Variation and oscillation for semigroups associated with discrete Jacobi operators

In this paper we prove weighted ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^p$$\end{document}-inequalities for variation and oscillation operators defined by semigroups of operators associated with discrete Jacobi operators. Also, we establish that certain maximal operators involving sums of differences of discrete Jacobi semigroups are bounded on weighted ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^p$$\end{document}-spaces. ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^p$$\end{document}-boundedness properties for the considered operators provide information about the convergence of the semigroup of operators defining them.


Introduction
The ρ-variational inequalities for bounded martingales were first studied by Lépingle in [24].These properties can be seen as extensions of Doob's maximal inequality and they give quantitative versions of the martingale convergence theorem.Generalizations of Lépingle's results can be found in [10,27] and [28].Bourgain ([10]) was the first in studying variational inequalities in ergodic theory.He rediscovered Lépingle's inequality and used it to establish pointwise convergence of ergodic averages involving polynomial orbits.The seminal paper [10] opened the study of variational inequalities in harmonic analysis and ergodic theory ( [11,12,18,19,21,22,25,26] and [27].Oscillation and variation estimates for semigroups of operators can be found, for instance, in [9,16,22,30] and [36].Let ρ > 0 and {a t } t>0 ⊂ C. We define the ρ-variation of {a t } t>0 , V ρ ({a t } t>0 ), by .
An important issue in this point is the measurability of these new functions.Comments about this property can be encountered after [11,Theorem 1.2].Our objective is to get L p -boundedness properties for the variations, oscillation and jump operators.As usual, in order to obtain L p -boundedness for the ρ-variation operator, we need to consider ρ > 2. This is the case when we work with martingales, see [22] and [29].The oscillation operator, which has exponent 2, can be a good substitute of the 2−variation operator.According to [25, (1.15)], we can see uniform λ-jump estimates as endpoint estimates for ρ-variations, ρ > 2.Moreover, it is proved in [25, Theorem 1.9] that the oscillation operator cannot be interpreted as an endpoint in the sense of inequality [25, (1.15)] for ρ-variations, ρ > 2.
Let {a j } j∈Z be an increasing sequence in (0, ∞) and {b j } j∈Z a bounded real sequence.
According to [7] and [20], we define, for every and the corresponding maximal operator, S * , by These operators can help us to complete the picture of the convergence properties of {T t } t>0 .By [20, Remark 1], we need to assume that the sequence {a j } j∈Z satisfies some extra condition (lacunarity, for instance) in order to obtain L p -boundedness properties for the operator S * .Our objective is to establish L p -inequalities for all above operators when {T t } t>0 is the discrete Jacobi heat semigroup.We now recall some definitions and properties about Jacobi polynomials that we will use along the paper.Let α, β > −1.For every n ∈ N 0 ∶= N ∪ {0}, we define the n−th Jacobi polynomial , see [35, p.67, formula (4.3.1)].We also consider p .
The sequence {p We define the difference operator J (α,β) as follows, , where The spectrum of the operator J (α,β) is [−2, 0] and, for every x ∈ [−1, 1], The operator J (α,β) is bounded from p (N 0 ) into itself, for every 1 ≤ p ≤ ∞.Furthermore, the operator J (α,β) is selfadjoint on 2 (N 0 ) and −J (α,β) is a positive operator in 2 (N 0 ).We denote by {W } t>0 the semigroup of operators generated by J (α,β) .We define the (α, β)-Fourier transform as follows Thus, F (α,β) is an isometry from 2 (N 0 ) into L 2 ((−1, 1), µ α,β ).We can write, for every t > 0, We can see that, for every t > 0, Gasper ([6, 14] and [15]) established the linearisation property for the product of Jacobi polynomials and his results can be transfered to the polynomials {p (α,β) n } n∈N 0 .Then, a convolution operator can be defined in the {p (α,β) n } n∈N 0 that is transformed by F (α,β) in the pointwise product.For every t > 0, W (α,β) t can be seen as a convolution operator.Askey ([5]) proved a power weighted transplantation theorem for Jacobi coefficients.Recently, Arenas, Ciaurri and Labarga ( [2]) extended Askey's result by considering the transplantation operator as a singular integral and weights in the Muckenhoupt class for (N 0 , P(N 0 ), µ d ).By taking as inspiration point the study of Ciaurri, Gillespie, Roncal, Torrea and Varona ( [13]) about harmonic analysis operators associated with the discrete Laplacian, Betancor, Castro, Fariña and Rodríguez-Mesa ( [8]) established weighted L p -inequalities for harmonic analysis operators in the discrete ultraspherical setting.They took advantage of the discrete convolution operator associated with the ultraspherical polynomials in the discrete context ([17]).Jacobi polynomials reduce to ultraspherical polynomials when α = β.Arenas, Ciaurri and Labarga ( [1,3,4]) extended the results in [8] to the Jacobi context.They needed to use a different procedure from the one employed in [8] for the ultraspherical setting because they can not use the convolution operator.Also, as in [8] and [13], scalar and vector-valued Calderón-Zygmund theory for singular integrals was a main tool.Maximal operators and Littlewood-Paley functions defined for the heat semigroup {W (α,β) t } t>0 were studied in [3] and [1], respectively.Riesz transforms associated with the discrete Jacobi operator J (α,β) were considered in [4].We now state our results.A real sequence {v n } n∈N 0 is said to be a weight when For every weight v on N 0 and 1 ≤ p < ∞, we denote by p (N 0 , w) the weighted p-Lebesgue space on (N 0 , P(N 0 ), µ d ) and by 1,∞ (N 0 , w) the (1, ∞)-weighted Lorentz space on (N 0 , P(N 0 ), µ d ).
a) The variation operator V ρ ({W Results in Theorem 1.1 had not been established for the semigroups generated by the discrete Laplacian and the ultraspherical operators.Now the results in the ultraspherical setting can be deduced from Theorem 1.1 when α = β.Moreover, it will be explained in Section 2 that our procedure in the proof of Theorem 1.1 allows us to prove the corresponding results for the semigroup generated by the discrete Laplacian. Calderón-Zygmund theory for vector-valued singular integrals ( [31] and [32]) will be a main tool in our proof of Theorem 1.1.We can not use the transplantation theorem as in [1] because, in contrast with the Littlewood-Paley functions, variation and oscillation operators are not related with Hilbert norms.We need to refine the arguments developed in [3] by using asymptotics for Jacobi polynomials and Bessel functions.
Ben Salem ([33]) solved an initial value problem associated with a fractional diffusion equation involving fractional powers of the Jacobi operator, (J (α,β) ) γ , and Caputo fractional derivatives in time.By using subordination, from Theorems 1.1 and 1.2 we can deduce the corresponding results when {W (α,β) t } t>0 is replaced by the semigroup of operators generated by (J (α,β) ) γ , γ > 0.
In the next section we prove Theorems 1.1 and 1.2.Throughout this paper, we will always denote by C and c positive constants that can change in each occurrence.
2. Proof of Theorem 1.1 First, we shall prove that V ρ ({W (1) = 0, n ∈ N 0 .We consider the operator J(α,β) defined by and the weight v (α,β) = {p We can write, for every where s } s>0 is Markovian.Furthermore, by using Jensen inequality we deduce that } t>0 is a diffusion semigroup in the Stein's sense ( [34]).
It is clear that g ρ = 0 if, and only if, g is constant.By identifying those functions that differ in a constant, ⋅ ρ is a norm in E ρ and (E ρ , ⋅ ρ ) is a Banach space.We can write ⋅ ρ is not a Hilbert norm.Then, a transplantation theorem can not be applied, in contrast with the case of Littlewood-Paley functions considered in [1].We are going to see that ( 2) First, we prove (2).According to [3, Lemma 5.1], we have that where, for k, l ∈ N, k ≥ 1 and t > 0, Let n, m ∈ N, n ≠ m.We decompose (n, m), t > 0, where Suppose that g ∶ (0, ∞) → C is a differentiable function.We can write We will use (4) several times in the sequel.According to [35, (7.32.6)], we have that ( 5) By using ( 4) and ( 5), we get On the other hand, since P (α+1,β+1) 0 Then, (5) leads to In [35,Theorem 8.21.12], it was established that where Here, c and are fixed positive numbers.By [24, (5.16.1)] we have that We define and l ∈ N.
Assume now that n > 1.By performing the change of variables x = cos θ, we can write Suppose that m > n.By (7) we get that Then, (7) and ( 8) lead to It follows that Thus, we conclude that We are going to see that where Again, since γ k ∼ k, k ∈ N, by using (8) we get Then, According to [24, (5.11.6)], we have that where g α (z) ≤ Cz −3 2 , z ≥ 1.
We define, We can write By using ( 9), we get Our next objective is to see that A straightforward manipulation leads to where η = απ 2 + π 4 and ρ = α + β + 1.We consider We shall prove that (11) By partial integration we obtain that We have that We also get We conclude that By proceeding in a similar way we can see that Note that the last inequality holds for every n, m ∈ N. Suppose that 1 < m − n < n.We decompose R t (n, m) as follows On the other hand, by proceeding as in the proof of (11), we can see that By combining all above estimates we prove that Also, the same arguments allow us to obtain that Thus, we have proved that Let now m ∈ N. According to [3, Lemma 5.1], we have that where By using (5), we get Then, since w Therefore, the proof of (2) is finished.By proceeding as in [3, pp. 13-14], we can see that in order to prove (3), it is sufficient to establish that ( 12) Then, n ≠ 0 ≠ m and m = 2 when n = 1.Assume also that (n, m) ≠ (1, 2).By using (4) and the arguments in [3, pp. 18-19] we can deduce that (12) holds once we will prove that (13) According to [3, Lemma 5.1 (a)], we get where, as in [3], We have that Then, Then, where t > 0 and r j n,m ≤ C n−m , j = 1, 2, 3. We have the following properties (a) Suppose that n = m + k, k ∈ N. It follows that ≥ km, and Suppose that g is a complex-valued function defined in (0, ∞).We defime .
By identifying each pair of functions g 1 and g 2 such that g 1 − g 2 is a constant, ⋅ O({t j } j∈N ) is a norm in th space F O({t j } j∈N ) of all complex functions g defined on (0, ∞) such that g O({t j } j∈N ) < ∞.Thus, (F O({t j } j∈N ) , ⋅ O({t j } j∈N ) ) is a Banach space.If g is a complex function which is differentiable in (0, ∞), we have that From the established estimates in subsection 2.1, we deduce that By using [8, Theorem 1.1], we conclude that the oscillation operator O({ W (α,β) t } t>0 , {t j } j∈N ) can be extended from p (N 0 , w) ∩ 2 (N 0 ) to p (N 0 , w) as a bounded operator (i) from p (N 0 , w) into itself, for every 1 < p < ∞ and w ∈ A p (N 0 ), (ii) from 1 (N 0 , w) into 1,∞ (N 0 , w), for every w ∈ A 1 (N 0 ).
In this section we shall prove the following result.Theorem 3.1.Let α, β ≥ −1 2. Assume that {a j } j∈Z is a ρ-lacunary sequence in (0, ∞) with ρ > 1 and {b j } j∈Z is a bounded sequence of real numbers.For every ) is bounded from p (N 0 , w) into itself, for every 1 < p < ∞ and w ∈ A p (N 0 ), and from 1 (N 0 , w) into 1,∞ (N 0 , w), for every w ∈ A 1 (N 0 ).Furthermore, for every 1 < p < ∞ and w ∈ A p (N 0 ), Proof.Let N ∈ Z.According to (15), we have that where C > 0 does not depend on N .The first term in the right hand side does not appear when n = 0.By using p -boundedness properties of discrete Hardy operators we can deduce that the corresponding properties for S {b j } j∈Z {a j } j∈Z ,N,glob ({W (α,β) t } t>0 ).The proof can be finished by using Theorem 3.1.
3.2.Some auxiliary results.In order to prove a Cotlar inequality for T (α,β) * , we need the following results.