time that u is a Bloch function, then we obtain the following equivalence. 2.16 Theorem: If u 6 8(H) and m is a positive integer, then Hulls z sup y’” 2 |D°u| z sup y”: [Dr'leuL H |a|=m H k=1 REMARKS: 1. The last quantity in 2.16 contains only first order partial derivatives with respect to the first n - 1 variables x1, . . . ,mn_1, whereas the second quantity contains partial derivatives with respect to 1:1, . . . ,xn_1 of order less or equal to m. This is the surprising fact mentioned in the abstract; it seems to suggest that for a Bloch function, the partial derivatives with respect to y are more important. 2. Because Hulls does not depend on m, we also deduce that for different integers m1, m2 sup y'”l |D°’u| z sup 3”"2 z |D°u|. H lal=m1 Ia|=mg 3. The condition |a| = m in 2.16 can be replaced by la] 3 m. 4. Unlike the case of 8.,(D), neither 2.11 nor 2.16 involves evaluating functions at some fixed point in H. BMO CONDITIONS: Here we’ll see that membership in 8 (H) is equivalent to satisfying certain BMO-type conditions. The realization that the Bloch space can be viewed as a BMO-type space first occurred in [CRW]; see also [A]. For completeness, we prove one of these results for 80(D). We let dA denote Lebesgue area measure on C, normalized so that A(D) = 1, and we let Da = {z E D: Iz — a| < (1 — |a|)/2}. Theorem: Let 1 S p < co and let f be a holomorphic function on D. Then 1 llflls z sup {3&7 l». |f(z) — f(a)|’dA}; , where the supremum is taken over all a E D. We now prove this theorem; the proof comes from [RUl]. (We will need some of these ideas in Chapter 2). Let a be in D and 2 be in Da. Fix p E [1,oo). The inequalities lf(2)-f(a)| s Allz-allf’(a+t(z-a))ldt al 1 Iz— < _ llflla/o l—|a+t(z—a)|dt Ilflls |/\ (the last inequality uses the fact that |z — al < (1 — |a|)/2) easily show that 060 sup (250—) j . |f(z) - “away S Hill. For the other inequality, we use the fact that 1.3 m0): 5 [Blind/1 for all f holomorphic on D (recall A(D) = 1). Now fix a E D and let r = (1 — |a|)/2. We apply 1.3 to g(z) = f(rz + a) — f(a), getting rlf’(a)l s /D wrz + a) — f(a)|dA(2) = ri, D. lf(z) — f(a)ldA(z) .1 _<. {,i, lo. we) — f(a)|’dA(z)}'. the conclusion of the theorem follows.D REMARK: The conclusion of this theorem is actually valid for all p E (0,00). 10 We now state a more ambitious version of the above theorem that we will prove in Chapter 2. We let B(a,r) = {z E R“ : Iz — a| < r} and fl(a,r) = B(a,r) n H. Given u E Ll(fl), we define l =— d. an lfll/QuV Here dV denotes the Lebesgue volume measure and III] denotes the volume of Q. 2.17 Theorem: Suppose u is a harmonic function on H that is volume integrable on every bounded subset of H. If 1 _<_ p < 00, then In 1 p 2.18 ||u||3~sup{lfi|/‘;Iu—u9|”dV} , where the supremum is taken over all 9 = Q(a,r) with a in the closure of H. Note that in this theorem, we don’t restrict Q to be a ball that stays away from 3H (as we did earlier for 8..(D)). For example, (I can be a ball tangent to 3H, or half a ball with center in 0H, or part of a ball with center in H. In the context of 8.,(D), the authors in [CRW] also allowed such general Q’s; however, our proof of 2.17 will not rely on the machinery developed in [CRW]. There is another BMO-condition that characterizes 8 (H), this one involving the Hardy-space norm. For 8.,(D), this first appeared in [RUl]. Recall that if u is a harmonic function on the open unit ball of R" and 1 _<_ p < 00, then the Hardy-space norm of u is 1 P Hull» = sup [/ |u(r<)|’da(C)] , 05r<1 S where 5 denotes the unit sphere of R" and do denotes the normalized surface area measure on 5, so that 0(5) = 1. 11 2.20 Theorem: If u is a harmonic function on H and l S p < 00, then IIUIIB 8 8UP IMO + TI) - “(a)llhr, where the supremum is taken over all a, r such that B(a,r) C H. In particular, Theorem 2.20 shows that if u E 8(H), then “u belongs to h” of every ball in H tangent to 6H”. HARMONIC CONJUGATES: Here we explore the notion of harmonic conjugates in higher dimensions. This subject is not as widely known as in the case n = 2. Specif- ically, we’ll address the following question: Given u 6 8 (H), will its harmonic conju- gates also be in 8(H)? We first make clear what we mean by harmonic conjugates in higher dimensions. Let u be a harmonic function on H. The functions v1, . . . , vn_1 on H are said to be harmonic conjugates of u if 1.4 (v1,...,v,,_1,u)=Vf for some harmonic function f on H. If such vj’s exist, then they must be harmonic, being partial derivatives of a harmonic function. Also 1.4 and the condition that f is harmonic are equivalent to the “generalized Cauchy-Riemann” equations 1.5 Dkvj = Djvk ; Dyvj = Dju 73-1 1.6 2 Djvj-l—Dyu = 0. .=1 Indeed, if 1.4 holds, then 1.5 will hold since f is twice continuously differentiable on H; also 1.6 easily follows from the fact that f is harmonic. Conversely, if 1.5 and 1.6 hold, then the differential form n 1 Zvjdxj +udy i=1 12 is closed. Therefore, since H is simply connected, 1.4 holds (see Theorem 8.4 of [Fl]) for some function f, which is harmonic by 1.6. The terminology “harmonic conjugates” comes from the case n = 2: By the Cauchy-Riemann equations, 1.5 and 1.6 hold if and only if u + ivl is holomorphic on H, as easily checked. Here of course the harmonic conjugate exists and is unique up to an additive constant. Thus, 1.4 is a natural generalization of the case n = 2. However, showing the existence of the harmonic conjugates in the case n > 2 requires more work. There is also a greater degree of non-uniqueness of harmonic conjugates when n > 2, as we show below. For now, we just state the following: 3.5 Theorem: If u is harmonic on H, then harmonic conjugates of it exist. We need more than just the existence of harmonic conjugates. We now return to our basic question: If u E 8(H), need the harmonic conjugates v1, . . . ,vn_1 belong to 8(H)? In the case n = 2 it is easy to see via the Cauchy-Riemann equations that if u is Bloch, then so is any of its conjugates. In the case n > 2 it is not true that all the conjugates of a Bloch function are Bloch. For example, letting u E 0, then v1(a:1,a:2,y) = 1:1 and v2(a:1,a:2,y) = —.2:2 are harmonic conjugates for u that are not Bloch (here f(m1,a:2,y) = (z? -- z§)/2). On the other hand, v1(.v1,:r2,y) = a and v2(a:1,:c2,y) = b (where a,b are constants) are also conjugates for u (here f (3:1, 9:2, y) = axl + (1.132) and v1, v; are this time Bloch. The next theorem shows that the above example is typical: Harmonic conjugates of a Bloch function may always be chosen to lie in the Bloch space. 13 3.12 Theorem: Let u E 8(H). Then there exist unique harmonic conjugates v1,.. .,v,,-1 of u on H such that v; E 8(H) and vJ-(0,1) = 0 for each j. Moreover, there exists a constant M, depending only on n, such that ”UjIIB S M IIuIIB for each j. This theorem will be very useful in studying the boundary behavior of the har- monic Bloch functions on H. BOUNDARY BEHAVIOR OF BLOCH FUNCTIONS: There are several well-known re- sults in the literature concerning the behavior of harmonic functions at an individual boundary point. Perhaps the best-known of these is due to Fatou. Let p be a com— plex Bore] measure on R, and let u denote the Poisson integral of u so that u is well defined and harmonic in the upper half-plane. Fatou showed that . l h 1'7 III—131+ $1.}. d” _ L implies 1.8 1,135: u(zy) = L, where L E C (see [Fa]). In general 1.8 does not imply 1.7, as Loomis showed in [L]. But in the same paper Loomis proved that 1.8 implies 1.7 if the measure [1 is assumed to be positive. It follows that 1.8 implies 1.7 if u = fdz, where f E L°°(R) and drr is the Lebesgue measure on R. However, 1.8 need not imply 1.7 if f E LP(R) and p < oo; in fact there is an f E 0““, L’(R) for which 1.8 does not imply 1.7. (We recall some classical terminology: The implication 1.7 =3» 1.8, which is true under a wide array of conditions, is called the “abelian” direction. The direction 1.8 => 1.7 is called the “tauberian” direction; it holds only when a certain condition—a “tauberian” 14 condition—is added. Thus positivity is the tauberian condition for the Loomis result.) W.Rudin generalized the Loomis result for positive measures to higher dimensions; he also showed that this result fails when L = +00 (see [Rl]). W.Ramey and D.Ullrich gave a different proof of Rudin’s result and extended it to the case dp = fdx, where f E BM 0. They also showed that in this latter case, 1.8 still implies 1.7 if f is real-valued and L = ioo (see [RU2]). The techniques used by Ullrich and Ramey in [RU2]—most importantly, dilation invariance and normal families—are available for the class of Bloch functions. Thus, it seems natural to ask about the relation between 1.7 and 1.8 for Bloch functions. But we run into an immediate difficulty: If u is a Bloch function, then u need not be the Poisson integral of a measure 11 on 0H. Indeed, if every Bloch function were a Poisson integral, then every Bloch function would have almost everywhere boundary limits by the Fatou Theorem. But it is well known that there are Bloch functions that fail to have almost everywhere boundary limits. For example, on D, f(2) = f: 22“ k=o is such a function (see [ACP])‘. (After a conformal mapping, we obtain a similar example for H.) Thus, in discussing the relation between 1.7 and 1.8 for Bloch functions, we need something on the boundary in place of the measure p. One possible way to arrive at a “boundary function” for a Bloch function is through the theory of distributions: 4.1 Theorem: If u is a Bloch function in the upper-half space, then jig; wumwwx exists for every smooth function (,0 with compact support. Moreover, this limit defines a distribution on the space of smooth test functions with compact support. 15 Of course when u is the Poisson integral of a measure p as above, then the dis- tribution of Theorem 4.1 is just the measure p. However, for our problem we need to go beyond the theory of distributions, because in 1.7 we are integrating p not against a smooth test function but against the characteristic function of an interval. This is what Ullrich did in [U]. More precisely, given a holomorphic function f on D, 0 0. We first show that the integrals Igu have good limiting behavior as h —) 0 if Q is nice enough. 16 4.2 Theorem: Suppose u E 8(H), Q is a bounded open set with (II-boundary, and 0 E (I. Then Inn = Ill—Ii!) 1311 exists and defines a bounded linear functional on 8(H). Moreover, there is a constant C, depending only on n, such that Illnll S C(1+Iaflltl-t|a|)(1+|1080’|), where d equals the diameter of Q. REMARKS: The operator norm lllgll is, of course, computed relative to the norm ||u||5 + Iu(0, l)| of 8(H). By pm we mean the surface area measure of 39. The proof of Theorem 4.2 uses the divergence theorem, as we’ll see in Chapter 4. Thus, the condition that 80 is 01 could easily be replaced by the weaker condition that an is piecewise C’. This suggests a natural question: What is the largest class of It’s for which Theorem 4.2 will hold? We have not been able to settle this question. To discuss the equivalence between 1.7 and 1.8 for 8 (H), we specialize to the case where Q is a ball centerd at the origin. Let us define the averages , 1 Aru — 11-13(1) l—B-r—l L'- ‘ll($, h)d$, where B, is the ball in R""1 centered at the origin of radius r. The proof of the above theorem will give us the estimate IIArUIl S C(1+|10s'"|)(||u||s + IU(0,1)|) for all r > 0. This estimate will be instrumental in proving the equivalence of 1.7 and 1.8 for harmonic Bloch functions on H, which is what the next theorem asserts. 17 4.14 Theorem: Ifu E 8(H) and L 6 {—00, +00], then [ir%u(0,y) = L if and only if [mtg/1m = L. Chapter 2 Basic Properties of Harmonic Bloch Functions The Cauchy-Riemann equations show that a real-valued harmonic function on D is Bloch if and only if it is the real part of a holomorphic Bloch function on D. Hence, most properties of real-valued harmonic functions on D can be obtained by studying the corresponding properties of holomorphic Bloch functions on D. Thus, in a way, the harmonic Bloch space on D, which we denote 8(D), is as well-known as 8.,(D). In this chapter we focus on the less-known harmonic Bloch space 8(H), where H: {2: (x,y):x E R”", y >0}. A real-valued harmonic function on H is said to be a Bloch function if sup y|V(u(:c,y))l < 00, where the supremum is taken over all (x, y) E H, and Vu denotes the gradient of u. The simplest Bloch functions on H are the real-valued bounded harmonic func- tions: For if u is harmonic and bounded on H, then by Cauchy’s estimates, there is a constant C such that C C [VU(2)[ —<- (1(2, CH) 2 3' 18 19 for all z = (x,y) e H (see 2.6 of [ABR]). As is well-known, the class of bounded harmonic functions on H is exactly the set of Poisson integrals of bounded measurable functions on 11""1 (see 7.14(b) of [ABR]). Thus u is bounded and harmonic on H if and only if u = P[f], where f E L°°(R”“) and 2.1 Plfl(:c.y)= [RM (Ix :ntl’fiti’fidt' Here cu is chosen so that / dt _ 1 6" nn-x (|t|2 +1)? _ ' We can easily check (for later purposes) that 2.1 defines a harmonic function on H whenever f is a measurable function on HP"1 such that |f(i)| 2.2 ————-dt . (an-1mm): < °° Examples of unbounded Bloch functions on H are the functions u(x,y) = log(x,2,+y2) k=1,...,n-1, as easily checked. These particular functions are, respectively, the Poisson inte- grals of the functions 2log|tk| E BMO(R""1). In fact, P[f] E 8(H) whenever f E BM 0(Rn'1). To see this, let f E BM 0. Then f satisfies 2.2 (see [FS]). More- over, setting u = P[ f], we know by [F8] that yIVu(x,y)|2dV is a Carleson measure on H. This means that there is a positive constant A such that 2.3 [C y|Vu(x,y)|2dV 5 Ah”"1 a,h for all a E R’"1 and h > 0. Here Cay. is the cylinder {Ix — a| < h} x (0, h). Now fix 2 = (x,y) E H, and let 3; = B(z,y/2). Because I'Vul2 is subharmonic, we have y’qu(z)l2 s y” {Viz—I [8’ |Vu(s,t)|2dsdt} _<_ 2" lel [B t|Vu(s,t)|2dsdt. 20 Note that [EA = cry" (where a is a constant depending only on 11). Also note that B, is contained in 0,3,”. Enlarging the domain of integration from B2 to C33,”, we see that inequality 2.3 implies that y|Vu(x,y)| is bounded on H; i.e, u E 8(H) as desired. (I thank Dr. Frazier for pointing out this simple way of seeing that P[f] E 8(H) whenever f E BMO.) However, not every harmonic Bloch function is the Poisson integral of a BMO function. Indeed, it is well-known that the holomorphic Bloch function on D 9(2) = Z 22" k=1 has finite boundary values nowhere on 8D. Hence, by Fatou’s theorem, g cannot be the Poisson integral of any f E L1(8D). (After a conformal mapping we obtain a similar example for H.) Recall in Chapter 1, we introduced the horizontal translations and dilations of a function defined on H, and asked the reader to check that the Bloch norm is invariant under these transformations. We will also need the vertical translations: If u is a function on H, then its vertical translate Thu is the function on H defined by nub, y) = U(w, y + h) for h > 0. If u E 8(H), then Thu is also in 8(H), and ”nulls _<_ Ilullg. Indeed, the harmonicity of mu is clear, and yllUllB y+ h yIVnu(x.y)l = yIVU(z.y+ h)l S S Hulls for all (x,y) E H. We’ll soon see that Thu equals P[u(-,h)], which will be important for our work later. But in order for P[u(-,h)] to make sense, we need a growth estimate on u. That’s the object of the next theorem, which shows that a Bloch function can grow 21 no faster than [log y| near 6H and 00. 2.4 Theorem: Ifu E 8(H), then IU($,y)| S IU(0,1)|+ ||u||8[1+ |logy|+ 21<>s(1+|fl¢|)l- PROOF: Fix (x,y) 6 H. We have that U |u(x.y) - no.1): = I /. D.u(z,s)dsl y u = IIUIlsllogyl- Hence, |u($,y)| S IU($,1)| + “Ullallosyl- To get an estimate on |u(x,1)|, we let r = 1 + le and use the triangle inequality x l x x |u(x, 1) _ "(0, 1)l S lur(;a ;) "' ur(;11)l+luf(:11)” 11,-“), 1)l + |u(01r) " "(091M Denote I, II, III respectively the three terms on the right hand side of the inequality. We then have 11 = |u(x,r) _ "(O’Tll u s sup mama sU—l'i’lzl swung; ‘6 [(W‘). (0.?” r 111 = |/ Dyu(0,t)dt| l S ||u||310gr. (Note that the dilation invariance of the Bloch norm was used in I ) The conclusion of the theorem follows.Cl 22 We are now able to prove an important property of the Bloch functions in the upper half-space: The vertical translates of a Bloch function are the Poisson integrals of their boundary values. More precisely, we have 2.5 Theorem: Ifu E 8(H), then W! = PHM» ’0] for every h > 0. For the proof of this theorem, we need the following three lemmas. 2.6 Lemma: If f is continuous on R""1 and 2.2 holds, then P[ f] can be extended continuously to the closure of H, with boundary values f. PROOF: This is a simple variation on the proof that the Dirichlet problem for H with bounded continuous boundary data is solvablefl 2.7 Lemma: Ifu is harmonic on R" and IU($)I S A(1+lflvl’") (1' E R") for some constant A, then u is a polynomial of degree S p. PROOF: Fix x E R" and let a be a multi-index. By Cauchy’s estimates, there is a positive constant Ca such that < CaA(1+ r?) _ rial ID“ "(ail for all r > le. Letting r go to infinity gives us D"u(x) = 0 for all a such that |a| > p. Thus, u is a polynomial of degree S p.El 23 2.8 Lemma: If f is a measurable function on R"" and there exists p 6 (0,1) such that |f(i)| S A+ Bltl” for some constants A and B, then there exist constants C and D such that lPlfl(z.y)| S C + Dl(z,y)l" ((x,y) 6 H). For the proof of this lemma we need the polar coordinate formula for integration on R": If g is a Borel measurable, integrable function on R", then 2.9 Ell—Bl 110ng = [:0 r"'1/Sg(r()do(C)dr. PROOF OF LEMMA 2.8: Let f be as in the statement of the lemma. The fact that f satisfies the condition 2.2 (and hence P[ f] is well defined on H) follows easily by using 2.9. We also have dt |P[fl(2,y)| s [R “(A+B""”’ "-1 (it2 + It -z|’)i y W "-1 (y2 + lt - avl’)i AB/ +c.1R for all (x,y) 6 H. We split the last integral into two parts, by integrating over {|t| < 2|xl} and over {|t| _>_ 2|xl}. In the first part, we replace |th in the numerator by |2x|P and'integrate over all of Rn'l. We then see that the first part is bounded by a constant times IxI". In the second part, we use the estimate 2 2) 2 BE 31 +lt-zl _y + 4 (valid for the domain of integration), then integrate over all of R""1 using the change of variables t = ys. We then see that the second part is bounded by a constant times y”. The conclusion of the lemma follows.Cl 24 PROOF OF THEOREM 2.5: Let u E 8(H) and fix h > 0. We easily check using the growth estimate 2.4 and the formula 2.9 that u(t, h) satisfies the condition 2.2. Define on H the harmonic functions v = P[u(-, h)], w = Thu — v. By Lemma 2.6, v extends to be continuous on the closure of H with boundary values u(-, h). Hence, w extends to be continuous on the closure of H with boundary values identically 0. We now extend w to all of R” by setting w(x, —y) = —w(x, y). By the Schwarz Reflection Principle, w is harmonic on R" (see 4.9 of [ABR]). We now show that w E 0 on R". We have by the growth estimate 2.4 that Mt. h)| S |u(0,1)l+llullsl1+l1°8 hI + 2108(1+|t|)l < A + Bltli for all t 6 RP“ (Here we choose the exponent 1/2 for no particular reason; any positive exponent less than 1 will do.) Therefore, by Lemma 2.8 wosMSA+mosw for all (x,y) E H. Also, by the growth estimate 2.4, IThU(x,y)l S IU(0,1)|+||u||Bl1+I103(y+h)|+2108(1+|$|)l s A + Bl(z,y)li for all (x,y) E H. Hence, wasnshwosn+was13A+anmt for all (x,y) E H. Therefore w is a polynomial of degree 0 (by Lemma 2.7), hence w E 0 on R" (since w E 0 on 6H). Thus 17.11 = P[u(-,h)] on H as desiredfl 25 In the remainder of this chapter, we characterize the Bloch functions on H in terms of their higher derivatives, and in terms of two bounded mean oscillation con- ditions. Here is first a simple characterization of the Bloch functions in terms of the last coordinate variable y. It shows that the constant functions are the only Bloch functions in H that don’t depend on y. 2.10 Lemma: Ifu E 8(H) doesn’t depend on y, then u is constant. PROOF: Our hypothesis implies that there exists a smooth function f defined on R"'1 such that u(x,y) = f(x) for all (x,y) E H. Therefore IVf($)l = IVU(1=,3I)| IIUIIB y S for all (x,y) E H. Letting y —-> 00, we see that Vf(x) = 0 for all x E Rn'l. Thus (since H is connected), f is constant, and therefore so is u.D Before we come to our characterization of the Bloch functions in terms of their higher derivatives, recall from chapter 1 that P". denotes the set of all harmonic polynomials of degree less or equal to m and that for any harmonic function u on H we define |lu+Pm|| =pg;,fmllu+plls- 2.11 Theorem: If u is a harmonic function on H and m is a positive integer, then ||u+Pm-1|| z 8}? 31'" Z lDa‘ul- |a|=m To prove the theorem we need the following two calculus-type lemmas. 26 2.12 Lemma: If a smooth function f defined on (0,00) satisfies $k+1|f(k’($)| S M for some integer lc 2 1, then there exists a polynomial pk_1 of degree S k — 1 such that 211(2) —p.-.(x)l s -’,:—f (. e (o , co». PROOF: By induction on k. Let k = 1. We show first that f (00) = limxnoo f (x) exists. For t S x we have |f(x)—f(t)l s [In 5 M Il-ll x t < w _ t This shows that f (00) exists. Because °° M _ < ’ < __ m2) f(00)| _. L Ifl _. x. we are done in the case I: = 1 (take p0 = f(oo)). Now suppose the lemma is true for k and x"+2|f("+1)(x)| S M. As above, this implies that f(“)(oo) exists. Furthermore, |f""(x) — f""(oo)l s j: |f"‘+"| s (Ki—4);? so that i xk+1|f(k)(z) _ fl")(oo)l _<_ +1. a. Hence _M_. (k) I: x"“|(f(:v)--——-f (°°"‘ )‘k’l k+l 1:! IA 27 Thus (by induction hypothesis), there is a polynomial p14 such that f(k) I: M ”(33) - ‘—_(§)i ‘Pk-ll S W Taking pk(x) = x“ f (“(00) / kl+ p1,-1 shows that the lemma is true for k+ 1. Therefore, the lemma is true for all positive integers [CD 2.13 Lemma: If u is a smooth function on H such that M=sup y'" 2 |D°u| < 00 H lal=m for some integer m 2 2, then there is a polynomial pm-1 of degree less or equal to m — 1 and a positive constant C, depending only on m and n, such that sup y“ 2 |D°(u-pm—1)| S CM- |a|=2 PROOF: By induction on m. The case m = 2 is clear (any linear polynomial will do). Now suppose the lemma is true for m and that spip y’”+1 E: |D°u| = M < oo. |a|=m+1 Fix x E 11"" and let 8 be a multi-index such that [HI = m. Define f(y) = D"u(x,y) for y > 0. We easily see that y'"“|f’(y)| S M for all y > 0. Therefore, because M is finite, f(oo) = L exists (see the proof of Lemma 2.11). Furthermore, L is independent of x. Indeed, the inequality MIrL‘l |Dfiu(z,y)- D”U(0.y)| S sap |D°u(2)l I-rl S ym+l 26 [(xvy)o (0.1!” for all (x,y) E H, shows that L =f(oo) = jig; Dfium) = ”119310 D511“), 3]). 28 It also follows from the fundamental theorem of calculus that y’"|f(y) - Ll = y'" IDBU(x,y) - LIS for all (x,y) 6 H. Hence y"‘|D”(u - qzs)(x, y))| S M for all (x,y) E H, where qg is a polynomial of degree less or equal to m such that Df’qg = L. Thus, 8:11) y'" ID"(u - qn)l S M. Letting q = Elfil=m Qg and C equal the number of possible multi-indices 8 such that [8| = m, we obtain 811p 11'" Z IDB(u - q)| S CM. H Ifil=m Hence (by induction hypothesis), there is a polynomial pm..1 such that 811p 31’ Z |D°(u-q-pm—1)| S CM- |a|=2 Taking pm = q — pm_1 shows that the Lemma is true for m + 1. Thus the Lemma is true for all integers m 2 2.0 If we add to the hypotheses of the above lemma the assumption that u is harmonic, we obtain 2.14 Corollary: Let u and pm_1 be as in Lemma 2.13. Ifu is harmonic on H, then pm_1 is harmonic on H. PROOF: Because u is harmonic, we obtain (from Lemma 2.13) 811p VINu - pm—1)l= sup y’IA(pm-1)| S CM- H H 29 Here A denotes the Laplacian: A = D: + - - - + D3,. Because A(pm-1) is a polynomial, the above inequality is possible only when A(pm_1) = 0.0 PROOF OF THEOREM 2.11: The case m = 1 is clear, so we assume that m > 1. By Cauchy’s Estimates, there is a positive constant C such that splp y'” z |D°’u| = SUP If" 2 |D°'(u+p)| l°l=m Ial=m S CH“ + File for all harmonic functions u on H and all p E P -1. Therefore, 811p y'" X |D°u| S Cllu + Pm-lll- H |a|=m For the other inequality, fix u as in the statement of the theorem and assume that sup y'” z: |D°u| = M < 00. H |a|=m We’ll show that there is a polynomial q E ’Pm-1 such that ”u — qllg is less than or equal to a constant multiple of M. To do that, we fix x E R""1 and j E {1, . . . ,n}. Define f(y) = Dj(u - pm_1)(x,y) for y > 0; pm_1 is as in 2.13. Then y’lf’(y)| = y’IDuDJ-(u — pm_1)(x,y)| S GM for all y > 0. Hence, f (00) = L,- exists. Moreover, L, is independent of x (as in the proof of 2.13). Furthermore, by the fundamental theorem of calculus, we obtain 2‘15 lej(u-pm—1)(z.y)-Lj| = lej(U(x,y)-pm—1(x,y)-Lj$j)l S CM for all (x,y) E H. Now take q($, y) = I’m-105,31)" 2 L198- i=1 30 Because u is harmonic, we obtain by 2.14 that pm-1 is harmonic. Thus q is harmonic. Therefore, q E ’Pm_1. Also, using 2.15, we easily obtain that IIu—qIIB S nCM. Thus, llu+7’ -1|l S 7108111) 31'" Z |D°U(x,y)l H Ialzm for all harmonic function u on H for which sup y'” 2 |D°’u| < oo. Ial=m For the rest of the harmonic functions u on H, we may take the same constant nC as above so that llu+ Pm—nll S "0821) y'" 2 ID"U(x.y)| |a|=m for all harmonic functions u on H .D If we know ahead of time that u 6 8(H), then we can say more. 2.16 Theorem: Let u be a Bloch function on H and let m be a positive integer. Then Halls z 82(1) 31’" Z ID°u(w,y)| “3213 ymZIDZ‘"D.-u(2,y)l- |a|=rn i=1 PROOF: Denote the three quantities of the theorem from left to right I, II, III. We’ll prove: I a: II, and I z III. We start by proving I z II. The fact that II is less or equal to a constant multiple of I follows from Cauchy’s Estimates. For the other inequality, we apply Theorem 2.11. Fix 6 > 0. Because ”u + ’Pm-1ll S IIuIIB < 00, there is a harmonic polynomial p( of degree less or equal to m — 1 such that |lu+Pe||s S €+|lu+P -1|| S e+ Csup y'” z |D°u(x,y)| < 00 (by Theorem 2.11). H lal=m 31 Hence, since ||p¢||3 S “u +p¢II5 + Hulls, we get pc is Bloch. Because 8(H) does not contain any nonconstant polynomial, pc must be constant. Thus, Hu + Pells = IIuIIB. Therefore, IIUHB S HOS}? y’" X |D°u(z.y)l- |a|=m Letting 6 go to 0 gives the desired inequality. The fact that III is less or equal to a constant multiple of I is again an easy consequence of Cauchy’s Estimates. The other inequality follows by an easy induction on m and by using the fundamental theorem of calculusfl We now look at two BMO-type conditions, each of which characterizes the Bloch functions on H. Recall that Q(a,r) = {[2 — a| < r} ('1 H. Here is the first one: 2.17 Theorem: Suppose u is a harmonic function on H that is volume integrable on every bounded subset ofH. If] S p < 00, then |~ P 1 2.18 ||u||3zsup{fiLIu—uglpdV} , where the supremum is taken over all (I = fl(a,r) with a in the closure of H. PROOF: Let u and p be as in the statement of the theorem. For a E H, put B, = B(a,a,,/2). Noting that u(a) = 118., for u harmonic on H, we have 1 p 2.19 HullesuP{I-I%I/B. |u—u(a)|PdV} , where the supremum is taken over all a E H. The proof of 2.19 is almost the same as the one given on page 9 of Chapter 1; we leave the easy changes to the reader. The equivalence 2.19 shows that ||u||3 is less than or equal to a constant times the right side of 2.18. For the other direction in 2.18, dilation and translation invariance will be crucial. After an appropriate dilation, as well as vertical and horizontal translations, we may 32 assume that Q = Q(a,r) touches 6H, that r = l, and that a lies on the y-axis. The triangle inequality gives 1 l 1 ' 1 ' {Wyn—awn} s{l-,,—I/alu—u(o.1)rdV} +Iun—u(0.1)l- Writing ug - u(O, 1) as (l/Ifll) fn(u — u(0,1))dV, and applying Jensen’s inequality, we see that the above is less than or equal to .1. r 2 {fill—If“ Iu _ u(0,1)|’dV} . Now with our assumptions on (I, we have 9 C C, where C = {le < 1} x (0,2). We also have III] 2 IBn(0,1)|/2. Thus the last expression displayed is less than or equal to a constant (depending only on n and p) times {fa [u _ u(0,1)|pdV}% . The growth estimate 2.4 and an easy integration now finishes the proof.D. The second BMO-type condition involves the Hardy-space norms || - |th on B (see page 10 of Chapter 1). 2.20 Theorem: If u is a harmonic function on H and l S p < 00, then II‘UIIB z 8UP IMO + m) - u(a)||hn where the supremum is taken over all a, r such that B(a,r) C H. For the proof of the theorem, we need the following lemma, which can be proved by using A.7 of [ABR] and then switching to polar coordinates. 2.21 Lemma: Up 6 (0,00), then f. P m (la-(C) < 00. Cu 33 PROOF OF THEOREM 2.20: Fix p 6 [1,00) and B(a,r) C H. Note that since B(a, r) C H, the radius r of the ball B(a,r) cannot exceed an, the n“ coordinate of a. It is clear that the function u(a + rx) - u(a) is well defined and harmonic on B. Now fix 3 6 [0,1) and C E S. We have |u(a + rsC) — u(a)| = l/ol Vu(a + tsrC) - rsttI [01 |Vu(a + tsr()||rsC|dt [1 rnuus d, o an-l-rant 1 S fi‘kdt. o 1+ant In the last inequality we used the fact that r S a... If (n _>_ 0, then obviously the last l/\ |/\ integral is less than or equal to ““llB- If (n > 0, the last integrand is dominated by Hulls/(l + (at), which implies that 103(1 + C.) (n We now have an estimate independent of s 6 [0,1). The definition of the hP-norm IU(a + NO - u(GUI S Hulls and Lemma 2.21 now show that there is a constant C, depending only on n and p, such that IIU(a + rx) - “(0)!th S CHulls- For the other inequality, note that if v 6 h”(B), then 2.22 IVv(0)| S C'llvllhn. where C is a positive constant depending only on n and p. (This follows from the Poisson integral representation formula of v.) Now fix a = (x, y) in H and apply 2.22 to v(z) = u(a + yz) —- u(a). We obtain, since |Vv(0)| = y|Vu(a)|, yIV"(a)l S CHM“ + 312) - “(OHM- Taking the sup on both sides of the above inequality yields the desired inequalityD Chapter 3 Conjugate Harmonic Functions THE UPPER HALF PLANE: If u is a real-valued harmonic function on H2, then a harmonic conjugate of u is any real-valued function v on H: such that u + iv is holomorphic on H2. It is well known that harmonic conjugates exist and are unique up to additive constants. Hence, there is only one such v as above that satisfies v(0, 1) = 0. It is immediate from the Cauchy-Riemann equations 61-22 8x_6y §E__§£ 0y_ ax’ that if u is a Bloch function, then so is v. Moreover, u and v have the same Bloch norm . THE UPPER HALF SPACE: Recall our definition from Chapter 1: Given a harmonic function u on H, the functions v1, . . . , vn_1 are said to be harmonic conjugates of u if 3.1 (v1,...,v,,_1,u)=Vf for some harmonic function f on H. The functions v1, . . . ,vn_1 are automatically harmonic, since they are partial derivatives of a harmonic function. Also, 3.1 and 34 35 the condition that f is harmonic are equivalent to the “generalized Cauchy-Riemann equations” 3.2 Dkvj = Div]: ; Dyvj = Dju 11—1 3.3 2 D,v,-+D,u = 0. i=1 When n = 2, by the Cauchy-Riemann equations, 3.2 and 3.3 hold if and only if u + iv is holomorphic on H. Thus, 3.1 is a natural generalization of the case n = 2. However, unlike the case n = 2, there is a greater degree of non-uniqueness of the vj’s, as we’ll see shortly. EXISTENCE OF HARMONIC CONJUGATES: Perhaps the best-known result concerning harmonic conjugates on H is the following: If yf(t) d, u(ay) = Plfl(:c,y) = c. R.-. (Ix —t|2 + w, , where f E L2(R”‘1), then the functions vj given by (“71' - tj)f(t)n dt 3.4 - , = 121(3 3!) Cu Rad (|x_t|2+y2)5_ are harmonic conjugates of u. Furthermore, each v,- is the Poisson integral of an L2 function on 11""1 (see page 78 of [8]). Unfortunately, not every harmonic function u on H is such a Poisson integral as above, so we need to do something else for such a function. 3.5 Theorem: If u is harmonic on H, then harmonic conjugates of u exist. PROOF: We need to find a solution for 3.2 and 3.3. The solutions for the second part of 3.2 are of the form v 3.6 v.,-(x,y)=/l Dyu(x,t)dt+ 2, harmonic conjugates for the same u may differ by more than just a constant. In fact, the proof of theorem 3.5 shows that adding any harmonic function g of x1, . . . , xn-1 to v1 will generate another set of n — 1 harmonic conjugates, say v1 + g, wl, . . . , wn_g. We can again add any harmonic function h of x2,. . . ,xn_1 to wl. It will generate new harmonic conjugates for u of the form v1 + g, to] + h, 31, . . . , sn_3, and so on. In the above example, the harmonic function added to v; is 2x1x2. Note that in the case n = 2, if we add a harmonic function of x1 (which must be of the form axl + b) to v1, then v1 + ax; + b is a harmonic conjugate of u if and only if a = 0, as easily checked. This is compatible with the well-known fact that when n = 2, the harmonic conjugate is unique up to an additive constant. HARMONIC CONJUGATES or BLOCH FUNCTIONS: Given u e 8(H), need the har- monic conjugates v1, . . . , v..-1 belong to 8 (H)? The answer is yes when n = 2; see the beginning of this chapter. In higher dimensions, the answer is no as we can easily see by taking u = 0. Here, 11 = 0 is Bloch but the harmonic conjugates v1(x1, x2, y) = x1 and v2(x1,x2,y) = ——x2 are not Bloch. Nevertheless, there are certain properties that any conjugate of a Bloch function must have. For example, we have control over the normal derivatives of the conju- gates. Indeed, since Dyvj = Dju, we have leuvj(-T,y)l S Hulls for all (x,y) E H and all j. We also have a uniform bound on some of the second partial derivatives of the harmonic conjugates. More precisely, there exists a positive 38 constant C such that 3-8 y’lDkavj(a=,y)l S CHalls for all (x,y) E H and allj E {1,...,n — 1} and k E {1,...,n}. Indeed, yZIDkavflx, 31“ = ylevavH-ta y)l |/\ y’leDquv, y)| (by 3-2) < C I lull 3 (Cauchy’s Estimates) for all (x,y) E H and allj E {1,..., n- l} and lc E {1,..., n}. We now begin to discuss the primary question of this chapter: Given u E 8(H), can we choose harmonic conjugates of u that are Bloch? Referring to the discussion above, suppose we know that for every x E R”‘1, Dim-(x,y) —> 0 as y —-+ 00. We can then fix a: E R"-1 and define on (0,00) the function f(y) = Dkvj(x,y). Then 3.8 implies that y’lf’(y)| S CIIuIIB. Therefore, since f(00) = 0, we get from the fundamental theorem of calculus that 3.9 y|f(y)l = yIDkvj($al/)l S Cllulls for all (x,y) E H; note that the constant C in 3.9 is the same as in 3.8 and hence is independent of u. Summarizing: To get Bloch conjugates for a Bloch function u on H, it suffices to find harmonic conjugates v1, . . . , vn_1 of u such that 3.10 ”111% Dka-(x, y) = 0, for all x E R"‘1 and for all k E {1,...,n} and j E {1,. . .,n -1}. Moreover, if 3.10 is satisfied, then (by 3.9) there is a positive constant C independent of u such that llvjlls S CIlullzs- 39 We now show that this can be done, essentially by using 3.4. 3.11 Lemma: For every h > 0, the functions v?(x.y)=cu ( ’ ’ 3"" (Ix—t|2+y2)r +t—I—(ItI2 1.1)?)110 h)dt are Bloch conjugates for Thu and there is a positive constant M, depending only on n, such that IIvflIa S MIIuIIB for all h > 0 and allj. REMARKS: The term tj/(Itl2 + 1)"/2 has been added here to ensure that the inte— grands belong to L1(R"‘l), and to ensure that vj-‘(0, 1) = 0. PROOF: Fix h > 0. The existence of the above integrals is checked by getting a common denominator for the integrand. A straightforward computation shows that vI‘,” ., v,’,‘_ _1 together with Thu satisfy 3. 2 and 3. 3. Therefore, of,” ., vf,’_ 1 are harmonic conjugates for 17.. To show that vI‘, . . . , v34 are Bloch, it suffices to show that 3.10 is satisfied. Fix j E {1, . . . ,n -- 1}. After differentiation under the integral sign and making the change of variable x — t = ys, we get C |u(x— ys h)I squ(x-y3ah)l 1),-v S — d‘ ’91“ J " ’ | I“,(:c y)| y Rh, “3'2 +1)% y 11..-: (M2 +1)-I- S 0 [21(1' - 3,3, h)ldt ? Ila-1 (Isl2 +1)i for all (x,y) E H. Using the growth estimate 2.4 in the above inequality and the estimate log(1 + Ix — y3|)Slog(1+IxI)+log(1+ y) + log(1 + Isl) for all x and s in R""1 and all y > 0, we easily see that there are two positive constants a and ,6 such that l l 1 l 1 + as; +le) +,ouyw) levj-‘(IMIN S a 40 for all (x,y) E H. Therefore, lim,,_.oo Djvj‘(x,y) = 0 for all x E R"‘1. Similarly, when k 7‘. j, hing...» Dka-‘(x,y) = 0 for all x E R”"1. Thus, 3.10 holds for vI‘,.. .,v,’:_1, n—l which implies that v? E 8(H). Furthermore, by 3.9 the Bloch norms of vI‘, . . . , v” are uniformly bounded by a constant multiple of Hull; (recall that IIThuIIB S IIuIIB).D We are now ready to prove the main result of this chapter. 3.12 Theorem: Let u E 8(H). Then there exist unique harmonic conjugates v1, .. . ,vn_1 of u on H such that v,- E 8(H) and vJ-(O, l) = 0 for each j. Moreover, there exists a constant M, depending only on n, such that IIvJ-IIB S M IIuIIB for each j. PROOF: Referring to vI‘, . . . , v24 of Lemma 3.11, we see that the vector-valued family of harmonic functions (of, . . . , v24) is uniformly bounded on compact subsets of H. (This follows because vf(0,1) = 0 for all h > 0 and all j, and since IIvI-‘IIB S M IIUIIB for all h > 0 and all j.) Therefore, there is a sequence (hk) that converges to 0 such that (vf‘ , . . . , 11:1,) converges uniformly on compact subsets of H to some vector of harmonic functions (v1, . . . , vn-1) (this is an easy generalization of 2.6 in [ABR]). Furthermore, the sequence of partial derivatives converges uniformly on compact subsets to the corresponding partial derivatives of (v1, . . . , vn-1) (see 1.19 of [ABR]). It follows easily by letting h], go to 0 in the generalized Cauchy-Riemann equations that v1, . . . ,vn_1 are harmonic conjugates of a. It is also an easy consequence of the uniform boundedness of the Bloch norms IIvj-“IIB, that each vj belongs to 8(H) and that IlvaIB S MIIuIIB for each j. Indeed, yIij(a=.y)l = ,Ijgoylei‘Was/H < ' ’9'- < ,ggouv, us _ Muuns. 41 for all (x, y) E H and all j. The fact that vJ-(O, 1) = 0 for allj is clear since vI-‘(0, 1) = 0 for all h > 0 and all j. Finally, the uniqueness of v1, . . . ,vn_1 follows from the fact that Dyvj = Dju and the fact that a Bloch function on H that doesn’t depend on y is necessarily a constant (see Lemma 2.10).D The last theorem is crucial for obtaining some results regarding the boundary behavior of the harmonic Bloch functions, as we’ll see in the next chapter. Chapter 4 Boundary Behavior of Bloch Functions Recall some of our discussion in Chapter 1: Although holomorphic Bloch functions on D need not have finite radial limits at any point on the boundary, they do have “average radial limits” over an interval on the boundary; this is what Ullrich showed in [U]. It easily follows that harmonic Bloch functions on D have this property. More- over, it terms of these averages, Ullrich obtained a necessary and sufficient condition for the existence of a radial limit at a given boundary point. The natural analogues of these results follow for harmonic Bloch functions in the upper half-plane after a conformal change of variables. The main purpose of this chapter is to explore these ideas in higher dimensions. But first, we prove that harmonic Bloch functions on H have “boundary values” in the sense of distributions. More precisely: 4.1 Theorem: If u is a Bloch function in the upper half-space, then [13,} R.-.“("” y)r(x)dx exists for every smooth function (,0 with compact support. Moreover, this limit defines a distribution on the space of smooth test functions with compact support. 42 43 PROOF: Let cp be as in the statement of the theorem. We define on (0,00) the function f(y) = [Rn-1 u($,y) 0: Because 1321 = 19(Thu), we have 4.6 Ilgul S 000077.11“), 1)I + IlThulIB) S CCn(|U(0,1)|+ ||u|l8103(1+ h) + Hulls) S CCn(1+103(1+ h))(IU(0,1)| + Halls) for all h > 0. The proof of Theorem 4.2 yields the following three corollaries: 4.7 Corollary: With 0 as in Theorem 4.2, we have ”1:;— In” s clan|(1+log(1+ d)) (h + hl log hl), where 0 < h < d and C is a constant depending only on n. PROOF: Fix it E B (H) We subtract the formula 4.4 from the formula 4.3 to get IISu - Ioul = | U. E n.(x)v.(x,y)ds(x)dyl. J=l 47 Hence, by the estimate 4.5, we obtain (since InJ-I S 1) n—l h llfiu-InUI s MIIuIIsz/o fanll+llosyl+21030+|xl)]ds(x)dy i=1 3 C |an|(1 + log(1 + d)) (h + h| log h|)||ulls. The desired conclusion followsD Note that Corollary 4.7 implies in particular that . h _ = mule In“ 0. The next result shows that the linear functional In is continuous in a stronger sense than that expressed in 4.2. 4.8 Corollary: Let Q be as in Theorem 4.2. If (mg) is a sequence of Bloch functions on H such that the norms IlukIIB are uniformly bounded and such that (u) converges to u uniformly on compact subsets of H, then [all = lim Ina}... k—eoo REMARK: The hypotheses here do not imply that uk —> u in the norm of B(H). For example, in Ba(D), the sequence fk(z) = 2" is uniformly bounded on D, hence uniformly bounded in the Bloch norm. This sequence converges to 0 uniformly on compact subsets of D, yet Imus > (1-(1— 1mm — l)“ ' " lc k ’ which is on the order of l/e for large lc. Thus, II kaIB 7‘» 0. (After a conformal mapping we obtain a similar example for H.) PROOF OF COROLLARY 4.8: We first observe that the function u is Bloch and that IIuIIB S sup IlukllB (the argument was done in the proof of 3.12). Hence, Igu 48 makes sense. Now let h < d. We have that Ilnu — InukI 5 I101! - [SUI + I161! - IgukI + I161”; — InukI for all k. The first term on the right of this inequality is small if h is small by 4.2. The second term converges to 0 as I: —) 00 by the uniform convergence of (uk) to u on compact subsets of H. For the third term, note that our hypotheses imply that there is a positive constant A such that Iuk(0,1)| + IlukIIB S A for all lc. Thus, using Corollary 4.7, we obtain I181”, — InukI S CAIBQIO + log(1 + d))(h + llI log III) for all k, which is small independently of k if h is small. The desired conclusion follows. D The next corollary states, roughly speaking, that for all y > 0, the average of u over yQ cannot be too far away from u(0,y). Again the dilation invariance of the Bloch norm will be used. 4.9 Corollary: Let Q be as in Theorem 4.2. Then 1 CC IWIWW - "(Dd/ll S fillulls for all y > 0. PROOF: Fix y > 0 and Q as in statement of the corollary. Easy manipulations show that 1 WIwm‘U = 1911”. Hence, because the Bloch function u, — u(O, y) vanishes at the point (0, 1), Theorem 4.2 implies that 49 1 1 IWIWO)“ — "(Oil/N = Ila—I100“ — "(Oil/DI —CC"IIu Ila Inl ” CC“ = — .0 I‘ll IlulIB l/\ We need to discuss the particular case where Q = B,, the open ball in R"-1 centered at the origin of radius r (B, has obviously smooth boundary). Recall from Chapter 1 the averages IBI/B u(x,h)dx. We deduce from Theorem 4.2 that Inn) Afu = Aru exists for all fixed r > 0. Also, the linear functional Ara is bounded on 8(H) for all 1 r > 0, and since there are positive constants an and flu such that IB,| = anrn' and IBB,I = flnr”‘2 for all r > 0, we easily obtain from 4.6 (here d = 2r) 410 ”Ala” S C(1+|108rl)(1+1°g(1+h))(llulls+|u(0,1)I) for all r > 0 and all h > 0, where C is a positive constant depending only on n. Thus, letting h —+ 0 in 4.10, we obtain llArull S C(1+I108"l)(llul|8 + |u(0,1)|) for all r > 0. We now prove a lemma involving the averages A,u that we will need for the main result of this chapter. Recall that en is the normalizing constant for the Poisson kernel (see 2.1). 50 4.11 Lemma: Ifu E 8(H), then yr" u(0,y) = nIBllcn [000 02+ +312)“;- A udr for all y > 0. PROOF: Fix u E B(H) and y > 0. Also fix h E (0,1) for the moment. Estimate 4.10 gives 4.12 “Alt!“ S C(1+|1<>gr|)(||u||ts+lu(0,1)I) for all r > 0 and all h E (0,1), where C is a constant depending only on 12. From the Poisson integral representation in Theorem 2.5, we have y u(t, h) 0, h = ———7 “( ”Jr ) 6" ran-lame)? We go to polar coordinates to obtain any + h) = (n -1)IBIIc.y/o°° -(-—",:'7:)—,- /Su(rc.h)da(<)dr for all y > 0. Now Afu = n 1 [G's-2 [S u(t(,h)do(()dt rn-l (again by going to polar coordinates). Therefore, “‘Afu) dr dr (r2 + yz)? °°dr“ u(o,y+h)=c.IBIIy/o ( Thus, integrating by parts in the formula of u(x, y + h), we obtain rfl h u(o,y+h)=n|B.|c,.y/°° (2+ y2)+ _flArudr. Note that the boundary terms in the integration by parts vanish by the estimate 4.12. This estimate also shows that the last integrand is bounded by a constant times (1 + I log rI)r" (.2 + W? 51 independently of h E (0,1). Because this last expression is integrable on (0, 00) for any y > 0, we can now let h -» 0 to obtain the desired result (by the Lebesgue Dominated Convergence Theorem).C| We now prove a result concerning the x-radialization of a given function on H: Given a continuous function u on H, its x-radialization R[u] is the function on H defined by Rluum) = [SU(IwIC,y)d0(C)- 4.13 Proposition: If u is harmonic on H, then RIu] is harmonic on H. PROOF: We use the converse of the mean value property (see 1.20 of [ABR]). The function RIu] is clearly continuous on H. Also, we can view RIu] as R[u1(x.y) = L. u(T(:c,y))dT, where 0,, denotes the group of orthogonal transformations on R“ that leave the y-axis invariant, and dT denotes the Haar measure on C“. Now, let 2 E H and let r > 0. Then stluKz + rodam = [9/6. u(T(z + rowdam = [G ls u(Tz + rT(())da(()dT (Fubini and linearity of T) = ./G.. lsu(Tz + rn)d0‘(17)dT (change of variables 17 = T(C)) = ./G.. u(Tz)dt (u is harmonic) = R[u](z). Thus RIu] is harmonic on H .D We are now ready to prove our main result of this chapter: An “abelian-tauberian” theorem characterizing the existence of a radial limit at a given boundary point in 52 terms of the functionals Aru. We’ll prove it at the origin, but at any other point (a, 0) of 0H, 7..., will take us back to the origin. 4.14 Theorem: Ifu E 8(H) and L E [—oo,+oo], then lirr6u(0,y) = L if and only if ling Ara = L. y—o r—v PROOF: We do the proof for the case n > 2, and we’ll indicate the necessary changes for the case n = 2. We first assume L = 0. Let us call the statement that lim,_.o A,u = 0 implies limyno u(0, y) = 0 the “abelian” implication; the other half of the theorm is the “tauberian” half. As one might think, the abelian direction is rather straightforward. Indeed, setting yr“ 0" + w”? Ky“) = "IBllcn in 4.11, we see that u(0,y) is the integral of A,u against the positive kernel K,(r), most of whose mass is near 0 for y small. (Loosely speaking, K y(r) is an approximate identity converging to the delta function at 0.) To deduce that u(0, y) -+ 0 if Ara —-) 0 is then a standard argument and we leave it to the reader. The tauberian half of the theorem is a normal families argument: Suppose ill?) u(0,y) = 0 but 113(1) Aru # 0. Then there exists 6 > 0 and a sequence of positive numbers r), -+ 0 such that IA,,u| > e for all Is. Now we consider the sequence of dilates uk(z) = u(rkz). First, observe that Auu = Aluk for all Is (this can easily be done by an adequate change of variables in h Ana). Hence, 4.15 IAlukI > 6 53 for all 1:. Second, because of the dilation invariance of the Bloch norm, (uh) is uniformly bounded on compact subsets of H. Indeed, this follows from the following inequalities |uk($,y)| S |uk(0,1)| + Iluklls(1 + |logy| + 21<>g(1+ |$|)) (estimate 24) = IU(0,rk)|+IIUI|B(1+ Hogs/H 2108(1+ |$|)) (IIUklls = IIUHB) _<_ C +||u||s(1+|103y|+ 2109;(1+|1=|)) (Ifigumd) = 0) for all (x,y) E H and all lc. Therefore, (uh) has a subsequence, which we still call (uh), that converges uniformly on compact subsets of H to a harmonic function u on H (see 2.6 of [ABR]). Examining the limit function, we have (since lim¢_.o u(0, t) = 0) v(0.y) = girgurdmy) = klim u(0,rky) = 0 for all y > 0. Now because the Bloch norms IIukIIB are uniformly bounded (by IIuIIB), it follows from Corollary 4.8, that Alv = [3.152, Aluk. Hence, (by 4.15) 4.16 IAle > e. To get a contradiction, we show that Alv = 0. To do that we first observe that since v(0,y) = 0 for all y > 0, we have R[v](0,y) = 0 for all y > 0. Also, RIv] is radial in x and harmonic on H (by 4.13). This is enough to give, by Proposition 2.11 of [RU 2], RM E 0 on H. Now we go to polar coordinates to obtain 54 AI.) .—. (n — 1) [01 r""2 [S v(r(, h)da(()dr = (n — 1) [)1 r"'2R[v](rq, h)dr (17 is any element on S) = 0 for all 0 < h < 1. Thus Alv = 0, contradicting 4.16. The case where L is a nonzero real number follows from the case L = 0 by considering the function u — L. The case L = :l:oo follows easily from Corollary 4.9. Indeed, if r is fixed, then letting fl = B in 4.9, gives (since r3 = B.) 1 IA,u — u(0,r)I = I-Ir—EIIugw — u(0,r)I |/\ CIIUIIB, where C is a constant depending only on n. Thus, lim,_.o Ara = 00 if and only if limyno u(0,y) = 00. The case n = 2 is easier, with the following change. Instead of using Proposition 2.11 of [RU 2], we use the following elementary fact: If u is a harmonic function on the upper half-plane and u E 0 on the y-axis, then u(—x,y) = —u(x,y).D BIBLIOGRAPHY [ACP] [Al [ABR] [CRW] [Fa] [F3] [F1] [L] BIBLIOGRAPHY J .M.Anderson, J.Clunie, Ch.Pommerenke, 0n Bloch functions and normal functions. J.Reine Angew. Math 270 (1974), 17-37. S.Axler, The Bergman space, the Bloch space, and commutators of multiplica- tion operators. Duke Mathematical Journal 53 (1986), 315-332. S.Axler, P.Bourdon, W.Ramey, Harmonic Functions Theory. Springer-Verlag, New York (1992). R.Coifman, R.Rochberg, G.Weiss, Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 (1976), 611-635. P.Fatou, Series trigonometriques et series de Taylor. Acta. Math. 30 (1906), 335-400. C.Fefferman, E.M.Stein, HP-spaces of several variables. Acta. Math. 129 (1972), 137-193. W.Fleming, Functions of Several Variables, 2nd Edition. Springer-Verlag, New York (1976). L.H.Loomis, The converse of the Fatou theorem for positive harmonic func- tions. Trans. Amer. Math. Soc. 53 (1943), 239-250. 56 57 [RUl] W.Ramey and D.Ullrich, Bounded mean oscillation of Bloch pull-backs. Math. Anna. 291 (1991), 591-606. [RU2] W.Ramey and D.Ullrich, 0n the behavior of harmonic functions near a boundary point. Trans. Amer. Math. Soc. 305. Number 1, (1988), 207-220. [R1] W.Rudin Tauberian theorems for positive harmonic functions. Neder. Akad. Wetensch. Proc. ser. A81 (1978), 376-384. [R2] W.Rudin, Functional Analysis. Mc Graw-Hill, New York (1973). [S] E.M.Stein, Singular Integrals and Difl'erentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, (1970). [U] D.Ullrich, Radial limits of Bloch functions in the unit disk. Bull. London Math. Soc. 18 (1986), 374-378. ‘IIIIIIIIIIIIIIIT