Almost Everywhere Convergence of Prolate Spheroidal Series

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Almost Everywhere Convergence of Prolate Spheroidal Series

Philippe Jaming, Michael Speckbacher

To cite this version:

Philippe Jaming, Michael Speckbacher. Almost Everywhere Convergence of Prolate Spheroidal Series.

2020. �hal-02436167�

Almost Everywhere Convergence of Prolate Spheroidal Series

Philippe Jaming^∗, Michael Speckbacher^1,∗

Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux 351, cours de la Lib´eration, 33405 TALENCE

France

Abstract

In this paper, we show that the expansions of functions from L^p-Paley- Wiener type spaces in terms of the prolate spheroidal wave functions con- verge almost everywhere for 1< p <∞, even in the cases when they might not converge in L^p-norm. We thereby consider the classical Paley-Wiener spaces P Wc^p ⊂ L^p(R) of functions whose Fourier transform is supported in [−c, c] and Paley-Wiener like spacesB^pα,c ⊂L^p(0,∞) of functions whose Hankel transformH^α is supported in [0, c]. As a side product, we show the continuity of the projection operatorP_c^αf :=H^α(χ_[0,c]· H^αf) fromL^p(0,∞) to L^q(0,∞), 1< p≤q <∞.

Keywords: Prolate spheroidal wave functions, almost everywhere

convergence, Paley-Wiener type spaces, Hankel transform, spherical Bessel functions

2010 MSC:42B10, 42C10, 44A15

1. Introduction

The prolate spheroidal wave functions (PSWF) form an orthonormal basis of L²(R) that is best concentrated in the time-frequency plane. They were first studied in the seminal work of Landau, Pollak, and Slepian [12, 13, 22, 21] at Bell Labs which inspired extensive research due to their efficiency

∗Corresponding author

Email addresses: philippe.jaming@u-bordeaux.fr (Philippe Jaming), speckbacher@kfs.oeaw.ac.at(Michael Speckbacher)

1M.S. was supported by the Austrian Science Fund (FWF) through an Erwin- Schr¨odinger Fellowship (J-4254).

as a tool for signal processing. They have since found further applications in various fields such as random matrix theory (e.g. [9, 14]), spectral estimation [23, 1] or numerical analysis (e.g. [26, 25]).

While the natural setting for prolate spheroidal wave functions isL²(R), the question of convergence of their series expansions arises inL^p(R). This issue has been solved by Barcel´o and Cordoba [2] who showed that the expansion of anL^p-band limited functionf in the PSWF-basis converges to f whenever 4/3 < p <4, and that this range of p’s is optimal. That result was recently extended in [3] to several natural variants of the prolates like the Hankel prolates.

The aim of this paper is to continue this work by investigating almost- everywhere convergence properties of PSWFs expansions of functions in L^p(R). This question is very natural in view of Carleson’s fundamental result about Fourier series [6] which was extended to several orthonormal bases of polynomials,e.g. by Pollard for Legendre series [17]. Their almost everywhere convergence results follow from very delicate analysis of the max- imal operator associated to the expansions considered. On the other hand, the situation is much better for expansions in terms of spherical Bessel func- tions (see[7, 8]) where almost-everywhere convergence is surprisingly simple once mean convergence has been established. The key here is the fast decay of Bessel functions with respect to its parameter. Our main result is to show that the situation is similar for PSWF series since the PSWF-basis can be nicely expressed in terms of spherical Bessel functions.

Let us now be more precise and introduce some notation before giving the exact statements. Forf ∈L¹(R), the Fourier transform is given by

fb(ξ) =F(f)(ξ) = Z

R

f(t)e^−2πiξtdt,

and the definition extends toL²(R) andS^′(R) (the space of tempered dis- tributions) in the usual way. The Paley-Wiener spaces of band-limited func- tions are denoted by

P W_c^p =n

f ∈L^p(R) : supp fb

⊆[−c, c]o ,

where the Fourier transform is to be understood in the distributional sense whenever p > 2. The projection Pcf =F⁻¹ χ_[−c,c]· F(f)

is a continuous operator fromL^p(R) to P Wc^p, 1< p <∞.

The Prolate Spheroidal Wave Functions (PSWFs) are eigenfunctions of an integral operator and, using the min-max theorem, can be defined by the

following extremal problem ψ_n,c:= argmax

(kfkL²(−1,1)

kfkL²(R)

:f ∈P W_c², f ∈span{ψ_k,c : k < n}^⊥ )

. The family (ψ_n,c)_n≥0 forms an orthonormal basis forP W_c² and satisfies also

Z ₁

−1

ψ_n,c(t)ψ_m,c(t)dt=λ_nδ_n,m, which is often referred to as double orthogonality.

The central object that we study in this paper is given by Ψ_Nf :=

XN n=0

hf, ψ_n,ciψ_n,c.

It was shown in [2] that Ψ_Nf → f, N → ∞, in L^p(R)-norm for every f ∈P Wc^p if and only if ⁴₃ < p <4. Our main contribution in this paper is that the series Ψ_Nf converges also almost everywhere, and that the range of convergence extends to 1 ≤ p < ∞. Moreover, for general functions f ∈L^p(R), Ψ_Nf converges almost everywhere to P_cf.

Theorem 1.1. If 1 < p < ∞, and f ∈ L^p(R), then Ψ_Nf → P_cf almost everywhere. Forp= 1, we have that Ψ_Nf →f almost everywhere for every f ∈P W_c¹.

Forf ∈L¹(0,∞), the Hankel transform is given by H^αf(x) :=

Z ∞ 0

f(y)√xyJα(xy) dy,

whereJα denotes the Bessel function of first kind and orderα > −¹2. Like the Fourier transform, the Hankel transform extends to a unitary operator onL²(0,∞) and in a distributional sense to L^p(0,∞),p >2. Similar to the Fourier transform, a function cannot be confined to a finite interval in both time and Hankel domain. See [4, 10, 18], for further uncertainty principles for the Hankel transform.

The Paley-Wiener type spacesB_α,c^p are defined as B^p_α,c:=

f ∈L^p(0,∞) : supp H^α(f)

⊆[0, c] ,

and the band-limiting projection operator is given byP_c^αf =H^α(χ_[0,c]·H^αf).

We will show in Theorem 4.1 that this operator is bounded on L^p(0,∞),

for 1 < p < ∞. Finally, the Circular (Hankel) Prolate Spheroidal Wave Functions (CPSWFs) were first introduced and studied in [19]. They are defined by

ϕ^α_n,c:= argmax

(kfkL²(0,1)

kfkL²(0,∞)

:f ∈B_α,c² , f ∈span{ϕ^α_k,c: k < n}^⊥ )

. The family (ϕ^α_n,c)n≥0 forms an orthonormal basis of B²_α,c. Note also that whenα = 1/2, these are usual PSFWs, more precisely, ϕ^1/2_n,c =ψ_2n,c. Again we are interested in expansions of f ∈ B^p_α,c with respect to the CPSWFs, i.e. in the convergence of

Φ_Nf :=

XN n=0

hf, ϕ^α_n,ciϕ^α_n,c.

It is shown in [3, Theorem 5.6] that Φ_Nf → f in L^p(0,∞)-norm for every f ∈Bα,c^p if and only if ³₄ < p <4. Like for the classical Paley-Wiener space, the series converges also almost everywhere and the range of convergence extends to 1≤p <∞ for functions in Bα,c^p .

Theorem 1.2. Let α > −¹₂. If 1< p <∞, then Φ_Nf → P_c^αf almost ev- erywhere for everyf ∈L^p(0,∞). Moreover, if p= 1, thenΦ_Nf →f almost everywhere for every f ∈Bα,c^p .

This paper is organized as follows. In Section 2, we collect some prop- erties on spherical Bessel functions and the Muckenhoupt class that we will need for the proof of our main results. We then prove Theorem 1.1 in Sec- tion 3, and conclude with illustrating how this proof has to be adapted to show Theorem 1.2 in Section 4.

As usual, we will writeA.B if there is a constantCsuch thatA≤CB andA∼B if A.B and B .A.

2. Preliminaries

2.1. Properties of spherical Bessel functions

In this section we define two families of spherical Bessel functions tailored to form orthonormal bases for the spacesP W_c² and B_α,c² respectively.

For α > −1/2, the Bessel functions of the first kind can be defined through the Poisson representation,see e.g. [15, 10.9.4],

J_α(x) = x^α π^1/22^αΓ(α+¹₂)

Z ₁

0

(1−t²)^α−12 cos(xt) dt.

Recall thatJ_α satisfies the pointwise bound

|J_α(x)|. |x|^α+1/2

2^αΓ(α+ ¹₂). (2.1)

It is shown in [2] and [3] that for 1< p <∞

x^−1/2J_n+1/2

L^p(R)∼









n^−1+1p when 1< p <4 n⁻³⁴ logn when p= 4 n^−56+3p1 when p >4

. (2.2)

Note that the nature of the right hand side shows thatx^−1/2J_2n+α+1

L^p(0,∞)

satisfies the same bounds though with different constants.

The classical (dilated) spherical Bessel functions are defined by jn,c(x) :=

r2n+ 1 2

J_n+1/2(cx)

√x . (2.3)

They satisfy the orthogonality relations Z

R

j_n,c(x)j_m,c(x) dx=δ_n,m, and their Fourier transforms are given by

dj_n,c(ξ) = (−1)ⁿ

r2n+ 1 πc P_n

ξ c

·χ_[−c,c](ξ),

whereP_n denotes the Legendre polynomial of degreen. Forp >1, one has that j_n,c ∈ L^p(R) and consequently j_n,c ∈ P Wc^p. Moreover, by (2.2) there exists γ_p < ¹₂ such that kj_n,ckL^p(R) . n^γp, for every 1 < p < ∞, and (2.1) implies that, forx fixed,

|j_n,c(x)|.

√2n+ 1 n!

c|x| 2

n

.n⁻². (2.4)

The second family of spherical Bessel functions given by k_n,c^α (x) :=p

2(2n+α+ 1)J_2n+α+1(cx)

√x , (2.5)

obeys the orthogonality relation Z _∞

0

k_n,c^α (x)k^α_m,c(x) dx=δ_n,m.

Their Hankel transforms are given by H^α(k_n,c^α )(x) =

r2(2n+α+ 1) c

x c

α+¹₂

P_n^(α,0)

1−2x c

2

χ_[0,c](x), (2.6) whereP_n^(α,0) denotes the Jacobi polynomials of degree n and parameter α, normalized so thatP_n^(α,0)(1) = Γ(n+1)Γ(α+1)^Γ(n+α+1) , see for example [20]. As before, for 1< p <∞, theL^p(0,∞) norms ofk_n,c^α are bounded likek_n,c^α

L^p(0,∞). n^γp for some 0≤γ_p <1. Hence,k^α_n,c∈B_α,c^p . By (2.1) we also get that

|k^α_n,c(x)|.

√2n+α+ 1 Γ(2n+α+ 1)

c|x| 2

2n

.n⁻², (2.7)

where the last estimate holds forxfixed with a constant that depends onx (which can be chosen uniformly ifx is in a compact set).

2.2. The Hilbert Transform on Weighted L^p-spaces

Let J ⊂ R be an interval. The class of Muckenhoupt weights A^p(J), 1< p <∞, consists of all functionsω :J →R₊ such that

[ω]_p:= sup

K

1

|K| Z

K

ω(x) dx 1

|K| Z

K

ω(x)^−qpdx ^p_q

<∞, where the supremum is taken over all finite length intervalsK ⊂J.

The Hilbert transform is defined as Hf(x) = 1

π Z

J

f(y) x−ydy,

where the integral has to be taken in the principal value sense.

Hunt, Muckenhaupt and Wheeden [11] showed that the Hilbert trans- form extends to a bounded linear operator on L^p(J, ω) → L^p(J, ω) if and only ifω is anA^p(J) weight, and the sharp dependence of the operator norm on [ω]_pwas established by Petermichl [16]. Let us also recall the well known fact that x^β ∈A^p(0,∞) if and only if−1< β < p−1.

3. Proof of Theorem 1.1

The prolate spheroidal wave functions (ψ_n,c)_n≥0 can be expressed in terms of the spherical Bessel functions (j_n,c)_n≥0 as

ψ_n,c=X

k≥0

bⁿ_k j_k,c.

It follows from [2, Eqs. (8) & (9)] that there exists n₀∈Nsuch that for all n≥n₀, the numbersbⁿ_k satisfy

(i) |bⁿ₀|.n⁻²,

(ii) fork≥1,|bⁿ_k|.n^−|k−n|.

Note that indexes of the base change coefficients in [2] are shifted, and that these estimates can also be obtained from [3]. It was proven in [3, Lemma 2.6] that if the above conditions are satisfied, thenkψn,ckL^p(R).n^γp. Lemma 3.1. If 1 < p <∞, then Ψ_Nf converges absolutely in every point (and uniformly on compact sets) to some functionh for every f ∈L^p(R).

Proof. Let us first rewrite ψ_n,c as ψ_n,c=X

k≥0

bⁿ_kj_k,c

=bⁿ₀j_0,c+bⁿ_nj_n,c+bⁿ_n−1j_n−1,c+bⁿ_n+1j_n+1,c+

n−2X

k=1

bⁿ_kj_k,c+ X

k≥n+2

bⁿ_kj_k,c. Subsequently, we estimate each of the summands separately. First, by (i) we have |bⁿ₀j_0,c(x)|.n⁻², and |bⁿ_nj_n,c(x)| ≤ |j_n,c(x)|.n⁻² by (2.4). If n≥2, the last two terms may be bounded using (ii) and (2.4) as

X

k≥n+2

|bⁿ_kj_k,c(x)|. X

k≥n+2

n^−|k−n|k⁻²≤n⁻²X

k≥0

n^−(k+2) .n⁻², and

n−2X

k=1

|bⁿ_kj_k,c(x)|.

n−2X

k=1

n^−|k−n|k⁻² ≤

n−1X

k=2

n^−k=n⁻²

n−3X

k=0

n^−k.n⁻². The third and fourth term may also easily be bounded using (ii):

|bⁿ_n−1j_n−1,c(x)|.n⁻¹(n−1)⁻².n⁻², and

|bⁿ_n+1j_n+1,c(x)|.n⁻¹(n+ 1)⁻².n⁻².

Therefore,|ψ_n,c(x)|.n⁻² which implies by H¨older’s inequality that XN

n=0

|hf, ψ_n,ciψ_n,c(x)| ≤ XN n=0

kfkL^p(R)kψ_n,ckL^q(R)|ψ_n(x)|

.kfkL^p(R)

XN n=0

n^γq−2 <∞.

✷

The following lemma is well-known to the majority of our readers. We will nevertheless include a proof here.

Lemma 3.2. If 1 ≤ p ≤ q < ∞, then P W_c^p ֒→ P W_c^q where the inclusion map is continuous and dense. Moreover, the projection P_c maps L^p(R) boundedly into P W_c^p if 1< p <∞.

Proof. Let ϑ₁, ϑ₂ be two functions from the Schwartz space such that supp(ϑb₁)⊆[−c, c], ϑb₂(ξ) = 1, for everyξ ∈[−c, c], and kϑ₁−ϑ₂kL¹(R)≤ε.

If f ∈ P Wc^p, then f ∗ϑ₂ =f and, by Young’s convolution inequality, one has

kfkL^q(R)=kf∗ϑ2kL^q(R) ≤ kfkL^p(R)kϑ2k_Lpq+p^pq−q(R). This shows that P Wc^p is continuously embedded into P Wc^q.

Let us now show the density. For every ε > 0, and f ∈ P W_c^q we may choose f_ε∈L^p(R)∩L^q(R), such thatkf−f_εkL^q(R)=ε. Again by Young’s convolution inequality we thus obtain

kf_ε∗ϑ₁−fkL^q(R) ≤ kf_ε∗ϑ₁−f∗ϑ₁kL^q(R)+kf ∗ϑ₁−f ∗ϑ₂kL^q(R)

≤ kf_ε−fkL^q(R)kϑ₁kL¹(R)+kfkL^q(R)kϑ₁−ϑ₂kL¹(R).ε.

Finally,f_ε∗ϑ₁ ∈P Wc^pas supp F(f_ε∗ϑ₁)

⊆supp(ϑb₁), andkf_ε∗ϑ₁kL^p(R)≤ kf_εkL^p(R)kϑ₁kL¹(R).

Concerning the continuity ofP_c, we observe that for almost every point x∈R,P_cf(x) may be written as the sum of Hilbert transforms

P_cf(x) =f ∗2csinc(2c·)(x) = Z

R

f(t)sin(2πc(x−t)) π(x−t) dt

= sin(2πcx) p.v.

Z

R

f(t)cos(2πct) π(x−t) dt

−cos(2πcx) p.v.

Z

R

f(t)sin(2πct) π(x−t)dt

= sin(2πcx)H f ·cos(2πc·)

(x)−cos(2πcx)H f·sin(2πc·) (x).

By the continuity of the Hilbert transform onL^p(R) in the range 1< p <∞, it thus follows that

kP_cfkL^p(R)≤ kH f·cos(2πc·)

kL^p(R)+kH f·sin(2πc·)

kL^p(R).kfkL^p(R).

✷

Proposition 3.3. If ⁴₃ < p < 4, then Ψ_Nf → P_cf almost everywhere for every f ∈L^p(R).

Proof. First, as P_c is continuous for ⁴₃ < p <4, it follows that hf, j_n,ci = hf, Pcjn,ci =hPcf, jn,ci and therefore ΨNf = ΨNPcf. Since ΨNPcf →Pcf in L^p(R) by [2, Theorem 1], there exists a subsequence (Ψ_NkP_cf)_k≥0 that converges almost everywhere toP_cf. From Lemma 3.1 we know that Ψ_Nf converges pointwise for everyx∈Rto a function h. Consequently,Pcf =h

almost everywhere. ✷

We have now gathered all ingredients to prove our main theorem.

Proof of Theorem 1.1. If 1 < p ≤ 2, then P_cf ∈ P W_c² for every f ∈ L^p(R) by Lemma 3.2. In this case, the result follows from Proposition 3.3 and the identity Ψ_Nf = Ψ_NP_cf. Similarly, the statement forp= 1 holds as P W_c¹ ⊂P W_c². If 2 < p < ∞, and f ∈L^p(R), then there exists a sequence (f_k)_k≥0 ⊂P W_c² by Lemma 3.2 such thatf_k→Pcf ∈P Wc^p andkf_kkL^p(R). kfkL^p(R). Without loss of generality, we may assume thatf_k →P_cf almost everywhere and defineX_f :={x∈R: lim_k→∞f_k(x) =P_cf(x)}, as well as

X_k :=n

x∈R: lim

N→∞Ψ_Nf_k(x) =f_k(x)o .

From Proposition 3.3 we know that eachR\X_k has Lebesgue measure zero and, consequently, so hasR\ T

k≥0X_k∩X_f

=S

k≥0(R\X_k)∪(R\X_f).

Using (2.4) and the estimate on the L^p-norms of j_n,c we derive X

n≥0

|hf_k, ψ_niψ_n(x)|.kfkL^p(R)

X

n≥0

n^γq−2<∞. Now, ifx∈T

k≥0X_k∩X_f, we may conclude by the dominated convergence theorem that

P_cf(x) = lim

k→∞f_k(x) = lim

k→∞ lim

N→∞Ψ_Nf_k(x) = lim

N→∞ lim

k→∞Ψ_Nf_k(x)

= lim

N→∞Ψ_NP_cf(x) = lim

N→∞Ψ_Nf(x).

✷

4. Adaptations for Circular PSWF

By a combination of several lemmas in [3] (in particular Lemmas 2.5, 5.3 &

5.4) one can deduce that the change of base coefficients bⁿ_m in ϕ^α_n,c= X

m≥0

bⁿ_m k^α_m,c

satisfy the conditions (i) and (ii). Hence, kϕ^α_n,ckL^p(0,∞) . n^γp, for some 0≤γ_p <1. The proof of Lemma 3.1 can therefore be transferred one-to-one to show that ΦNf converges pointwise to a functionhfor everyf ∈L^p(0,∞), 1< p <∞.

The analogue of Lemma 3.2 needs some preparation. In particular, we need to show the continuity of the projection operatorP_c^α. In [24] the author shows that for the following convention of the Hankel transform

F^αf(x) := x^−α/2 2

Z _∞

0

f(t)J_α √ xt

t^α/2dt,

the projectionF^α(χ_[0,c]·F^α)(x) acts as a continuous operator on the weighted Lebesgue spacesL^p((0,∞), x^α) whenever 4_2α+3^α+1 < p <4_2α+1^α+1. Although dif- ferent definitions of the Hankel transform can be related to each other and theirL²-theory coincides, this is no longer true for generalL^p-spaces,p6= 2.

For instance, when considering theL^p (0,∞), x^α

-convergence of spher- ical Bessel expansions, Varona [24] showed that these series converge inL^p if and only if maxn

4

3,4_2α+3^α+1o

< p <minn

4,4_2α+1^α+1o

. The restriction with respect toα however disappears in [3] due to a different convention for the Hankel transform andL^p(0,∞)-convergence holds if and only if ⁴₃ < p <4.

As a second example, we show in the following that for the convention chosen in this paper, the projection P_c^α is continuous for 1 < p <∞. We will follow the arguments of the proof of [24, Theorem 2].

Theorem 4.1. If α >−¹₂, and 1< p≤q <∞, then

kP_c^αfkL^q(0,∞).kfkL^p(0,∞), f ∈L^p(0,∞).

Proof. Let us first write downP_c^α explicitly P_c^αf(x) =H^α χ_[0,c]· H^αf

(x) = Z _c

0 H^αf(t)√

txJ_α(tx) dt

= Z c

0

Z ∞ 0

f(y)√

tyJ_α(ty)dy√

txJ_α(tx) dt

= Z ∞

0

f(y)√xy Z c

0

J_α(tx)J_α(ty)tdtdy

= Z _∞

0

f(y)√xycyJ_α−1(cy)J_α(cx)−xJ_α−1(cx)J_α(cy)

x²−y² dy,

where we have used the explicit expression for Lommel’s integrals [5, p. 101]

to derive the last equality. By a change of variables one then obtains P_c^αf(√

x)

= Z _∞

0

f(√y)(xy)^1/4c

√yJ_α−1(c√y)J_α(c√ x)−√

xJ_α−1(c√

x)J_α(c√y) x−y

dy 2y^1/2

=W₁f(x)−W₂f(x), where

W1f(x) := πcx^1/4J_α(c√x)

2 H

f(√y)y^1/4Jα−1(c√y) (x), and

W₂f(x) := πcx^3/4J_α−1(c√ x)

2 H

f(√y)y^−1/4J_α(c√y) (x).

Note that both x^−1/2, and x^p/2−1/2 are Muckenhoupt A^p(0,∞) weights by the remark in Section 2.2. Hence, the Hilbert transform is continu- ous on the respective weighted Lebesgue spaces for every 1 < p < ∞. As x^1/4J_α(c√

x).1, we thus have Z ∞

0 |W1f(x)|^px^−1/2dx. Z ∞

0

H

f(√y)y^1/4Jα−1(c√y) (x)

^px^−1/2dx .

Z _∞

0

f(√y)y^1/4J_α−1(c√y)^py^−1/2dy .

Z ∞

0 |f(√y)|^py^−1/2dy= 2 Z ∞

0 |f(y)|^p dy, as well as

Z _∞

0 |W₂f(x)|^px^−1/2dx. Z _∞

0

H

f(√y)y^−1/4J_α(c√y)

(x)^px^p/2−1/2dx .

Z ∞ 0

f(√y)y^−1/4Jα(c√y)

^py^p/2−1/2dy .

Z _∞

0 |f(√y)|^py^−1/2dy= 2 Z _∞

0 |f(y)|^p dy.

This proves the case p = q using kP_c^αfkL^p(0,∞) ≤ kW₁f(x²)kL^p(0,∞) + kW2f(x²)kL^p(0,∞) and a change of variables.

Now, notice that for every 1≤p≤ ∞one has

kP_c^αfkL^∞(0,∞).kχ_[0,c]H^αfkL¹(0,∞).kH^αfkL^p(0,∞).

Setting p = 2 gives kP_c^αfkL^∞(0,∞) . kfkL²(0,∞) by Plancherel’s theorem, andp=∞ yields kP_c^αfkL^∞(0,∞) .kfkL¹(0,∞). It thus follows by the Riesz- Thorin theorem that kP_c^αfkL^∞(0,∞) . kfkL^p(0,∞), for every 1 ≤ p ≤ 2.

Moreover, as kP_c^αfkL^p(0,∞) . kfkL^p(0,∞), 1 < p < ∞, it follows again by the Riesz-Thorin theorem that

kP_c^αfkL^q(0,∞).kfkL^p(0,∞), 1< p≤2, p≤q <∞. (4.8) Since the adjoint ofP_c^α:L^p(0,∞)→L^q(0,∞) is given byP_c^α:L^q′(0,∞)→ L^p′(0,∞), we conclude that also

kP_c^αfk_L^p′_(0,∞).kfk_L^q′_(0,∞), 1< q^′ ≤p^′, 2≤p^′ <∞. (4.9) Combining (4.8) and (4.9) thus yields the continuity of P_c^α : L^p(0,∞) → L^q(0,∞) for the whole scale 1< p≤q <∞. ✷ Corollary 4.2. If 1 < p ≤ q < ∞, then B_α,c^p ֒→ B_α,c^q with the inclusion map being continuous and dense. Moreover, if1≤p≤ ∞, thenB¹_α,c⊂Bα,c^p . Proof. If f ∈ B_α,c^p one has P_c^αf = f and consequently kfkL^q(0,∞) . kfkL^p(0,∞) by Theorem 4.1, which shows that Bα,c^p is continuously embed- ded into Bα,c^q . Furthermore, if f ∈B_α,c¹ , then f ∈ Bα,c^p asf ∈ L¹(0,∞)∩ L^∞(0,∞)⊂L^p(0,∞).

It remains to show that the inclusion is dense. This however follows from the density properties of L^p(0,∞) and Theorem 4.1: Take f ∈ Bα,c^q

and a sequence (f_k)_k≥0⊂L^p(0,∞)∩L^q(0,∞) such thatf_k→f inL^q(0,∞).

By the continuity of the projection operator, it follows that (P_c^αf_k)_k≥0 ⊂ Bα,c^p ∩Bα,c^q and

kf −P_c^αf_kkL^q(0,∞)=kP_c^α(f −f_k)kL^q(0,∞).kf−f_kkL^q(0,∞)→0.

✷ Thus, as in Proposition 3.3, it follows from [3, Theorem 5.6] that Φ_Nf converges almost everywhere toP_c^αf for everyf ∈L^p(0,∞) if ³₄ < p <4.

This leaves us with all ingredients in place to prove Theorem 1.2 which is shown along the same line of argument as in the proof of Theorem 1.1.

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