Multipliers of Laplace transform type for ultraspherical expansions

Math. Nachr. 281, No. 7, 978 – 988 (2008) / DOI 10.1002/mana.200610654

Multipliers of Laplace transform type for ultraspherical expansions

Teresa Martinez∗1

1 Departamento de Matematicas, Faculdad de Ciencias, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Received 8 February 2006, revised 20 January 2007, accepted 21 February 2007Published online 5 June 2008

Key words Ultraspherical polynomials, multipliersMSC (2000) Primary: 42C05, Secondary: 42C15

We define and investigate the multipliers of Laplace transform type associated to the differential operatorLλf(θ) = −f ′′(θ) − 2λ cot θf ′(θ) + λ2f(θ), λ > 0. We prove that these operators are bounded inLp((0, π), dmλ) and of weak type (1, 1) with respect to the same measure space, dmλ(θ) = (sin θ)2λdθ.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

We consider the ultraspherical polynomials of type λ, with any λ > 0, Pλn (x) defined as the coefficients in the

expansion of the generating function(1 − 2xω + ω2

)−λ =∑∞

n=0 ωnPλ

n (x) (see [17] for further details). Itis known that the set

{Pλ

n (cos θ) : n ∈ N}

is orthogonal and complete in L2[0, π] with respect to the measuredmλ(θ) := sin2λ θdθ. More precisely, we have∫ π

0

Pλn (cos θ)Pλ

m(cos θ) dmλ(θ) = δn,m21−2λπΓ(λ)−2 Γ(n+ 2λ)(n+ λ)n!

= δn,m/γn. (1.1)

It is also known that the functions Pλn (cos θ) are eigenfunctions of the operator Lλ

Lλf(θ) = −f ′′(θ) − 2λ cot θf ′(θ) + λ2f(θ), (1.2)

with eigenvalues µn = (n+ λ)2. We associate to every function f in L2(dmλ) its ultraspherical expansion

f(θ) =∞∑

n=0

anPλn (cos θ), where an = γn

∫ π

0

f(θ)Pλn (cos θ)dmλ(θ). (1.3)

In this paper we are interested in multipliers of Laplace transform type for this kind of expansions. Following[16, p. 58], given a complex-valued function m defined on the set {n+ λ : n ≥ 0}, the multiplier operator Tm

is defined as

Tmf(θ) =∞∑

n=0

m(µ1/2

n

)anP

λn (cos θ). (1.4)

The multiplier Tm is of Laplace transform type if m is of the form

m(ξ) = ξ

∫ ∞

0

κ(t)e−ξt dt, ξ ∈ C,

where κ is a bounded measurable function in (0,∞). Then,m is also said to be of Laplace transform type. To beprecise, the theorem we prove (in Section 5) is the following.

∗ e-mail: [email protected]

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 281, No. 7 (2008) 979

Theorem 1.1 Every multiplier of Laplace transform type Tm defined as in (1.4) is bounded in Lp(dmλ) forp ∈ (1,∞) and of weak type (1, 1) with respect to the measure dmλ.

The constant C such that ‖Tmf‖L ≤ C‖f‖L

(L = Lp, p > 1 or L = L1,∞) depends on λ and on ‖κ‖L∞.

An antecedent of Theorem 1.1 can be found in [16, p. 58]. In this result the boundedness of the multiplier inLp(X, dν) where X is a compact group endowed with its Haar measure, is obtained for every p, 1 < p < ∞,by means of the boundedness in Lp of certain Littlewood–Paley square functions. These square functions areassociated to the Poisson semigroup of a Laplacian inX . In particular, the Laplacian is a second order differentialoperator that commutes with the first order differentials. So, the techniques used in this proof are not valid inour case, since Lλ does not commute with the derivative. There can be found another result in [16, p. 121] formultipliers of Laplace transform type. In this other result, it is proven that the multipliers are also bounded inLp, 1 < p < ∞, without imposing any conditions on the underlying measure space. The proof of this resultinvolves the boundedness of certain square functions, which is achieved after the application of Rota’s theorem.One of the hypothesis needed to apply that theorem is that the Poisson semigroup defining the square functionis markovian (that is, it maps constants into constants). In our case, see Section 2 for the details, we havePr(1)(θ) = Pr

(Pλ

0

)(θ) = rλ, and therefore the hypothesis in Stein’s theorem are not satisfied. Nevertheless,

we could apply this result to the semigroup Pr = r−λPr and obtain that any Laplace transform type multiplier

associated to Pr,

Tmf(θ) =∞∑

n=0

m(n)anPλn (θ)

is bounded in Lp(dmλ). We have

Tmf(θ) =∞∑

n=0

m(n+ λ)anPλn (θ) =

∞∑n=0

m(n+ λ)m(n)

m(n)anPλn (θ) = T Tmf(θ)

and it would be necessary to prove that the multiplier T is bounded in all Lp in order to be able to derive ourtheorem from Stein’s result.

There is a number of works studying multiplier and transplantation theorems for Jacobi or ultrasphericalseries. In the paper [13] Muckenhoupt and Stein studied ultraspherical expansions from a harmonic analysispoint of view. For any function as in (1.3) they define the “harmonic” extension of f as

f(r, θ) =∞∑

n=0

anrnPλ

n (cos θ). (1.5)

They proved that the multipliers Tf(θ) =∑∞

n=0m(n)anPλn (cos θ) such that |m(n)| ≤M and

∑nk=1 k|m(k)−

m(k+1)| ≤Mn, for every n ≥ 1, are bounded in Lp(dmλ) for 2λ+1λ+1 < p < 2λ+1

λ . This range of the p’s cannotbe improved and the functionsm of Laplace transform type do not match the second of those hypothesis. Similartechniques were also used in [10] to define the conjugate Jacobi series and the conjugate function associated toJacobi expansions. The author proved, in the same spirit than Muckehnhoupt and Stein, the strong type (p, p)of the conjugate series and the conjugate function, and the weak type (1, 1) of the conjugate series, by way of acareful study of the properties of the kernels.

In [12, Theorem 1.14], Muckenhoupt gives very general conditions under which a multiplier for Jacobi ex-pansions is bounded in Lp, 1 < p < ∞. In [5] the boundedness in Lp of some Jacobi multipliers is stated,under certain conditions on the differences of the multiplier sequence. This type of conditions appear also in[3], where Connett and Schwartz study the boundedness in Lp, 1 < p < ∞, and from L1 into L1,∞ of themultipliers for ultraspherical expansions. The conditions given there deal with the differences of the multipliersequence {mn} of an order which depends on the value of λ. The method of the proof in [3], which involvesarguments of a highly technical nature, consists in identifying the multiplier as the limit of Calderon–Zygmundoperators that satisfy the Hormander conditions uniformly. Similar conditions on the multiplier sequence ap-pear in [4], to define families of spaces of multiplier sequences closed under interpolation. In [1] the particularcase of the multipliers of Laplace transform type is studied following the lines of [16]. Under the parallel con-ditions in the ultraspherical expansion case (that involve the differences of the multiplier sequence of an order

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980 Martinez: Multipliers for ultraspherical expansions

that depends on the particular value of λ) the boundedness in Lp of the multiplier is obtained in a restrictedrange of p’s, 1 < p < ∞. All the results mentioned so far refer to sufficient conditions for the boundedness ofmultipliers. In [8] there appear also necessary conditions for the boundedness in Lp, 1 < p < ∞, of the mul-tipliers for ultraspherical expansions, in such a way that they are comparable to the sufficient conditions. Since∣∣ξNm(N)(ξ)

∣∣ ≤ CN‖κ‖∞( ∫∞

0 tN−1e−t dt +∫∞0 tNe−t dt

)for m a function of Laplace transform type, we

immediately have supN (n+ 1)N∆Nmn <∞, for all N . This assumption is stronger than the ones in the worksof Muckenhoupt and Stein, Connett and Schwartz, Gasper and Trebels.

In the present work, Theorem 1.1 states both the strong (p, p), p > 1, and weak (1, 1) results mentionedabove in the particular, smooth case of the Laplace transform type multipliers. This boundedness propertieshold for every λ > 0 and for every p, 1 < p < ∞ (in the case of strong boundedness), without imposing anyextra condition on the function m other than being of Laplace transform type. The proof of Theorem 1.1 isachieved by means of a general method that has been widely used by many authors in various contexts (see e.g.[6, 7, 11, 14, 15]) and that, in the context of ultraspherical expansions, was first explored in [2]. The general ideaof the method for an operator T is producing a partition of T into its “local” and its “global” parts according totwo regions: the “local region”, where roughly speaking we can use that the Lebesgue measure dφ and dmλ(φ)are equivalent, and its complementary, that we call “global region” (see (3.1) and (3.2)). In the particular versionof the method related to our operator Tm, by a “heritage” theorem (Theorem 3.4), we prove that the local part ofour operator inherits the boundedness in L2 with respect to dmλ from the boundedness of Tm. Next, for operatorswith kernels supported in the local part (“local operators”), we prove (see Lemma 3.5) that the boundedness withrespect to the Lebesgue measure is equivalent to the boundedness with respect to dmλ(φ). We use these resultsto conclude that the local part of Tm is bounded in L2(dθ). The crucial step is observing that the local part ofthe multiplier operators treated in this paper are Calderon–Zygmund operators, and thus, by the classical theoryfor them, we obtain that they are bounded on Lp(dφ) for every p in the range 1 < p < ∞ and of weak type(1, 1) with respect to the Lebesgue measure. Next, again by Lemma 3.5, we conclude that the local part of Tm isbounded in Lp(dmλ) for p ∈ (1,∞) and of weak type (1, 1) with respect to dmλ. In the global part we just usethat the kernel of the operator Tm is bounded by a positive and nice kernel.

The structure of the paper is as follows. In Section 2 we define the multipliers of Laplace transform type andprove that they are given by a kernel in the Calderon–Zygmund sense. In Section 3, we introduce the “generalmachinery” needed to handle the “local” and the “global” parts of the operators. Section 4 is devoted to state theestimates on the kernels of the operators needed to get the proof of Theorem 1.1, which is given in Section 5.

2 Multipliers of Laplace transform type

The multiplier operators defined as in (1.4) are bounded in L2(dmλ) if and only if the functionm is bounded and

the norm of Tm is the supremum of∣∣m(µ1/2

n

)∣∣. Clearly, any multiplier of Laplace transform type satisfies thiscondition, and therefore the definition (1.4) makes sense for every f ∈ L2(dmλ). We are interested in findingan expression of Tm in terms of a kernel in the Calderon–Zygmund sense. With this aim, following [16] let usdefine for any function as in (1.3), the Poisson integral of f as

Pe−tf(θ) = e−t√

Lλf(θ) =∞∑

n=0

ane−t(n+λ)Pλ

n (cos θ).

Observe that, by the change r = e−t, for any Pλn (cos θ), we have that Pr

(Pλ

n

)(θ) = rn+λPλ

n (cos θ). Also, we

easily see that for any f ∈ L2(dmλ), Prf(θ) = rλf(r, θ). The results in [13] for f(r, θ) (see 1.5) state that it isgiven by a kernel, P (r, θ, φ) which can be computed explicitly. Hence, for any θ ∈ [0, π] we have

Prf(θ) =∫ π

0

rλP (r, θ, φ)f(φ) dmλ(φ) (2.1)

where

P (r, θ, φ) =λ

π

(1 − r2

) ∫ π

0

sin2λ−1 t(1 − 2r(cos θ cosφ+ sin θ sinφ cos t) + r2

)λ+1dt. (2.2)

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com

Math. Nachr. 281, No. 7 (2008) 981

We also define√LλP

λn (cos θ) = (n + λ)Pλ

n (cos θ). On the other hand, performing the change of variablesr = e−t in the formula for m, we see that a functionm is of Laplace transform type if

m(ξ) = ξ

∫ 1

0

ψ(r)rξ−1 dr,

where ψ(r) = κ(− log r). Hence, for any operator as in (1.4), we get that

TmPλn (θ) = m(n+ λ)Pλ

n (cos θ)

= (n+ λ)∫ 1

0

ψ(r)r(n+λ)−1Pλn (cos θ) dr

=∫ 1

0

ψ(r)√Lλr

√LλPλ

n (cos θ)dr

r

=∫ 1

0

ψ(r)∂rPrPλn (cos θ) dr

Let us call the polynomial function any finite linear combination of ultraspherical polynomials, that is, any f ofthe form f =

∑Nn=0 anP

λn . By the former expression, for any polynomial function f we have that

Tmf(θ) =∫ 1

0

ψ(r)√Lλr

√Lλf(θ)

dr

r=∫ 1

0

ψ(r)∂rPrf(θ) dr. (2.3)

This formula holds in a weaker sense for every function in L2(dmλ).Lemma 2.1 Equation (2.3) holds for every f ∈ L2(dmλ), where the integrals converge in the weak topology

of the operators in L2(dmλ).

P r o o f. To get the first equality, let us denote by 〈·, ·〉 the interior product in L2(dmλ), and observe that forany functions f =

∑∞n=0 anP

λn (cos θ) and g =

∑∞n=0 bnP

λn (cos θ) in L2(dmλ), we get

〈Tmf, g〉 =∞∑

n=0

m(n+ λ)anbn∥∥Pλ

n

∥∥2

L2(dmλ)

=∞∑

n=0

∫ 1

0

ψ(r)(n + λ)rn+λ−1 dr anbn∥∥Pλ

n

∥∥2

L2(dmλ).

Since∑

n |an| |bn|∥∥Pλ

n

∥∥2

L2(dmλ)≤ ‖f‖L2(dmλ)‖g‖L2(dmλ), we can use Fubini’s theorem to interchange the

order of the integral an the sum, and we get

〈Tmf, g〉 =∫ 1

0

ψ(r)∞∑

n=0

(n+ λ)rn+λ−1anbn∥∥Pλ

n

∥∥2

L2(dmλ)dr =

∫ 1

0

ψ(r)⟨√

Lλr√

Lλf, g⟩ dr

r.

To get the second equality, observe that the operator ∂rPrf =∑∞

n=0(n + λ)rn+λ−1anPλn (cos θ) gives a well

defined function in L2(dmλ) for any f ∈ L2(dmλ), since for each r ∈ (0, 1),∣∣(n+λ)rn+λ

∣∣ ≤ C for any n ≥ 0.In particular, this implies that ∂rPr is bounded in L2(dmλ). Hence, the second equality in 2.3 holds since for fand g in L2(dmλ), one has⟨

1r

√Lλr

√Lλf, g

⟩=

∞∑n=0

(n+ λ)rn+λ−1anbn∥∥Pλ

n

∥∥2

L2(dmλ)= 〈∂rPrf, g〉.

Before we proceed further let us introduce some notation. Let us write

Dr = 1 − 2ra+ r2, with a = cos θ cosφ+ sin θ sinφ cos t. (2.4)



Note that Dλ+1r is the denominator of the integrand in the right-hand side of the equation for the Poisson kernel

(2.2). With this notation, we write

∂rP (r, θ, φ) = −λπ

2r∫ π

0

(sin t)2λ−1

Dλ+1r

dt+λ

π(1 − r2)∂r

∫ π

0

(sin t)2λ−1

Dλ+1r

dt. (2.5)

In the following lemma we show that Tm is defined by a kernel in the sense of Calderon–Zygmund.

Lemma 2.2 Given f ∈ L2(dmλ), for θ outside the support of f we have that

Tmf(θ) =∫ π

0

M(θ, φ)f(φ) dmλ(φ), (2.6)

where

M(θ, φ) =∫ 1

0

ψ(r)[λrλ−1P (r, θ, φ) − 2λ

πrλ+1

∫ π

0

(sin t)2λ−1

Dλ+1r

dt

− λ(λ+ 1)π

rλ(1 − r2

) ∫ π

0

(sin t)2λ−1∂rDr

Dλ+2r

dt

]dr.

(2.7)

P r o o f. Let us observe that

Tmf(θ) =∫ 1

0

ψ(r)∂r

∫ π

0

rλP (r, θ, φ)f(φ) dmλ(φ) dr (2.8)

can be written as (2.6) with

M(θ, φ) =∫ 1

0

ψ(r)∂r

(rλP (r, θ, φ)

)dr, (2.9)

after applying Fubini’s theorem in (2.8). According to (2.4), for negative a, Dr ≥ 1, and for positive a, a ≤cos(θ − φ). Also, Dr is a quadratic function of r whose absolute minimum is reached at the point r = a, and ifθ does not belong to the support of f , we have that Dr ≥ 1 − cos(θ − φ)2 ≥ C. In particular, this implies thatthe integrands (with respect to dt) in the terms of the sum in the right-hand side of (2.7) are bounded above bya constant times (sin t)2λ−1. Since ψ is a bounded function, this implies that the whole integrand (with respectto dr × dt× dφ) in the right-hand side of (2.8) is bounded above by a constant times rλ−1(sin t)2λ−1|f(φ)| andthus it belongs to L1(dmλ × dr × dt) for f ∈ L2([0, π]; dmλ) with support not containing θ. Hence, Fubini’stheorem can be applied, and by interchanging the order of integration in (2.8) we get (2.6) with M having theform given in (2.7).

3 General machinery

This section is essentially developed in [2]. We state here the main results for the convenience of the reader, andrefer to that work for the proofs.

Define a sequence of points θi and the family of intervals Bi in[0, π

2

]for a fixed, small δ > 0 (that will be

the same throughout the paper), as

θ0 =π

2, θi+1 =

θi

1 + δ, Bi =

{φ :

11 + δ

<θi

φ< 1 + δ

}.

Let us also consider intervals B(θi, δ′) =

{φ : (1 + δ′)−1 < θi/φ < 1 + δ′

}. One can easily extend

(by

symmetry with respect to π2

)the above family to a set of intervals covering (0, π), that will be denoted in the

same way {Bi}∞i=0.

Lemma 3.1 For the collection of intervals {Bi}∞i=0 defined above we have

1. The collection {Bi}∞i=0 of closed intervals covers (0, π).


Math. Nachr. 281, No. 7 (2008) 983

2. There exists δ0 such that the intervals B(θi, δ0) are pairwise disjoint.

3. For every n ≥ 1, if 1 + δ0 = (1 + δ)n, then the collection {B(θi, δ0)} has bounded overlap.

4. For every n ≥ 1, if 1 + δ0 = (1 + δ)n, then there exists a constant C depending only on δ, n, such that forevery measurable set E ⊂ B(θi, δ0), we have

1Cθ2λ

i |E| ≤ mλ(E) ≤ C θ2λi |E|.

Let us denote by Nt the following region: consider the part of the region Mt ={(θ, φ) : 1

1+t <θφ < 1 + t

}under the diagonal of the square [0, π]2 with negative slope, and define Nt to be the union of this region andits reflected with respect to the point (π/2, π/2). Let us consider the regions Nt1 and Nt2 , with t1 = δ and1 + t2 = (1 + δ)3. Observe that both regions Nt1 and Nt2 are contained in Mt2 . These regions will define the“local” and “global” parts of the operators.

We will consider linear operators T mapping the space of polynomial functions into the space of measurablefunctions on [0, π], satisfying the following assumptions:

(a) T either extends to a bounded operator on Lq(dmλ) for some 1 < q < ∞, or of weak type (1,1) withrespect to dmλ,

(b) there exists a measurable function K , defined in the complement of the diagonal in [0, π] × [0, π], suchthat for every function f and all θ outside the support of f ,

Tf(θ) =∫ π

0

K(θ, φ)f(φ) dmλ(φ),

(c) for all (θ, φ), θ = φ, the functionK verifies∣∣K(θ, φ)(sinφ)2λ∣∣+ ∣∣K(θ, φ)(sin θ)2λ

∣∣ ≤ C |θ − φ|−1,∣∣∂θ

[K(θ, φ)(sinφ)2λ

]∣∣+ ∣∣∂φ

[K(θ, φ)(sin φ)2λ

]∣∣ ≤ C |θ − φ|−2.

Take a function ϕ ∈ C∞([0, π] × [0, π]) such that 0 ≤ ϕ ≤ 1, ϕ(θ, φ) = ϕ(π − θ, π − φ) in order to preservethe symmetry of the problem, and such that

|∂θϕ(θ, φ)| + |∂φϕ(θ, φ)| ≤ C|θ − φ|−1

if θ = φ and

ϕ(θ, φ) =

{1 for (θ, φ) ∈ Nt1 ,

0 for (θ, φ) ∈ Nt2 ,

where t1 and t2 are some constants satisfying t1 < δ and 1+ t2 > (1 + δ)3 for δ as in Lemma 3.1. We define theglobal and local parts of the operator T by

Tglobf(θ) =∫K(θ, φ)(1 − ϕ(θ, φ))f(φ) dφ, (3.1)

Tlocf(θ) = Tf(θ) − Tglobf(θ). (3.2)

Lemma 3.2 Let dν be any positive measure in [0, π], let {fj} be a sequence of functions and define f =∑j 1Bjfj, where {Bj : j ∈ N} is the collection of intervals in Lemma 3.1. Then, for all λ > 0,

ν{θ : |f(θ)| > k} ≤∑

j

ν{θ ∈ Bj : |fj(θ)| > k/2}, (3.3)

and for any 1 ≤ q <∞

‖f‖Lq(dν) ≤ 2

(∑j

∫Bj

|fj(θ)|q dν(θ))1/q

. (3.4)



Lemma 3.3 Let T be a linear operator mapping polynomial functions into the space of measurable functionson [0, π] verifying (a). Given the covering {Bj}∞j=0, we define the operator

T 1f(θ) =∑

j

1Bj (θ)∣∣T (1B′

jf)(θ)∣∣, where B′

j = B(θj , δ′), 1 + δ′ = (1 + δ)2.

Then, T 1 is also bounded from L1(dmλ) into L1,∞(dmλ) or from Lq(dmλ) into Lq(dmλ), 1 < q < ∞, as thecase might be.

The following theorem is one of the main tools in the proofs of the results in the paper.

Theorem 3.4 Under the assumptions (a), (b) and (c) made on T above, the operator Tloc inherits from Teither Lq(dmλ)-boundedness or the weak type (1, 1) with respect to mλ, as the case might be.

Similar arguments as the ones shown in the above proof for T 1 lead to the following lemma.

Lemma 3.5 Assume that an operator S defined on polynomial functions satisfies (b), with a kernel supportedin Nt2 . Then, strong type (p, p) for the Lebesque measure and the mλ measure are equivalent. The same holdsfor weak type (1, 1).

4 Estimates on the kernel of Tm

4.1 The local part

In this section we study the behavior of the kernel of the multiplier M(θ, φ) defined in (2.7). We prove that itverifies the size estimates of (c), which, together with the boundedness of the operator in L2(dθ), say that thelocal part of Tm is a Calderon–Zygmund operator.

Lemma 4.1 There exists a constant C > 0 such that for θ = φ in the local region Nt2 , we have∣∣M(θ, φ)(sin φ)2λ∣∣ ≤ C |θ − φ|−1,∣∣∂θ

[M(θ, φ)ϕ(θ, φ)(sin φ)2λ]∣∣+ ∣∣∂φ

[M(θ, φ)ϕ(θ, φ)(sin φ)2λ]∣∣ ≤ C |θ − φ|−2.

The constant C is of the form C = C(λ)‖κ‖L∞

P r o o f. Since P (r, π− θ, π− φ) = P (r, θ, φ), then we have M(π− θ, π− φ) = M(θ, φ). Therefore, if weget the bounds in the region (θ, φ) ∈ [0, π/2] × [0, π] we get the same inequality in the region [π/2, π] × [0, π].Thus, we will restrict ourselves to prove the inequalities for (θ, φ) ∈ [0, π/2] × [0, π] in the local region. Forthose values of (θ, φ), we get the following estimates, that will be used throughout the proof:

sin θ ∼ sinφ, |θ − φ| ∼ | sin(θ − φ)| ≤ C sinφ, 1 − cos(θ − φ) ∼ |θ − φ|2.

The proof of these inequalities is trivial, although the argument differs when φ ∈ [0, π/2] than when φ ∈ [π/2, π].Since Dr ≥ (1 − r)2 and ψ is a bounded function, according to (2.7) and (2.9) we get

|M(θ, φ)| ≤ C

∫ 1/2

0

rλ−1 dr + C + C

∫ 1

1/2

∣∣∂r

(rλP (r, θ, φ)

)∣∣ dr.By (2.7) we can split the integrand of the third term above as

∣∣∂r

(rλP (r, θ, φ)

)∣∣ ≤∑3i=1Ni(r, θ, φ), where

N1(r, θ, φ) = C

∫ π/2

0

(sin t)2λ−1

Dλ+1r

dt,

N2(r, θ, φ) = C

∫ π

π/2

(sin t)2λ−1

Dλ+1r

dt,

N3(r, θ, φ) = C(1 − r)∫ π

0

(sin t)2λ−1|∂rDr|Dλ+2

r

dt.


Math. Nachr. 281, No. 7 (2008) 985

Let us denoteσ = sin θ sinφ,

∆r = 1 − 2r cos(θ − φ) + r2 = (1 − r)2 + 2r(1 − cos(θ − φ)).(4.1)

With this, from (2.4) we write

Dr = 1 − 2ra+ r2 = ∆r + 2rσ(1 − cos t),where

a = cos θ cosφ+ σ cos t = cos(θ − φ) − σ(1 − cos t).

We will use the following estimate

I2λ−1λ+1 =

∫ π2

0

t2λ−1

(∆r + rσt2)λ+1dt ≤ C

∆rrλσλ, (4.2)

which can be easily obtained by the change of variables t = ∆12r (rσ)−

12u. By using that sinx ∼ x and 1−cosx ∼

x2 for x ∈ [0, π/2], it is not difficult to obtain

N1(r, θ, φ) +N2(r, θ, φ) ≤ CI2λ−1λ+1 ,

where in the case of N2 we have firstly made the change of variables π − x = t. For the term N3, observe thatfor r ∈ [1/2, 1]

(1 − r)|∂rDr| ≤ C(1 − r)[|1 − r| + |1 − cos(θ − φ)| + |σ(1 − cos t)|]

≤ C

[(1 − r)2 +

12[(1 − r)2 + 2r(1 − cos(θ − φ))]

√1 − cos(θ − φ)

+12[(1 − r)2 + 2rσ(1 − cos t)

]√σ(1 − cos t)

]≤ C Dr.

(4.3)

Thus, after applying the same change of variables π − x = t used above for N2, we get that N3(r, θ, φ) ≤C I2λ−1

λ+1 . This gives (and observe that with the only restriction r ∈ [1/2, 1]) that∣∣∂r

(rλP (r, θ, φ)

)∣∣ ≤ C I2λ−1λ+1 . (4.4)

The next step is integrating in r and using (4.2) and the fact that that in the local region, sin |θ−φ| ∼ |θ−φ| and1 − cos(θ − φ) ∼ |θ − φ|2. By the change of variables u = 1−r

|θ−φ| we obtain∫ 1

1/2

∣∣∂r

(rλP (r, θ, φ)

)∣∣ dr ≤ C

σλ

∫ 1

1/2

dr

(1 − r)2 + |θ − φ|2 ≤ C

σλ|θ − φ| .

Since (sinφ)2λ ∼ σλ in the local part, one gets∣∣M(θ, φ)(sin φ)2λ∣∣ ≤ C(sin φ)2λ

σλ|θ − φ| ≤ C

|θ − φ| .

For the estimates concerning the derivative, let us observe first that

∂θ

[M(θ, φ)ϕ(θ, φ)(sin φ)2λ]

= ∂θM(θ, φ)ϕ(θ, φ)(sin φ)2λ + M(θ, φ)∂θϕ(θ, φ)(sinφ)2λ.

The same arguments as in the proof of Lemma 2.2 allows us to put the derivatives inside the integrals. For thesecond term, we simply use the estimates already obtained for

∣∣M(θ, φ)(sinφ)2λ∣∣ and the hypothesis for ϕ. The

first term involves

∂θM(θ, φ) =

� 1

0

ψ(r)λ

π

�− (λ+ 1)rλ−1(λ(1 − r2) − r2)

� π

0

(sin t)2λ−1∂θDr

Dλ+2r

dt

− (λ+ 1)rλ�1 − r2� � π

0

(sin t)2λ−1

Dλ+1r

∂θ∂rDr Dr − (λ+ 2)∂θDr∂rDr

D2r

dt

�dr.

By using (4.3), and also the following estimates for 1/2 ≤ r ≤ 1,


986 Martinez: Multipliers for ultraspherical expansions∣∣∣∣∂θDr

Dr

∣∣∣∣ ≤ 2r sin |θ − φ|2r(1 − cos(θ − φ))

+2r cos θ sinφ(1 − cos t)2r sin θ sinφ(1 − cos t)

≤ C

|θ − φ| ,∣∣∣∣∂θ∂rDr

Dr

∣∣∣∣ =∣∣∣∣1/r∂θDr

Dr

∣∣∣∣ ≤ C

|θ − φ| ,(4.5)

we easily get that∣∣∂θM(θ, φ)ϕ(θ, φ)(sin φ)2λ∣∣ ≤ C +

C(sinφ)2λ

|θ − φ|∫ 1

1/2

I2λ−1λ+1 dr ≤ C

|θ − φ|2 .

The derivative in φ has three terms,

∂φ

(Mϕ (sinφ)2λ)

= ∂φMϕ (sinφ)2λ + M ∂φϕ (sin φ)2λ + Mϕ 2λ(sinφ)2λ−1 cosφ.

The first two terms are treated similarly, by using the parallel estimates to (4.3) and (4.5). For the last one wesimply apply the first size estimate of the lemma and observe that in the local part 1/ sinφ ≤ C/|θ − φ|.

4.2 The global part

Using the symmetry of the kernel of Tm, M(θ, φ) = M(π − θ, π − φ), we may restrict ourselves to considerthe global part of Tm only for θ ∈ [0, π/2]. Observe that for θ ∈ [0, π/2], we can give a bound above for Tm,glob

with three integrals as follows, where c = (1 + δ)3:

|Tm,globf(θ)| ≤∫ π

0

|M(θ, φ)|(1 − ϕ(θ, φ))|f(φ)| dmλ(φ)

=∫{θ>cφ}

+∫{π

2 >φ>cθ}+∫

[π/2,π]

= I + II + III.

(4.6)

For the last integral, observe that there exists an ε > 0 such that in the intersection [0, π/2] × [π/2, π] with theglobal region, |θ − φ| > ε. Therefore,Dr ≥ 1 − (cos(θ − φ))2 ≥ Cε. Thus |M(θ, φ)| ≤ ∫ 1

0Cε

(rλ−1 + 1

)dr,

as can be easily seen from (2.7), and we get

III ≤ Cε

∫[π/2,π]

|f(φ)| dmλ(φ) ≤ Cδ‖f‖L1(dmλ),

that is, this part is bounded in all Lp(dmλ), 1 ≤ p ≤ ∞. For the remaining terms, we will use that from (4.4)one can achieve the following inequality

|M(θ, φ)| ≤∫ 1/2

0

C(rλ−1 + 1) dr + C

∫ 1

1/2

I2λ−1λ+1 dr

≤ C + C

∫ 1

1/2

dr

((1 − r)2 + (1 − cos(θ − φ)))λ+1

≤ C

(1 − cos(θ − φ))λ+1/2,

(4.7)

where the last estimate can easily be obtained by the change of variable u = 1−r√1−cos(θ−φ)

. Let us now handle the

integral I. If θ > cφ and θ, φ ∈ [0, π/2], then sin(θ − φ) ∼ sin θ and 1 − cos(θ − φ) ∼ |θ − φ|2 ∼ sin(θ − φ)2.Thus, by these properties, (4.7) implies

|M(θ, φ)| ≤ C +C

(sin θ)2λ+1≤ C

(sin θ)2λ+1for θ > c φ.

In this inequality, the constants appearing are of the form C = C(λ)‖κ‖L∞ . We will state the boundedness ofthis operator in the following lemma.


Math. Nachr. 281, No. 7 (2008) 987

Lemma 4.2 The operator

M1f(θ) =∫ π

0

M1(θ, φ)f(φ) dmλ(φ),

where

M1(θ, φ) =1

(sin θ)2λ+11{θ>cφ}∩[0,π/2]2(θ, φ)

for c > 1, is of weak type (1, 1) and strong type (p, p) for 1 < p <∞ with respect to the measure dmλ.

P r o o f. First we prove the weak type. Since the operator is linear, we can restrict ourselves to prove the weaktype (1, 1) for positive functions with ‖f‖L1(dmλ) = 1. In this case, |M1f(θ)| ≤ 1

(sin θ)2λ+1 . Take any k ≤ 2.Then,

mλ

({θ : M1f(θ) > k

}) ≤ mλ([0, π]) ≤ C

k.

For k > 2, observe that 1(sin θ)2λ+1 > k if and only if θ < arcsin

((1/k)1/(2λ+1)

), since in that region sin θ has

a well defined increasing inverse. Also, in this region (θ ≤ π/2 − ε), there exists C such that 0 < C < cos θ.Therefore,

mλ

({θ : M1f(θ) > k

})= mλ

({θ : θ < arcsin

((1/k)1/(2λ+1)

)})≤ 1C

∫ arcsin((1/k)1/(2λ+1)

)0

cos θ(sin θ)2λ dθ

=1Ck

.

Now we have to prove that M1 is bounded on L∞(dmλ), and then the result will follow from Marcinkiewiczinterpolation theorem. This boundedness holds, since

sup0≤θ≤π/2

∫ π2

0

(sinφ)2λ

(sin θ)2λ+11{θ>cφ}(θ, φ) dφ <∞.

To give a bound for II, observe that in the region where φ > c θ and θ, φ ∈ [0, π/2], we have that

1 − cos(θ − φ) ∼ |θ − φ|2 ∼ (sin(θ − φ))2

and also | sin(θ − φ)| ∼ sinφ. Since here (4.7) still holds, we obtain

|M(θ, φ)| ≤ C

(sinφ)2λ+1for φ > c θ.

Lemma 4.3 The operator

M2f(θ) =∫ π

0

M2(θ, φ)f(φ) dmλ(φ),

where

M2(θ, φ) =1

(sinφ)2λ+11{φ>cθ}∩[0,π/2]2(θ, φ)

for c > 1 is of weak type (1, 1) and strong type (p, p) for 1 < p <∞ with respect to the measure dmλ.

P r o o f. To prove weak type (1,1) we argue as in the previous lemma, since in this region sinφ ≥ C sin θ.The second part holds, because M2 is the adjoint of M1 in L2(dmλ).



5 Proof of Theorem 1.1

Let us define the global and local parts of the multiplier Tm according to (3.1) and (3.2). The boundedness of theglobal part was obtained in Subsection 4.2.

Consider the local part Tm,loc. Since Tm is an operator bounded in L2(dmλ), by Theorem 3.4 its local partis also bounded in L2(dmλ). Observe that the first estimate in Lemma 4.1 implies assumption (c) for Tm, sincein the local part sin θ ∼ sinφ. By Lemma 3.5, Tm,loc is bounded in L2(dφ). Since ϕ is a bounded function, theestimates of Lemma 4.1 estate that Tm,loc is a (local) Calderon–Zygmund operator. By classical results, it followsthat this operator is bounded in Lp(dφ) for p ∈ (1,∞) and from L1(dφ) into L1,∞(dφ). A new application ofLemma 3.5 gives the boundedness of Tm,loc in Lp(dmλ) for p ∈ (1,∞) and the weak type (1, 1) with respect tothe measure mλ.

Acknowledgements The author was partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP and by BFM grant 2002-04013-C02-02.

The author would like to thank the referees for their valuable suggestions, that contributed to improve the manuscript.

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Multipliers of Laplace transform type for ultraspherical expansions