Lusin area integrals related to Jacobi expansions

We investigate mixed Lusin area integrals associated with Jacobi trigonometric polynomial expansions. We prove that these operators can be viewed as vector-valued Calder\'on-Zygmund operators in the sense of the associated space of homogeneous type. Consequently, their various mapping properties, in particular on weighted $L^p$ spaces, follow from the general theory.


Introduction
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) Calderón-Zygmund operators. That research included such operators as higher order Riesz transforms, multipliers of Laplace and Laplace-Stieltjes transform types, Jacobi-Poisson semigroup 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 in [13, p. 187] in the present paper, we study mixed Lusin area integrals from a similar perspective. These objects have more complex structure than those mentioned above, therefore their treatment is considerably more involved and demands more effort and additional technical tools. We point out that analysis in various Jacobi settings received a considerable attention in recent years, see for instance [4,9,11,12,13] as well as numerous other references given there.
In the last years Lusin area type integrals attracted attention of many mathematicians. These operators, sometimes called conical square functions, were also studied in some contexts of orthogonal expansions, see e.g. [1,2,3,5,8,10,15,16,21]. In the classical situation these objects turn out to be not only interesting on their own right, but also have significant applications. For instance, a variant of Lusin area integral was used by Segovia and Wheeden [19] to characterize potential spaces on R d , d ≥ 1. Inspired by that paper, the authors of [2] showed, among other things, that a similar characterization is also possible in case of some Schrödinger 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.
In this article we study two kinds of mixed Lusin area integrals S α,β M,N and S α,β M,N (see Section 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 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.
Our main result, see Theorem 2.1 below, says that the mixed Lusin area integrals of both kinds can be viewed as vector-valued Calderón-Zygmund operators in the sense of the associated 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.
Now we introduce the notion of (higher order) derivatives associated with our setting. The natural first order derivative δ emerges from the factorization notice that δ * is the formal adjoint of δ in L 2 (dµ α,β ). The choice of δ as the first order derivative 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 δ N and D N = . . . δδ * δδ * δ N components , N ≥ 0 (by convention, δ 0 = D 0 = Id), respectively. We point out that the derivative δ N is used in [4,11,13] whereas D N appears in [4,9] and have its roots in the so-called symmetrization procedure proposed in [14]. Now we are ready to introduce the central objects of our study in this paper. We define the mixed Lusin area integrals as follows where M, N ∈ N are such that M + N > 0, Γ(θ) is the cone with vertex at θ ∈ (0, π), (note that the exact aperture of this cone is insignificant for our developments and that we may replace Γ by Γ = Γ ∩ (−π, π) × (0, ∞)) and V α,β t (θ) is the µ α,β measure of the ball (interval) centered at θ and of radius t, restricted to (0, π). More precisely, Observe that the formulas defining S α,β M,N f and S α,β M,N f , understood in a pointwise sense, are valid for any f ∈ L p (wdµ α,β ), w ∈ A α,β p , 1 ≤ p < ∞; see the comment concerning the smoothness of H α,β t f (θ) above.
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 α,β , | · |). We will need a slightly more general definition of the 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.
It is well known that a large part of the classical theory of Calderón-Zygmund operators remains valid, with appropriate adjustments, when the underlying space 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 α,β , | · |) associated with a Banach space B, then its mapping properties in weighted L p spaces follow from the general theory.
Obviously, the mixed Lusin area integrals S α,β M,N and S α,β M,N are nonlinear. However, they can be written as where the function Ω α,β is given by This, in turn, shows that these operators can be viewed as vector-valued linear operators taking values in B = L 2 (Γ, t 2M +2N −1 dηdt). Further, note that the formal computation suggests that the kernels associated with S α,β M,N and S α,β M,N are given by The main result of the paper reads as follows.  investigated in [13] and [9], respectively. This, however, was recently established, even on weighted L p , 1 < p < ∞, spaces, in [13, Corollary 5.2] and [9, Theorem 3.2 and Proposition 3.9], respectively.
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 L 2 (dµ α,β )boundedness of the operators in question and the standard estimates for the kernels S α,β M,N (θ, ϕ) and S α,β M,N (θ, ϕ). The latter fact is justified in Theorem 2.3 below. The details are fairly standard and thus left to the reader. Theorem 2.3. Assume that α, β > −1, and let M, N ∈ N be such that M + N > 0. Then the kernels S α,β M,N (θ, ϕ) and S α,β M,N (θ, ϕ) satisfy the standard estimates with the Banach spaces B = L 2 (Γ, t 2M +2N −1 dηdt) and with any γ ∈ (0, 1/2] satisfying γ < α ∧ β + 1.
The proof of Theorem 2.3, which is the most technical part of the paper, is located in Section 3. An important consequence of Theorem 2.1 is the following result.
Proof. The fact that S α,β M,N and S α,β M,N extend to bounded operators on L p (wdµ α,β ), w ∈ A α,β p , 1 < p < ∞, and from L 1 (wdµ α,β ) to L 1,∞ (wdµ α,β ), w ∈ A α,β 1 follows from the (vector-valued) 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.
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) estimate for S α,β 1,0 = S α,β 1,0 with all A α,β 1 weights admitted. The assumptions imposed in [7] are indeed satisfied, since the Jacobi-heat kernel of the Jacobi-heat semigroup exp(−tJ α,β ) t>0 possesses the so-called Gaussian bound. The latter was recently established independently by Coulhon

Kernel estimates
This section is devoted to proving standard estimates for the kernels S α,β M,N (θ, ϕ) and S α,β M,N (θ, ϕ) related to the Banach spaces B = L 2 (Γ, t 2M +2N −1 dηdt) 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 admissible 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 authors to elaborate a technique of estimating various kernels expressible via H α,β t (θ, ϕ) and in 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 all α, β > −1. This is caused by the fact that for α, β ≥ −1/2 the above mentioned integral representation for H α,β t (θ, ϕ) is less complicated. However, in the sequel we would like to treat 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 α be the probability measure on the interval [−1, 1] defined by and in the limit case dΠ −1/2 is the sum of point masses at −1 and 1 divided by 2. Further, let and put Furthermore, for W, s ∈ R fixed, we consider a function Υ α,β W,s (t, θ, ϕ) defined on (0, π] × (0, π) × (0, π) as follows.
Proof. We first note that the estimates obtained in [13,Corollary 3.5] actually hold uniformly in t ∈ (0, T 0 ], where T 0 > 0 is an arbitrary but fixed constant. Therefore, the estimate for the first term in question is just this strengthened version of the above mentioned result since δ N = ∂ N . We now focus on the remaining term. Notice that the Jacobi-Poisson semigroup H α,β t = exp(−t √ J α,β ), t > 0, satisfies the differential equation . This forces that the Jacobi-Poisson kernel H α,β t (θ, ϕ) also satisfies this equation with respect to θ, which, together with the identity J α,β = D 2 + λ α,β 0 , leads to the equation Iterating this we infer that for t > 0 and θ, ϕ ∈ (0, π) we get where N = χ {N is odd} and c n are some constants. Now, using the already justified estimate for the first term in question and the fact that for any −∞ < W ≤ W ′ < ∞, s ∈ R fixed we have Υ α,β W,s (t, θ, ϕ) Υ α,β W ′ ,s (t, θ, ϕ), t ∈ (0, π], θ, ϕ ∈ (0, π) (this easily follows from the boundedness of q(θ, ϕ, u, v)), we obtain the desired conclusion for the second term in question.
We need the following generalization of the above lemma.
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 L p ((0, 1), t W −1 dt) appearing there by L p ((0, T 0 ), t W −1 dt) with any T 0 > 0 fixed), which played a crucial role in showing standard estimates for various kernels investigated in the just mentioned paper. This lemma provides an important connection between estimates emerging from Lemma 3.2 and the standard estimates related to the space of homogeneous type ((0, π), dµ α,β , | · |).
Proof. Since δ N = ∂ N , the norm finiteness of the first expression in 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].
In the proof of this lemma we will use the following estimate. For a fixed ξ ≥ 1 we have Proof of Lemma 3.7. Since the function Ω α,β (θ, η, t) stabilizes for t ≥ π and the constraint 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 Therefore we have reduced the proof of the lemma to showing that for j = 1, 2 we have for each fixed γ ∈ (0, 1/2] satisfying γ < α ∧ β + 1. We will treat I 1 and I 2 separately. Let f (θ) = sin θ+η 2 2α+1 cos θ+η 2 2β+1 . Applying the estimate (7) specified to ξ = 1/(2γ) and then the Mean Value Theorem we obtain where θ is a convex combination of θ and θ ′ , which depends also on η. An elementary computation gives .
Taking into account (6) we obtain Using once again (6) and (9) we get The relevant bound for the first term is straightforward. To analyze the second one it is enough to use the estimates The third term can be bounded by means of the inequality and the fact that 2α + 2 > 2γ. This finishes proving Lemma 3.7.