Analysis related to all admissible type parameters in the Jacobi setting

We derive an integral representation for the Jacobi-Poisson kernel valid for all admissible type parameters $\alpha,\beta$ in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the Jacobi setting, which works for all possible $\alpha$ and $\beta$. As a consequence, we can prove that several fundamental operators in the harmonic analysis of Jacobi expansions are (vector-valued) Calder\'on-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory. The new Jacobi-Poisson kernel representation also leads to sharp estimates of this kernel. The paper generalizes methods and results existing in the literature, but valid or justified only for a restricted range of $\alpha$ and $\beta$.


Introduction
This paper is a continuation and completion of the research performed recently in [7] by the first and second authors. Given parameters α, β > −1, consider the Jacobi differential operator This operator, acting initially on C 2 c (0, π), has a natural self-adjoint extension in L 2 (dµ α,β ), whose spectral decomposition is discrete and given by the classical Jacobi polynomials. Various aspects of harmonic analysis related to the Jacobi setting has been studied in the literature. This line of research goes back to the seminal work of B. Muckenhoupt and E.M. Stein [6], in which the ultraspherical case (α = β) was investigated. Later several other authors contributed to the subject, see [7,Section 1] and also the end of [7, Section 2] for a detailed account and references.
The main result of [7] is restricted to α, β ≥ −1/2. It states that several fundamental operators in the harmonic analysis of Jacobi expansions, including Riesz transforms, imaginary powers of the Jacobi operator, the Jacobi-Poisson semigroup maximal operator and Littlewood-Paley-Stein type square functions, are (vector-valued) Calderón-Zygmund operators in the sense of the space of homogeneous type ([0, π], dµ α,β , | · |). Here | · | stands for the ordinary distance. Consequently, the mapping properties of these operators follow from the general theory. The proofs in [7] rely that the operators in question can be interpreted as Calderón-Zygmund operators and giving, as a consequence, their L p mapping properties. Finally, Section 6 is devoted to sharp estimates of the Jacobi-Poisson kernel.
Throughout the paper we use a fairly standard notation with essentially all symbols referring to the space of homogeneous type ([0, π], dµ α,β , | · |). Since the distance in this space is the Euclidean one, the ball denoted B(θ, r) is simply the interval (θ − r, θ + r) ∩ [0, π]. 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.
The last term in (1) is nonzero when α + β < −1. As we shall see later, there are important cancellations between the two terms in (1) for large t.
We remark that in Theorem 2.1 it does not matter whether one integrates over the open interval (0, 1) or over (0, 1], even when the measure is dΠ −1/2 . But in the sequel, it will be more convenient to use (0, 1]. Next we restate the formula of Theorem 2.1 in order to obtain a more suitable representation of H α,β t (θ, ϕ) for the kernel estimates in Section 4. Recall that for −1 < α < −1/2, Π α (u) is an odd function, which is negative for u > 0. It can easily be verified that the density |Π α (u)| defines a finite measure on [−1, 1]. In fact, we have the following.
Proof. These three quantities are even in u, and we need consider only u ∈ (0, 1). It is enough to observe that then |Π α (u)| ≃ (i) If α, β ≥ −1/2, then Proof. Item (i) is just (3). To prove the remaining items, we combine Theorem 2.1, Lemma 2.2 and symmetries of the quantity Ψ α,β E (t, θ, ϕ, u, v), its derivatives in u and v, and the measures involved. We give further details in case of (ii), leaving similar proofs of (iii) and (iv) to the reader.
Assume that −1 < α < −1/2 ≤ β. Since dΠ β is a symmetric probability measure on [−1, 1] and has no atom at 0, formula (7) reduces to Then, expressing Ψ α,β E via Ψ α,β and making use of the symmetry of dΠ β , we see that In I 1 we integrate by parts in the u variable, which is legitimate in view of Lemma 2.2. Observe that the integrand in I 1 vanishes for u = 1 and that Π α (0) = 0. We get Inserting the definition of the symmetrization Ψ α,β E , one easily finds that The conclusion follows.
Remark 2.4. All the representations of H α,β t (θ, ϕ) contained in Proposition 2.3 are positive in the sense that each of the double integrals (there are one of these in (i), two in (ii) and in (iii), and four in (iv)) is nonnegative.

Preparatory results
In this section we gather various technical results, altogether forming a transparent and convenient method of proving standard estimates for kernels defined via the Jacobi-Poisson kernel. The essence of this technique is a uniform way of handling double integrals against products of measures of type dΠ γ and Π γ (u) du, independently of the integrands. Then the expressions one has to estimate contain only elementary functions and are relatively simple.
The result below, which is a generalization of [7, Lemma 4.3], plays a crucial role in our method to prove kernel estimates. It provides a link from estimates emerging from the integral representation of H α,β t (θ, ϕ), see Proposition 2.3, to the standard estimates related to the space of homogeneous type ([0, π], dµ α,β , | · |).
Note that for any α, β > −1 fixed, the µ α,β measure of the interval B(θ, |θ − ϕ|) can be described as follows, see [7,Lemma 4.2], Notice also that the right-hand sides of the estimates in Lemma 3.1 are always larger than a positive constant, uniformly in θ, ϕ ∈ [0, π]. This fact will be used in the sequel without further mention. To prove Lemma 3.1, we need item (b) in the lemma below. This is a generalization of [7, Proof. Part (a) is proved in [8]. Part (b) can easily be deduced from (a) since the integral to be estimated is controlled by the same integral with κ = 0.
To prove (9), it is convenient to distinguish two cases.
The remaining part of this section embraces various technical results, which will allow us to control the relevant kernels by means of the estimates from Lemma 3.1. To state the next lemma and also for further use, we introduce the following notation. We will omit the arguments and write briefly q instead of q(θ, ϕ, u, v), when it does not lead to confusion. For a given parameter λ ∈ R, we define the auxiliary function To prove this lemma, we need two preparatory results. One of them is Faà di Bruno's formula for the N th derivative, N ≥ 1, of the composition of two functions (see [5] for the related references and interesting historical remarks). With D denoting the ordinary derivative, it reads where the summation runs over all j 1 , . . . , j N ≥ 0 such that j 1 + 2j 2 + . . . + N j N = N . Further, in the proof of Lemma 3.3 we will make use of the following bounds given in [7].
Proof of Lemma 3.3. Given λ ∈ R, we introduce the auxiliary function We first reduce our task to showing the estimate where c λ is a constant, possibly negative. Using Faà di Bruno's formula (10) with f (t) = cosh t 2 − 1 + q and either g( where the C λ,j are constants, possibly zero. Differentiating these identities with respect to θ, ϕ, u, v and then applying (11) and the relations cosh we see that Now by the boundedness of q and the inequality forced by the constraint j 1 + . . . + (M + 1)j M +1 = M + 1, we get the asserted estimate. Thus it remains to prove (11). We assume that N ≥ 1. The simpler case N = 0 is left to the reader. Taking into account the relations see [7,Section 4], and using Faà di Bruno's formula with f (θ) = cosh t 2 − 1 + q and g(x) = x −λ , we get where c λ,j are constants. Further, keeping in mind that L, R, K ∈ {0, 1} and applying repeatedly Leibniz' rule, we see that ∂ L ϕ ∂ N θ Ψ λ (t, q) is a sum of terms of the form constant times 1 where the indices run over the set described by the conditions j i ≥ 0, j 1 + . . . + N j N = N , l 1 , l 2 , l 3 ≥ 0, l 1 + l 2 + l 3 = L and the exponents of q − 1 and ∂ θ q are nonnegative. Similarly, where also r 1 , . . . , r 5 ≥ 0, r 1 + . . . + r 5 = R, l 1 + l 2 ≥ r 2 , l 3 ≥ r 5 . Finally, since the derivative ) is a sum of terms of the form constant times 1 (cosh t 2 − 1 + q) λ+ i j i +l 1 +r 1 +k 1 Here we must add the conditions k 1 , . . . , k 5 ≥ 0, k 1 + . . . + k 5 = K and replace l 1 + l 2 ≥ r 2 , l 3 ≥ r 5 by l 1 + l 2 ≥ r 2 + k 2 , l 3 ≥ r 5 + k 5 . We shall estimate all the factors in this product from above. Since t ≤ 1, we can replace cosh t 2 − 1 + q by t 2 + q. The quantities q and ∂ ϕ ∂ θ q are bounded. Further, we apply Lemma 3.4 to get To deal with the resulting exponent of 1/(t 2 + q), we observe that cf. (12). Using also the estimates Notice that 2k 1 + k 2 + k 4 ∈ {0, K, 2K}, and similarly 2r 1 + r 2 + r 4 ∈ {0, R, 2R}. This observation leads directly to (11). The proof of Lemma 3.3 is complete.
Lemma 3.6. Let γ ∈ R and η ≥ 0 be fixed. Then Proof. This is elementary. For γ = 0, one has The next lemma will be frequently used in Section 4 to prove the relevant kernel estimates. Only the cases p ∈ {1, 2, ∞} will be needed for our purposes. Other values of p are also of interest, but in connection with operators not considered in this paper.
Proof. It is enough to prove the desired estimate without the term 1 in the left-hand side. Further, since |θ − ϕ| 2 q, it suffices to consider the case s = 0. We prove the estimate when −1 < α, β < −1/2. The remaining cases are left to the reader; they are simpler since then α + β + 3/2 > 0 and one needs Lemma 3.6 only with γ > 0.
The next lemma and corollaries are long-time counterparts of Corollary 3.5 and Lemma 3.7.
To prove this, it is more convenient to employ the series representation of H α,β t (θ, ϕ) rather than the formulas from Proposition 2.3.
Proof of Lemma 3.8. For α, β > −1, t > 0 and θ, ϕ ∈ [0, π] we have Denote the sum in (13) by S. To estimate S and its derivatives, we will need suitable bounds for ∂ N θ P α,β n (θ), N ≥ 0. It is known (see [12, (7.32.2)]) that Combining this with the identity (cf. [12, (4 In view of these facts, the series in (13) can be repeatedly differentiated term by term in t, θ and ϕ, and we get the bounds Since the other term in (13) is trivial to handle, the conclusion follows.
A strengthened special case of Corollary 3.9 will be needed when we estimate kernels associated with multipliers of Laplace-Stieltjes type.
We will show that the following kernels, with values in properly chosen Banach spaces B, satisfy the standard estimates.
(I) The kernel associated with the Jacobi-Poisson semigroup maximal operator, where X is the closed separable subspace of L ∞ (dt) consisting of all continuous functions f on (0, ∞) which have finite limits as t → 0 + and as t → ∞. Observe that H α,β t (θ, ϕ) t>0 ∈ X, for θ = ϕ, as can be seen from Proposition 2.3 and the bound q (θ − ϕ) 2 , and the series representation (see the proof of Lemma 3.8).
We shall see that the same holds also in the vector-valued cases we consider. Then the derivatives in (19) are taken in the weak sense, which means that for any and similarly for ∂ ϕ . If these weak derivatives ∂ θ K(θ, ϕ) and ∂ ϕ K(θ, ϕ) exist as elements of B and their norms satisfy (19), the scalar-valued case applies and (16) and (17) follow. The result below extends to all α, β > −1 the estimates obtained in [7,Section 4] for the restricted range α, β ≥ −1/2. Moreover, here we also consider multipliers of Laplace and Laplace-Stieltjes transform type, which were merely mentioned in [7] and which cover as a special case the imaginary powers of J α,β (or J α,β Π 0 when α + β + 1 = 0) investigated there. In the proof we tacitly assume that passing with the differentiation in θ or ϕ under integrals against dt or dν(t) is legitimate. In fact, such manipulations can easily be verified by means of the dominated convergence theorem and the estimates obtained in Corollary 3.5 and Lemma 3.8.
Since φ is bounded, to prove the gradient estimate it is enough to verify that ∇ θ,ϕ ∂ t H α,β t (θ, ϕ) The case of K α,β ν (θ, ϕ). To show the growth condition it is enough, by the assumption (18) concerning the measure ν, to check that The first estimate above is an immediate consequence of Corollary 3.10 (applied with N = L = 0). On the other hand, the remaining bound is just part of the growth condition for H α,β (θ, ϕ), which is already justified. Taking (18) into account, to verify the gradient estimate (19), it suffices to show that Again, an application of Corollary 3.10 (with either N = 1, L = 0 or N = 0, L = 1) produces the first bound. The second one is contained in the proof of the gradient estimate for H α,β (θ, ϕ).
The proof of Theorem 4.1 is complete.
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 [10,11]. In particular, if T is a Calderón-Zygmund operator in the sense of ([0, π], dµ α,β , | · |) associated with a Banach space B, then its mapping properties in weighted L p spaces follow from the general theory. Let be the Jacobi-Poisson semigroup. For α, β > −1 consider the following operators defined initially in L 2 (dµ α,β ).
The formulas defining H α,β * and g α,β M,N are understood pointwise and are actually valid for general functions f from weighted L p spaces with Muckenhoupt weights. This is because for such f the integral defining H α,β t f (θ) is well defined and produces a smooth function of (t, θ) ∈ Since all the necessary ingredients are contained in [7] and in the present paper, we leave further details to interested readers.
Proof. The part concerning R α,β N and M α,β here the integrand in each double integral is nonnegative, and the one corresponding to η = 1 is dominating. Thus restricting the set of integration to (1/2, 1] 2 and making use of Lemma 2.2, we write The proof of Theorem 6.1 is complete.