Weighted two-parameter Bergman space inequalities

Publ. Mat. 47 (2003), 161–193 WEIGHTED TWO-PARAMETER BERGMAN SPACE INEQUALITIES J. Michael Wilson Abstract For f,afunction defined on Rd1×Rd2, take uto be its biharmonic extension into Rd1+1 +×Rd2+1 +.Inthis paper we prove strong sufficient conditions on measures µand weights vsuch that the inequality (∗)Rd1+1 +×Rd2+1 + |∇1∇2u|qdµ(x1,x 2,y 1,y 2)1/q ≤Rd1×Rd2 |f|pvdx 1/p will hold for all fin a reasonable test class, for 1 <p≤2≤q<∞. Our result generalizes earlier work by R. L. Wheeden and the author on one-parameter harmonic extensions. We also obtain sufficient conditions for analogues of (∗)tohold when the entries of ∇1∇2uare replaced by more general convolutions. 1. Introduction In an earlier paper [WhWi], Richard L. Wheeden and the author studied the following weighted norm inequality for the Poisson integral u(x, y)(x∈Rd,y>0) of a function f: Rd+1 + |∇u(x, y)|qdµ(x, y)1/q ≤Rd |f|pvdx 1/p .(1.1) In this inequality, ∇denotes the full gradient in Rd+1 +:∇=(∂/∂x1,..., ∂/∂xd,∂/∂y); Rd+1 +is the usual upper half space Rd×(0,∞); µis a positive Borel measure defined on Rd+1 +; and vis a non-negative function in L1 loc(Rd). We studied this inequality primarily for pand qin the 2000 Mathematics Subject Classification. 42B25. Key words. Bergman spaces, weighted norm inequalities, Littlewood-Paley theory. 162 J. M. Wilson range 1 <p≤q<∞.For the case in which q≥2, we proved sufficient conditions on µand v(depending on p,q, and d) for the inequality (1.1) to hold for all f∈∪ 1≤r<∞Lr(Rd,dx). The argument in [WhWi]began with the observation that (1.1) is aspecial case of a more general inequality. Let hbeasmooth function with decay at infinity (precisely how much decay will be specified later), defined on Rd.Fory>0, set hy(x)=y−dh(x/y), the usual L1-dilation. If we set u(x, y)=f∗hy(x), then any component of ∇u(x, y) can be written as f∗(y−1φy)(x), where φis smooth, has some decay, and in addition satisfies Rd φdx=0.(1.2) This said, we may now shift our attention to an arbitrary smooth φ with decay (how much, again, to be specified presently), and which satisfies (1.2), and we may ask: What conditions on µand vensure that Rd+1 + |f∗(y−1φy)(x)|qdµ(x, y)1/q ≤Rd |f|pvdx 1/p (1.3) holds for all fin our test class? In this paper, we are concerned with two-parameter generalizations of (1.1) and (1.3), and especially the latter. What does “two-parameter” mean? Let Rd=Rd1×Rd2.Fori=1,2, let φibe smooth functions with good decay, defined on Rdi, and which satisfy Rdiφidxi=0. Inour two-parameter problem, we look for sufficient conditions on measures µ, defined on Rd1+1 +×Rd2+1 +, and non-negative weights v∈L1 loc(Rd1×Rd2), which are sufficient for the inequality (1.4) Rd1+1 +×Rd2+1 + |f∗(y−1 1(φ1)y1)·(y−1 2(φ2)y2)(x1,x 2)|qdµ(x, y) 1/q ≤Rd1×Rd2 |f|pvdx 1/p to hold for all f. (Here we are using ‘(x, y)’ to stand for ‘(x1,x 2,y 1,y 2).’) When we write (φi)yi(xi), we mean, of course, y−di iφi(xi/yi). In the case where the φi’s are the kernels that “generate” the components of the Poisson kernel (in their respective upper half spaces!), such a result Two-Parameter Bergman Inequalities 163 would yield a sufficient condition for the inequality (1.5) Rd1+1 +×Rd2+1 + |∇1∇2u|qdµ(x1,x 2,y 1,y 2)1/q ≤Rd1×Rd2 |f|pvdx 1/p , where uis f’s biharmonic extension and ∇idenotes the full gradient in the (xi,y i)variables. Thus, ∇1∇2uisa(d1+1)×(d2+1)matrix of functions, and |∇1∇2u|can be taken to be the square root of the sum of the squares of its entries. In this paper we prove sufficient conditions for inequality (1.4), valid for 1 <p≤2≤q<∞and for a certain class of φi’s. This class includes the kernels that generate the x-derivatives of the Poisson kernels, but not, alas, the y-derivatives. The reason for this troubling gap is that, while the convolution kernels for the x-derivatives of the d-dimensional Poisson kernel decay to order (1 + |x|)−d−2, the corresponding y-derivative kernel only decays like (1 + |x|)−d−1. Unfortunately, our general oneparameter result (Theorem 1.1 below) requires decay like (1 + |x|)−d−2. In [WhWi], the authors treated the y-derivative by means a trick combining harmonicity and the Poisson kernel’s semigroup property. The whole trick is given on [WhWi, pp. 955–959], but in a nutshell it’s this. Our duality argument (which works so well with the x-derivatives) requires that we obtain good Littlewood-Paley estimates for a certain function Tg(x), expressed as a weighted integral of ∂Py(x−t)/∂y over (t, y)∈Rd+1 +.Forthe x-derivatives, the corresponding integrals involved ∂Py(x−t)/∂xi, and we got our Littlewood-Paley estimates by convolving with ψη(·), where ψwasasmooth, compactly-supported function with cancellation. The extra decay in the ∂Py(x−t)/∂xi’s let us bound the resulting integrals nicely. Lacking that decay for the y-derivative, R. L. Wheeden and the author convolved Tg with ∂Py(·)/∂y.Byharmonicity and the semigroup property, the resulting integral could be expressed in terms of second partials in the xderivatives —for whose kernels we do have good bounds. In our final section we drag this additional argument in to obtain a sufficient (but not so good) condition for the y-derivatives in the bi-space setting as well. Before stating our main theorem, we should state the one-parameter result from [WhWi] that motivated it. Even this earlier result is fairly technical, and the two-parameter result is, in our opinion, liable to be 164 J. M. Wilson completely indigestible to a reader who has not seen the one-parameter version first. The one-parameter result. As is traditional in this business, we begin with cubes Q⊂Rd.We use (Q)todenote the sidelength of Q, and |Q|is its Lebesgue measure. We denote the Euclidean center of Qby xQ.Byˆ Qwe mean the set ˆ Q={(x, y)∈Rd+1 +:x∈Q, 0<y<(Q)}, the so-called “Carleson box” sitting above Q.Weuse T(Q)todenote the “top half” of ˆ Q: T(Q)={(x, y)∈Rd+1 +:x∈Q, (Q)/2≤y<(Q)}. One more definition: If η≥0, σ∈L1 loc(Rd)isanon-negative weight, and Q⊂Rdis a cube, we set σ∗(Q, η)≡Q σ(x) logη(e+σ(x)/σQ)dx,(1.6) where σQ=(1/|Q|)Qσ,σ’s average over Q. Equation (1.6) defines an Orlicz-type norm that shows up in weighted Littlewood-Paley theory [W1], [W2], and whose properties underlie the results in [WhWi] as well as those of the present paper. Theorem 1.1. Let mbeanon-negative integer. Let φ∈C ∞(Rd)have φ=0.Letφalso satisfy |φ(x)|≤(1 + |x|)−d−2−mand |∇φ(x)|≤(1 + |x|)−d−3−mfor all x∈Rd.Letv∈L1 loc(Rd)beanon-negative weight and let µbeapositive Borel measure on Rd+1 +.Let1<p≤2≤q<∞. Set σ=v1−p, where pis the dual exponent to p.Letη>p /2. There is a positive constant C=C(η,p,q,d, m)such that the following is true: If there exists a weight wsatisfying σ∗(Q, η)≤Q w(1.7) and (1.8) µ(T(Q))1/q Rd logp/q (e+|x−xQ|/(Q))w(x) ((Q)+|x−xQ|)(d+2+m)p/qdx1/p ≤C(Q)d+1−(d+2+m)/q for all cubes Q⊂Rd, then (1.3) holds for all f∈∪ 1≤r<∞Lr(Rd,dx). Remark. The reader can see what we mean by indigestibility. Two-Parameter Bergman Inequalities 165 Remark. The theorem, as stated in [WhWi], actually gives a sufficient condition for the range 1 <p≤q<∞, with q≥2. We have stated this limited form of the theorem to make it more closely resemble Theorem 1.3 below. The restriction in Theorem 1.3 comes about because our method of proof, in two parameters, requires p≥2. This is related to another difference between Theorem 1.1 and the corresponding result in [WhWi]. The theorem in [WhWi]does not contain the hypothesis (1.7). Rather, it speaks of pairs of weights (‘p-pairs’) (σ, w) for which walso satisfies (1.8). However, as is pointed out in [WhWi, p. 949] and in [W1], a pair that satisfies (1.7) is ap-pair. Unfortunately, we have no good characterization of p-pairs (for p=2)inthe two-parameter setting. We express Theorem 1.1 in this fashion in order to make its statement look more like those of Theorem 1.3 and Theorem 5.3 (see below). Remark. If σbelongs to the Muckenhoupt A∞class, then (1.7) holds for w=cσ, where cdepends on η,d, and the A∞“box specs” of σ.Inthat case, (1.8) amounts to saying that µand σcannot put too much mass too near any cube Q. Since σis big when vis small, this is a quantitative wayofsaying that vcannot be too small near points where µis “large”. Theorem 1.1 is a restatement of this fact for v’s whose corresponding σ’s are not in A∞. The two-parameter result. We begin here with rectangles R=Q1×Q2, where the Qiare cubes in Rdi.Weuse |R|to mean the Lebesgue measure of R.Weset T(R)= T(Q1)×T(Q2) and ˆ R=ˆ Q1׈ Q2, where T(Qi) and ˆ Qiare as defined above. We will be using the next definition so often that it merits its own formal statement: Definition 1.2. Let η≥0beanumberand let σ∈L1 loc(Rd1×Rd2) be a non-negative weight. If R⊂Rd1×Rd2is rectangle, we set σ(R, η)=R σ(x) logη(e+σ(x)/σR)dx, where σR=(1/|R|)Rσdenotes σ’s average over R. Remark. The only difference between Definition 1.2 and the one given earlier is that Definition 1.2 applies to rectangles. Our main result, which we prove in Section 4, is: 166 J. M. Wilson Theorem 1.3. Let m1and m2be non-negative integers. For i=1,2, let φi∈C ∞(Rdi)have φi=0.Letthe φialso satisfy |φi(xi)|≤ (1 + |xi|)−di−2−miand |∇φi(xi)|≤(1 + |xi|)−di−3−mifor all xi∈Rdi. Let v∈L1 loc(Rd)be a non-negative weight and let µbe apositive Borel measure on Rd1+1 +×Rd2+1 +.Let1<p≤2≤q<∞. Set σ=v1−p, where pis the dual exponent to p.Letη>p and let >0. There is a positive constant C, C=C(η,,p, q, d1,d 2,m 1,m 2), such that the following is true: If there exists a weight wsatisfying σ(R, η)≤R w(1.9) and µ(T(R))1/qRd1×Rd21 ((Q1)+|x1−xQ1|)(d1+2+m1−)p/q ×1 ((Q2)+|x2−xQ2|)(d2+2+m2−)p/q ×w(x)dx1/p ≤C(Q1)d1+1−(d1+2+m1−)/q(Q2)d2+1−(d2+2+m2−)/q (1.10) for all rectangles R=Q1 ×Q2, then (1.4)holds for all f∈∪1≤r<∞Lr(Rd ,dx). Remark. Note the absence of log’s in the numerator and the extra ’s in the denominator of the two-parameter condition (1.10). The rest of the paper is laid out as follows. In Section 2 we state and prove certain results from weighted Littlewood-Paley theory which we will need in the proof of Theorem 1.3. In Section 3 we state a technical result from [WhWi], concerning convolutions of smooth functions with specified amounts of decay and cancellation, and we apply this result to prove a lemma (Lemma 3.2). Lemma 3.2 is a pointwise substitute for a series of integral inequalities used in [WhWi]toprove Theorem 1.1. This pointwise result is part of what lets us prove our two-parameter theorem without having a full-blooded, two-parameter weighted-norm theory of the Littlewood-Paley square function; it is also where the extra ’s in (1.10) will come from. In Section 4 we prove Theorem 1.3. In Section 5 we state and prove a sufficiency result for the biharmonic Poisson kernel. Two-Parameter Bergman Inequalities 167 2. Littlewood-Paley theory The basis of all of our arguments is the Calder´on-Torchinsky decomposition lemma. Let ψi(i=1,2) be real, radial, C∞functions defined on Rdi, that satisfy: 1) ψi=0; 2) supp ψi⊂{xi:|xi|≤1}⊂Rdi; 3) for any ξ∈Rdi\{0}, ∞ 0 |ˆ ψi(tξ)|2dt t=1. For yi>0, we let (ψi)yi(xi)=yi−diψi(xi/yi). If y1and y2are positive numbers and x=(x1,x 2)∈Rd1×Rd2,wedefine y=(y1,y 2) and set Ψy(x)=(ψ1)y1(x1)·(ψ2)y2(x2). The Calder´on-Torchinsky lemma consists in the following observation: if f∈L2(Rd1×Rd2), then, by Fourier inversion, f(x)=Rd1+1 +×Rd2+1 + (f∗Ψy(t)) ·Ψy(x−t)dt1dt2dy1dy2 y1y2 (2.1) as a distribution [CF]. It is easy to show that, for f∈L2, the (vector-valued) integral (2.1) actually converges to fin the L2norm. If fis smooth and decays rapidly at infinity, then the integral (2.1) converges uniformly and pointwise, and can be cut up and rearranged at will. We will use this freedom in the following way. Let R=Q1×Q2⊂Rd1×Rd2be a double-dyadic rectangle, that is, a Cartesian product of dyadic cubes Qi⊂Rdi, and let T(R)=T(Q1)×T(Q2)bethe corresponding “top half” of its twoparameter Carleson box, as defined above. (It is important to note that the family {T(R)}Rtiles Rd1+1 +×Rd2+1 +.) With suitable (and quite weak) hypotheses on f,wemay re-write the integral formula (2.1) as a sum: f= RT(R) (f∗Ψy(t)) ·Ψy(x−t)dt1dt2dy1dy2 y1y2 = R bR(x). Each of these functions bRhas support contained in ˜ R(the concentric triple of R), is smooth (it inherits this from the ψi’s), and has cancellation in the x1and x2directions; that is to say, for each fixed x∗ 1∈Rd1, Rd2 b(x∗ 1,t)dt =0, 168 J. M. Wilson and analogously for each fixed x∗ 2∈Rd2; this cancellation property is also inherited from the ψi’s. The meaning of the Calder´on-Torchinsky lemma is that any (essentially arbitrary) function can be written as a sum of smooth, compactly supported functions that have cancellation. We can go further. Let us say that a function aR(x)isadapted to a rectangle R=Q1×Q2⊂ Rd1×Rd2if: a) supp aR⊂R; b) aRis infinitely differentiable; c) for each x∗ 1∈Rd1,∇x2aR(x∗ 1,·)∞≤(Q2)−1|R|−1/2; d) for each x∗ 2∈Rd2,∇x1aR(·,x ∗ 2)∞≤(Q1)−1|R|−1/2; e) ∇x1∇x2aR∞≤(Q1)−1(Q2)−1|R|−1/2; f) for each x∗ 1∈Rd1,Rd2aR(x∗ 1,t)dt =0; g) for each x∗ 2∈Rd2,Rd1aR(t, x∗ 2)dt =0. Each of the functions bRobtained above can be expressed as λ˜ Ra˜ R, where each a˜ Ris adapted to ˜ R, and the λ˜ R’s are complex numbers satisfying: |λ˜ R|≤CT(R) |f∗Ψy(t)|2dt1dt2dy1dy2 y1y21/2 , for some constant Cthat depends on Ψ (which, recall, depends on d1 and d2) but not on f. Let us say that a function fis in standard form if there is a finite family, G,oftriples of double-dyadic rectangles, such that f(x)= R∈G λRaR(x), where the λR’s are real numbers and each aRis adapted to R. (Notice that the ‘tildes’ have been “absorbed” into the R’s.) We will use Littlewood-Paley theory to bound certain functions in standard form on weighted spaces. We will measure the “badness” of our weights via the Orlicz-type norm σ(R, η) given in Definition 1.2. When η>0, the ratio σ(R, η)/Rσmeasures the extent to which σ’s mass gets concentrated on a small part of R(that is, a subset with small Lebesgue measure compared to |R|). In particular, the ratio is uniformly bounded (for any η>0) if and only if σis a two-parameter A∞weight. The proof of Theorem 1.3 depends on this result from [W2]: Two-Parameter Bergman Inequalities 169 Theorem 2.1. Let η>2. There is a constant C=C(η,d1,d 2)so that the following holds: If σ∈L1 loc(Rd1×Rd2)is any non-negative weight and f=R∈G λRaRis any function in standard form, then Rd1×Rd2 |f|2σdx≤C R∈G |λR|2 |R|σ(R, η). Theorem 2.1 has an immediate consequence. For f=R∈G λRaRin standard form, set ˜ S(f)(x)= R∈G |λR|2 |R|χR(x)1/2 . (This is one of many variants of the Lusin square function.) The next corollary follows by rearranging sums. Corollary 2.2. Let σand wbe weights such that, for some η>2, σ(R, η)≤Rwfor all rectangles R⊂Rd1×Rd2.Forany fin standard form, Rd1×Rd2 |f|2σdx≤C(η,d1,d 2)Rd1×Rd2 ˜ S2(f)w dx. In the one-parameter setting, both Theorem 2.1 and Corollary 2.2 have Lpanalogues for p=2[W1]. Precisely, by applying the oneparameter version of the Calder´on-Torchinsky lemma, we can write an essentially arbitrary fas a sum f=QλQa(Q), indexed over the dyadic cubes Q⊂Rd, where the λQ’s are numbers and the a(Q)’s are smooth functions satisfying: a) supp a(Q)⊂˜ Q, the concentric triple of Q; b) ∇a(Q)∞≤(Q)−1|Q|−1/2; c) a(Q)=0. We define an analogous one-parameter square function: S(f)(x)≡  Q |λQ|2 |Q|χ˜ Q(x)  1/2 . Now let η>p/2(0<p<∞), and suppose that σand ware two weights in L1 loc(Rd) satisfying Q σ(x) logη(e+σ(x)/σQ)dx ≤Q w(x)dx 176 J. M. Wilson Corollary 3.4. Let γ>0and 0<<1. There is a constant C= C(γ,,d1,d 2,m 1,m 2)so that, for all rectangles R=Q1×Q2⊂Rd1×Rd2 and all integers k1and k2,  R=Q 1×Q 2×(Q i)=2ki(Qi) B(R,R)γχ˜ R(x1,x 2) ≤C×(1 + |k1|)γ2−|k1|γ ×(1 + |k2|)γ2−|k2|γ ×(Q1)(d1+m1+2−)γ×(Q2)(d2+m2+2−)γ ×1 ((Q1)+|x1−xQ1|)d1+m1+2−((Q2)+|x2−xQ2|)d2+m2+2−γ . 4. Proof of Theorem 1.3 We rephrase our weighted norm inequality in a dual form. Set σ= v1−p. Let φ1and φ2satisfy the respective hypotheses of Theorem 1.3. If g:Rd1+1 +×Rd2+1 +→ Cis bounded, Borel measurable, and compactly supported, we define: ˜ Tg(x1,x 2)=Rd1+1 +×Rd2+1 + g(t1,t 2,η 1,η 2) ×[η−1 1(φ1)η1(t1−x1)η−1 2(φ2)η2(t2−x2)] dµ(t1,t 2,η 1,η 2). This integral converges absolutely for all x∈Rd1×Rd2because of our special assumptions on g(note that the support of gstays away from ∂(Rd1+1 +×Rd2+1 +)). The operator ˜ Tis the adjoint of the operator that takes finto f∗(y−1 1(φ1)y1)·(y−1 2(φ2)y2)(x1,x 2). Inequality (1.4) will hold for all f∈∪ 1≤r<∞Lr(Rd1×Rd2,dx)if Rd1×Rd2 |˜ Tg(x)|pσdx 1/p ≤Rd1+1 +×Rd2+1 + |g(t, y)|qdµ(t, y)1/q for all these g.Wewill prove Theorem 1.3 by showing that that is what happens (given hypotheses (1.9) and (1.10)). Two-Parameter Bergman Inequalities 177 For i=1,2, we can write φi=ρ(1) i+ρ(2) i,(4.1) where supp ρ(1) i⊂{xi:|xi|≤1}⊂Rdi, each Rdiρ(j) i=0,and Rdiρ(2) iPi(x)dx =0for all polynomials Pi(in the xivariables) of degree ≤mi+1;wedothis by, essentially, throwing mi+1 of φi’s moments “onto” ρ(1) i. When we do this, the functions ρ(1) iget one good property (compact support), while the non-compactly supported ρ(2) i’s get lots of cancellation. Using our decompositon (4.1), we may write ˜ Tg as a sum of four terms: ˜ Tg = 2  k,j=1 ˜ T(k,j)g, where ˜ T(k,j) g(x1,x 2)= Rd1+1 +×Rd2+1 + g(t1,t 2,θ 1,θ 2) ×[θ−1 1(ρ(k) 1)θ1(t1−x1)θ−1 2(ρ(j) 2)θ2(t2−x2)] dµ(t1,t 2,θ 1,θ 2). Now, the piece ˜ T(1,1)g, from its very formulation, is equal to a function in standard form. We can dispose of it quickly. We write: ˜ T(1,1) g(x1,x 2)= Rd1+1 +×Rd2+1 + g(t1,t 2,θ 1,θ 2) ×[θ−1 1(ρ(1) 1)θ1(t1−x1)θ−1 2(ρ(1) 2)θ2(t2−x2)] dµ(t1,t 2,θ 1,θ 2) = RT(R) g(t1,t 2,θ 1,θ 2) ×[θ−1 1(ρ(1) 1)θ1(t1−x1)θ−1 2(ρ(1) 2)θ2(t2−x2)] dµ(t1,t 2,θ 1,θ 2). (4.2) The sum is over all double-dyadic rectangles R, but only finitely many terms are not identically zero, because gand the ρ(1) i’s have compact supports. It is clear that each summand in (4.2), as a function of x, has support contained in its respective ˜ R. These functions also inherit smoothness and cancellation from the ρ(1) i’s. Thus we may write the 178 J. M. Wilson sum as  R λ˜ Rb˜ R(x1,x 2), where each b˜ Ris adapted to ˜ Rand the λ˜ R’s satisfy |λ˜ R|≤CT(R) |g|dµ(t, y)(Q1)−1(Q2)−1|R|−1/2 ≤CT(R) |g|qdµ(t, y)1/q µ(T(R))1/q(Q1)−1(Q2)−1|R|−1/2 with a constant Cthat depends on the di’s and the ρ(1) i’s. Take η>p ,asinthe hypotheses of Theorem 1.3, and suppose that w is a weight satisfying (1.9) for all rectangles R.ByTheorem 2.3, Rd1×Rd2 |˜ T(1,1)g|pσdx≤C R |λ˜ R|2 |R|σ(˜ R, η)2/pp/2 ≤C R |λ˜ R|2 |R|w(˜ R)2/pp/2 . Since q≤2, the last quantity is less than or equal to C R |λ˜ R|q |R|q/2w(˜ R)q/pp/q .(4.3) The hypothesis (1.10) on wimplies (after an elementary estimate) µ(T(R))q/qw(˜ R)q/p(Q1)−q(Q2)−q|˜ R|−q≤C. Therefore, our bound on λ˜ Rimplies that (4.3) is less than or equal to C RT(R) |g|qdµ(t, y)p/q =CRd1+1 +×Rd2+1 + |g|qdµ(t, y)p/q , which is exactly what we want. Thus, the ˜ T(1,1)gterm is okay. The terms ˜ T(1,2)g,˜ T(2,1)g, and ˜ T(2,2)ginvolve non-compactlysupported kernels, and require different arguments. This is where we will use Lemma 3.1. It is obvious that ˜ T(1,2)gand ˜ T(2,1)gare the same kind of animal, and so we need only treat one of them. It will turn out Two-Parameter Bergman Inequalities 179 that the argument that handles ˜ T(2,2)gcan also be used, with minor modifications, on ˜ T(1,2)g. Therefore we shall deal with ˜ T(2,2)gfirst. Our argument is modeled closely on that of [WhWi]. Let κbe the dual exponent to p/2 (which, recall, is ≥1), and let h∈Lκ(σ)be non-negative, satisfy hLκ(σ)=1,and be chosen so that Rd1×Rd2 |˜ T(2,2)g|pσdx=Rd1×Rd2 |˜ T(2,2)g|2hσdx p/2 .(4.4) We seek a good a priori bound, independent of h, for the right-hand side of (4.4). The function ˜ T(2,2)gis bounded, smooth, and has good decay at infinity. If we let Ψybe as defined at the beginning of Section 2, then by a standard approximation argument (essentially Fatou’s Lemma), combined with Theorem 2.1, we may write: Rd1×Rd2 |˜ T(2,2)g|2hσdx≤C R |ΛR|2 |R|(hσ)( ˜ R, η), where ηis any number larger than 2, and ΛR=T(R) |˜ T(2,2)g∗Ψy(t)|2dt1dt2dy1dy2 y1y21/2 . As in the proof of Theorem 2.3, we can dominate (hσ)( ˜ R, η)byaconstant times ˜ R hσφ (˜ R)dx, where φ(˜ R)is positive and satisfies 1 |˜ R|˜ R exp([φ(˜ R)]1/η)dx ≤6. With the φ(˜ R)’s now fixed, let us define ν(R)=˜ R hσφ (˜ R)dx. Then: Rd1×Rd2 |˜ T(2,2)g|2hσdx≤C R |ΛR|2 |R|ν(R), and it is this last object which we must bound. We need to know how big Λ(R) can get (or doesn’t get). Let us make the convention that “(x, y)∈Rd1+1 +×Rd2+1 +” means “x=(x1,x 2); xi∈Rdi;y=(y1,y 2); yi>0”; and analogously, when we write “(x, y)∈ 180 J. M. Wilson T(R)”, with R=Q1×Q2,wemean that (xi,y i)∈T(Qi). For (t, θ)∈ Rd1+1 +×Rd2+1 +, set Πθ(t)=(ρ(2) 1)θ1(t1)·(ρ(2) 2)θ2(t2). The “ρ(2)” functions satisfy the cancellation and decay hypotheses required of the φi’s in the statement of Lemma 3.1. The discussion following the lemma shows that if (t, θ)∈T(R)=T(Q1)×T(Q2) and (x, y)∈T(R)=T(Q 1)×T(Q 2), then θ−1 1θ−1 2|Ψy∗Πθ(t−x)|≤Ca1(Q 1,Q 1)·a2(Q 2,Q 2) =Cβ(R,R). Since |˜ T(2,2)g∗Ψy(x)|≤Rd1+1 +×Rd2+1 + |g(t, θ)|θ−1 1θ−1 2|Ψy∗Πθ(t−x)|dµ(t, θ) = R(T(R) |g(t, θ)|θ−1 1θ−1 2|Ψy∗Πθ(t−x)|dµ(t, θ), we at once get that Λ(R)=T(R) |Ψy∗T(2,2)g(x)|2dx1dx2dy1dy2 y1y21/2 ≤C|R|1/2 R β(R,R)G(R), where we have set G(R)=T(R) |g(t, θ)|dµ(t, θ). (We refer the reader to [WhWi, pp. 942–943] for a detailed discussion of this argument in the one-parameter setting.) If we now define Γ(R)=µ(T(R))1−q, then H¨older’s inequality implies  R G(R)qΓ(R)1/q ≤Rd1+1 +×Rd2+1 + |g|qdµ(t, θ)1/q . Two-Parameter Bergman Inequalities 181 On the other hand, the preceding discussion implies that Rd1×Rd2 |˜ T(2,2)g|p σdx 1/p ≤C R |Λ(R)|2 |R|ν(R) 1/2 ≤C  R R β(R,R)G(R)2 ν(R)   1/2 . (4.5) Our goal now is to show that, under the hypotheses of Theorem 1.3, the inequality   R R β(R,R)G(R)2 ν(R)  1/2 ≤C R G(R)qΓ(R)1/q obtains for all non-negative, finitely-supported sequences {G(R)}R. In other words, we have reduced our problem to showing that the “kernel” β(R,R) maps boundedly from the sequence space q(Γ(R)) into the sequence space 2(ν(R)). We shall prove this boundedness in the same way as in [WhWi], i.e., by means of the Riesz-Thorin Interpolation Theorem. We shall need two endpoint estimates, ∞→ ∞and 1→ 2/q (recall that 2/q≥1). In order to make these estimates (particularly the first) go through smoothly, let us redefine our problem, by setting G(R)=Y(R)|ˆ R|, and having {Y(R)}be the sequence that is acted on. This change-of-variable requires that we replace the kernel β(R,R) with B(R,R). In addition, we must replace the “weight” Γ(R)by|ˆ R|qΓ(R). This done, we now need to show that the kernel B(R,R) maps boundedly ∞→ ∞and 1(|ˆ R|qΓ(R)) → 2/q(ν(R)). ∞→ ∞.This is equivalent to having RB(R,R)≤Cfor all R, and this inequality will follow if we have, for i=1,2, and all dyadic cubes Q i⊂Rdi,  Qi Ai(Q i,Q i)≤C. This is proved in [WhWi], though with slightly different notation from what we have here. For the sake of completeness (and ease of reading), we shall give a proof that uses our present notation. 182 J. M. Wilson Let us write the sum as (I)i+(II)i+(III)i, where (I)i= Qi Qi⊂˜ Q i Ai(Q i,Q i) (II)i= Qi:Qi⊂ ˜ Q i (Qi)≤(Q i) Ai(Q i,Q i) (III)i= Qi:Qi⊂ ˜ Q i (Qi)>(Q i) Ai(Q i,Q i). Now: (I)i≤C Qi Qi⊂˜ Q i (Qi) (Q i)di+mi+2 log(e+(Q i)/(Qi)) ≤Cδ Qi Qi⊂˜ Q i |Qi| |Q i|1+δ , for some δ>0, since di+mi+2>d i. But it is easy to see [WhWi] that this last sum is ≤Cδ,di.Somuchfor (I)i. (II)i: (II)i= ∞  k=0  Qi:Qi⊂ ˜ Q i (Qi)=2−k(Q i) Ai(Q i,Q i) ≤C ∞  k=0 xi/∈˜ Q i (Q i)(2−k(Q i))mi+2 (2−k(Q i)+|x−xQ i|)di+mi+3 dxi ≤C ∞  k=0 (Q i)(2−k(Q i))mi+2(Q i)−mi−3 ≤C ∞  k=0 2−k(mi+2) ≤C. Two-Parameter Bergman Inequalities 183 (III)i: (III)i= ∞  k=1  Qi:Qi⊂ ˜ Q i (Qi)=2k(Q i) Ai(Q i,Q i) ≤C ∞  k=1 Rdi (Q i)(2k(Q i))mi+2 (2k(Q i)+|x−xQ i|)di+mi+3 dxi ≤C ∞  k=1 2−k ≤C. The ∞→ ∞bound has been proved. Now for the 1→ 2/qbound. By Minkowki’s inequality for double integrals,   R R B(R,R)Y(R)2/q ν(R)  q/2 ≤ R R B(R,R)2/qν(R)q/2 Y(R). Therefore, the 1→ 2/qbound will follow if  R B(R,R)2/qν(R)q/2 ≤C|ˆ R|qΓ(R),(4.6) holds for all R, for some constant C. Inequality (4.6) will turn out to be an easy consequence of Lemma 3.2 and the hypotheses of Theorem 1.3. 184 J. M. Wilson Proof of Inequality (4.6):  R B(R,R)2/qν(R) = k1,k2 R=Q 1×Q 2 (Q j)=2ki(Qj) B(R,R)2/q˜ R hφ(˜ R)σdx ≤C k1,k2Rd1×Rd2      R=Q 1×Q 2 (Q j)=2kj(Qj) B(R,R)2/qχ˜ R(x)     hφ(˜ R)σdx ≤C k1,k2     Rd1×Rd2      R=Q 1×Q 2 (Q j)=2kj(Qj) B(R,R)p /q (φ(˜ R))p /2χ˜ R (x)      σdx      2/p . (4.7) Inequality (4.7) is true because of H¨older’s inequality (recall the normalization on h) and the fact that, for each fixed pair (k1,k 2), no point of Rd1×Rd2lies in more than C(d1,d 2) rectangles ˜ Rwith the specified dimensions. The reasoning from Theorem 2.3 tells us that, for each ˜ R, (φ(˜ R))p/2χ˜ R(x)σdx≤Cσ(˜ R,ηp /2), which, by taking ηsufficiently close to 2, we may assume is ≤Cw(˜ R). Thus, we may dominate (4.7) by C k1,k2     Rd1×Rd2      R=Q 1×Q 2 (Q j)=2kj(Qj) B(R,R)p/qχ˜ R(x)     wdx      2/p . Two-Parameter Bergman Inequalities 185 Because of Corollary 3.4, this is less than or equal to: C k1,k2(1 + |k1|)(1 + |k2|)2−(|k1|+|k2|)2/q ×Rd1×Rd21 ((Q1)+|x1−xQ1|)d1+m1+2− ×1 ((Q2)+|x2−xQ2|)d2+m2+2− ×(Q1)(d1+m1+2−)(Q2)(d2+m2+2−)p/q wdx 2/p ≤CRd1×Rd21 ((Q1)+|x1−xQ1|)d1+m1+2− ×1 ((Q2)+|x2−xQ2|)d2+m2+2− ×(Q1)(d1+m1+2−)(Q2)(d2+m2+2−)p/q wdx 2/p , which (see again the hypotheses of Theorem 1.3) is assumed to be less than or equal to Cµ(T(R))−2/q(Q1)2(d1+1)(Q2)2(d2+1). When we raise this to the power q/2, the result is less than or equal to Cµ(T(R))−q/q|ˆ R|q=CΓ(R)|ˆ R|q, which is what we wanted. Therefore, the T(2,2) term is okay. We can handle the term ˜ T(1,2)gby modifying the preceding argument just a little. First, observe that, if f∈L2(Rd1×Rd2), then f(x1,x 2)=Rd2+1 + [f(x1,·)∗(ψ2)y2(t2)·(ψ2)y2(x2−t2)]dt2dy2 y2 (4.8) in L2. The meaning of (4.8) is that we take the convolution of fwith (ψ2)y2“in the x2variable” (leaving x1fixed), and then convolve that with (ψ2)y2again, much as we do in the original Calder´on-Torchinsky formula (2.1). The proof of (4.8) comes by Fourier inversion, where we 192 J. M. Wilson holds for all mixed partials such that j2=0, and for all f∈∪ 1≤r<∞Lr(Rd,dx). The symmetric result holds for j2=0and j1=0. When j1=j2=0, the inequality analogous to (5.3) is: µ(T(R))1/q Rd1×Rd21 ((Q1)+|x1−xQ1|)(d1+1−)p/q ×1 ((Q2)+|x2−xQ2|)(d2+1−)p/q ×w(x)dx1/p ≤C(Q1)d1+1−(d1+1−)/q(Q2)d2+1−(d2+1−)/q =C|ˆ R|1/q(Q1)−/q(Q2)−/q. References [CF] S.-Y. A. Chang and R. A. Fefferman,Acontinuous version of duality of H1with BMO on the bidisc, Ann. of Math. (2) 112(1) (1980), 179–201. [St] E. M. Stein,“Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals”, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993. [WhWi] R. L. Wheeden and J. M. Wilson,Weighted norm estimates for gradients of half-space extensions, Indiana Univ. Math. J. 44(3) (1995), 917–969. [W1] J. M. Wilson,Weighted norm inequalities for the continuous square function, Trans. Amer. Math. Soc. 314(2) (1989), 661–692. [W2] J. M. Wilson, Some two-parameter square function inequalities, Indiana Univ. Math. J. 40(2) (1991), 419–442. [W3] J. M. Wilson,Atwo-parameter “Bergman space” inequality, Proc. Amer. Math. Soc. 125(3) (1997), 755–762. [W4] J. M. Wilson,Asemi-discrete Littlewood-Paley inequality, Studia Math. (to appear). Two-Parameter Bergman Inequalities 193 Department of Mathematics and Statistics University of Vermont Burlington, VT 05405 U.S.A. E-mail address:[email protected] Primera versi´o rebuda el 21 de febrer de 2002, darrera versi´o rebuda el 29 d’octubre de 2002.