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LUSIN AREA INTEGRALS RELATED TO JACOBI EXPANSIONS TOMASZ Z. SZAREK 7 Abstract. We investigate mixed Lusin area integrals associated with Jacobi trigonometric 1 polynomialexpansions. Weprovethattheseoperatorscanbeviewedasvector-valuedCalderoń- 0 Zygmund operators in the sense of the associated space of homogeneous type. Consequently, 2 their various mapping properties, in particular on weighted Lp spaces, follow from the general n theory. a J 9 ] 1. Introduction A C One of the principal aims of the papers [11] and [13] was to prove that several fundamental . harmonic analysis operators in Jacobi trigonometric polynomial expansions are (vector-valued) h t Calderón-Zygmund operators. That research included such operators as higher order Riesz a m transforms, multipliers of Laplace and Laplace-Stieltjes transform types, Jacobi-Poisson semi- [ group maximal operator and Littlewood-Paley-Stein type mixed g-functions. This article is a continuation and completion of the research performed in [11, 13]. Motivated by the comment 1 v in [13, p.187] in the present paper, we study mixed Lusin area integrals from a similar perspec- 9 tive. These objects have more complex structure than those mentioned above, therefore their 8 treatment is considerably moreinvolved and demandsmoreeffort andadditional technical tools. 2 2 We point out that analysis in various Jacobi settings received a considerable attention in recent 0 years, see for instance [4, 9, 11, 12, 13] as well as numerous other references given there. . 1 Inthelastyears Lusinareatypeintegrals attracted attention ofmanymathematicians. These 0 operators, sometimes called conical square functions, were also studied in some contexts of or- 7 1 thogonal expansions, see e.g. [1, 2, 3, 5, 8, 10, 15, 16, 21]. In the classical situation these objects : v turn out to benot only interesting on their own right, butalso have significant applications. For i instance, a variant of Lusin area integral was used by Segovia and Wheeden [19] to characterize X potential spaces on Rd, d 1. Inspired by that paper, the authors of [2] showed, among other r a ≥ things, that a similar characterization is also possible in case of some Schro¨dinger operators’ frameworks. We point out that quite recently some variant of Lusin area integral was investigated in the Ornstein-Uhlenbeck context in [8, 10, 16] in connection with the Gaussian Hardy space theory point of view. Thus our motivation to study mixed Lusin area integrals in Jacobi expansions comes also from their potential applications in further research. α,β α,β In this article we study two kinds of mixed Lusin area integrals S and (see Section M,N SM,N 2 for the definitions), which come from two different notions of higher order derivatives. The first one is simply a composition of the first order derivative and was for instance implemented in [11, 13]. The second one was used in [9] and has its roots in the so-called symmetrization procedure proposed by Nowak and Stempak in [14]. We note that, in some aspects, the latter notion seems to be closer to the classical theory on the Euclidean spaces. To see this compare 2010 Mathematics Subject Classification: primary 42C05; secondary 42C10 Key words and phrases: Jacobi polynomial, Jacobi expansion, Jacobi operator, Jacobi-Poisson kernel, Jacobi- Poisson semigroup, square function, Lusin area integral, Calderoń-Zygmund operator. Research supported bythe National Science Centre of Poland, project no. 2012/05/N/ST1/02746. 1 2 T.Z.SZAREK for example [4, Proposition 2.5] with [4, Remark 3.8], where two kinds of higher order Riesz transforms are studied in the Jacobi context. Ourmainresult,seeTheorem2.1below,saysthatthemixedLusinareaintegralsofbothkinds can beviewed as vector-valued Calderón-Zygmundoperators inthesenseoftheassociated space of homogeneous type. Consequently, their mapping properties follow from the general theory. The main difficulty connected with the Calderón-Zygmund theory approach is showing the related kernel estimates. Here the starting point is the method of proving standard estimates established in [11] for α,β 1/2 and extended in [13] to all admissible α,β > 1. We ≥ − − point out that the crucial role in that method plays a convenient integral representation for the Jacobi-Poisson kernel obtained in [13]. Nevertheless, to treat Lusin area integrals, which have more complex nature than the operators considered in the above mentioned papers, some generalization of this technique is needed. The latter is of independent interest and is inspired by similar tools elaborated recently in other settings of orthogonal expansions, see [5, 15, 21]. The paper is organized as follows. In Section 2 we introduce the context of Jacobi expansions and define the mixed Lusin area integrals. Further, we state the main result (Theorem 2.1) and reduce its proof to showing the standard estimates for the related kernels (Theorem 2.3). We conclude this section by giving comments connected with our main result. Section 3 contains preparatory facts and lemmas, which finally allows us to prove the relevant kernel estimates. This is the largest and the most technical part of the paper. Throughout the paper we use a fairly standard notation with essentially all symbols referring to the space of homogeneous type ((0,π),µ , ), where is the Euclidean norm and µ α,β α,β |·| |·| is a measure on (0,π) defined below. In particular, for θ (0,π) and r > 0 the ball B(θ,r) is ∈ simply the interval (θ r,θ +r) (0,π). By f,g we mean fgdµ whenever the − ∩ h idµα,β (0,π) α,β integral makes sense. Further, by Lp(wdµ )= Lp((0,π),wdµ ) we understand the weighted α,β α,β R Lp space with w being a non-negative weight on (0,π). Furthermore, for 1 p < we denote ≤ ∞ α,β by A the Muckenhoupt class of A weights connected with the space ((0,π),µ , ), for the p p α,β |·| α,β definition of A see for instance [11, p.720]. p When writing estimates, we will frequently use the notation X . Y to indicate that X CY ≤ with a positive constant C independent of significant quantities. We shall write X Y when ≃ simultaneously X . Y and Y . X. 2. Preliminaries and statement of main result As in [11, 13], we consider expansions into Jacobi trigonometric polynomials. For parameters α,β > 1 the normalized Jacobi trigonometric polynomials are given by − α,β(θ)= cα,βPα,β(cosθ), θ (0,π), n 0, Pn n n ∈ ≥ α,β α,β where c are suitable normalizing constants, and P are the classical Jacobi polynomials n n defined on the interval ( 1,1), see Szego˝’s monograph [20] or [11, 13]. It is well known that the − system α,β : n 0 is an orthonormal basis in L2(dµ ), where µ is a measure on the n α,β α,β {P ≥ } interval (0,π) defined by the density θ 2α+1 θ 2β+1 dµ (θ)= sin cos dθ, θ (0,π). α,β 2 2 ∈ (cid:16) (cid:17) (cid:16) (cid:17) α,β Further, are eigenfunctions of the Jacobi differential operator n P d2 α β+(α+β+1)cosθ d α+β +1 2 α,β = − +λα,β, where λα,β = . J −dθ2 − sinθ dθ 0 0 2 (cid:18) (cid:19) LUSIN AREA INTEGRALS RELATED TO JACOBI EXPANSIONS 3 More precisely, 2 α+β+1 α,β α,β = λα,β α,β, λα,β = n+ , n 0. J Pn n Pn n 2 ≥ (cid:18) (cid:19) We denote by the same symbol α,β the natural self-adjoint and non-negative extension in J L2(dµ ) given by α,β ∞ α,βf = λα,β f, α,β α,β J n Pn dµα,βPn n=0 X (cid:10) (cid:11) on the domain consisting of all f L2(dµ ) for which the above series converges in L2(dµ ). α,β α,β ∈ The Jacobi-Poisson semigroup generated by √ α,β is expressible via the spectral series − J ∞ α,βf = e−t√λαn,β f, α,β α,β, t 0, f L2(dµ ). Ht Pn dµα,βPn ≥ ∈ α,β n=0 X (cid:10) (cid:11) Further, it has the integral representation π α,βf(θ)= Hα,β(θ,ϕ)f(ϕ)dµ (ϕ), θ (0,π), f L2(dµ ), Ht t α,β ∈ ∈ α,β Z0 ∞ Hα,β(θ,ϕ) = e−t√λαn,β α,β(θ) α,β(ϕ), t > 0, θ,ϕ (0,π). t Pn Pn ∈ n=0 X α,β It is worth noting that the series/integral defining f(θ) converges pointwise and produces Ht a smooth function of (t,θ) (0, ) (0,π) for any f Lp(wdµ ), w Aα,β, 1 p < ; for α,β p ∈ ∞ × ∈ ∈ ≤ ∞ details see [11, Section 2]. Furthermore, the series defining the so-called Jacobi-Poisson kernel α,β H (θ,ϕ) is well defined in a pointwise sense for all t > 0, θ,ϕ (0,π) and produces a smooth t ∈ function of (t,θ,ϕ) (0, ) (0,π)2. We point out that the exact behavior of this kernel, as ∈ ∞ × well as its various derivatives with respect to t,θ and ϕ, is hidden behind subtle oscillations. To overcome this problem we will use a convenient method of estimating various kernels defined α,β via H (θ,ϕ) established recently in [11] under a restriction α,β 1/2 and then extended in t ≥ − [13] to all admissible α,β > 1. − Now we introduce the notion of (higher order) derivatives associated with our setting. The natural first order derivative δ emerges from the factorization α,β = δ∗δ+λα,β, J 0 where d d θ θ δ = , δ∗ = (α+1/2)cot +(β +1/2)tan ; dθ −dθ − 2 2 notice that δ∗ is theformal adjoint of δ in L2(dµ ). Thechoice of δ as thefirstorder derivative α,β is motivated by mapping properties of fundamental harmonic analysis operators in the Jacobi framework; see [11, Remark 2.6] where it is shown that the choice of δ∗ would be inappropriate. On the other hand, the proper choice of higher order derivatives is a much more subtle matter. In the sequel we will consider two choices. Precisely, we can iterate δ or interlace δ with δ∗, which leads to δN and DN = ...δδ∗δδ∗δ, N 0 ≥ N components (by convention, δ0 = D0 = Id), respect|ivel{y.z W}e point out that the derivative δN is used in [4, 11, 13] whereas DN appears in [4, 9] and have its roots in the so-called symmetrization procedure proposed in [14]. 4 T.Z.SZAREK Now we are ready to introduce the central objects of our study in this paper. We define the mixed Lusin area integrals as follows dµ (η)dt 1/2 Sα,β f(θ)= t2M+2N−1 ∂MδN α,βf(η) 2χ α,β , θ (0,π), M,N (cid:18)ZΓ(θ) t Ht {η∈(0,π)} Vtα,β(θ) (cid:19) ∈ (cid:12) (cid:12) (cid:12) (cid:12) dµ (η)dt 1/2 α,β f(θ)= t2M+2N−1 ∂MDN α,βf(η) 2χ α,β , θ (0,π), SM,N (cid:18)ZΓ(θ) t Ht {η∈(0,π)} Vtα,β(θ) (cid:19) ∈ (cid:12) (cid:12) where M,N N are such that M(cid:12) +N > 0, Γ(θ) i(cid:12)s the cone with vertex at θ (0,π), ∈ ∈ Γ(θ)= (θ,0)+Γ, Γ = (η,t) R (0, ) : η < t , ∈ × ∞ | | (note that the exact aperture of this cone is in(cid:8)significant for our developme(cid:9)nts and that we may α,β replace Γ by Γ = Γ ( π,π) (0, )) and V (θ) is the µ measure of the ball (interval) ∩ − × ∞ t α,β centered at θ and of radius t, restricted to (0,π). More precisely, e α,β V (θ)= µ B(θ,t) , t > 0, θ (0,π). t α,β ∈ ObservethattheformulasdefiningSα,β(cid:0)f and(cid:1)α,β f,understoodinapointwisesense,arevalid M,N SM,N for any f Lp(wdµ ), w Aα,β, 1 p < ; see the comment concerning the smoothness of α,β p ∈ ∈ ≤ ∞ α,β f(θ) above. Ht To obtain the boundedness result for the mixed Lusin area integrals in question we will prove that they can be viewed as vector-valued Calderón-Zygmund operators in the sense of the space of homogeneous type ((0,π),µ , ). We will need a slightly more general definition of the α,β |·| standard kernel, or rather standard estimates, than the one used in [11, 13]. More precisely, we will allow slightly weaker smoothness estimates as indicated below, see for instance [5, 15, 21] where the Lusin area integrals were treated in some other orthogonal expansions settings. Let B be a Banach space and let K(θ,ϕ) be a kernel defined on (0,π) (0,π) (θ,ϕ) :θ = ϕ × \{ } and taking values in B. We say that K(θ,ϕ) is a standard kernel in the sense of the space of homogeneous type((0,π),µ , )if itsatisfies theso-called standard estimates, i.e., thegrowth α,β |·| estimate 1 (1) K(θ,ϕ) B . , θ = ϕ, k k µ (B(θ, θ ϕ)) 6 α,β | − | and the smoothness estimates θ θ′ γ 1 (2) K(θ,ϕ) K(θ′,ϕ) B . | − | , θ ϕ > 2θ θ′ , k − k θ ϕ µ (B(θ, θ ϕ)) | − | | − | (cid:18)| − |(cid:19) α,β | − | ϕ ϕ′ γ 1 (3) K(θ,ϕ) K(θ,ϕ′) B . | − | , θ ϕ > 2ϕ ϕ′ , k − k θ ϕ µ (B(θ, θ ϕ)) | − | | − | (cid:18) | − | (cid:19) α,β | − | for some fixed γ > 0. Notice that the right-hand side of (1) is always larger than the constant 1/µ (0,π), which will be used frequently in the sequel without further mention. Further, α,β notice that the bounds (2) and (3) imply analogous estimates with any 0 < γ′ < γ instead of γ. Moreover, observe that in these formulas the ball (interval) B(θ, θ ϕ) can be replaced by | − | B(ϕ, ϕ θ ), in view of the doubling property of µ . α,β | − | A linear operator T assigning to each f L2(dµ ) a strongly measurable B-valued function α,β ∈ Tf on (0,π). Then T is said to be a (vector-valued) Calderón-Zygmund operator in the sense of the space ((0,π),µ , ) associated with B if α,β |·| (A) T is bounded from L2(dµ ) to L2(dµ ), α,β B α,β LUSIN AREA INTEGRALS RELATED TO JACOBI EXPANSIONS 5 (B) there exists a standard B-valued kernel K(θ,ϕ) such that π Tf(θ)= K(θ,ϕ)f(ϕ)dµ (ϕ), a.a. θ / suppf, α,β ∈ Z0 for every f L∞((0,π)). ∈ Here integration of B-valued functions is understood in Bochner’s sense, and L2(dµ ) is the B α,β Bochner-Lebesgue space of all B-valued dµ -square integrable functions on (0,π). α,β It is well known that a large part of the classical theory of Calderón-Zygmund operators remains valid, with appropriate adjustments, when the underlyingspace is of homogeneous type and the associated kernels are vector-valued, see for instance [17, 18]. In particular, if T is a Calderón-Zygmund operator in the sense of ((0,π),µ , ) associated with a Banach space B, α,β |·| then its mapping properties in weighted Lp spaces follow from the general theory. α,β α,β Obviously, the mixed Lusin area integrals S and are nonlinear. However, they can M,N SM,N be written as Sα,β f(θ)= ∂MδN α,βf(θ+η)χ Ω (θ,η,t) , θ (0,π), M,N t Ht {θ+η∈(0,π)} α,β L2(Γ,t2M+2N−1dηdt) ∈ q SMα,,βNf(θ)= (cid:13)(cid:13)∂tMDNHtα,βf(θ+η)χ{θ+η∈(0,π)} Ωα,β(θ,η,t)(cid:13)(cid:13) L2(Γ,t2M+2N−1dηdt), θ ∈ (0,π), q where the fun(cid:13)ction Ω is given by (cid:13) (cid:13) α,β (cid:13) sin θ+η 2α+1 cos θ+η 2β+1 Ω (θ,η,t) = 2 2 , η R, t > 0, θ,θ+η (0,π). α,β Vα,β(θ) ∈ ∈ (cid:0) (cid:1) t (cid:0) (cid:1) This, in turn, shows that these operators can be viewed as vector-valued linear operators taking values in B = L2(Γ,t2M+2N−1dηdt). Further, note that the formal computation suggests that α,β α,β the kernels associated with S and are given by M,N SM,N α,β α,β α,β α,β S (θ,ϕ) = S (θ,ϕ) , (θ,ϕ) = (θ,ϕ) , M,N M,N,η,t (η,t)∈Γ SM,N SM,N,η,t (η,t)∈Γ where M,N N are su(cid:8)ch that M +N(cid:9)> 0, and (cid:8) (cid:9) ∈ Sα,β (θ,ϕ) = ∂MδNHα,β(ψ,ϕ) χ Ω (θ,η,t), M,N,η,t t ψ t ψ=θ+η {θ+η∈(0,π)} α,β q SMα,,βN,η,t(θ,ϕ) = ∂tMDψNHtα,β(ψ,ϕ)(cid:12)(cid:12) ψ=θ+ηχ{θ+η∈(0,π)} Ωα,β(θ,η,t). q The main result of the paper reads as follows.(cid:12) (cid:12) Theorem 2.1. Let α,β > 1 and M,N N be such that M +N > 0. Then the mixed Lusin − ∈ area integrals Sα,β and α,β can be viewed as vector-valued Calderoń-Zygmund operators in M,N SM,N the sense of ((0,π),µ , ) associated with the Banach spaces B = L2(Γ,t2M+2N−1dηdt). α,β |·| The proof of Theorem 2.1 splits naturally into the following two results. Proposition 2.2. Let α,β > 1 and M,N N be such that M+N > 0. Then the mixed Lusin − ∈ area integrals Sα,β and α,β are bounded on L2(dµ ). Moreover, Sα,β and α,β , viewed as M,N SM,N α,β M,N SM,N vector-valued operators, are associated in the Calderón-Zygmund theory sense with the kernels Sα,β (θ,ϕ) and α,β (θ,ϕ), respectively. M,N SM,N Proof. By Lemma 3.6 below we see that proving the L2(dµ )-boundedness of Sα,β and α,β α,β M,N SM,N α,β α,β + is equivalent to showing an analogous property for the related g-functions g and G M,N M,N investigated in [13] and [9], respectively. This, however, was recently established, even on (cid:0) (cid:1) weighted Lp, 1 < p < , spaces, in [13, Corollary 5.2] and [9, Theorem 3.2 and Proposition ∞ 3.9], respectively. 6 T.Z.SZAREK To prove the kernel associations one can proceed as in [21, Proposition 2.5 on pp.1528-1529], where similar fact was proved for the first order Lusin area integrals in the Laguerre-Dunkl context. The crucial ingredients needed in the reasoning are the just explained L2(dµ )- α,β α,β boundedness of the operators in question and the standard estimates for the kernels S (θ,ϕ) M,N α,β and (θ,ϕ). Thelatter factisjustifiedinTheorem2.3below. Thedetails arefairlystandard SM,N and thus left to the reader. (cid:3) Theorem 2.3. Assume that α,β > 1, and let M,N N be such that M + N > 0. Then − ∈ the kernels Sα,β (θ,ϕ) and α,β (θ,ϕ) satisfy the standard estimates with the Banach spaces M,N SM,N B = L2(Γ,t2M+2N−1dηdt) and with any γ (0,1/2] satisfying γ < α β+1. ∈ ∧ TheproofofTheorem2.3,whichisthemosttechnicalpartofthepaper,islocatedinSection3. An important consequence of Theorem 2.1 is the following result. Corollary 2.4. Let α,β > 1 and M,N N be such that M +N > 0. Then the mixed Lusin − ∈ area integrals Sα,β and α,β are bounded on Lp(wdµ ), w Aα,β, 1 < p < , and from M,N SM,N α,β ∈ p ∞ L1(wdµ ) to L1,∞(wdµ ), w Aα,β. α,β α,β ∈ 1 Proof. The fact that Sα,β and α,β extend to bounded operators on Lp(wdµ ), w Aα,β, M,N SM,N α,β ∈ p 1 < p < , and from L1(wdµ ) to L1,∞(wdµ ), w Aα,β follows from the (vector-valued) ∞ α,β α,β ∈ 1 Calderón-Zygmund theory. Therefore it suffices to justify that these extensions coincide with the original definitions. The latter, however, can be done by using arguments similar to those mentioned in the proof of [11, Corollary 2.5]. Further details are left to the reader. (cid:3) We note that some mapping properties for the first order vertical Lusin area integrals, i.e. with M = 1 and N = 0, follow from results established in a general framework of spaces of homogeneous type. Using [7, Theorem 1.1] we obtain, in particular, weighted weak type (1,1) α,β α,β α,β estimate for S = with all A weights admitted. The assumptions imposed in [7] are 1,0 S1,0 1 indeed satisfied, since the Jacobi-heat kernel of the Jacobi-heat semigroup exp( t α,β) − J t>0 possesses the so-called Gaussian bound. The latter was recently established independently by (cid:8) (cid:9) Coulhon, Kerkyacharian, Petrushev in [6, Theorem 7.2] and by Nowak, Sjögren in [12, Theorem A]. 3. Kernel estimates α,β α,β ThissectionisdevotedtoprovingstandardestimatesforthekernelsS (θ,ϕ)and (θ,ϕ) M,N SM,N related to the Banach spaces B = L2(Γ,t2M+2N−1dηdt) and asserted in Theorem 2.3. We point out that a prominent role in our proof is played by the method of proving kernel estimates established in [11] under the restriction α,β 1/2 and then generalized in [13] to all ad- ≥ − missible type parameters α,β > 1. In those papers a convenient integral representation for − the Jacobi-Poisson kernel was established, see [13, (1) and Proposition 2.3]. This allowed the α,β authors to elaborate a technique of estimating various kernels expressible via H (θ,ϕ) and in t consequence to show that several fundamental harmonic analysis operators in the Jacobi context are (vector-valued) Calderón-Zygmund operators. However, the Lusin area integrals have more complex structure than the operators investigated in [11, 13] and thus to treat them we need to establish further generalization of this interesting method. To achieve this, we will adapt some intuitions and ideas from the contexts of the Dunkl harmonic oscillator [21] and the Dunkl Laplacian [5], where analogous techniques were developed. We emphasize that the analysis related to the restricted range of α,β 1/2 is much simpler than the one concerning ≥ − LUSIN AREA INTEGRALS RELATED TO JACOBI EXPANSIONS 7 all α,β > 1. This is caused by the fact that for α,β 1/2 the above mentioned integral − ≥ − α,β representation for H (θ,ϕ) is less complicated. However, in the sequel we would like to treat t all α,β in a unified way. To prove the kernel estimates stated in Theorem 2.3 we will need some preparatory results, which are gathered below. Some of them were obtained in the previous papers [11, 13], but we recall them here for the sake of reader’s convenience. To state them we shall use the same notation as in [13]. For α > 1/2, let dΠ be the probability measure on the interval [ 1,1] α − − defined by Γ(α+1) dΠ (u) = (1 u2)α−1/2du, α √πΓ(α+1/2) − and in the limit case dΠ is the sum of point masses at 1 and 1 divided by 2. Further, let −1/2 − dΠ , K = 0 −1/2 dΠ = dΠ = , α > 1, α,K (α+1)K−(1−K)/2 (dΠα+1, K = 1 − and put θ ϕ θ ϕ q(θ,ϕ,u,v) = 1 usin sin vcos cos , θ,ϕ (0,π), u,v [ 1,1]. − 2 2 − 2 2 ∈ ∈ − Furthermore, for W,s R fixed, we consider a function Υα,β(t,θ,ϕ) defined on (0,π] (0,π) ∈ W,s × × (0,π) as follows. (i) For α,β 1/2, ≥ − dΠ (u)dΠ (v) α,β α β Υ (t,θ,ϕ) := . W,s x (t2+q(θ,ϕ,u,v))α+β+3/2+W/4+s/2 (ii) For 1< α < 1/2 β, − − ≤ Kk θ ϕ α,β Υ (t,θ,ϕ) :=1+ sin +sin W,s 2 2 K=0,1k=0,1,2(cid:18) (cid:19) X X dΠ (u)dΠ (v) α,K β . x × (t2+q(θ,ϕ,u,v))α+β+3/2+W/4+Kk/2+s/2 (iii) For 1< β < 1/2 α, − − ≤ Rr θ ϕ α,β Υ (t,θ,ϕ) :=1+ cos +cos W,s 2 2 R=0,1r=0,1,2(cid:18) (cid:19) X X dΠ (u)dΠ (v) α β,R . x × (t2+q(θ,ϕ,u,v))α+β+3/2+W/4+Rr/2+s/2 (iv) For 1< α,β < 1/2, − − Kk Rr θ ϕ θ ϕ α,β Υ (t,θ,ϕ) :=1+ sin +sin cos +cos W,s 2 2 2 2 K,R=0,1k,r=0,1,2(cid:18) (cid:19) (cid:18) (cid:19) X X dΠ (u)dΠ (v) α,K β,R . x × (t2+q(θ,ϕ,u,v))α+β+3/2+W/4+Kk/2+Rr/2+s/2 Notice that for any τ R we have Υα,β(t,θ,ϕ) = Υα,β (t,θ,ϕ), which will frequently be ∈ W,s W−2τ,s+τ used in the sequel. Now we are ready to state the following lemma, which was partially proved in [13, Corollary 3.5]. 8 T.Z.SZAREK Lemma 3.1. Let M,N N and L,P 0,1 be fixed. Then, ∈ ∈{ } ∂L∂P∂MδNHα,β(θ,ϕ) + ∂L∂P∂MDNHα,β(θ,ϕ) . Υα,β (t,θ,ϕ) ϕ θ t θ t ϕ θ t θ t 2M+2N,L+P uniformly i(cid:12)n t (0,π] and θ,ϕ (cid:12)(0,π(cid:12)). (cid:12) (cid:12) ∈ ∈(cid:12) (cid:12) (cid:12) Proof. We firstnote that the estimates obtained in [13, Corollary 3.5] actually hold uniformly in t (0,T ], where T > 0 is an arbitrary but fixed constant. Therefore, the estimate for the first 0 0 ∈ term in question is just this strengthened version of the above mentioned result since δN = ∂N. We now focus on the remaining term. Notice that the Jacobi-Poisson semigroup α,β = exp( t√ α,β), t > 0, satisfies the differen- Ht − J tialequation ∂2 α,βf(θ)= α,β α,βf(θ). Thisforces thattheJacobi-Poisson kernelHα,β(θ,ϕ) tHt J Ht t also satisfies this equation withrespectto θ,which, together withthe identity α,β = D2+λα,β, J 0 leads to the equation D2Hα,β(θ,ϕ) = (∂2 λα,β)Hα,β(θ,ϕ), t > 0, θ,ϕ (0,π). θ t t − 0 t ∈ Iterating this we infer that for t > 0 and θ,ϕ (0,π) we get ∈ ⌊N/2⌋ (4) ∂L∂P∂MDNHα,β(θ,ϕ) = c (λα,β)⌊N/2⌋−n∂L∂P+N∂M+2nHα,β(θ,ϕ), ϕ θ t θ t n 0 ϕ θ t t n=0 X where N = χ and c are some constants. Now, using the already justified estimate for {N isodd} n the first term in question and the fact that for any < W W′ < , s R fixed we have −∞ ≤ ∞ ∈ Υα,β(t,θ,ϕ) . Υα,β (t,θ,ϕ), t (0,π], θ,ϕ (0,π) W,s W′,s ∈ ∈ (this easily follows from the boundedness of q(θ,ϕ,u,v)), we obtain the desired conclusion for the second term in question. (cid:3) We need the following generalization of the above lemma. Lemma 3.2. Let M,N N and L,P 0,1 be fixed. Then, ∈ ∈{ } ∂L∂P∂MδNHα,β(ψ,ϕ) + ∂L∂P∂MDNHα,β(ψ,ϕ) . Υα,β (t,θ,ϕ) ϕ ψ t ψ t ψ=θ+η ϕ ψ t ψ t ψ=θ+η 2M+2N,L+P unif(cid:12)ormly in (η,t) Γ, t (cid:12)(0,π] a(cid:12)nd(cid:12)θ,ϕ (0,π) satisfying θ(cid:12)+η (cid:12)(0,π). (cid:12) ∈ ∈(cid:12) (cid:12) (cid:12) ∈ (cid:12) ∈(cid:12) To prove Lemma 3.2 we shall use the following result. Lemma 3.3. For all (η,t) Γ, and θ,ϕ (0,π) with θ+η (0,π), and all u,v [ 1,1], we ∈ ∈ ∈ ∈ − have t2+q(θ+η,ϕ,u,v) t2+q(θ,ϕ,u,v). ≃ Proof. We first show the estimate (.). To prove this, it suffices to verify that q(θ+η,ϕ,u,v) . t2+q(θ,ϕ,u,v), (η,t) Γ, θ,ϕ,θ+η (0,π), u,v [ 1,1]. ∈ ∈ ∈ − Observe that the following comparability holds, see [11, (23)], q(θ,ϕ,u,v) (θ ϕ)2 +(1 u)θϕ+(1 v)(π θ)(π ϕ), θ,ϕ (0,π), u,v [ 1,1]. ≃ − − − − − ∈ ∈ − Further, since (θ+η ϕ)2 . (θ ϕ)2+η2 (θ ϕ)2+t2, − − ≤ − (1 u)(θ+η)ϕ (1 u)θϕ+(1 u)tϕ, − ≤ − − (1 v)(π θ η)(π ϕ) (1 v)(π θ)(π ϕ)+(1 v)t(π ϕ), − − − − ≤ − − − − − LUSIN AREA INTEGRALS RELATED TO JACOBI EXPANSIONS 9 it is enough to check that tϕ .t2+(θ ϕ)2+θϕ, t > 0, θ,ϕ (0,π) − ∈ t(π ϕ) .t2+(θ ϕ)2+(π θ)(π ϕ), t > 0, θ,ϕ (0,π). − − − − ∈ This, in turn, follows from the relation (θ ϕ)2 + θϕ θ2 + ϕ2 for θ,ϕ (0,π), and by a − ≃ ∈ reflection in π/2 reason. Therefore we proved (.). Now, the bound (&) is a straightforward consequence of the first one. Precisely, t2+q(θ,ϕ,u,v) = t2+q((θ+η)+( η),ϕ,u,v) . t2+q(θ+η,ϕ,u,v) − since ( η,t) Γ and θ,θ+η (0,π). (cid:3) − ∈ ∈ Proof of Lemma 3.2. By Lemma 3.1 it suffices to show that for each W,s R fixed, we have ∈ Υα,β(t,θ+η,ϕ) . Υα,β(t,θ,ϕ), (η,t) Γ, θ,ϕ,θ+η (0,π). W,s W,s ∈ ∈ We prove this estimate when 1 < α,β < 1/2; the proofs of the remaining cases are similar − − and even simpler, and hence are omitted. To proceed, notice that for any κ 0 fixed we have ≥ θ+η ϕ κ θ ϕ κ sin +sin . sin +sin +tκ, (η,t) Γ, θ,ϕ,θ+η (0,π), 2 2 2 2 ∈ ∈ (cid:16) θ+η ϕ(cid:17)κ (cid:16) θ ϕ(cid:17)κ cos +cos . cos +cos +tκ, (η,t) Γ, θ,ϕ,θ+η (0,π). 2 2 2 2 ∈ ∈ (cid:16) (cid:17) (cid:16) (cid:17) These can easily be justified with the aid of the relations x x (5) sin x, cos π x, x (0,π), 2 ≃ 2 ≃ − ∈ and the fact that η < t. Using these bounds, Lemma 3.3 and then the obvious inequality | | t (t2+q(θ,ϕ,u,v))1/2, we get ≤ θ ϕ Kkε1 θ ϕ Rrε2 Υα,β(t,θ+η,ϕ) . 1+ sin +sin cos +cos W,s 2 2 2 2 K,XR=0,1k,rX=0,1,2ε1,Xε2=0,1(cid:18) (cid:19) (cid:18) (cid:19) dΠ (u)dΠ (v) tKk(1−ε1)+Rr(1−ε2) α,K β,R x × (t2+q(θ,ϕ,u,v))α+β+3/2+W/4+Kk/2+Rr/2+s/2 θ ϕ Kkε1 θ ϕ Rrε2 . 1+ sin +sin cos +cos 2 2 2 2 K,XR=0,1k,rX=0,1,2ε1,Xε2=0,1(cid:18) (cid:19) (cid:18) (cid:19) dΠ (u)dΠ (v) α,K β,R . x × (t2+q(θ,ϕ,u,v))α+β+3/2+W/4+Kkε1/2+Rrε2/2+s/2 α,β Since kε ,rε 0,1,2 , the last quantity above is comparable to Υ (t,θ,ϕ) and the conclu- 1 2 ∈ { } W,s sion follows. (cid:3) The next result is a slightly modified special case of [13, Lemma 3.7] (note that the norm estimate in [13, Lemma 3.7, p.201, l.2] is still valid if we replace Lp((0,1),tW−1dt) appearing there by Lp((0,T ),tW−1dt) with any T > 0 fixed), which played a crucial role in showing 0 0 standard estimates for various kernels investigated in the just mentioned paper. This lemma providesanimportantconnectionbetweenestimatesemergingfromLemma3.2andthestandard estimates related to the space of homogeneous type ((0,π),dµ , ). α,β |·| Lemma 3.4 ([13, Lemma 3.7]). Let W 1 and s 0 be fixed. Then the estimate ≥ ≥ 1 1 kΥαW,β,s(t,θ,ϕ)kL2((0,π),tW−1dt) . θ ϕs µ (B(θ, θ ϕ)) α,β | − | | − | 10 T.Z.SZAREK holds uniformly in θ,ϕ (0,π), θ = ϕ. ∈ 6 Note that for any α,β > 1 fixed, the µ measure of the ball (interval) B(θ,r) can be α,β − described as follows, cf. [11, Lemma 4.2], r(θ+r)2α+1(π θ+r)2β+1, r (0,π) (6) µ (B(θ,r)) − ∈ , θ (0,π). α,β ≃ (1, r π ∈ ≥ The next lemma is a long-time counterpart of Lemma 3.4 and was partially proved in [13, Corollary 3.9]. Lemma 3.5. Let α,β > 1, M,N N be such that M +N > 0, and let L,P 0,1 and − ∈ ∈ { } W 1 be fixed. Then ≥ sup ∂L∂P∂MδNHα,β(θ,ϕ) + ∂L∂P∂MDNHα,β(θ,ϕ) < . θ,ϕ∈(0,π) ϕ θ t θ t ϕ θ t θ t L2((π,∞),tW−1dt) ∞ (cid:13) (cid:16)(cid:12) (cid:12) (cid:12) (cid:12)(cid:17)(cid:13) (cid:13) (cid:13) Proof(cid:13). Since δN(cid:12)= ∂N, the norm finite(cid:12)nes(cid:12)s of the first expression(cid:12) i(cid:13)n question is a direct consequence of [13, Corollary 3.9]. The conclusion for the second term follows by combining the identity (4) with [13, Corollary 3.9]. (cid:3) The next three lemmas (Lemmas 3.6–3.8) allow us to control certain expressions involving α,β α,β the function Ω appearing in the definitions of the kernels S (θ,ϕ) and (θ,ϕ). α,β M,N SM,N Lemma 3.6. Assume that α,β > 1. Then − χ Ω (θ,η,t)dη = 1, t > 0, θ (0,π). {θ+η∈(0,π)} α,β ∈ Z|η|<t Proof. Simple exercise. (cid:3) Lemma 3.7. Assume that α,β > 1. Then there exists γ = γ(α,β) (0,1/2] such that − ∈ |θ−θ′| 2γ 2 , t (0,π] Z|η|<tχ{θ+η,θ′+η∈(0,π)}(cid:12)qΩα,β(θ,η,t)−qΩα,β(θ′,η,t)(cid:12) dη . |(cid:16)θ−tθ′|(cid:17)2γ, t ≥∈ π , (cid:12) (cid:12) uniformly in t > 0 and θ(cid:12),θ′ (0,π). Moreover, the e(cid:12)stimate holds with any γ (0,1/2]  ∈ ∈ satisfying γ < α β+1. ∧ In the proof of this lemma we will use the following estimate. For a fixed ξ 1 we have ≥ (7) x y ξ . xξ yξ , x,y 0. | − | | − | ≥ Proof of Lemma 3.7. Since the function Ω (θ,η,t) stabilizes for t π and the constraint α,β ≥ θ,θ+η (0,π) forces η < π, we may assume that 0< t π. Further, from now on we restrict ∈ | | ≤ our attention to θ θ′ t. Otherwise, an application of Lemma 3.6 shows that the left-hand | − | ≤ side in question is controlled by a constant and the conclusion trivially follows. Using the estimate (7) with ξ = 2 we obtain 2 χ{θ+η,θ′+η∈(0,π)} Ωα,β(θ,η,t) Ωα,β(θ′,η,t) − . χ{−θ∧θ′<η<(cid:12)(cid:12)πq−θ∨θ′} Ωα,β(θ,η,qt) Ωα,β(θ′,η(cid:12)(cid:12),t) (cid:12) − (cid:12) (cid:12) sinθ+η 2α+1 cos θ+η 2β+(cid:12)1 sin θ′+η 2α+1 cos θ′+η 2β+1 (cid:12) 2 2 (cid:12) − 2 2 ≤ χ{−θ∧θ′<η<π−θ∨θ′}(cid:12)(cid:12)(cid:0) (cid:1) (cid:0) (cid:1) Vtα,β((cid:0)θ′) (cid:1) (cid:0) (cid:1) (cid:12)(cid:12) (cid:12) (cid:12)