Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions

We apply a symmetrization procedure to the setting of Jacobi expansions and study potential spaces in the resulting situation. We prove that the potential spaces of integer orders are isomorphic to suitably defined Sobolev spaces. Among further results, we obtain a fractional square function characterization, structural theorems and Sobolev type embedding theorems for these potential spaces.


Introduction
This article is motivated by the recent results of Nowak and Stempak [25] and the author's papers [16,17]. In [16] we investigated Sobolev and potential spaces related to discrete Jacobi function expansions. The main achievement of [16] is a suitable definition of the Sobolev-Jacobi spaces so that they are isomorphic with the potential spaces with appropriately chosen parameters. The article [16] is a continuation and extension of a similar study conducted in the setting of ultraspherical expansions by Betancor et al. [4]. The other author's paper [17] contains further investigations of the Jacobi potential spaces. The most important outcome of [17] is a characterization of the potential spaces by means of suitably defined fractional square functions. The research in [17] was inspired by another paper of Betancor et al. [5], in which a general technique of using square functions in analysis of potential spaces associated with discrete and continuous orthogonal expansions was developed.
On the other hand, in [25] Nowak and Stempak proposed a symmetrization procedure in a context of general discrete orthogonal expansions related to a second-order differential operator L, a 'Laplacian'. This procedure, combined with a unified conjugacy scheme established in an earlier article by the same authors [24], allows one to associate, via a suitable embedding, a differential-difference 'Laplacian' L with the initially given orthogonal system of eigenfunctions of L so that the resulting extended conjugacy scheme has the natural classical shape. In particular, the related 'partial derivatives' decomposing L are skew-symmetric in an appropriate L 2 space and they commute with Riesz transforms and conjugate Poisson integrals. Thus the symmetrization procedure overcomes the main inconvenience of the theory postulated in [24], that is the lack of symmetry in the principal objects and relations resulting in essential deviations of the theory from the classical shape. The price is, however, that the 'Laplacian' L and the associated 'derivatives' are not differential, but differential-difference operators. It was shown in [25] that the symmetrization is supported by a good L 2 theory. Moreover, in [15] the author verified that further support comes from the L p theory, at least when the Jacobi polynomial context is considered.
In the present paper we apply the above mentioned symmetrization procedure to the setting of Jacobi function expansions considered in [16,17]. Then we define and study the associated potential spaces and Sobolev spaces. As the main results, we establish an isomorphism between these spaces (Theorem 3.3) and characterize the potential spaces by means of suitably defined fractional square functions (Theorems 4.2 and 4.3). Among further results, we prove structural and embedding theorems for the potential spaces, in particular we obtain a counterpart of the classical Sobolev embedding theorem in the Jacobi setting (see Theorems 5.1 and 5.3). All of this extends the results from [16,17] to the symmetrized situation.
The general strategy we use to prove the results in the symmetrized setting relies on two steps. In the first step we exploit symmetries of the operators under consideration in order to reduce the analysis essentially to the initial non-symmetrized case. Then the second step consists in taking advantage of the results already existing in the literature, mostly from author's previous articles [16,17]. Even though the general line of reasoning is relatively easy, some details occur to be rather technical and complex.
An important aspect, and in fact also a partial motivation of our research, is the suggestion from [25, Section 5] that the symmetrization could have a significant impact on developing the theory of Sobolev spaces related to orthogonal expansions. This concerns, in particular, higher-order 'derivatives' leading to appropriate Sobolev spaces. It turns out, however, that in our symmetrized framework the relevant higher-order 'derivatives' are not constructed from the first-order 'derivative' (see Proposition 3.4), as one would perhaps expect after reading the optimistic comments in [24,Section 5]. Thus these derivatives are even more exotic than the variable index derivatives that are suitable in the initial non-symmetrized Jacobi setting. So it seems that the symmetrization brings no improvement in dealing with Sobolev spaces, at least in the Jacobi setting considered. This makes a noteworthy contrast to the conjugacy scheme which benefits a lot from the symmetrization.
An interesting study of variable exponent Sobolev spaces for Jacobi expansions is contained in the recent paper by Almeida, Betancor, Castro, Sanabria and Scotto [1]. The results of [1] are related to those in the author's papers [16,17], but were obtained independently. In particular, there is a partial overlap in characterizations of the Jacobi potential spaces via fractional square functions obtained in [17] and [1]. We thank one of the referees for bringing [1] to our attention.

Notation
Throughout the paper, we use a fairly standard notation with essentially all symbols referring either to the measure space ((−π, π), dθ) or the restricted space ((0, π), dθ). Given a function f on (−π, π), we denote by f + its restriction to the subinterval (0, π), and by f even and f odd its even and odd parts, respectively, We let whenever the integrals make sense. For 1 ≤ p ≤ ∞, p denotes its conjugate exponent, 1/p + 1/p = 1. When writing estimates, we will 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.
In this work we shall consider the setting related to the larger interval (−π, π) equipped with Lebesgue measure. An application of the symmetrization procedure from [25] to the context of L α,β brings in the following symmetrized Jacobi 'Laplacian' and the associated 'derivative': is the reflection of f , and D α,β and D * α,β are given on (−π, π) by (2.1). Note that, due to the reflection occurring in D α,β , we deal here with a Dunkl type operator. For more details concerning Jacobi-Dunkl operators we refer to [10], see also [14,Section 7].
Also, the following remark is in order. Formally, the space underlying the symmetrized setting is the sum (−π, 0) ∪ (0, π). Nevertheless, often it can (and will) be identified with the interval (−π, π), since for some aspects of the theory the single point θ = 0 is negligible. A typical example here are L p inequalities which "do not see" sets of null measure. On the other hand, some objects in the symmetrized situation may not even be properly defined at θ = 0 (the latter may in addition depend on the parameters of type), hence this point must be excluded from some considerations like, for instance, continuity or smoothness questions. That is why in what follows several times (−π, π)\{0} appears rather than (−π, π).
The orthonormal basis in L 2 (−π, π) arising from the symmetrization procedure applied to the system of Jacobi functions is where φ α,β n are even extensions (denoted still by the same symbol) to (−π, π) of the Jacobi functions. More precisely, where c α,β n are suitable normalizing constants, P α,β n are the classical Jacobi polynomials as defined in Szegő's monograph [29], and Observe that D α,β f is an odd (even) function if f is even (odd). Consequently, Φ α,β n is even (odd) if and only if n is an even (odd) number. By using [16, formula (5)] we find that Notice that when α ≥ −1/2, all Φ α,β n , n ≥ 0, are continuous functions on (−π, π); on the other hand, for α < −1/2 and n even a singularity at θ = 0 occurs. It is a nice coincidence that in our setting Φ α,β n are essentially φ α,β k or φ α,β k with shifted parameters. Roughly speaking, this makes the analysis in the symmetrized situation reducible to the analysis in the initial, nonsymmetrized setting. In general, and even in other Jacobi contexts (see, e.g., [15]), things are more complicated.
According to [25,Lemma 3.5], each Φ α,β n is an eigenfunction of the symmetrized Jacobi operator. More precisely, where we use the notation n = n+1 2 introduced in [25] (here · denotes the floor function). Thus L α,β , considered initially on C 2 c ((−π, π)\{0}), has a natural self-adjoint extension to L 2 (−π, π), denoted by the same symbol, and given by on the domain Dom L α,β consisting of all functions f ∈ L 2 (−π, π) for which the defining series converges in L 2 (−π, π); see [25, Section 4]. Next, we gather some facts about potential operators associated with L α,β . Let σ > 0. We consider the Riesz type potentials L −σ α,β assuming that α + β = −1 (when α + β = −1, the bottom eigenvalue of L α,β is 0) and the Bessel type potentials (Id +L α,β ) −σ with no restrictions on α and β. Clearly, these operators are well defined spectrally and bounded in L 2 (−π, π). The spectral decomposition of L −σ α,β is given by Splitting f into its even and odd parts, we can write This is the decomposition of L −σ α,β f into its even and odd parts, since the two terms in (2.3) are even and odd functions, respectively. Clearly, an analogous decomposition holds for (Id +L α,β ) −σ . We shall use these facts in the sequel.
Proof . We consider only the Riesz type potentials since the arguments for the Bessel type potentials are parallel. Define the restricted operators acting initially on the smaller space L 2 (0, π): Thus the assertion we must prove is equivalent to the following: (L −σ α,β ) + e and (L −σ α,β ) + o , defined initially on L 2 (0, π), extend simultaneously to bounded operators from L p (0, π) to L q (0, π) if and only if 1 q ≥ 1 p − 2σ; moreover, these operators extend simultaneously to bounded operators from L p (0, π) to L ∞ (0, π) if and only if α, β ≥ −1/2 and 1 p < 2σ. Now it is enough to observe that, in view of (2.2) and the identity λ α,β n+1 = λ α+1,β+1 n , the operators (L −σ α,β ) + e and (L −σ α,β ) + o coincide, up to the constant factor 1/2, with the Riesz type potentials L −σ α,β and L −σ α+1,β+1 related to L α,β and investigated in [19]. The conclusion then follows by [ The extensions from Proposition 2.1 are unique provided that p < ∞. In this case we denote them by still the same and common symbol L −σ α,β . It is worth noting that all these extensions are actually realized by an integral operator with a positive kernel. But this fact is irrelevant for our purposes, therefore we omit further details. Denote S α,β := span Φ α,β n : n ≥ 0 .
Proof . It is enough to observe that the lemma holds for f ∈ S α,β and then use the density of S α,β (see Lemma 2.2) in the dual space (L p (−π, π)) * = L p (−π, π).
In order to give a suitable definition of Sobolev spaces in the symmetrized setting we need to understand the structure of the potential spaces. The following result describes the symmetrized potential spaces in terms of the potential spaces related to the initial, non-symmetrized situation. The latter spaces are defined similarly as L p,s α,β (−π, π), see [16,17] for details.

Sobolev spaces
Our aim in this section is to establish a suitable definition of Sobolev spaces in the symmetrized setting. Here "suitable" means existence of an isomorphism between the Sobolev spaces and the potential spaces with properly chosen parameters. Note that such an isomorphism gives also a characterization of the potential spaces with some parameters in terms of appropriate higher-order 'derivatives'.
Here D (k) α,β is a suitably defined differential-difference operator of order k playing the role of higher-order derivative, with the differentiation understood in a weak sense; we use the convention D α,β = D α+k−1,β+k−1 • · · · • D α+1,β+1 • D α,β is inappropriate as well. Counterexamples for these choices are discussed at the end of this section.
Although Proposition 3.4 shows that the spaces W p,m α,β and L p,m α,β (−π, π) do not coincide in general, one might still wonder what the relation between them is, if any. The answer is given by the next result.
The proof of Proposition 3.5 involves higher-order Riesz transforms of the following form. For k ≥ 1 integer, let Clearly, R k α,β is well defined on S α,β . But we also need to know that each R k α,β , k ≥ 1, extends to a bounded operator on L p (−π, π). Lemma 3.6. Let α, β > −1, p ∈ E(α, β) and k ≥ 1. Then the operator extends uniquely to a bounded linear operator on L p (−π, π).
Assuming that this result holds, we now give a short proof of Proposition 3.5. Lemma 3.6 will be shown subsequently.
Proof of Proposition 3.5. We may assume that α + β = −1, since treatment of the opposite case is analogous. Let f ∈ L p,m α,β (−π, π). Then f = L −m/2 α,β g for some g ∈ L p (−π, π). By the L p -boundedness of L −(m−k)/2 α,β (see Proposition 2.1) and Lemma 3.6, for any 0 ≤ k ≤ m we have where R k α,β stands for the extension provided by Lemma 3.6 (with the natural interpretation R 0 α,β = Id). The second identity above is easily justified when g ∈ S α,β , and then it carries over to general g by continuity. The conclusion follows.
It remains to prove Lemma 3.6. The argument relies on a multiplier-transplantation theorem due to Muckenhoupt [18], see [16,Lemma 2.1]. Here we merely sketch the proof, leaving the details to interested readers.
Proof of Lemma 3.6. Assume that α+β = −1 (the opposite case is similar) and take f ∈ S α,β . We have This is the decomposition of R k α,β f into its even and odd parts, respectively, or vice versa, depending on whether k is even or odd. Since (see the proof of Proposition 2.5) it suffices to show the bounds Here (3.1) is contained in [16,Proposition 4.2], since the underlying operator coincides with the Riesz transform R k α,β considered in [16]. So it remains to verify (3.2). Taking into account [16, formulas (5) and (6)] one finds that

Characterization of potential spaces via fractional square functions
In this section we give necessary and sufficient conditions, expressed in terms of suitably defined fractional square functions, for a function to belong to the potential space L p,s α,β (−π, π). For the sake of brevity, we restrict our main attention to the case α + β = −1. Nevertheless, after a slight modification the result is valid also when α + β = −1. This issue is discussed at the end of this section.
This leads to the following characterization of the symmetrized potential spaces.
Theorem 4.2. Let α, β > −1 be such that α + β = −1 and let p ∈ E(α, β). Fix 0 < γ < k with k ∈ N. Then f ∈ L p,γ α,β (−π, π) if and only if f ∈ L p (−π, π) and g γ,k α,β f ∈ L p (−π, π). Moreover, In the remaining part of this section we deal with the case α + β = −1, which is not covered by Theorem 4.2. Actually, only a slight modification is needed, and this is connected with the fact that for α + β = −1 the potential spaces are defined via the Bessel type potentials (Id +L α,β ) −s/2 . The main idea of what follows is taken from [17,Section 4]. Here we give only a general outline and state the relevant result. The details consist of a combination of the facts and results described at the end of [17,Section 4] and the arguments already used in this section. This is left to interested readers.

Structural and embedding theorems for potential spaces
We now show some results revealing relations between the symmetrized potential spaces with different parameters and also establishing mapping properties of certain operators with respect to the potential spaces. At the end of this section we state an analogue of the classical Sobolev embedding theorem for the symmetrized potential spaces. The results of Section 3 suggest the following alternative definition of Riesz transforms in the symmetrized setting. For k ≥ 1 integer, we let α,β (Id +L α,β ) −k/2 , α + β = −1.
Notice that R k α,β is well defined on S α,β . In the structural theorem below both R k α,β and D (k) α,β are understood as operators given initially on S α,β .
Parts (iv) and (v) of Theorem 5.1, as stated, do not hold when k is an odd number. Roughly speaking, this is because in such a case D α,β f (θ). Another interesting question is whether there are any inclusions between potential spaces with different parameters of type. It turns out that in general the answer is negative.
We finish this section with a counterpart of the classical Sobolev embedding theorem. The statement below is a direct consequence of an analogous result in the non-symmetrized situation [17, Theorem 3.2] and Proposition 2.5. We leave the details to the interested readers. Theorem 5.3. Let α, β > −1, p ∈ E(α, β) and 1 ≤ q < p(α, β).

Sample applications of potential spaces
The study of symmetrized potential spaces performed in the previous sections reveals that the symmetrized objects inherit many of the properties of their non-symmetrized prototypes. Furthermore, most of the proofs were based on the arguments relying on suitable decompositions of the operators into their even and odd symmetric parts. This actually reduced our problems to the non-symmetrized setup. Therefore, it comes as no surprise that both theories, symmetrized and non-symmetrized, have parallel applications. Below we present some results which illustrate the utility of the symmetrized potential spaces. The proofs combine the symmetry arguments that have already appeared in this paper with the results from [17,Section 5]. We leave them to the reader.
Then exp(itL α,β )f , understood spectrally, is a solution to this problem. The next result shows that the theory of symmetrized Jacobi potential spaces can be used to study pointwise almost everywhere convergence of this solution to the initial condition.