A CHOICE OF SOBOLEV SPACES ASSOCIATED WITH ULTRASPHERICAL EXPANSIONS

We discuss two possible definitions for Sobolev spaces associated with ultraspherical expansions. These definitions depend on the notion of higher order derivative. We show that in order to have an isomorphism between Sobolev and potential spaces, the higher order derivatives to be considered are not the iteration of the first order derivatives. Some discussions about higher order Riesz transforms are involved. Also we prove that the maximal operator for the Poisson integral in the ultraspherical setting is bounded on the Sobolev spaces. 2000 Mathematics Subject Classification. Primary: 42C05; Secondary: 42C15.

Given β > 0 we consider the fractional integral L − β 2 λ , see (6).The ultraspherical potential space, L p λ,β , is defined as the range of the operator L − β 2 λ on the space L p (0, π).We endow L p λ,β with the norm induced by the usual norm in L p (0, π), that is, λ g and g ∈ L p (0, π).Given m ∈ N, the ultraspherical Sobolev space, W p λ,m , is defined as follows We shall write in an abridge form ( 4) We make the convention D (0) = Id.Along the paper the parameter λ is a fixed number, hence we believe that the no appearance of λ in D (k) doesn't produce confusion to the reader.We equipped W p λ,m with the norm || • || W p λ,m given by ( 5) We study some properties of the spaces L p λ,m and W p λ,m that lead to the main result of the paper.Theorem 1.Let λ > 0, 1 < p < ∞ and m ∈ N. Then W p λ,m = L p λ,m .Simplifying matters, the key behind the proof of this theorem is to establish an inequality of the type D (k) f p ∼ L k/2 λ f p , or equivalently D (k) L −k/2 λ g p ∼ g p .In other words, two facts have to be proved: first, the boundedness in L p (0, π) of the operators D (k) L −k/2 λ ; secondly, a certain inverse process that roughly gives g p ≤ D (k) L −k/2 λ g p .For the inverse process we use auxiliary operators, see Proposition 5.In the proof of these boundedness we use, among other tools, a Muckenhoupt multiplier transplantation result, see Lemma 1.The operators D (k) L −k/2 λ play the role of the "higher order Riesz transforms".These last thoughts are contained in Section 3.
Other higher order Riesz transforms associated with the operator L λ , were defined in [5] , where and D λ is the operator defined in (1).These Riesz transforms would suggest to define the Sobolev space as the subspace of functions f ∈ L p (0, π) such that D k λ f ∈ L p (0, π), k = 0, 1, . . ., m.This Sobolev space is denoted by W p λ,m and endowed with the norm given by In Section 4 and continuing with this line of thought, we ask if a result like Theorem 1 is possible for these Sobolev spaces, namely are the spaces W p λ,m and L p λ,m isomorphic?We prove that although the Riesz transforms, , are bounded in L p (0, π), the inverse process that is needed for a theorem like the Theorem 1 doesn't work.In fact the Theorem 2 below will be proved.We should mention here that we give a different proof of the boundedness of the Riesz transforms from the one given in [5].
Theorems 1 and 2 suggest that the adjective "Sobolev" has to be given to the spaces W p λ,m .Finally some properties of these Sobolev spaces W p λ,m are analyzed.In Section 3 it is shown that if W p m (I), I a real interval, denotes the classical Sobolev space on I, then ), for every 0 < a < b < π (Proposition 3).This fact and the procedure developed by Kinnunen [11], allow us to prove in Section 5 the following theorem about the maximal operator for the Poisson integral, P λ * .
Throughout this paper by C we always represent a suitable positive constant that can change in each occurrence.For every 1 ≤ p < ∞, we denote as usual by p ′ the conjugate exponent of p.
Negative powers of the operator L λ can be defined as ultraspherical multipliers as follows.If β > 0 the operator L −β λ is given by ( 6) where It is clear that, for every β > 0, L −β λ defines a bounded operator from L 2 (0, π) into itself.
Let S λ be the linear space generated by the system {ϕ λ n } n∈N of ultraspherical functions.This linear space plays an important role in our study.S λ is a dense subspace on L p (0, π), 1 ≤ p < ∞, see [19,Lemma 2.3].
λ is bounded and one to one on L p (0, π).
Boundedness of ultraspherical multipliers has been investigated by several authors (see [7], [15], [6] and [14]).In particular by using the following lemma one can prove the boundedness in This lemma is an easy consequence of [14,Corollary 17.11], and it will be useful in the sequel.
Proposition 1 gives sense to the potential spaces,

Ultraspherical Sobolev spaces
A standard procedure shows that the space Next we establish properties of these Sobolev spaces.
The function Our objective now is to prove Theorem 1. Previously we need to establish the boundedness of some operators.Lemma 2. Let λ > 0, 1 ≤ p ≤ ∞ and k ∈ N. Then the projector operator P k defined by is bounded on L p (0, π).
Proof: It is sufficient to note that where p ′ denotes the conjugate exponent of p.
Proof: According to (12) we can write We denote by g the function Let us choose J ≥ 2(2λ + k + 2).Applying Lemma 1, we get that the operator R k λ,1,J defined by can be extended to L p (0, π) as a bounded operator.Also, by proceeding as in the proof of Lemma 2 we can establish that the operator R k λ,1 −R k λ,1,J is a bounded operator from L p (0, π) into itself.Thus, we conclude that R k λ,1 can be extended to L p (0, π) as a bounded operator from L p (0, π) into itself.Proposition 5. Let λ > 0, 1 < p < ∞ and k ∈ N. We define on S λ+k the operator λ,2 can be extended to L p (0, π) as a bounded operator from L p (0, π) into itself.
Proof: Let f ∈ S λ+k .According to (16) for j = k, we have that By proceeding as in the proof of Proposition 4 we can see that R λ,2 can be extended to L p (0, π) as a bounded operator from L p (0, π) into itself.
We can write, for every f ∈ S λ , In the next proposition we prove the boundedness of the inverse of can be extended as a bounded operator from L p (0, π) into itself.
Proof: Fix J ≥ 4(λ + 1).By proceeding as in the proof of Proposition 4 we first take a function g J such that for n large enough.In order to establish the boundedness property for the operator T k λ , it is then sufficient to prove that the operator can be extended to L p (0, π) into itself.Moreover, Lemma 2 allows us to reduce the boundedness of T k λ,J to functions f ∈ S λ,k .Indeed, suppose that (25) Since f 1 ∈ S λ,k , by (25) and Lemma 2 we get that Finally, to prove the boundedness of the operator T k λ,J on S λ,k we apply Lemma 1.

Proof of Theorem 1:
Then, according to Propositions 5 and 6, since R m λ,1 g 2 = R m λ,1 g, we get On the other hand, Propositions 4 and 1 lead to Thus (26) is established.

Alternative definition of Sobolev spaces
As we said in the introduction the n-th order Riesz transform associated with the operator L λ is given by We observe that By using Propositions 4 and 5 we will prove that R k λ can be extended to L p (0, π) as a bounded operator, for every k ∈ N and 1 < p < ∞.As it was mentioned our proof is different from the one presented in [5]. where for certain constants c ℓ,k m and d ℓ,k m .Here, N ℓ,k depends on ℓ and k as it is indicated in the following table: Hence, for every θ ∈ (0, π), Let f ∈ S λ .From ( 27) and taking into account ( 12) and ( 18), it deduces that, for every k ∈ N, By using estimations (28) and (29) we then obtain that, for each θ ∈ (0, π), when k is even, and in the case that k is odd.Hence, according to Propositions 1 and 4, the boundedness of the operator R (k) λ will be established when we prove that, for every λ > 0 and j ∈ N, the operator is bounded from L p (0, π) into itself.We proceed by induction on j.Indeed, let λ > 0 and 1 < p < ∞.Note firstly that Suppose now that the operator C λ,j is bounded on L p (0, π), for every λ > 0 and j = 0, . . ., s, where s ∈ N. Let us see that the operator C λ,s+1 is bounded on L p (0, π), with λ > 0 fixed.By using Leibniz rule it follows that, for every f ∈ S λ+s+1 , where the operator T λ,s is defined by We observe that ( 12) and ( 18) lead to Also, a straightforward manipulation shows that From this estimation, and by using the induction hypothesis and Proposition 4 we deduce that the operator Hence, from (31) and Proposition 5, it follows that the operator T λ,s is bounded from L p (0, π) into itself.We can write (sin θ) where θ ∈ (0, π), q ∈ C ∞ (R), q(0) = 0 and q(π) = 0.By choosing δ > 0 such that q(θ) = 0 for θ ∈ Thus, the proof is finished.
Finally we can give the proof of Theorem 2, see Introduction, which establishes the relation between the spaces W p λ,m and W p λ,m .
Proof of Theorem 2: Assume that f ∈ W p λ,m .By Theorem 1 there exists g ∈ L p (0, π) such that f = L − m 2 λ g and g p is equivalent to f W p λ,m .Then, by Propositions 7 and 1, for every k = 0, 1, . . ., m, we have that

A maximal operator on the ultraspherical Sobolev spaces
Kinnunen [11] (see also [12] and [13]) proved that the Hardy-Littlewood maximal operator is bounded in the classical Sobolev space W p 1 (R n ) for every 1 < p < ∞.In this section we prove that the maximal operator associated with the Poisson integral for the operator L λ is bounded from W p λ,1 into itself, for every 1 < p < ∞.The maximal operator associated with {P λ r } 0≤r<1 is defined by This operator P λ * is bounded from L p (0, π) into itself, for every 1 < p < ∞ and from L 1 (0, π) into L 1,∞ (0, π) [18, Theorem 2.2].Now we can give the proof of Theorem 3.
Proof: Let f ∈ W p λ,1 and let {r j } ∞ j=1 be an enumeration of the rational numbers in (0, 1).Then, we have The sequence F k = max 1≤j≤k P λ rj (f ) k∈N is bounded in W p λ,1 .W p λ,1 , after renorming, is isometrically isomorphic to a closed subspace of L p (0, π) × L p (0, π).Then W p λ,1 , after renorming, is reflexive and the closed unit ball in W p λ,1 is sequentially compact in the weak topology.Hence, there exists a subsequence {F k l } l∈N of {F k } k∈N and a function g ∈ W p λ,1 such that F k l → g, as l → ∞, in the weak topology of W p λ,1 and g W p λ,1 ≤ C f W p λ,1 .Since F k l (θ) → P λ * (f )(θ), as l → ∞, for θ ∈ (0, π), and {F k l } l∈N is increasing, we conclude that P λ * (f ) = g and the proof of Theorem 3 is finished.The argument in the proof of the last theorem can be used in those cases in which formulas ( 12) and ( 18) produce identities like (19).This is the case of the maximal operator associated with partial sums for ultraspherical expansions.In fact the following theorem can be proved with these ideas and [9, Theorem F].