This is to certify that the dissertation entitled 7 (\YlMOHlL Ber Mun Funfi‘tlotfi on Half: - spams presented by [Hem Ha 3“ YE has been accepted towards fulfillment of the requirements for Pk . D degree in MRS WMz (7% Majorrpofessor Date Z/J’}/7}L MS U is an Affirmative Action/Equal Opportum'ly Institution 0-12771 —__———— —~— v-7. ——~—— -———————— LIBRARY M‘Chlgan State University PLACE II RETURN BOXtonmovothbcMekoum yum TO AVOID FINES Mum on or baton dd. duo. DATE DUE DATE DUE DATE DUE l - - I |:|——_| 3—7 LAD - MSU leAnNflmdtveActlaVEin Opportunity lrutltwon HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES By HEUNG SU YI A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1994 ABSTRACT HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES BY Heung Su Yi In this paper, we extensively study harmonic Bergman functions on the upper half spaces. First we show some interesting properties and examples of Bergman func- tions. Next we solve the duality problem of harmonic Bergman spaces and then we find “tame” dense subspaces of them. We also show that for any harmonic Bergman function, there are unique harmonic Bergman conjugates having an ap- propriate norm-bounds from the fact that any Bergman function can be stated in terms of its derivatives. Finally we show that the Bergman norm is equivalent to a “tangential derivative” norm as well as to a “normal derivative” norm and we also show that similar results hold for harmonic Bloch spaces which can be considered as the appropriate limit of harmonic Bergman spaces. To my parents, to my sister and brother, to my beautiful wife and to my lovely son Juhyun. iii ACKNOWLEDGEMENTS I deeply thank Dr. Wade Ramey, my dissertation advisor, for his help, adv-ice, knowledge and encouragement during my study at Michigan State University. I would like to thank Dr. Sheldon Axler for helping me with Latex. And I also thank Dr. JoelShapiro, Dr. John McCarthy and Dr. Jeanne Kerr for agreeing to be on my committee and for their excellent teaching. Finally, I wish good luck to my office-mate Youngkwon Song. iv Contents 1 Introduction 2 Basic Properties and Examples 3 The Bergman Projection 4 Nonorthogonal Projections 5 The Harmonic Bloch Space and the Dual of b1 6 Harmonic Conjugates Bibliography 14 21 29 41 48 Chapter 1 Introduction The upper half-space H = Hn is the open subset of R" given by H = {(13,31) 6 R"III/>0}, where we have written a typical point z E R" as 2 = (:r,y), with a: E Rn"l and y E R. Let p E [1, 00). We are interested in harmonic functions u on H such that 1/? Ilullp = (/Hlurdv) < oo. Here dV denotes Lebesgue volume measure on H, which we may also write as dz, dw, etc. The collection of all such functions will be denoted b” = b”(H). We refer to the Banach spaces b” as harmonic Bergman spaces. (That they are Banach spaces is proved in Proposition 8.3 of [ABR].) Harmonic Bergman spaces have not been studied as extensively as their holomor- phic counterparts. One property of the holomorphic Bergman spaces that we will sorely miss is their invariance under multiplication by bounded holomorphic func- tions; nothing comparable holds for harmonic Bergman spaces. Furthermore, most work on Bergman spaces—even in the holomorphic case—has been done for bounded domains. As we will see, the unboundedness of H causes some difficulties; for exam- ple, the Bergman kernel is not integrable over H. Thus, unlike the case of the disk or 1 ball, the Bergman projection of an L°°-function on H is not even defined. Finding the dual space of bl—not an easy task even the classical settings—will therefore involve some extra work. Another problem we have on H is that of finding “tame” dense subspaces of I)" (especially when p = 1). On the ball, of course, we can always choose the space of harmonic polynomials. On H it is not so clear what to do; finding such dense subspaces in this context will be a recurrent theme throughout the paper. Fortunately, not all the news about H is bad. For example, H is invariant under both horizontal translations and dilations—transformations that preserve harmonic functions. (The horizontal translations on H correspond to rotations on the unit ball, but there is nothing on the ball analogous to the dilation structure of H.) Moreover, H is a product domain, giving us a way to integrate by parts that will be crucialfor many of our results. Let us summarize the main results of this paper. In Chapter 2 we look at some basic properties and examples. For example, we consider the question of when the Poisson integral of a function on 8H belongs to b”. The case p = 2 is the simplest and here we give a complete answer (in terms of the Fourier transform); perhaps surprisingly, the Poisson integral of a function in L1 n L2(8H) always belongs to b2 when n > ‘2, but hardly ever when n = 2. In Chapter 3 we show that the Bergman projection, initially defined as the or- thogonal projection of L2 onto (72, extends to a bounded projection of LP onto b” for the range 1 < p < oo. Letting q denote the index conjugate to p, we then easily show that the dual space of b” can be identified with b" for this range of p-values. (The dilation structure of H allows for an easier proof than for the analogous result on, say, the unit disk.) In Chapter 4 we study certain nonorthogonal projections of LP onto I)". These projections are in many ways better behaved than is the Bergman projection, and they will be useful in many contexts. For example, they allow us to see that the Bergman norm can be stated in terms of derivatives. The latter property gives us a nice way to see that the harmonic Bloch space (definition below) is the appropriate limit of b? as p ——> 00. As suggested above, finding the dual space of b1 is problematic. In Chapter 5 we will modify the Bergman projection, obtaining an appropriate bounded linear map of L°° into the harmonic Bloch space. We then prove that the dual space of b1 is the harmonic Bloch space modulo constants. The proviso “modulo constants” arises because of the til-cancellation property: If u 6 12‘, then [H u(w) dw = 0, as we show in Chapter 2. The last chapter is devoted to harmonic conjugation. Recall that harmonic conju- gation is bounded in the Hardy-space LP-norm when 1 < p < 00, but not when p = 1. However, for the unit disk and other settings, it is well-known that harmonic conju- gation is bounded in the Bergman-space LP-norm for all p 6 [1,00). We prove the analogous result for the upper-half space in all dimensions: If u E b”, with p E [1, 00), then there exist unique harmonic conjugates v1, . . . , vn_1 of u that belong to b” (with appropriate norm-bounds). Recalling that Bergman-space norms can be restated in terms of derivatives, we use the conjugation results to show that the bP-norm is equiv- alent to a “tangential derivative” norm as well as to a “normal derivative” norm. A similar result holds for the Bloch-norm. These last results were unanticipated by us and appear to be new (even in the context of the unit disk). Chapter 2 Basic Properties and Examples As the title indicates, this chapter is devoted to exploring some basic properties and examples in the harmonic Bergman spaces 1)". (One fundamental object of study that we defer until the next chapter is the Bergman kernel.) Unless otherwise stated, the full range 1 _<_ p < 00 is intended when discussing b”, and all functions are assumed to be complex valued. Any unexplained notation will be as in [ABR]. For an open (I C R", we let b”(fl) denote the collection of harmonic functions u on {I such that 1/p [lullbmm = (/9 Iulp (IV) < 00. The standard estimate obtained from the mean-value property for harmonic functions shows that if u E bP(Q), then 1 IIUIlbvm) 2: < M )I — V(B)1/P d(z,60)"/P (2.1) for all z E Q, where B denotes the unit ball in R“ and C! denotes euclidean distance. Inequality (2.1) implies that convergence in the b”(Q)-norm implies uniform conver- gence on compact subsets of O, that b”(Q) is a Banach space, and that for fixed 2 6 Q, the map 2 ——> u(z) is a bounded linear functional on b"(Q). (For these and related results see [ABR], Chapter 8.) Cauchy’s estimates lead from (2.1) to the inequality a ”Ullbvm) ID "(2“ S Cd(z, 0S2)n/p+|al’ (2'2) where a is a multi-index and C is a constant that depends only on n, p, and a. (We will follow the practice of allowing C to denote a constant whose value may change from line to line; at appropriate places we will specify the parameters on which C' depends.) For most of the paper we are concerned with the setting (I = H. Here we write I)”, ||u||p, and LP in place of b”(H), HUHWH)» and LP(H,dV). Note that if u E b", then (2.2) becomes a U ID u(:c,y)| s Cyfl/pl’t.’ (2.3) for all (33,31) 6 H. For u a function on H and 6 > 0, let Tgu denote the function on H defined by 7511(2) = u(z + (O, 6)). We will need the following fact: If u E I)”, then Tgu ——> u in the norm of b1D as 6 -> 0. (Proof: If f E LP, then Taf —> f in L” as 6 —> 0, as we see by proving this first for continuous f with compact support.) Taking a = 0 in (2.3) shows that if u E b”, then u is bounded on each proper half-space contained in H, hence is the Poisson integral of its boundary values on each such half-space. In other words, T521 2 P[U(,6)] on H for each 6 > 0. Here Plf](z) = f,” P(z,s)f(s)ds denotes the Poisson integral of f (defined for any f E LP(0H), l S p S 00), where ds denotes Lebesgue measure on 8H (which we may also denote by dx, etc.). The function P is the Poisson kernel for H, defined for (z, s) E H X 6H by 2 y PW) z nV(B) (Irv - 3:2 + yanfl’ where z = (:r,y). Recall that Ian P(z,s) ds = 1 for all z E H, and that P(-,s) is positive and harmonic on H for each s 6 3H. The Poisson integral gives us a nice way to derive an important property of Bergman functions. Theorem 2.1 Ifu E b”, then the integrals fa” |u(:r,y)|” d2: increase as y decreases. Proof: Suppose 0 < y1 < y;. Then U(x,y2) = P[U(*,yl)l(x,y2 — 3/1). Applying Jensen’s inequality, we get IU(rv,y2)|” S P[|1t(-,y1)|”]($,y2 — 3/1)- Integrating with respect to .r and then using Fubini’s theorem yields (recall that fa” P(z,s)ds :1) f8” lumar dx 3 f,” lu(x,y1)l”dx, as desired. I The last proof actually shows that the conclusion of Theorem 2.1 holds if we only assume that u is harmonic on H and that u is bounded on every proper half- space contained in H. Note also that Theorem 2.1 implies that if u E b”, then Tgu E h” = h”(H) for every 6 > 0, where hp is the Hardy LP-space of functions v harmonic on H such that 1/p anhp=sup(/ lv(w,y)l”dx) 0 8H We now rove the b1 cancellation property mentioned in the introduction: If P u E bl, then [H u(w) dw = 0. In fact more is true, as the next result shows. Theorem 2.2 (The bl-cancellation property): Ifu 6 b1, then d =0 Amway) iv for every y > 0. Proof: Theorem 2.1 shows that u(-, y) E L1(8H) for every y > 0. Fix 6 > 0. Then Tsu=P[u(-,6)], so that [Hrrsu(w)dw = [000 f3}! f8” P((:r,y),s)u(s,6) dsdxdy = Améflu(s,6)dsdy, where we have used F ubini’s theorem and the fact that fa” P((:r,y),3) dz: = 1. Be- cause the inner integral in the last line is independent of y, we must have / u(s,6)ds = 0 8H as desired. . I Poisson Integrals Contained in b” Finding functions in b” that display specified boundary behavior is not as simple as for the Hardy-space h”. In the latter setting, we simply design an appropriate LP-function on 8H and then take its Poisson integral. This suggests the following question: Given f E L”(0H), when does P[f] lie in b”? In the case p = 2 we give a complete answer (Theorem 2.3 below) in terms of the Fourier transform. For any f E L2, we let f denote the Fourier transform of f. Here (and in the rest of the 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) and B(f) satisfy cA(f) S B(f) S CA(f) for all f under consideration. Theorem 2.3 For f E L2(BH), —1 1/ 2 IlPlflllz z (fmlf in )l lxl dx) . (2.4) Proof: Letting f E L2(8H), we can view P[f](:r,y) as a convolution over 8H: Purl-.31): P. * f(x) = [H as: — smads, where the definition of P3, should be clear from context. We thus have jW/M your) )Izdxdy = [000 [WM]? 11*f)($)|2dxdy /, (”lawn (army, f” |P[f1(w)|2 dw Now, modulo some constants (depending only on n and the normalization of the Fourier transform), we have Py(:r) = e‘yla"l (see [StW], page 16). Reversing the order of integration in the last integral above now gives the desired result. I Theorem 2.3 shows that an interesting dichotomy occurs between the cases n = 2 and n > 2. Let f E L1 O L2(0H). Then f is continuous on 8H. Integrating in polar coordinates, we see for such an f that the right side of (2.4) is always finite when n > 2, but is finite when n = ‘7 only if f(0) = 0, i.e., only if fan f(:c) d3: = 0. We have not obtained necessary and sufficient conditions for Poisson integrals of Lp-functions to lie in b” for any p ¢ 2. However, the following result handles the case of P[f] when f has compact support, at least for 1 < p < oo. Theorem 2.4 Let p 6 (1,00), let f E Lp(0H), and assume that f has compact support. (a) pr > n/(n -— 1), then P[f] E b”. (b) If1< p S n/(n — 1), then P[f] E b” ifand only iffaH f(a:)d:t = 0. Proof: Note that if f E LP(6H), then standard Hardy-space theory shows that l: l... lPlfl($,y)|”dxdy s 12],,” lfl” < so for any R > 0. (True even if p = 1.) Thus we only need to worry about whether P[f] E LP({|z| > R} n H) for R large. Let A = f3” f(:c)d:r, set c = 2/(nV(B)), and let K denote the support of f. For large 2 = (3:,y), we have c s y — y ds /,.f< )(lz—sl" Izlnl 3/ < .— _ Clzln'H/ |f(s)|ds, Imus) — cA 3’ / ~ where C is a constant depending on n and K. For large B > 0, the function y/Izl'“H belongs to Lp({|z| > R} n H) for all p >1, while y/Izl" E Lp({|z| > R} n H) if and only if p > n / (n —— 1). Both parts of the theorem now follow easily. I The case p = 1 seems difficult. The following theorem indicates that the cancella- tion condition [3” f (1:) dr = 0 is far from sufficient to guarantee that P[ f] 6 b1, even if f is smooth and has compact support. Theorem 2.5 (n = 2) : [ff 6 L1(R), f is odd, f is not identically 0, and f 2 0 on (0,00), then P[f] is not in ()1. Proof: Let :1: > 0. Then mm) = i /_: f(s)( ds 10 = _/:of( )((x—s)2+x2_(x+s:):2+x2) d8 1 41:23 z E/o “We —- .)2 + wax + .)2 + x2) “’3' (2'5) By replacing s by .733 in (2.5), we have 0— .)2+1)((1+ s)2+12)d3 P[f](rv,x) = ; /°° f(:vs)( IV 1_4-07r./ f(:rs) )sds 4 a: = d . 107rx2/0 f(8)3 3 Because f is not identically 0, we have P[f](:r,a:) __>_ C/ar2 for large at. A similar estimate holds for rays from the origin close to the ray y = :6. Thus for large 2 lying in a sector of H, we have |P[f1( (2 )2! % which implies P[f] is not in b1. I Positive Functions in I)? Theorem 2.6 The space 12” contains a positive function if and only ifp > n/(n — 1). Proof: If p > n/(n — 1), we can let f be any nice positive function with compact support on 8H. The function P[ f] is then positive and harmonic on H, and Theorem 4 (a) shows that P[f] E I)”. Now suppose u is positive and harmonic on H and that u E b”. By Theorem 7.24 in [ABR], um) = Cy + Plfll($ay) 11 for all (x, y) E H, where c is a nonnegative constant and p is a positive Borel measure MS) fem (1+ Isl)n < 00' We clearly have c = 0, or else u is not in b”. Because it is not the zero measure, we on 8H such that can choose a compact set K C 8H such that p( K) > 0. We then easily see that 2 y nV(B) (Izl + R) 11(2) > np(K) on H, where R = max{|a:| : a: E K}. Thus, because u E b”, we must have p > n/(n —1). I Distinctness of the Spaces b? Theorem 2.7 pr 75 q, then I)” does not contain (2". Proof: Suppose to the contrary that b’0 C b". Because convergence in any Bergman space implies uniform convergence on compact subsets, the closed graph theorem shows then that the identity map from b” to b9 is continuous. Thus there exists a constant C such that IIUIlq S CHalli» (2-6) for all u E b”. To show that the last inequality fails, we consider first the case n > 2. With 2 = (:r,y), define u(z) = D: Izl2’" for z E H. Being a partial derivative of the harmonic function Izlz’", u is harmonic (on R" \ {0}). It is easy to see that 12 for some homogeneous polynomial f of degree 3. (Note: The polynomial f is itself harmonic, but we do not need this fact here.) For any 6 > 0, we use homogeneity to see that _ |f(z+(0.6))|” “mu: — l” 12+ (0,6)|(n+4)pd WP / |f(z+(0,1))|” d 6(n+4)p H |z+(0,1)|(n+4)p Thus, ||Tgu||p = C(n,p)6("/P)'"‘1, which implies ”7'6““? : C(n,p,q)6n/q-n/p “7511“,, for all 6 > 0. Because p 95 q, the right side above is not a bounded function of 6. Thus (2.6) fails, and therefore I)” is not contained in b", as we wanted to show. The proof when n = 2 is the same, except that the function Izlz‘" should be replaced by log I2 |. I Derivatives of Bergman Functions If 01 is a multi-index, a 75 0, then there exist functions 21 E b” for which Dau ¢ b”. (We can prove this with the closed graph theorem by an argument similar to the one in the last section.) This phenomenon occurs because of possible bad behavior of Dan near BH—not near oo—as the next result shows. Theorem 2.8 Let Q be a half-space properly contained in H. Ifu E b”, then Dan E bP(Q) for every 0. Proof: We can set Q = {z E H : 2,, > 6} for some 6 > 0, and assume u E b”. Let a be a multi-index. For 2: E Q, let B; denote the open ball with center 2 and radius 6; note that 82 C H. By (2.2) we have a C ID “(zll S mllullbrvwz), 13 where C = C(n,a,p). For (z,w) E Q x H, set x(z,w) = 1 if |z — wI < 6 and x(z, w) = 0 otherwise. Then by Fubini’s theorem we have a 0” AID u(z)|p dz S mfg/”lu(w)lpx(z,w)dw dz < CP ”V B 6” — Wllullp ( ) 9 giving the conclusion of the theorem. I The next result shows how differently Bergman functions on H behave when compared with Bergman functions on bounded domains. Theorem 2.9 Ifu E b” and a is a multi-index, then Dan E 0 if and only ifu E 0. Proof: We clearly need only worry about the direction “=>”, which we prove by induction on Io]; note that there is nothing to prove when a = 0. Assuming this direction has been proved for multi—indices of order m 2 0, suppose that where |a| = m. Then Dan is independent of the jth coordinate variable. Let Q be any half-space properly contained in H. By Theorem 2.8, Dan E b"(SI). But Fubini’s theorem shows that if v E bp(fl) and v is independent of one of the coordinate variables, then 1) E 0. We conclude that Dan E 0 on (I, which implies Dan E 0 on H. By our induction hypothesis, u E 0, as desired. I Chapter 3 The Bergman Projection The easiest Bergman space to understand is b2. This is because I)2 is a Hilbert space, so that we have the entire Hilbert-space machinery at our disposal. Fix 2 E H. Because point evaluation is a bounded linear functional on ()2, there exists a unique function R(z, ) E I)” such that u(z) = /H u(w)R(z, w) dw for all u 6 ()2. The function R(z, w) is called the reproducing kernel or the Bergman kernel for b“. Standard results (see Chapter 8 in [ABR]) show that R(z,w) = R(w, z) and that R(z,w) is real valued; thus we can remove the complex conjugate in the integral above. Because b2 is a closed subspace of the Hilbert space L2, there is a unique orthogonal projection II of L2 onto b“. We call II the Bergman projection. It is easy to check that IIf(z) = /Hf(w)R(z,w)dw (3.1) for all f E L”. Any orthogonal projection is self~adjoint, but because R(z,w) is real valued, we have more: [Hmngdv = [H mg) W 14 15 for all f,g 6 L2. By Theorem 8.22 in [ABR], 4 n(z,, + ion)2 — Iz — El“ R(z, 1.0) = nV(B) IZ _ wln+2 a (3’2) where ti: = (101,. . . , wn_1, —wn). From this formula we see that |R(z, mi 3 C_ (3.3) lz - wl" for all 2,11) 6 H, where C = C(n). The last estimate shows, after integrating in polar coordinates, that R(z, ) E b9 for all q 6 (1,00]. Thus the integral in (3.1) is well defined whenever f 6 L” for p E [1, oo). Extending the definition of II to every LP, 1 S p < 00, we see that IIf is a harmonic function on H whenever f belongs to one of these LP-spaces. We next claim that II is the identity on b”, i.e., that u(z) = /H u(w)R(z,w) dw (3.4) holds for all u E b” and for all p E [1, 00). This would be easy if we knew that b'0 D b2 was dense in 1)". While this is true (and will follow as a corollary of results we obtain later), we do not know an elementary proof. Fortunately, the proof of Theorem 8.22 in [ABR] can be used nearly word for word to prove (3.4); we omit the argument here. (It resembles several arguments we give in the next chapter.) We now show that for 1 < p < 00, II is a bounded projection from LP onto bp. The following lemma will be useful in proving this. Lemma 3.1 Ifl < p < 00, then there is a constant C = C(n,p) such that -1/p / w"- dw=Cz;1/” HIz—w|" for allz E H. 16 Proof: Fix 2 E H. Letting w = (33,31), we have ‘l/P oo ‘l/P /H y dw : L] z" + y d3: dy. (3.5) Iz—w|" o zn+y aHIz—wl" Note that the inner integral in the right side of (3.5) equals nV(B) _nV(B) 2 8HP(2+(0,y),$)dx— 2 , where P is the Poisson kernel for H. Hence, after applying change of variable y +—-—) zny, we see that the right side of (3.5) equals Cz;1/P, where _nV(B) °°y‘1/” 0- 2 f. This completes the proof. I Theorem 3.2 Ifl < p < 00, then II is a bounded projection of LP onto b". Proof: We only need to check the boundedness of II. Let f E L”. Then for z E H, (3.3) implies 1 l2 - wl" IHf(z)| .<_ c/HIflwn dw 1 I2 - wl" = C [H lf(w)ly‘/”" y-l/de, where q denotes the index conjugate to p and w = (:12, 3]). Applying Holder’s inequality to the functions 1/pq -1/pq 31 y |f(1v)| |z—2D|"/P’ lz—tiiln/‘I’ we have 1/ -1/p P/q IIIf(z)|”SC(/Hlf(w)l”lzy_ wldw) (fHIZ’L de) - Hence, from Lemma 3.1, we get yl/q —1/ p < p q . [Emmy dz_C/H/H|f(w)| |z_wlndw 2,, dz 17 By applying Fubini’s Theorem and then Lemma 3.1 once again, we have I. IHf(z)|"dz s C [H warm. as desired. (Here C = C(n,p).) I REMARK: In the proof of the last theorem, we have used ideas that can be found [Ax] and [Z], which originate in [ShW] and [FR]. The dilation structure of H gives us a simpler proof than those of analogous results in [FR] and [Ax]. Although II is the identity operator on b1, it does not map L1 into 6‘. To see this, let f denote the characteristic function of some small ball centered at (0,1), divided by the volume of this ball. By the mean value property for harmonic functions, lIf(z) = R(z,(0,1)). In a thin cone centered along the y-axis, 1 1+ [2]" |R(z,(0,1))l z 7 which shows that IIf is not in b1. The Dual ofb” for 1< p < 00 The main purpose of this section is to prove that for 1 < p < 00, the dual of the harmonic Bergman space I)” can be identified with b", where q denotes the index conjugate to p. It is clear that if v 6 b“, then the mapping u +———+ /H u v dV (3.6) is a bounded linear functional on b” whose norm is at most ]|v||q. Furthermore, distinct functions in b9 induce distinct linear functionals in b”. To see this, suppose v E b9 and that v induces the zero-functional on b". Because R(z, -) 6 b", we have 22(2) = f” R(z,w)v(w)dw = 0 18 for every 2 E H by (3.4). Le, 22 E 0 as desired. Therefore, to prove our duality result, we only need to show that every bounded linear functional on b” is of the form (3.6), with an appropriate estimate on "22”,. Theorem 3.3 Ifl < p < 00, then (0’)" “z’ b", where q denotes the index conjugate to p. Proof: We first show [Hmngdv = [Hf(119)dV (3.7) for all f E L" and for all g 6 L9. Note that (3.7) is true whenever p = q = 2. Since L2flL' is dense in L” for every r 6 (1,00), (3.7) follows from the p =2 2 case by a simple limiting argument using Héilder’s inequality and Theorem 3.2. Now suppose A is a bounded linear functional on b”. By the Hahn-Banach Theo- rem, there is a function f E L9 such that ”A” = ||f||q and A(u)=/Hude for all u 6 bp. Because u = IIu for all u E b”, we use (3.7) to conclude that Au = [Humfmv for all u E b”. Thus A is induced by the bq-function IIf. Finally, ||1If||q S CIIfllq = C ”A”, where C = C (n,q). This, together with the remarks made at the beginning of this section, completes the proof. I Dense Subspaces of b” for 1< p < 00 It is frequently useful to know that there is a dense subspace of b? consisting of functions that vanish to high order at 00. For this purpose, let a be a multi-index and set Mo, = span{D°‘R(z,w) : z E H}, 19 where “span” denotes the linear span over C. (We are thinking of D"R(z,w) as a function of w.) Because DSRWJU) = (-1)°"+"'+°""D§'R(z.w), where a = (01, . ..,a,,), we need not specify the variable of differentiation in the definition of M a. To estimate the size of the derivatives of R(z, w), observe that D§R(z,w) — M" _ “7) _ [Z __ wln+2+2|a|’ where f0, is a homogeneous polynomial of degree 2 + Ia]; this is straightforward to verify by induction. Therefore IDSR(z.w)l < C(n’“) — m (3'8) for all z,w E H. From (3.8) we see that if u 6 Ma, then there is a constant C such that l (1+lwl,..... |u(w)| S C for all w E H. Thus the next theorem, which is a corollary of the duality result in the last section, gives us dense subspaces of b” whose members vanish at 00 to high order. Theorem 3.4 Each Mo, is dense in l)” for 1 < p < 00. Proof: Fix a and l < p < 00. Suppose v E 1)“ satisfies / uv (IV = 0 H for all u 6 Ma. Then A D:R(z,w)v(w) dw = D: [H R(z,w)v(w) dw = D°v(z) = 0 20 for all z 6 H. This implies that v E 0 by Theorem 2.9. Hence by a standard corollary of the Hahn-Banach Theorem and Theorem 3.3, Ma must be dense in b”. I Interestingly, the space Mo, is not always dense in b1. Note that M0 is not even contained in bl, while Mo, is contained in b1 if a 94 0. As we show later, if 0 ¢ 0, an interesting difference arises between the cases 12 > 2 and n = 2: When n > 2, MO, is not dense in b1 for most choices of a; when n = 2, Mo, is always dense in ()1. We next show that functions in b” can be approximated on H by bP-functions defined on much larger half-spaces. For this purpose, set H. = {(23.31) E R” = y > -5} for 6 > 0. Theorem 3.5 Ifl < p < 00, then bP(H5) is dense in b” for each 6 > 0. Proof: Fix 6 > 0. Because R(z, ) E b”, we easily see that R(z + (0,6), ) E bP(H5) for any 2 E H. Now suppose v E bq E (5’)" and that v vanishes on bP(H5). Then v(z + (0,5)) = j” R(z + (0,6),w)v(w) dw = 0 for all z E H, which implies v E 0 on H. By a standard corollary of the Hahn-Banach Theorem, b”(H5) must be dense in (2”. I REMARK: After obtaining the dual space of b1 in Chapter 5, we will be able to show that b1(H5) is dense in b1 for every 6 > 0. Note the contrast with the Hardy spaces hp: Ifl < p < 00, then h”(H5) is dense in h”, while this fails ifp = 1 (there is nothing in hl(H5) close to P(-, 0), because the latter is generated by a singular measure). Chapter 4 Nonorthogonal Projections In this chapter, we study certain nonorthogonal projections of LP onto b? for p 6 [1,00). (We call them “nonorthogonal” because they are not orthogonal when acting on the Hilbertospace L2.) Unlike the Bergman projection, these projections are bounded on L1. The Ll-boundedness will enable us, finally, to obtain dense subspaces of b1 whose members vanish to high order at 00; this will be useful in finding the dual space of b1 (Chapter 5). Additionally, the new projections lead to certain “derivative norms” on the Bergman spaces; they will also play a role in the harmonic conjugation results of Chapter 6. Let m denote a nonnegative integer and let p E [1, 00). For f 6 LP and z E H, define 11mm) = cm [H f(w)y"‘D;"R(z.w)dw. where cm = (-—2)’”/m! and w = (:r,y). Note that by (3.8), y’”D3‘R(z,-) E b" for all q E (l, 00], so that the above integral is well-defined and harmonic on H for all f E L", 1 S p < 00. Note also that H0 is the Bergman projection II. We study II", by exploiting the connection between the Bergman and Poisson kernels. For 2, w E H, the “extended Poisson kernel” is the function _ 2 2,, + 7.0,, _ nV(B) |z — 212]". 21 P(z,w) 22 Note that P(z, (x,y)) = P(z + (0,y),:c) for all z, (1:,y) E H, which implies P(-,w) is a positive harmonic function on {2 E R" : 2,, > —w,,} for each w E H. Furthermore, by (3-2). 23P(z, w) 0w" R(z,w) = — (4.1) for all z, w E H. Thus the definition of IIm can be rewritten umflz) = —2c.. [H f(w)y"‘D;"+‘P(z.w)dw- Theorem 4.2 below is the main result of this chapter. We first prove the following lemma, which contains an identity we need in proving Theorem 4.2. Lemma 4.1 Let 1 S p < 00 and let 6 > 0. Ifu E bP(H5), then 21(2) = —2c.,, L[D;"+1u(w)]ymP(z,w) dw (4.2) for all z E H. (Here w = (r,y).) Proof: Fix 2 E H. Because u E b”(H5), we have a C |D u(:c,y)l S (y+6)n/p+lal on H for every 0:. This estimate guarantees that the integrand in (4.2) belongs to L1. It also shows that DZ‘HU is bounded on H, so that we may apply the Poisson integral formula. Thus the integral in (4.2) equals [0... y'" a” DL"“U(:E. y)P(z + (0.11)...) dxdy = /°° ymD;"+1u(z + (0,211)) dy. Integrating by parts m times, we see that the last term equals 1 0° 1 — Dyu(z + (0,2y)) dy = cm 0 —2cm u(z). This completes the proof. I 23 REMARK: The idea of integrating by parts in the manner of the above proof—an idea we will use many times in this paper—is inspired by the proof of Theorem 8.22 in [ABR]. Theorem 4.2 Ifl S p < 00 and m > 0, then IIm is a bounded projection of L” onto 1)”. Proof: First, we show that H", is the identity operator on b”. We do this by induction. Because no is the Bergman projection, we have nothing to prove in the m = 0 case. Assuming that Hm_1 is the identity on b” for some m 2 1, we first deal with the case it E b”(H5) for some 6 > 0. Fix 2 E H. Letting w = (x,y), we have IImu(z) = cm/Hu(w)ymD;”R(z,w)dw = —2c.. l... [0”..(.,y)ymp;n+1p(.,(.,y))dyd.. (4.3) After integrating by parts in the inner integral in (4.3), we have IImu(z) = 2mcm/3H/0 u(:r,y)ym'lD;"P(z,(.r,y))dydx +2c.. [a H [0 °°iD.u(x.y)1ymD;“P(z. (z,w) dy d... (4.4) The first term above equals 221(2) from the induction hypothesis. To evaluate the second term, we integrate by parts m times in the inner integral, obtaining 2cm(-1)"‘::%( 7," ) (FEW/3H AmID?‘j+IU($.y)Iym’jP(z, (17.31))dydm. Hence, (4.4) equals —u(z) from Lemma 4.1, which implies IImu = u. Assuming that lIm_1 is the identity on b”, we have shown that IImu = u whenever u E b”(H5) for some 6 > 0. To handle the general 11 E I)”, apply the above to T511 and let 6 ——> 0. This completes the induction and hence the proof that each IIm is the identity on every b”. 24 We now prove the boundedness of IIm. Because C l2 - “a?!" lymD;"3(z.w)| S 3 we can show the boundedness of Hm for the range 1 < p < 00 by the exactly same method we used in the proof of Theorem 3.2. Hence, we only need to consider the p = 1 case. Let f E L1. Then IIHmfII. s C//|f(w)l|2 ——dwdz g CAI/[f(wfl ———_———_yln+l dwdz. Therefore by F ubini’s Theorem and the estimate 1 -——dz < /H |z -—w|n+1 we get ||IIme1 S CHle as desired. (Here C = C(n,m,p).) I o 3’ (4.5) That operators analogous to H2 can be bounded on L1 goes back to [Ah]. Dense Subspaces Revisited The boundedness of II", on LP leads easily to the existence of useful dense sub- spaces of b” for the full range 1 S p < 00. (Recall that in Chapter 3 we showed that the spaces Ma and bP(H5) are dense in I)” only for the range 1 < p < 00.) For m a positive integer, we let Dm denote the vector space of functions 11 harmonic on H that satisfy lu(z) < C __ 4.6 |—l+|z|"‘ ( ) for all z E H. Theorem 4.3 Ifl S p < 00 and m > 0, then Dn+m is a dense subspace of b”. 25 Proof: The space PM", is a clearly contained in I)”. To prove density, let u E I)”. Choose compact sets K1, K2, . . . C H such that K,- C K,“ and H = U Kj. Setting u,- equal to 21 times the characteristic function of K j, we easily check that Hmuj E ’Dn+m from (3.8). Because IIHmuJ' -u|lp = IIHm(Uj-U)||p S IIHmll IIUJ' - ullp -* 0 as j -—) 00, we have PM”, dense in l)” as desired. I Derivative Norms on b" The main result in this section is the following theorem. Theorem 4.4 Ifl S p < 00 and m 2 0, then HUllp "3 Z IlymD°UI|p *5 IlymDL“UI|p lal=m for all u E b”. Note the proviso “u E If” in Theorem 4.4; the norm equivalence stated there would fail if we allowed u to vary over all harmonic functions on H. To prove Theorem 4.4, we only need to show that there are positive constants Cl, C2 (depending only on n, m,p) such that 0. Z llymDaullp s Ilullp s Czllymnrun. (4.7) lal=m for all u E b”. We prove (4.7) one inequality at a time. Theorem 4.5 Ifl S p < 00 and m 2 0, then there is a positive constant C = C(n,m,p) such that Z llymD°UI|p S CllUllp . (4-8) |a|=m for all u E l)”. 26 Proof: Let u E b” and let oz be a multi-index with IQ] = m > 0 (if m = 0, then (4.8) is clearly true). Then by Theorem 4.2, II1u(z) = u(z) = —2/H u(w)wannR(z,w) dw (4.9) for z E H. Differentiating through the integral in (4.9) and using the estimate (3.8), we have |D°u( (|) 0, and fix k 2 0. With w = (x,y), we have Ho(y"DSU)(z) = -2 fa” fooolDSU(x.y)ly"DyP(/Z.(x.y))dyd:v- After integrating by parts in the inner integral above, we obtain IIo(ykD:u)(Z) = 2L”/OOO[D:+1u(x,y)]ykP(z,(:c,y))dyda: +2k f,” [0 °°[D’:u(x.y)1y'°-1P 0. 28 For an arbitrary u E b”, Theorem 4.5 shows that we can make our usual limiting argument for T511 as 6 —> 0 to obtain (4.11) for u. The proof of the lemma is complete. The following theorem completes the proof of Theorem 4.4. Theorem 4.7 Ifl S p < 00 and m 2 0, then there is a positive constant C = C(n,m,p) such that IIUIlp S CllymDZ‘UIIP for all u E b”. Proof: If u E I)”, then yng‘u E LP by Theorem 4.5. Hence by Lemma 4.6 and Theorem 4.2, we have IIuIIp : ”(CmH/Cl) “1(3/mDLnulllp _<- Cl|ymD;"u]|p for all u E b”. I REMARK: 1. We naturally expect the LP-norm to approach the L°°-norm as p —) 00. Thus the norm equivalence expressed in Theorem 4.4 suggests that the harmonic Bloch space (defined in Chapter 5)-——rather than the space of bounded harmonic functions—is the “right” limit of b” as p -—i 00. 2. After we obtained the results in this section, Sheldon Axler pointed out to us that the material here is analogous to 4.2.7—4.2.10 in [Z]. Chapter 5 The Harmonic Bloch Space and the Dual of b1 In this chapter we identify the dual space of ()1. To motivate some of the definitions to come, suppose A E (b1)“. By the Hahn-Banach Theorem, there is a function f E L°° such that A(u)=/Hude for all u E b1. Now u 2 Hu, and so we could hope that A(u) z/H(Ilu)de=/Hu(IIf)dV for all such u. Unfortunately, lI f is not always well-defined for f E L°° (recall R(z, ) is not in L1). However, note that the bl-cancellation property (Chapter 2) implies that constants will play no role in the dual space of ()1. Thus, if we expect the dual space of b1 to be a space of functions on H, it must be a space of functions that vanish at some prescribed point of H. This leads to a natural modification of the operator II, as we now describe. We first set 20 = (0,1). For z,w E H, define ~ R(z,w) = R(z,w) -—— R(zo,w). 29 30 Because of cancellation at infinity, the kernel R(z, ) belongs to L1 (as we show below). Thus we can define II on L°° by lIf(z) = f” f(w)R(z,w)dw. (5.1) As we shall see, II maps L°° into the harmonic Bloch space. A harmonic function u on H is called a Bloch function if IIUIIB = SUpyIVU($,y)l < 00. where the supremum is taken over all (:1:,y) E H and Vu denotes the gradient of u. (If u is complex valued, then we use the Cn-norm to calculate IVuI.) We let 8 denote the collection of Bloch functions on H and let B denote the subspace of functions in B that vanish at 20. The space B is a Banach space under the Bloch norm I] ”3. h To show that II maps L°° into [3 (actually onto 8), we need an estimate on R(z, w). We obtain this via two inequalities for real numbers (one of which is not so obvious). Let’s start with the easier one: If m is a positive integer, then for any b,d > 0, _1_ _ 1] Ir —— dml 5m dm bmdm |m(b — d)(bm"l + film—1)] < . - bmdm (5 2) < m|b—d| (—1—+—1—) (53) - bmd 5.1m ’ ' where we used the Mean Value Theorem in (5.2). The second inequality we need is given in the following lemma. Lemma 5.1 For each positive integer m, there is a constant C = C(m) such that if 0| _ C- ( _1 _ + _1_) |2—20|" I2—-w]”[zo—w| |z-w||zo--w|” /\ C ( 1 1 S _ _ + _ . |2 — 20]" [z — w|"(1+ wn) (2,, + wn)|zo — w|" Hence, as in (3.5) we have /H|u(2)R R(,zw)|de——§—_——/oo 1 dw |2——20|" o (2,,+w,,)(1+wn) Note that the last integral equals (log 2,,)/(2,, — 1), which is comparable to (1 + llog 2n|)/(1 + 2,.). Using the idea in (3.5) once again we have //H|u(2) R(,)zwldwszC/Ooo Lfll—(ig—zzii—‘dzn <00, (1+zn)2 where C = C(n,u). I Theorem 5.5 HA 5 (b‘)", then there is v e 8’ such that A = f d (u) Huv V for all u E Du“. Moreover, “our; S CIIAII, where C = C(n). 34 Proof: We know that A is given by some f E L°° as in (5.8), with ”A” = ||f||oo. Let U = IIf. Then by Theorem 5.3, v E B and ”out; S Cllfllco = C||A||. If u E ’DnH, then LuvdV = /Hu(2)/Hf(w)R(z,w) dw dz = /Hf(w)/H u(z)R(z,w) d2 dw. (5.10) (Switching the order of integration above is permissible by Lemma 5.4.) Note that the inner integral in (5.10) equals u(w) by the bl-cancellation property. Thus (5.10) equals A(u), completing the proof. I We next show that every '0 E B induces a bounded linear functional on b1. We first review some additional properties of Bloch functions: If v E B and 6 > 0, then llTs'UlIB S IIUHB; (5-11) 75v 2 P[v(-,6)] on H; (5.12) |D$v(9¢5y)l S CIIUHB/ym, m =1.---. (5-13) Inequality (5.11) is clear, (5.12) is proved in [Aj], and (5.13), with C = C(n,m), follows from Cauchy’s estimates. We now extend the domain of II to the set of all functions f for which the integrand in (5.1) belongs to L‘. From (5.9), we then see that flu is harmonic on H whenever v E B. The identity expressed in the next lemma will be helpful in finishing our proof that (b1)" E B. (This lemma is the B—analogue of Lemma 4.6.) Lemma 5.6 va E B, then v = c,,,II(ymD;”v) for m 2 0. Proof: We have already observed that B(ymDZ‘v) is well defined when m = 0; property (5.13) shows that this expression is well defined if m > 0. 35 To prove the lemma we start with v = 7521 for some it E B; note that v E B, but 2) need not be in B. For such a function 2), we follow the strategy of Lemma 4.1. Thus we will show 12(2) — v(zo) = —2cm L[D$+lv(w)]ym P(z,w)dw, (5.14) where P(2,w) = P(z, w) — P(zo,w) and w = (2,3,1). Note that the integrand in (5.14) belongs to L1 by (5.13) and the estimate |P(2,w)l S C/Iz — ml" (harder to prove than (5.3) but easier than Lemma 5.1). The integral in (5.14) equals [Om gm fa” D;"+1v(w)P(z + (O,y),.r)d:r dy. (5.15) Because (5.13) implies that D;"+1v is the Poisson integral of its boundary values on every half-space contained in H, (5.15) equals [0 y... (031+va + (0,231)) - D;"“v(zo + (0.220)) 4% (5.16) We estimate the integrand in (5.13) with some simple calculus: IDZ'HNZ + (0.231)) - D;"+1v(20 + (0.231)” S Equation (5.14) follows after integrating by parts m times in (5.16). Returning to the statement of Theorem 5.6, consider the m = 0 case. By (4.1), ~ Hv = -2 / /°°..(..,y)p,r(.,(.,y))dyd.. an 0 Integrating by parts in the inner integral, we have n 2 = 2] ,01’5 , d v( ) Mm- ) (2 as) a: 2//°°D ,P,, dd. + a” 0 Mr 31) (2 (x 31)) y a: From (5.12) and the m = 0 case of (5.14), we see that the last expression equals v(2) — v(20). 36 Summarizing, we have shown that if v E B, then II(7'5v)(z) = 1'5’U(Z)— 7'5’U(Zo). for 6 > 0. Using (5.9) and letting 6 —* 0, we see that the dominated convergence theorem gives IIv(2) = 12(2). We are done with the proof of the theorem in the case m = 0. To obtain the theorem for m > 0 we proceed by induction, recalling that Lemma 4.1 has been proved for all m. The proof from here is so similar to the analogous passage in the proof of Lemma 4.6 that we omit the details. I The m = 0 case of the last theorem shows that II is the identity on B. The m = 1 case implies that II is a surjection of L°° onto B. We can now show that every '0 E B induces a bounded linear functional on Dn-l-l, hence on b1. Theorem 5.7 Hz) E B, then the map A defined by A(u)=/HuvdV is a bounded linear functional on Dn+1(under the b1 -norm). Moreover, ”A“ S 2||v||3. Proof: The linear map A is well-defined by (5.9). From the m = 1 case of Theorem 5.6, we have '0 = —2I~I(yDyv). Hence, [HuvdV = —2/Hu(2)/H[Dyv(w)]yR(z,w) dw d2, (5.17) where w = (1:,y). Because [IyDyvlloo S ||v||3, Lemma 5.4 allows us to switch the order of integration in (5.17). So (5.17) becomes —2/ [D,,( v( w)]y/u(2 (,2 w) )dzdwz—2/H [Dyv( w()]yu(w )dw 37 where we have again used the til-cancellation property. We therefore have I [Haven/I s 2 f” ID.v(w)|ylu(w)|dw 2Ilvllzsllulln |/\ completing the proof. I By combining Theorem 5.5 and Theorem 5.7, we obtain the main result of this chapter. Theorem 5.8 (b1)"‘ E B. Proof: We have a map : B —* (b‘)" given by v t———) A”, where A” is the map induced by v as in Theorem 5.7. That theorem shows that the linear map is well defined and bounded, while Theorem 5.5 shows that (I) is onto. To complete the proof, we only need to show is one-to-one (by the open mapping theorem). For this purpose, suppose A, is the zero functional on b1 for some v E B. Because R(z, -) E 73”“, the m = 0 case of Lemma 5.6 shows 0 = AU(R(2, )) = 115(2) = 5(2) for every 2 E H. This implies (I) is one-to-one as desired. I Dense Subspaces of b1 Because (b1)"' E B, we can now understand when the subspaces Mo, and bl(H5) are dense in b1. (The analogous density problems for b”, 1 < p < 00, were discussed in Chapter 3; see Theorems 3.4 and 3.5) The next two propositions are relevant to the case of Mo. Proposition 5.9 (n = 2) : III) E B, then Dav E 0 if and only ifv E 0. 38 Proof: One direction is clear. For the other direction, we proceed by induction on [a]; there is nothing to prove when a = 0. Assuming the proposition holdsfor multi-indices of order m, suppose DjDav E 0, where [a] = m. Then Dav depends on only one variable. But by (5.13), Dav is “Bloch” on each proper half-space of H2. Because a Bloch function depending on only one variable is constant, we see that Dav is constant on H2. Using (5.13) again, we see that this constant must be zero. By our induction hypothesis, v E 0, completing the proof. I Proposition 5.10 (n > 2) : (a) va E B and on = [a], then Dav E 0 if and only ifv E 0. (b) If [a] > an, then there is a nonzero v E B such that Dav E 0. Proof: To prove (a), note that because every derivative of a Bloch function tends to zero at infinity in the “y directions”, any harmonic Bloch function that does not depend on y must be constant. A simple induction argument now finishes the proof. To prove (b), note that if la] > an, then a,- 74 0 for somej 76 n. Letting v denote any nontrivial function in B that is independent of the jth coordinate variable, we arrive at a nonzero v E B with Dav E 0. I Corollary 5.11 (a) Ifn = 2, then Mo, is dense in b1 for all a 5£ 0. (b) lfn > 2, then Mo, is dense in b1 if and only if [a] 2 an > 0. Proof: If a is a nonzero multi-index, then I) E B E (b1)" vanishes on Mo, C ’13,,“ if and only if 0 = [H D:R(2,w)v(w) dw = D“ f” R(z, w)v(w) dw = Dav(2) for all z E H. The proof now follows easily from the Hahn-Banach Theorem and the last two propositions. I 39 We now show that b1(H5) is dense in b1. As one might expect, Theorem 5.8 is the key, although the proof is different from that of Theorem 3.5. Theorem 5.12 For each 6 > 0, b1(H,5) is dense in b1. Proof: Fix 6 > 0 and z E H. Suppose v E B E (b1)"‘ and that the corresponding linear functional vanishes on b‘(H5). For nonzero a, D§R(z, ) E Du“, which implies D§R(2 + (0, 6), ) E b1(H5) fl Du“. Thus = a 6 d 0 AD, R(z + (0, ),w)v(w) w = D“ ” 0 6 , d z/HR(2+( , )w)v(w) w = D°v(2 + (0,6)) for all a 76 0. It follows that the power series of v at the point 2 + (0,6) vanishes except for the constant term. Therefore 1), being harmonic, is constant on H. Because v E B, we have v E 0 on H. By a standard corollary of the Hahn-Banach Theorem, b1(H5) must be dense in b‘. I Higher-Derivative Norms on B Lemma 5.6 implies the following analogue for the Bloch space of Theorem 4.4 Theorem 5.13 If m is a positive integer, then llvlls E Z llymDavlloo E Ilymevlloo Ial=m for 5115 e 8'. Proof: We only need to show that there are two positive constants C1,C2 (de- pending only on m and n) such that C. E III/“D“vlloo < llvlls s cznymnrvn... (5.18) lal=m 40 for all v E B. The first inequality is a simple consequence of Cauchy’s estimates and the definition of the Bloch norm. To prove the second inequality, let '0 E B and assume oz is a multi-index with [al = m. With w = (2:, y), Lemma 5.6 implies 12(2) 2 cm [H ymDZ‘v(w)R(z,w) dw for z E H. Differentiating through the integral and using estimate (3.8), we obtain l2,,D,-v(2)| S CznffllymD$v(w)D,,R(2,w)|dw dw H [z — w|n+1 |/\ CznllymDZ‘vlloo < CllymDanIIooa where C depends only on m and n. This shows ||v||5 S CllymDZ‘vlloo, completing the proof. I REMARK: Theorem (5.13) strengthens the conclusion obtained in Theorem 2.16 of [Aj]- Chapter 6 Harmonic Conjugates Given a harmonic function u on H, the functions v1,...,vn_1 on H are called harmonic conjugates of u if (v1,...,vn_1,u) = Vf (6.1) for some harmonic function f on H. We refer to the vector (121,. . . ,vn_1,u) in (6.1) as a conjugate system. If (6.1) holds, then v1,...,vn_1 are partial derivatives of a harmonic function, so they are harmonic on H. Also, (6.1) and the condition that f be harmonic is equivalent to the following “generalized Cauchy-Riemann equations”: Dkvj = Djvk; Dnvj = Dju 11-] Z Div,- + Dnu = 0 i=1 In particular, if n = 2, the pair of harmonic functions (v, u) is a conjugate system if and only if u + iv is holomorphic on H2. If u is harmonic on H, then harmonic conjugates of u always exist. Unfortunately, they are far from unique. (When n > 2, harmonic conjugates for a given u may well differ by more than a constant.) We refer to [Aj] for more on this. 41 42 Below we show that if 1 S p < 00 and u E b”, then there are unique harmonic conjugates v1, . . . ,vn_1 of u belonging to b”. Furthermore, “1),-”p S C||u||p for some constant C = C(n,p). Note that we exclude the p = 00 case. To see why, consider 71-] Way) = Z; achtant’rj/y). which is bounded and harmonic on H. Then a straightforward computation shows that every harmonic conjugate 12,- of u is of the form vJ-(x, y) = 10gb} +312) +wj(a:), so that 1),-(3:, y) —> 00 as y ——) 00 for each fixed 2:. Thus none of the conjugate functions of u is bounded. (The natural conjugates of bounded harmonic functions are Poisson integrals of BMO-functions; see [FS].) On the other hand, note that we get to include the case p = 1, in contrast to‘the Hardy space theory (recall that the Riesz transforms are not bounded on L1(3H)). That conjugation can be bounded in the Ll-norm on Bergman-type spaces was first observed in [FR]. (See also [Ax] and [Z].) Theorem 6.1 Let p E [1, co) and u E I)”. Then there are unique harmonic conjugates 121,. . . ,vn_1 ofu on H such that v,- E 1)”. Moreover, there is a constant C = C(n,p) such that ”1),-Hp S C||u||p for each j. Proof: For eachj=1,...,n — 1, set 1),-(z): 2]” u(w)wanJR(z,w) dw for all z E H. Clearly each v,- is harmonic on H. Note that DwJR(z,w) = —DZJR(z,w) for j = 1,. . . ,n — 1. Differentiating through the integral sign, we easily see Dkvj = Dj'Uk, for j,k =1,...,n — 1. Also, Dnvj 2 DZ]. (—2/Hu(w)wnDan(z,w) dw) . 43 Because Dan(z, w): Du," R(z, w), we see that Dav, = Djfllu = Dju. Finally, Zva(z )+Du(z )=—2/Hu( w)yAzR(,zw)dwE 0. (Here, A; = D; + - . - + Din.) Thus, v1, . . . ,vn-1 are harmonic conjugates of 11. Because IDsz(z,w)| satisfies the same estimate as IDwnR(z,w)|, we obtain the estimate ||vj||p S C Hall? in exactly the same way that we proved III was bounded in the LP-norm. (See the proof of Theorem 4.2.) For uniqueness, suppose u1,. . . , un_1 are also harmonic conjugates of u such that uj E b” for each j. Then by Theorem 4.7, lle _ qulp— < 0“an n(vj uj-lllp By noticing Dn(vj — u,) = Dj(u - u) E 0, we have v,- = u, as desired. ' I Tangential Derivative Norms Recall that Theorem 4.4 asserts that the Bergman norm is equivalent to a “deriva— tive norm” involving only differentiation in the normal direction. Using results from the previous section, we can now show that the Bergman norm is equivalent to a tangential derivative norm. Theorem 6.2 Ifl S p < 00 and m 2 0, then 2 llymDOUHp % IIUIlp |a|=m 011:0 for all u E 1)”. Proof: By Theorem 4.4, we only need to show that there is C = C(n,p) such that HymDyullp S 0 Z “ymDaqu. (6'2) |a|=m On=o 44 First, consider the case m = 21:: for some k > 0. (If m = 0, then (6.2) is clearly true.) Since u is harmonic, 71-1 2 ——2: ? Dyu— DJu. i=1 Hence, n-l lDZ‘uI = IDSkuI = I 2 D?1 ---DJ2-ku| S C 2 |D°u|. jl ----- jk=1 Inlay; (In: This shows (6.2), because the sum in (6.2) is finite. Ifm = 2k +1 for some k 2 0, then 71-] DZ‘u = — Z ngDjvj. 1:1 Hence, we have 11—] k 2 IlymD$UI|p S C 2: III/“Dy Djvjllp i=1 n—l Z Z llymDavjllp i=1 Ial=m |/\ n—l S 0 Z llymDZ‘vJ-llp, j=1 where we used the Theorem 4.4 applied to 2),. Note that Dyvj = Dju. Therefore, n-l llymDL”UI|p S CZIIyijDSkUHp i=1 n—1 n-1 CZ Z llyijDflmekullp i=1 1'1 .---.J'k=1 S 0 Z IlymDaullp. | 2, at least when p = 2. For convenience, we will show that the equivalence n—l “allz “ Z llvjllz (6-3) i=2 fails. We recall some basic results from Hardy space theory ([3]): The map f i——-+ R, f is bounded on L2((9H), where R,- f denotes the jth Riesz transform of f. If f E L2(0H), then (R f)($)= Tl:— | fiv( ) on 0H; furthermore the functions P[ij] are harmonic conjugates of P[ f] Let f E L2(8H). Because 2 ~ ‘ 2 —1 ||P[f1l|2 ~ fafilf(:v)| le dx (Theorem 2.3), we see that if P[ f] E 62, then each P[Rj f] E 62, and thus the functions P[Rj f] must be the harmonic conjugates of P[ f] specified by Theorem 6.1. To see that (6.3) fails, note that from the above discussion we have ZIIPIR fllli ~ 2],,” ——dx j=2 jl_T—l() / (le? — as) |f(:v)| d... 8H Ital2 lxl 46 But we can easily choose f such that the last integral is small when compared with lflxllz ~ 2 f... ——da: ~ “Pm”. lxl It follows that (6.3) fails, as we wished to show. Harmonic Conjugation on B The following theorem was proved in [Aj]. Theorem 6.4 If u E B, then there are unique harmonic conjugates v1,...,vn_1 of u on H such that vj E 3, Moreover, there is a constant C = C(n) such that ||va|3 S CIIuIIB for each j. REMARK: We could prove this theorem using the ideas of the proof of Theorem 6.1 as a guide; however, the technical details for 6 seem more difficult. The following results can be proved in exactly the same way we proved Theorem 6.2 and Corollary 6.3. So we omit the proofs. Theorem 6.5 Ifm > 0, then )3 llymD°UI|oo % IIUIIB |a|=m an =0 for all u E 3. Corollary 6.6 For u E 6, 11-1 IIUIIB “ Z “villa, i=1 Where '01,. . . , vn_1 are the harmonic conjugates ofu given by Theorem 6.4. 47 We can also show that no proper subset of {v1,...,vn-1} will do in the last Corollary when n > 2. In fact, proving that n—l Hulls z X: lIvJ-lls i=2 fails is much easier than proving (6.3) fails: We simply choose any nontrivial u E [3" that depends only on $1 and y. We then have n—1 n—1 n—1 Z IIvJ-Ils e Z IIyDyvjll... = z; llyDJ-ulloo = 0. .___2 :2 i=2 Bibliography [Aj] H. Ajmi, Harmonic Bloch Functions on the Upper Half Space, Ph. D. Thesis, Michigan State University, 1992 [Ah] L. Ahlfors, Some Remarks on Teichmiiller’s Space of Riemann Surfaces, Annals of Math. 74 (1961), 171-191 [Ax] S. Axler, Bergman Spaces and Their Operators, Surveys of Some Recent Results in Operator Theory Vol. 1, Pitman Research Notes in Math. 171 1988, 1-50 [ABR] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer-Verlag, New York, 1992 [F S] C. Fefferman and E. Stein, H P-Spaces of Several Variables, Acta. Math. 129 (1972), 137-193 [FR] F. Forelli and W. Rudin, Projections on Spaces of Holomorphic Functions in Balls, Indiana University Mathematics Journal 24 (1974), 593-602 [ShW] A. Shields and D. Williams, Bounded Projections, Duality, and Multipli- ers in Spaces of Analytic Functions, Trans. Amer. Math. Soc. 162 (1971), 287-302 [S] E. Stein, Singular Integrals and Diflerentiability Properties of Functions, Princeton University Press, Princeton, 1970 [StW] E. Stein and C. Weiss, Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971 [Z] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc., New York and Basel, 1990 48 HICHIGRN STRTE UNIV. LIBRQRIES l l l llllll lll llll lll llllll llll llll |l| llll lll lllll lll l l l l 31293010152126