CALDER ´ON-ZYGMUND OPERATORS ASSOCIATED TO ULTRASPHERICAL EXPANSIONS

ULTRASPHERICAL EXPANSIONS

DARIUSZ BURACZEWSKI, TERESA MARTINEZ, AND JOS´E L. TORREA

Abstract. We define the higher order Riesz transforms and the Littlewood- Paley g-function associated to the differential operator Lλf (θ) = −f^′′(θ) − 2λ cot θf^′(θ)+λ²f (θ). We prove that these operators are Calder´on-Zygmund operators in the homogeneous type space¡(0, π), (sin t)^2λdt¢. Consequently, L^p weighted, H¹− L¹and L^∞− BMO inequalities are obtained.

1. Introduction intro

B. Muckenhoupt and E. Stein, in[5], defined and studied the versions of some^MS objects to classical Fourier Analysis (conjugate functions, maximal functions, g-functions and multipliers) for the system of the ultraspherical polynomials.

Their approach follows the lines of the classical Fourier Analysis in the torus.

In particular, the relationship between Fourier series, analytic functions and harmonic functions plays an essential role. For instance, their definition of the conjugate function was via a boundary value limit of certain “conjugate harmonic function” which satisfies the appropriate Cauchy-Riemann equations.

The technique involved the definition of the “harmonic extension” including a careful analysis of its kernel. Then they built a conjugate harmonic function and proved the existence of a boundary value function, the conjugate function.

They got L^p boundedness for p in the range 1 < p < ∞ and some substitutive inequality in the case p = 1. This method was followed later by different authors when defining classical operators for orthogonal expansions. In ^MS[5] they did not study the kernel of the conjugate function.

Five years later, there appeared Stein’s celebrated monograph ^st1[7], where

“maximal functions”, “g-functions”, “Riesz transforms”, and “multipliers” were also defined. He used systematically a point of view based on an analysis of a Laplacian on compact Lie groups. He studied the “heat” and “Poisson”

semigroups associated to the Laplacian and, from them, he derived the rest of the operators by using some spectral formulas.

Date: March 13, 2005.

2000 Mathematics Subject Classification. 42C05,42C15.

Key words and phrases. Ultraspherical polynomials, Calder´on-Zygmund operators.

The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. First author was supported in part by KBN grant 1P03A01826. Second and third authors were partially supported by BFM grant 2002-04013- C02-02.

1

A natural question arising from[5] is studying some other classical operators^MS in the context of the system of the ultraspherical polynomials. In particular, our aim was to study higher order Riesz transforms. In order to define those operators, it seems that the natural procedure to follow is the one suggested in the monograph ^st1[7], and we do so in this paper. The method used to prove the boundedness of higher order Riesz transforms could be applied to a more general class of operators. For instance, we present in this paper its application to study the boundedness of the Littlewood-Paley g-function, although more operators fit in this technique (e.g. multipliers of Laplace transform type). More concretely, we find that all these operators are naturally Calder´on-Zygmund operators in a space of homogeneous type. Therefore, we get as a byproduct of the general theory L^p, H¹, BM O boundedness and weighted inequalities for them (see Theorem 1.4).

By using a different method, the (first order) Riesz transform was studied in [1]. It was defined followingBMTU [7] and, among other results, the L^{st1 p}-boundedness for p ∈ (1, ∞) and the weak type (1, 1) were obtained.

We consider the ultraspherical polynomials P_n^λ(x), λ > 0, defined as the coefficients in the expansion of the generating function (1 − 2xω + ω²)^−λ = P_∞

n=0ωⁿP_n^λ(x) (see[8] for further details). It is known that the set {P^Szego n^λ(cos θ) : n ∈ N} is orthogonal and complete in L²[0, π] with respect to the measure dmλ(θ) = (sin θ)^2λdθ. The functions P_n^λ(cos θ) are eigenfunctions of the operator Lλ

op (1.1) L_λf (θ) = −f^′′(θ) − 2λ cot θf^′(θ) + λ²f (θ),

with eigenvalues µn = (n + λ)². Every function f in L²(dmλ) has the ultra- spherical expansion f (θ) = P_∞

n=0anP_n^λ(cos θ)kPn^λk⁻¹L²(dmλ). Following [7], we^st1 define its Poisson integral as

our poisson (1.2) P f (e^−t, θ) = e^−t√Lλf (θ) = X∞ n=0

ane^−t(n+λ)P_n^λ(cos θ)kPn^λk⁻¹L²(dmλ).

The calculus formula s^−a= _Γ(a)¹ R_∞

0 e^−tst^a−1dt can be used to define definition Llambda-05 (1.3) L^−l/2_λ f (θ) = 1

Γ(l) Z _∞

0

e^−t√Lλf (θ)t^l−1dt, l ≥ 1, f ∈ L²(dmλ).

On the other hand, it is easy to check that Lλ is formally self-adjoint on the space L²(dmλ) and that it factorizes as Lλf (θ) = (−∂θ^∗∂θ + λ²)f (θ), where

∂_θ^∗ = ∂θ+2λ cot θ is formally adjoint to ∂θ, i.e., h∂θ^∗f, giL²(m^λ) = −hf, ∂^θgiL²(m^λ). Following[7], the Riesz transform (l = 1) and the higher order Riesz transforms^st1 (l > 1) are defined as

R^l_λf (θ) = ∂_θ^l(Lλ)^−l/2f (θ), l ≥ 1.

Also, the Littlewood-Paley g-function is defined as Gf(θ) =

µ Z 1 0

r log1

r¡|∂^rP f (r, θ)|²+ |∂^θP f (r, θ)|²¢ dr

¶1/2

. For these operators, we have the following result.

theorem principal Theorem 1.4. The operators R^l_λf , for any natural l ≥ 1, and Gf are bounded in L^p(w dmλ), 1 < p < ∞ for any weight w in the Muckenhoupt class A^p with respect to the measure dmλ and of weak type (1, 1) with respect to the measure w dmλ for w ∈ A¹. Also, they map L^∞ into BM O(dmλ) and H¹(dmλ) into L¹(dmλ) boundedly.

In fact, this theorem is a consequence of the general theory for Calder´on- Zygmund operators in spaces of homogeneous type and the following theorems.

principal Riesz Theorem 1.5. For any natural l ≥ 1, the operators Rλ^l are Calder´on-Zygmund operators in the homogeneous type space ([0, π), dmλ).

For the g-function, we study separately the operators G¹f (θ) =

µ Z 1 0

r log1 r

¯¯∂rPrf (θ)¯

¯

2dr

¶1/2

eq g1 (1.6)

G²f (θ) = µ Z 1

0

r log1 r

¯¯∂θPrf (θ)¯

¯

2dr

¶1/2

. eq g2 (1.7)

theorem principal g Theorem 1.8. There exist vector valued Calder´on-Zygmund operators in the homogeneous type space ([0, π), dm_λ) mapping scalar-valued functions into func- tions with values in L²((0, 1), dr), T_Gⁱ, i = 1, 2, such that

Gⁱf (θ) = kTGⁱf (θ)kL²((0,1),dr).

The structure of the paper is as follows. In section 2, we state the technical Theorem 2.4, as an intermediate step in the proofs of Theorems 1.5 and 1.8.

In section 3 we verify the hypothesis of Theorem 2.4 for the higher order Riesz transforms, and in section 4 we do the corresponding checking for the g-function.

Let us mention just a word about notation. Throughout the paper, the letter C will denote a constant whose value may vary from line to line, and let us call polynomial function to any finite linear combination of ultraspherical polynomials, that is, any f of the form

eq polynomial (1.9) f =

N

X

n=0

anP_n^λ

kPn^λk^L2(dmλ), a_n= Z π

0

f (θ) P_n^λ(cos θ)

kPn^λk^L2(dmλ)dm_λ(θ) 2. Preliminaries and technical tools technical lemmas

Following[3] and^C [4], a space of homogeneus type (X, ρ, µ) is a set X together^CW with a quasimetric ρ and a positive measure µ on X such that for every θ ∈ X and r > 0, µ(B(θ, r)) < ∞, and such that there exists 0 < C < ∞ such that

for every θ ∈ X and r > 0 µ(B(θ, 2r)) ≤ Cµ(B(θ, r)). In our case, X = [0, π]

with the metric given by the absolute value, and the measure is dm_λ, which is doubling, what can be easily checked.

We say that a kernel K : X × X\{x = y} → C is a standard kernel if there exist ε > 0 and C < ∞ such that for all x 6= y ∈ X and z with ρ(x, z) ≤ ερ(x, y),

|K(x, y)| ≤ C

µ(B(x, r)), where r = ρ(x, y), and equation CZ kernel 1 (2.1)

|K(x, y) − K(z, y)| + |K(y, x) − K(y, z)| ≤ ρ(x, z) ρ(x, y)

C µ(B(x, r)). equation CZ kernel 2 (2.2)

Thus, a Calder´on-Zygmund operator (with associated kernel K) is a linear op- erator T bounded in L² such that, for every f ∈ L² and x outside the support of f ,

T f (x) = Z

X

K(x, y)f (y) dµ(y).

It is known that any Calder´on-Zygmund operator as above is bounded in L^p(w dµ), for 1 < p < ∞ and any weight w in the Muckenhoupt class A^p with respect to the measure dµ. They also map L¹(w dµ) into L^1,∞(w dµ) for any weight w in the Muckenhoupt class A1 with respect to the measure dµ, L^∞(µ) boundedly into BM O(dµ) and H¹(dµ) into L¹(dµ) (see[2]).^Cal

For B a Banach space, vector-valued Calder´on-Zygmund operators, T from L²(dµ) into L²_B(dµ), are defined in the same way as scalar valued ones, but considering K : X × X\{x = y} → B instead of a scalar valued kernel, and taking B-norms in (2.1) and (2.2) instead of absolute values. The boundedness results mentioned above also hold in the vector valued case (see ^RT[6]).

drawing1

θ φ

π 2

π 2 π 3

φ = 2θ/3 φ = 3θ/2

The symmetry with respect to π/2 in the kernels of our opera- tors will play an important role in the proofs. Also, it will be use- ful to have in mind the picture of the area where we are placing the variables θ and φ. Our first step is studying in detail the behavior of the measure of B(θ, |θ − φ|).

lema medida bola Lemma 2.3. There exists a constant C > 0 such that for any θ ∈ [0, π/2],

mλ(B(θ, |θ − φ|)) ≤ C







|θ − φ|¡ sin(θ ∨ φ)¢^2λ if φ ∈ [0, π/2]

φ if φ ∈ [π/2, π], φ > ³2θ

|θ − φ|¡ sin(θ ∨ φ)¢^2λ if φ ∈ [π/2, π], φ < ³2θ.

stronger conditions Theorem 2.4. Let B be a Banach space, and T : L²(dmλ) → L²B(dmλ) be an operator given by integration against a kernel K in the Calder´on-Zygmund sense, such that

i) If |θ − φ| ≥ π/6, then

kK(θ, φ)kB ≤ C and

k∂^θK(θ, φ)kB+ k∂^φK(θ, φ)kB ≤ C.

ii) For every (θ, φ) either belonging to [0, π/2] × [0, π/2] or to [0, π/2] × [π/2, π] ∩ {φ ≤ ³2θ}, we have

kK(θ, φ)kB≤ C

|θ − φ|

1

(sin(θ ∨ φ))^2λ, and strong kernel 1 (2.5)

k∂^θK(θ, φ)kB+ k∂^φK(θ, φ)kB≤ C

|θ − φ|²

1

(sin(θ ∨ φ))^2λ. strong kernel 2 (2.6)

iii) The kernel K is symmetric (K(π−θ, π−φ) = K(θ, φ)) or antisymmetric (K(π − θ, π − φ) = −K(θ, φ)).

Then, T is a Calder´on-Zygmund operator in (0, π) with ρ(x, y) = |x − y|.

Proof. Lemma 2.3 and (2.5) clearly imply (2.1) in the region stated in ii).

On the other hand, for (θ, φ) ∈ [0, π/2] × [π/2, π] ∩ {φ ≥ ³₂θ}, we have π/6 ≤

|θ − φ| ≤ π, and therefore, by i), kK(θ, φ)kB ≤ C ≤ ^Cπ_φ . Using again Lemma 2.3, (2.5) and the symmetry condition iii), we obtain (2.1) easily.

By using Lemma 2.3 and (2.6), we get equation final kernel 2 (2.7) k∂^θK(θ, φ)k^B+ k∂^φK(θ, φ)k^B≤ C

|θ − φ|

1

mλ(B(θ, |θ − φ|))

for θ and φ in the region stated in ii). For (θ, φ) ∈ [0, π/2] × [π/2, π] ∩ {φ ≥ ³₂θ}, we have π/6 ≤ |θ − φ| ≤ π, and therefore, by i), k∂^θK(θ, φ)kB+ k∂^φK(θ, φ)kB≤ C ≤ _|θ−φ|φ^C . By Lemma 2.3, we have (2.7) in (θ, φ) ∈ [0, π/2] × [π/2, π] ∩ {φ ≥

3

2θ}. The symmetry condition iii) implies that (2.7) holds for any (θ, φ) ∈ [0, π] × [0, π]. By standard calculations, this inequality implies condition (2.2)

for ε = 1/2 ¤

3. Proof of Theorem 1.5 section Riesz

Given a polynomial function f , we have R_λ^lf (θ) = ∂_θ^l(Lλ)^−l/2f (θ) = ∂_θ^l

à _N X

n=0

an

(n + λ)^k

P_n^λ(cos θ) kPn^λkL²(dm^λ)

! rt (3.1)

=

N

X

n=0

an

(n + λ)^l

∂_θ^lP_n^λ(cos θ) kPn^λk^L2(dmλ).

The case l = 1 was extensively studied in [1]. There it was proved its bound-^BMTU edness in L^p for p ∈ (1, ∞) and its weak type (1, 1), as well as the fact that it is a principal value, the boundedness of the maximal operator, etc.. In order to prove Theorem 1.5, we first prove the boundedness in L² of the higher order Riesz transforms.

subsection l2 bdd

3.1. L² boundedness of higher order Riesz transforms. We shall see that for any polynomial function (1.9), we have kR^lλf kL²(dm^λ) ≤ C kfkL²(dm^λ) with a constant independent of N . First of all, let us observe that without loss of generality, we can consider only polynomial functions f such that there only appear P_n^λ(cos θ) for n ≥ l + 1 in their expansion. Since any linear operator is bounded on finite dimensional spaces, we have that the restriction of R_λ^l to the span of {P0^λ, . . . , P_l^λ} is bounded, and it only remains to check the case when n ≥ l + 1.

The case l = 1 was treated in ^BMTU[1], and it was seen to be bounded in L²(dmλ).

Lets us explore the case l = 2. In this case, for any n ≥ 3 R²_λP_n^λ(cos θ) = 1

(n + λ)²∂_θ²P_n^λ(cos θ)

= −2λ

(n + λ)² cos θP_n−1^λ+1(cos θ) + 4λ(λ + 1)

(n + λ)² (sin θ)²P_n−2^λ+2(cos θ).

riesz differential (3.2)

On the other hand, from (1.1) we obtain ∂_θ² = −L^λ − 2λ cot θ ∂^θ + λ² and therefore

riesz2 operator (3.3) R²_λ = ∂_θ²L⁻¹_λ = −Id − 2λ cot θ ∂^θL⁻¹_λ + λ²L⁻¹_λ . With this expression,

action riesz operator (3.4) R_λ²P_n^λ = −Pn^λ+ 4λ²

(n + λ)² cos θP_n−1^λ+1+ λ²

(n + λ)²P_n^λ. Mixing (3.2) and (3.4) one has

2λ(2λ + 1)

(n + λ)² cos θP_n−1^λ+1(cos θ) eq fundamental (3.5)

= P_n^λ(cos θ) − λ²

(n + λ)²P_n^λ(cos θ) + 4λ(λ + 1)

(n + λ)² (sin θ)²P_n−2^λ+2(cos θ).

In particular, with this calculation we get from (3.4) R_λ²P_n^λ(cos θ) = −1

2λ + 1P_n^λ(cos θ) + 1 2λ + 1

1

(n + λ)²P_n^λ(cos θ) +8λ²(λ + 1)

2λ + 1

1

(n + λ)²(sin θ)²P_n−2^λ+2(cos θ),

and therefore

R²_λf (θ) = −1 2λ + 1

N

X

n=3

an

kPn^λk^L2(dmλ)P_n^λ(cos θ)

+ 1

2λ + 1

N

X

n=3

an

kPn^λkL²(dm^λ)

1

(n + λ)²P_n^λ(cos θ) +8λ²(λ + 1)

2λ + 1

N

X

n=3

an

kPn^λk^L2(dmλ) 1

(n + λ)²(sin θ)²P_n−2^λ+2(cos θ), Clearly, the first two sums are operators bounded in L²(dm_λ). For the third one, we will use the following lemma.

lemma boundedness Lemma 3.6. For every k ≥ 0, the operator T acting on polynomial functions T f (θ) =

N

X

n=k+1

a_n kPn^λk^L2(dmλ)

1

(n + λ)^k(sin θ)^kP_n−k^λ+k(cos θ) is bounded in L²(dm_λ).

Proof. By using the orthogonality of (sin θ)^kP_n−k^λ+k in L²(dmλ), and kPn^λk²L²(dmλ) = 2^1−2λπΓ(λ)⁻²Γ(n + 2λ)

(n + λ)n!, (see ^Szego[8]), we obtain

°

°

°

°

N

X

n=k+1

an

kPn^λkL²(dmλ)

1

(n + λ)^k(sin θ)^kP_n−k^λ+k

°

°

°

°

2 L²(dmλ)

=

N

X

n=k+1

a²_n kPn^λk²L²(dmλ)

kP_n−k^λ+kk²L²(dm^λ+k)

(n + λ)^2k ≤ C

N

X

n=k+1

a²_n= C kfk²L²(dm^λ).

¤ We use induction in l to prove the boundedness of R^l_λ for any l ≥ 1. Let us assume that R^k_λ is bounded in L²(dmλ) for k ≤ l − 1. ¿From (3.3), for l ≥ 3, we have

R^l_λ = ∂_θ^lL^−l/2_λ = ∂_θ^l−2R_λ²L^−(l−2)/2_λ lgeq2 riesz operator (3.7)

= −∂θ^l−2L^−(l−2)/2_λ − 2λ∂θ^l−2cot θ ∂_θL^−l/2_λ + λ²∂_θ^l−2L^−l/2_λ

= −R^l−2λ − 2λ∂θ^l−2cot θ ∂θL_λ^−l/2+ λ²R^l−2_λ L⁻¹_λ .

By the induction hypothesis, first and last operators are bounded in L²(dmλ).

It remains to show that the second term is also bounded.

lemma derivatives Lemma 3.8. For any k ≥ 0,

∂_θ^k¡ cos θP_n−1^λ+1(cos θ)¢ =

k+1

X

a=1

X

A, B:

A+B=a

CAB(cos θ)^A(sin θ)^BP_n−a^λ+a(cos θ)

where CAB ∈ R may be zero.

Proof. We will prove the result by induction in k. For k = 0 it is clearly true, with C10 = 1 and C01 = 0. Let us suppose that the formula holds until k − 1.

For k, we have

∂_θ^k¡ cos θP_n−1^λ+1(cos θ)¢ = ∂θ k

X

a=1

X

A, B:

A+B=a

CAB(cos θ)^A(sin θ)^BP_n−a^λ+a(cos θ)

and it is enough to see that any term ∂θ¡(cos θ)^A(sin θ)^BP_n−a^λ+a(cos θ)¢ is a sum of terms of the form (cos θ)^A˜(sin θ)^B˜P_n−˜a^λ+˜a(cos θ) con 1 ≤ ˜a ≤ k + 1, ˜A + ˜B = ˜a.

And this holds, since

∂θ¡(cos θ)^A(sin θ)^BP_n−a^λ+a(cos θ)¢ = A(cos θ)^A−1(sin θ)^B+1P_n−a^λ+a(cos θ) + B(cos θ)^A+1(sin θ)^B−1P_n−a^λ+a(cos θ)

− 2(λ + a)(cos θ)^A(sin θ)^B+1P_n−(a+1)^λ+(a+1)(cos θ).

¤ Thus, for a polynomial function f =PN

n=l+1a_nkPn^λk_L⁻¹²_(dmλ)P_n^λ, we can write

∂_θ^l−2cotθ ∂θL^−l/2_λ f (θ) =

N

X

n=l+1

an

kPn^λkL²(dmλ)

1

(n + λ)^l∂_θ^l−2¡ − 2λ cos θP_n−1^λ+1(cos θ)¢

= Xl−1

a=1

X

A, B:

A+B=a

CAB N

X

n=l+1

a_n kPn^λk^L2(dmλ)

1

(n + λ)^l(cos θ)^A(sin θ)^BP_n−a^λ+a(cos θ) middle operator 2 (3.9)

=

l−1

X

a=1

X

A, B:

A+B=a

CABT_{A, B, a}^l−1 f (θ).

Thus, the boundedness of this operator is done with the following lemma.

boundedness TABa Lemma 3.10. For any l ≥ 2, the operators TA, B, a^l−1 appearing in (3.9) with non zero coefficients are bounded in L²(dmλ).

Proof. We will proceed by induction in A: in the case A = 0, we have to prove that for every a ≤ l − 1,

T_{0, a a}^l−1 f (θ) =

N

X

n=l+1

an

kPn^λk^L2(dmλ) 1

(n + λ)^l cos θ(sin θ)^aP_n−a^λ+a(cos θ)

is bounded in L²(dmλ). This holds by Lemma 3.6. Now assume that the induction hypothesis is true until A − 1, that is, for any ˜A ≤ A − 1, we have that for every a ≤ l − 1, the operator TA, a− ˜^l−1˜ A, af (θ) is bounded in L²(dmλ). For A ≥ 1, by (3.5) with n − (a − 1) and λ + (a − 1) instead of n and λ, one can write

1

(n + λ)^l(cos θ)^A(sin θ)^BP_n−a^λ+a = C

(n + λ)^l−2(cos θ)^A−1(sin θ)^B×

× µ

P_n−(a−1)^λ+(a−1)+ C

(n + λ)²P_n−(a−1)^λ+(a−1)+ C

(n + λ)²(sin θ)²P_n−(a+1)^λ+(a+1)

¶

= C

(n + λ)^l−2(cos θ)^A−1(sin θ)^BP_n−(a−1)^λ+(a−1)+ C

(n + λ)^l(cos θ)^A−1(sin θ)^BP_n−(a−1)^λ+(a−1)

+ C

(n + λ)^l(cos θ)^A−1(sin θ)^B+2P_n−(a+1)^λ+(a+1). The first two terms give rise to operators bounded in L²(dmλ), by the induction hypothesis. For the third factor, if a+1 ≤ l−1, then by the same reason it is also a bounded operator. But we want to prove the boundedness for any a ≤ l − 1, and thus it remains to prove the case a = l − 1. Since ∂θ^l−2cot θ ∂θL^−l/2_λ P_n^λ =

−2λ∂θ^l−2¡ cos θP_n−1^λ+1¢, an operator T_A,B,l−1^l−1 in which there appear terms with P_n−(l−1)^λ+(l−1) in the left-hand of expression (3.9), necessarily comes from the factors in which the derivatives ∂_θ^l−2 act on the polynomial P_n−1^λ+1 and not in any other term (otherwise we would obtain P_n−a^λ+a with a < l − 1). Therefore, the only terms with non-zero coefficients coming by applying (3.5) in the right-hand of (3.9) must sum up a constant times the operator

T1, l−2, l−1^l−1 f (θ) =

N

X

n=l+1

a_n kPn^λk^L2(dmλ)

1

(n + λ)^l cos θ(sin θ)^l−2P_n−(l−1)^λ+(l+1)(cos θ).

By using (3.5) with n − (l − 2) and λ + (l − 2) instead of n and λ, we can write 1

(n + λ)^lcos θ(sin θ)^l−2P_n−(l−1)^λ+(l+1)

= C(sin θ)^l−2 (n + λ)^l−2

µ

P_n−(l−2)^λ+(l−2)+ C

(n + λ)²P_n−(l−2)^λ+(l−2)+ C

(n + λ)²(sin θ)²P_n−l^λ+l

¶

= C(sin θ)^l−2

(n + λ)^l−2P_n−(l−2)^λ+(l−2)+ C(sin θ)^l−2

(n + λ)^l P_n−(l−2)^λ+(l−2)+ C(sin θ)^l (n + λ)^lP_n−l^λ+l. By Lemma 3.6, the operators are bounded in L²(dmλ).

kernel of the riesz

3.2. Kernel of the Riesz transfom. Observe that by (1.2), P f (e^−t, θ) = e^−tλf (e^−t, θ), where f (r, θ) = P_∞

n=0anrⁿP_n^λ(cos θ)kPn^λk⁻¹L²(dmλ) is defined by Muckenhoupt and Stein in ^MS[5]. They compute explicitly the kernel P (r, θ, φ) of

f (r, θ):

p1 (3.11) P f (r, θ) = r^λ Z π

0

P (r, θ, φ)f (φ)dm_λ(φ), θ ∈ [0, π]

where r = e^−t and prtf (3.12) P (r, θ, φ) = λ

π Z π

0

(1 − r²) sin^2λ−1t

¡1 − 2r(cos θ cos φ + sin θ sin φ cos t) + r²¢λ+1dt.

Before continuing further, let us state some useful notation.

notation (3.13)

σ = sin θ sin φ, a = cos θ cos φ + σ cos t = cos(θ − φ) − σ(1 − cos t),

∆r = 1 − 2r cos(θ − φ) + r² = (1 − r)²+ 2r(1 − cos(θ − φ)),

∆ = ∆₁, D_r = 1 − 2ra + r² = ∆_r+ 2rσ(1 − cos t).

lemma kernels Lemma 3.14 (Lemma 2 in[1]). Given f ∈ L^{BMTU 1}(dmλ) and l ≥ 1, for almost every θ ∈ [0, π], we have that

equation Wlambda (3.15)

(Lλ)^−2lf (θ) = Z π

0

W_λ^l(θ, φ)f (φ) dmλ(φ), where W_λ^l(θ, φ) = 1

Γ(l) Z 1

0

r^λ−1¡ log1 r

¢_l−1

P (r, θ, φ) dr.

Given f ∈ L¹(dm_λ), l ≥ 1 and θ outside the support of f, we have that R^l_λf (θ) =

Z π 0

R^l_λ(θ, φ)f (φ) dmλ(φ), where Rlambda (3.16)

R_λ^l(θ, φ) = 1 Γ(l)

Z 1 0

r^λ−1¡ log1 r

¢_l−1λ

π(1 − r²) Z π

0

(sin t)^2λ−1∂_θ^l¡ 1 D_r^λ+1¢ dt.

Proof. L^−1/2_λ is defined in (1.3). Then, to get (3.15) it is enough to apply Fubini’s theorem. To prove (3.16), it is enough to justify the differentiation inside the integral sign. See ^BMTU[1] for the details. ¤ Now we shall see that the kernel R^l_λ(θ, φ) satisfies the hypothesis of The- orem 2.4. Since P (r, π − θ, π − φ) = P (r, θ, φ), then R^lλ(π − θ, π − φ) = (−1)^lR^l_λ(θ, φ). Therefore, R^l_λ(θ, φ) satisfies condition iii) in Theorem 2.4. Ob- serve that ∂_θ^l+1(D^−λ−1_r ) and ∂φ∂_θ^l(D^−λ−1_r ) are quotients with a bounded func- tion in the numerator and a certain power of Dr in the denominator. Since for r ∈ (0, 1/2), Dr ≥ C and for r ∈ (1/2, 1) and |θ − φ| > π/6, Dr ≥ C, we get

|R^lλ(θ, φ)| + |∂^θR^l_λ(θ, φ)| + |∂^φR^l_λ(θ, φ)|

≤ C Z 1/2

0

r^λ−1¡ log1 r

¢l−1

dr + C Z 1

1/2

r^λ−1¡ log1 r

¢l−1

dr ≤ C.

With this we obtain condition i) in Theorem 2.4. In order to prove condition ii), we need a careful analysis of ∂_θ^l³

1 D^rλ+1

´. Recall our notation (3.13) and let

us call

b = ∂_θa = − sin(θ − φ) − cos θ sin φ(1 − cos t), and observe ∂θb = a. We have the following lemma to describe ∂_θ^l³

1 D^λr⁺¹

´ sum more precisely:

Lemma 3.17.

∂^l_θ³ 1 D^λ+1_r

´=X cl,k,i,j

r^i+jaⁱb^j D_r^λ+1+k and cl,k,i,j 6= 0 only if:

conditions derivative (3.18) k = 1, ..., l, i + j = k, j ≥ 2k − l.

Proof. For l = 1, ∂_θ^l³

1 D^λr⁺¹

´ = ^Crb

Dr^λ+2 and therefore only c1,1,0,1 is nonzero. As- sume that the lemma is true for l. Since

derivative Mlijk (3.19) ∂θ

³r^i+jaⁱb^j D^λ+1+k_r

´= c1

r^i+jaⁱ⁻¹b^j+1 D^λ+1+k_r + c2

r^i+jaⁱ⁺¹b^j−1 D^λ+1+k_r + c3

r^i+j+1aⁱb^j+1 D^λ+1+(k+1)r

(we assume c1 = 0 if i = 0 and c2 = 0 if j = 0), we easily see that all these

expressions satisfy (3.18) for l + 1. ¤

As a consequence we may write: R^l_λ(θ, φ) = P cl,k,i,jM_n,k,i,j(θ, φ), where c_l,k,i,j are as in the last lemma and

Ml,k,i,j(θ, φ) = Z 1

0

Z π 0

r^λ+i+j−1¡ log1 r

¢_l−1

(1 − r²)aⁱb^jsin^2λ−1t D_r^λ+1+k dtdr.

The next step is checking that each of these terms verifies condition ii) of The- orem 2.4. This will be done in two lemmas.

Lemma 3.20. There exists a constant C > 0 such that, for every θ ∈ [0, π/2]

and 2θ/3 ≤ φ ≤ 3θ/2, we have

|M^l,k,i,j(θ, φ)| ≤ C

|θ − φ|

1 (sin φ)^2λ, equation r5 (3.21)

|∂^θMl,k,i,j(θ, φ)| + |∂^φMl,k,i,j(θ, φ)| ≤ C

|θ − φ|² 1 (sin φ)^2λ. equation r6 (3.22)

Proof. Let us start with the first inequality. Since for r ∈ (0, 1/2), D^r ≥ C, and for r ∈ (1/2, 1), log¹r ≤ C(1 − r), and also by using that |a| ≤ C, |b| ≤

sin |θ − φ| + sin φ(1 − cos t), we have

|Ml,k,i,j(θ, φ)| ≤ C Z 1/2

0

r^λ+k−1¡ log1 r

¢l−1

dr + C

Z 1 1/2

Z π

0 (1 − r)^l(sin |θ − φ| + sin φ(1 − cos t))^j

D_r^λ+k+1 (sin t)^2λ−1dt dr eq m sin derivar (3.23)

≤ C + C Z 1

1/2

Z π

0 (1 − r)^l(sin |θ − φ|)^j(sin t)^2λ−1 D^λ+k+1_r dt dr + C

Z 1 1/2

Z π/2

0 (1 − r)^l(sin φ(1 − cos t))^j(sin t)^2λ−1

D_r^λ+k+1 dt dr

+ C Z 1

1/2

Z π

π/2(1 − r)^l(sin φ)^j(sin t)^2λ−1

D^λ+k+1_r dt dr = C + I + II + III.

For the first term, we use that j ≥ 2k − l, and thus l ≥ 2k − j, and then we perform successively the changes of variables x = p σ

∆^rt and u = ^1−r√_∆.

I ≤ C Z 1

1/2(1 − r)^2k−j(sin |θ − φ|)^j Z π/2

0

t^2λ−1

(∆r+ σt²)^λ+k+1 dt dr

= C Z 1

1/2(1 − r)^2k−j(sin |θ − φ|)^j

Z ^π₂√ ^σ

∆r

0

¡q

∆^r

σ x¢2λ−1q

∆^r σ dx

∆^λ+k+1_r (1 + x²)^λ+k+1 dr

≤ (sin |θ − φ|)^jC Z 1

1/2

(1 − r)^2k−jdr σ^λ∆^k+1_r

≤ C(sin |θ − φ|)^j σ^λ

Z _2√∆¹

0

(√

∆u)^2k−j√

∆ du

∆^k+1(1 + u²)^k+1 dr

≤ C(sin |θ − φ|)^j

σ^λ∆^j/2+1/2 ≤ C

σ^λ∆^1/2 ≤ C

|θ − φ|

1

(sin(θ ∨ φ))^2λ,

where in the last two inequalities we have used that for 2θ/3 ≤ φ ≤ 3θ/2,

equivalences local (3.24) 1 − cos(θ − φ) ∼ (sin |θ − φ|)², sin θ ∼ sin φ, | sin(θ − φ)| ≤ C sin φ.

The proof of these inequalities is trivial, although the argument differs when φ ∈ [0, π/2] than when φ ∈ [π/2, π]. Analogously we may estimate II and III.

By (3.19) and analogous arguments as in (3.23), we have

|∂θM_l,k,i,j(θ, φ)| ≤ C + C Z 1

1/2

Z π

0 (1 − r)^l |b|^j+1

D_r^λ+k+1(sin t)^2λ−1dt dr +C

Z 1 1/2

Z π

0 (1 − r)^l |b|^j−1

D_r^λ+k+1(sin t)^2λ−1dt dr +C

Z 1 1/2

Z π

0 (1 − r)^l |b|^j+1

D_r^λ+k+2(sin t)^2λ−1dt dr

= C + A + B + D.

For the first integral, we have that A ≤ C

Z 1 1/2

Z π

0 (1 − r)^l |b|^j+1

D_r^λ+k+1(sin t)^2λ−1dt dr ≤ C

|θ − φ|²

1 (sin(θ ∨ φ))^2λ as it was proven by the estimates of I, II and III above. Observe that if j = 0, the second term would not appear. So, we may proceed to estimate B assuming j ≥ 1, and obtaining

B ≤ C Z 1

1/2(1 − r)^l Z π

0

(sin |θ − φ| + sin φ(1 − cos t))^j−1

D^λ+k+1_r (sin t)^2λ−1dt dr

≤ C Z 1

1/2

Z π/2

0 (1 − r)^l(sin |θ − φ|)^j−1(sin t)^2λ−1 D_r^λ+k+1 dt dr +C

Z 1 1/2

Z π/2

0 (1 − r)^l(sin φ(1 − cos t))^j−1(sin t)^2λ−1

D_r^λ+k+1 dt dr

+C Z 1

1/2

Z π

π/2(1 − r)^l(sin φ)^j−1(sin t)^2λ−1 D^λ+k+1_r dt dr

≤ C

σ^λ∆ ≤ C

|θ − φ|²

1

(sin(θ ∨ φ))^2λ,

where the last two inequalities follows as in I, II and III above. Observe that D ≤ C

Z 1

1/2(1 − r)^l Z π

0

(sin |θ − φ| + sin φ(1 − cos t))^j+1

D_r^λ+k+2 (sin t)^2λ−1dt dr The same arguments drive to the bound

D ≤ C

|θ − φ|²

1

(sin(θ ∨ φ))^2λ.

The proof of (3.22) for ∂φ follows the same lines. ¤ parte global riesz Lemma 3.25. For every θ, φ ∈ [0, π/2] and φ outside the region 2θ/3 ≤ φ ≤

3θ/2, we also have (3.21) and (3.22).

Proof. In the following calculations we will use that for θ, φ ∈ [0, π/2], then 1 − cos(θ − φ) ∼ |θ − φ|² ∼ (sin |θ − φ|)². Also, for φ outside the region

2θ/3 ≤ φ ≤ 3θ/2, sin |θ − φ| ∼ sin(θ ∨ φ) and |b| ≤ C sin |θ − φ|. Therefore, by the same techniques applied in (3.23), for M_l,k,i,j(θ, φ) we have

|M^l,k,i,j(θ, φ)| ≤ C + C Z 1

1/2(1 − r)^2k−j |b|^j

∆^λ+k+1_r dr

≤ C + C(sin |θ − φ|)^j Z 1

1/2

(1 − r)^2k−j

∆^λ+k+1_r dr

≤ C + C(sin |θ − φ|)^j Z _2√∆¹

0

(√

∆u)^2k−j√

∆ du

∆^λ+k+1(1 + u²)^λ+k+1 dr

≤ C + C(sin |θ − φ|)^j

∆^λ+j/2+1/2 ≤ C

|θ − φ|

1

(sin(θ ∨ φ))^2λ. We proceed analogously for the derivative

|∂^θMl,k,i,j(θ, φ)| ≤ C + C Z 1

1/2

Z π

0 (1 − r)^l |b|^j+1

D^λ+k+1_r (sin t)^2λ−1dt dr + C

Z 1 1/2

Z π

0 (1 − r)^l |b|^j−1

D_r^λ+k+1(sin t)^2λ−1dt dr + C

Z 1 1/2

Z π

0 (1 − r)^l |b|^j+1

D_r^λ+k+2(sin t)^2λ−1dt dr ≤ C

|θ − φ|²

1

(sin(θ ∨ φ))^2λ.

¤ 4. Proof of Theorem 1.8

section g function r

Let us write Gⁱf (θ) = kTGⁱf (θ)kL²((0,1),dr), i = 1, 2 where T_Gⁱ is the operator mapping a scalar valued functions into L²(dr)-valued functions given by equation Tg 1 (4.1) T_G¹f (θ) =

r r log1

r ∂rP f (r, θ), T_G²f (θ) = r

r log1

r ∂θP f (r, θ).

subsection bddness of g

4.1. Boundedness in L²(dmλ) of the g-functions. For any polynomial func- tion f ∈ L²(dmλ), the operator ∂rP f =PN

n=0(n+λ)r^n+λ−1anP_n^λkPn^λk⁻¹ gives a well defined function in L²(dmλ), since for each fixed r ∈ (0, 1), |(n + λ)r^n+λ| ≤ C for any n ≥ 0. Thus, we can write

T_G¹f =

∞

X

n=0

r r log1

r (n + λ)r^n+λ−1anP_n^λkPn^λk⁻¹ =

∞

X

n=0

g¹(n)anP_n^λkPn^λk⁻¹,

where g¹(n) = q

r log¹_r (n + λ)r^n+λ−1 belongs to L²(dr) uniformly in n. In particular, this implies that G¹ is bounded in L². To get the boundedness of G² in L² we proceed similarly. Observe that for any polynomial f ∈ L²(dmλ),

∂_θP f =

N

X

n=1

r^n+λa_n(−2λ) sin θP_n−1^λ+1kPn^λk⁻¹

We can write T_G²f =

N

X

n=1

r r log1

r r^n+λ an

n + λ(−2λ) sin θP_n−1^λ+1kPn^λk⁻¹

= X∞ n=0

g²(n) an

n + λ(−2λ) sin θP_n−1^λ+1kPn^λk⁻¹ = Tg²(Rλf ),

where Rλ is the Riesz transform operator (see [1]) and T^BMTU g² is the multiplier associated to the orthogonal system in L²(dmλ) given by the functions hn(θ) = sin θP_n−1^λ+1(cos θ) with coefficients g²(n) = q

r log¹_r (n + λ)r^n+λ−1. The coeffi- cients are uniformly bounded in L²(dr). This, together with the L²-boundedness of the Riesz transform, see ^BMTU[1] for the details, give that G² is bounded in L²(dm_λ).

4.2. Kernel of the g-function. In the next lemma, we find the vector-valued kernel in the Calder´on-Zygmund sense of T_Gⁱ, i = 1, 2.

kernel in CZ sense Lemma 4.2. For every f ∈ L¹(dmλ) and θ outside the support of f , we have equation Tg 2 (4.3) T_Gⁱf (θ) =

Z π 0

τ_rⁱ(θ, φ)f (φ) dmλ(φ), i = 1, 2, where

τ_r¹(θ, φ) = r

r log1 r

λ π

·

λr^λ−1P (r, θ, φ) − 2r^λ+1 Z π

0

(sin t)^2λ−1 D_r^λ+1 dt

− (λ + 1)r^λ(1 − r²) Z π

0

(sin t)^2λ−1∂rDr

D^λ+2_r dt

¸ equation Tg1 kernel (4.4)

τ_r²(θ, φ) = r

r log1 r

λ(λ + 1)

π r^λ(1 − r²) Z π

0

(sin t)^2λ−1∂θDr

D_r^λ+2 dt equation Tg2 kernel (4.5)

are kernels taking values in L²(dr).

Proof. By the definition of T_Gⁱ, in order to get (4.3) we only need to put the derivative inside the integrals in the expressions (4.1). With our usual notation (3.13), we write

partial P (4.6) ∂rP (r, θ, φ) = −λ π2r

Z π 0

(sin t)^2λ−1

D^λ+1_r dt + λ

π(1 − r²)∂r

Z π 0

(sin t)^2λ−1 D_r^λ+1 dt For negative a, Dr ≥ 1, and for positive a, a ≤ cos(θ − φ). Therefore, if θ does not belong to the support of f , we have that Dr ≥ 1 − cos(θ − φ)² ≥ C.

This implies that for each r the remaining integrands in the right hand of the equations in (4.1) belong to L¹(dmλ × dt) for f ∈ L¹(dmλ). This shows (4.4). Also, if θ does not belong to the support of f , we have that |τr¹(θ, φ)| ≤ Cq

r log¹_r r^λ−1 and |τr²(θ, φ)| ≤ C, which are functions in L²(dr). ¤

Our aim is to prove that the kernels of T_G¹ verify the hypothesis of Theorem 2.4. We have already seen that they are bounded in L²(dm_λ). Observe that since P (r, π − θ, π − φ) = P (r, θ, φ), then the symmetry condition iii) holds.

On the other hand, for |θ − φ| > π/6 and r ∈ (1/2, 1), then Dr ≥ 1 − (cos(θ − φ))² ≥ C. For r ∈ (0, 1/2), clearly D^r ≥ C. Then kτr¹(θ, φ)kL²(dr) ≤ Ckq

r log¹_r (r^λ−1+ 1)k^L2(dr) = C and kτr²(θ, φ)k^L2(dr) ≤ C, as can be easily seen from (4.4) and (4.5). Thus condition i) in Theorem 2.4 also holds.

Checking condition ii) requires a bit more of work.

local estimates Lemma 4.7. There exists a constant C > 0 such that, for every θ ∈ [0, π/2]

and 2θ/3 ≤ φ ≤ 3θ/2, we have (for i = 1, 2)

kτrⁱ(θ, φ)kL²(dr) ≤ C

|θ − φ|

1

(sin(θ ∨ φ))^2λ, local estimate g 1 (4.8)

k∂^θτ_rⁱ(θ, φ)kL²(dr)+ k∂^φτ_rⁱ(θ, φ)]kL²(dr) ≤ C

|θ − φ|²

1

(sin(θ ∨ φ))^2λ. local estimate g 2 (4.9)

Proof. When r ∈ (0, 1/2), D^r ≥ (1 − r)² ≥ C and therefore |τr¹(θ, φ)| ≤ Cq

r log¹_r (r^λ−1+C) and |τr²(θ, φ)| ≤ C. For r ∈ (1/2, 1), let us observe that, by (4.4), we can split |τr¹(θ, φ)| according to the following sum |∂^r(r^λP (r, θ, φ))| ≤ P3

i=1Ni(r, θ, φ), where N₁(r, θ, φ) = C

Z π/2 0

(sin t)^2λ−1

D_r^λ+1 dt, N₂(r, θ, φ) = C Z π

π/2

(sin t)^2λ−1 D_r^λ+1 dt, N3(r, θ, φ) = C(1 − r)

Z π 0

(sin t)^2λ−1|∂rD_r| D_r^λ+2 dt.

We will use the following estimate ialphabeta (4.10) I_λ+1^2λ−1 =

Z ^π₂

0

t^2λ−1

(∆r+ rσt²)^λ+1dt ≤ C

∆rr^λσ^λ, which can be easily obtained by the change of variables t = ∆

1

r2(rσ)⁻¹²u. By using that sin x ∼ x, 1 − cos x ∼ x² and 1 + cos x ≥ Cx² for x ∈ [0, π/2], it is not difficult to obtain the following estimate,

N1(r, θ, φ) + N2(r, θ, φ) ≤ CIλ+1^2λ−1,

where in the case of N2 we have firstly made the change of variables π − x = t.

For the term N3, observe that for r ∈ [1/2, 1]

partial dr (4.11) (1−r)|∂^rDr| ≤ C(1−r)[|1−r|+|1−cos(θ −φ)|+|σ(1−cos t)|] ≤ C D^r. Thus, after applying the same change of variables π − x = t used above for N², we get that N3(r, θ, φ) ≤ C Iλ+1^2λ−1. This gives (and observe that with the only restriction r ∈ [1/2, 1]) that

bound for partial p (4.12) |∂^r(r^λP (r, θ, φ))| ≤ C Iλ+1^2λ−1.