MAXIMAL FUNCTIONS

MAXIMAL FUNCTIONS

AHMAD AL-SALMAN

Received 16 January 2006; Revised 12 April 2006; Accepted 13 April 2006

We establishL^pestimates for certain class of maximal functions with kernels inL^q(Sⁿ⁻¹).

As a consequence of suchL^p estimates, we obtain theL^pboundedness of our maximal functions when their kernels are inL(logL)^1/2(Sⁿ⁻¹) or in the block spaceB^0,_q^−1/2(Sⁿ⁻¹), q >1. Several applications of our results are also presented.

Copyright © 2006 Ahmad Al-Salman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and statement of results

LetRⁿ,n≥2, be then-dimensional Euclidean space and let Sⁿ⁻¹ be the unit sphere in Rⁿequipped with the normalized Lebesgue measuredσ. For nonzeroy∈Rⁿ, we will let y= |y|⁻¹y. LetΩbe an integrable function on Sⁿ⁻¹that is homogeneous of degree zero onRⁿand satisfies the cancelation property

Sⁿ⁻¹Ω(y)dσ(y)=0. (1.1)

Consider the maximal functionᏹΩ, ᏹΩ(f)(x)=sup

h∈U

Rⁿ f(x−y)|y|⁻ⁿh|y|

Ω(y)d y, (1.2) whereUis the class of allh∈L²(R+,r⁻¹dr) withhL²(_R+,r⁻¹dr)≤1.

The operator ᏹΩ was introduced by Chen and Lin [7]. They showed that ᏹΩ is bounded on L^p(Rⁿ) for all p >2n/(2n−1) provided that Ω∈Ꮿ(Sⁿ⁻¹). Recently, we have been able to show that theL^p(Rⁿ) boundedness ofᏹΩ still holds for all p≥2 if the conditionΩ∈Ꮿ(Sⁿ⁻¹) is replaced by the more natural and weaker conditionΩ∈ L(logL)^1/2(Sⁿ⁻¹) [2]. Moreover, we showed that if the conditionΩ∈L(logL)^1/2(Sⁿ⁻¹) is replaced by any condition in the form Ω∈L(logL)^r(Sⁿ⁻¹) for some r <1/2, thenᏹΩ

might fail to be bounded onL².

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 56272, Pages1–17 DOI10.1155/JIA/2006/56272

On the other hand, whenΩlies inB_s^0,−1/2(Sⁿ⁻¹),s >1, which is a special class of block spaces B_q^κ,υ(Sⁿ⁻¹) (seeSection 5 for the definition), we were able to show thatᏹΩ is bounded on L^p for all p≥2 [3]. Moreover, we showed that the condition Ω∈ B^0,_s^−1/2(Sⁿ⁻¹),s >1 is nearly optimal in the sense that the exponent−1/2 cannot be re- placed by any smaller number for theL² boundedness ofᏹΩto hold. We remark here that block spaces have been introduced by Jiang and Lu to improve previously obtained L^pboundedness results for singular integrals [7]. It should be noted here that the relation between the spacesB^0,_s^−1/2(Sⁿ⁻¹) andL(logL)^1/2(Sⁿ⁻¹) is unknown.

However, it is known that L^q(Sⁿ⁻¹) is properly contained in L(logL)^1/2(Sⁿ⁻¹)∩ B^0,_s^−1/2(Sⁿ⁻¹) for all q, s > 1. Moreover, it is not hard to see that every Ω in L(logL)^1/2(Sⁿ⁻¹)∪B^0,_s^−1/2(Sⁿ⁻¹) can be written as an infinite sum of functions inL^q(Sⁿ⁻¹).

This gives rise to the question whether the results pertaining theL^pboundedness ofᏹΩin [2,3] can be obtained via certain correspondingL^pestimates with kernels inL^q(Sⁿ⁻¹). It is one of our main goals in this paper to consider such problem. It should be pointed out here that a positive solution for this problem will not only make life easier when dealing with kernels inL(logL)^1/2(Sⁿ⁻¹) orB^0,_s^−1/2(Sⁿ⁻¹), but also will pave the way for extending several results that are known when kernels are inL^q(Sⁿ⁻¹).

Our work in this paper will be mainly concerned with the following general class of maximal functions:

ᏹΩ,P(f)(x)=sup

h∈U

Rⁿe^iP(y)f(x−y)|y|⁻ⁿh|y|

Ω(y)d y, (1.3) whereP:Rⁿ→Ris a real-valued polynomial.

Clearly, ifP(y)=0, thenᏹΩ,P=ᏹΩ. For the significance of considering integral op- erators with oscillating kernels, we refer the readers to consult [1,4,11,16,19,22–24], among others.

Our result concerningL^pestimates with kernels inL^q(Sⁿ⁻¹) is the following theorem.

Theorem 1.1. LetΩ∈L^q(Sⁿ⁻¹),q >1, be a homogeneous function of degree zero onRⁿ withΩ1≤1. LetP:Rⁿ→Rbe a real-valued polynomial of degreed. LetᏹΩ,Pbe given by (1.3). Then

ᏹΩ,P(f)_p≤

1 + log^1/2e+Ωq

Cp,qfp (1.4)

for allp≥2, whereC_p,q=(2^1/q/(2^1/q−1))C_p. Here 1/q=1−1/qandC_pis a constant that may depend on the degree of the polynomialPbut it is independent of the functionΩ, the indexq, and the coefficients of the polynomialP.

We remark here that the constantCp,q inTheorem 1.1satisfiesCp,q→ ∞asq→1⁺. That is, the constantC_p,q diverges when q tends to 1. This behavior ofC_p,q is natural since, by [2, Theorem B(b)], the special operatorᏹΩ=ᏹΩ,0is not bounded onL²if the functionΩis assumed to satisfy only the sole condition thatΩ∈L¹(Sⁿ⁻¹) (i.e.,q=1).

By a suitable decomposition of the functionΩand an application ofTheorem 1.1, we prove the following theorem which is a proper extension of the corresponding result in [2].

Theorem 1.2. Suppose thatΩ∈L(log⁺L)^1/2(Sⁿ⁻¹) satisfying (1.1). LetP:Rⁿ→Rbe a real-valued polynomial. Then ᏹΩ,P is bounded onL^p(Rⁿ) for all p≥2 withL^p bounds independent of the coefficients of the polynomialP.

We should point out here that an alternative proof ofTheorem 1.2can be obtained by observing thatCp,q≈Cp/(q−1), whereCp,qis the constant inTheorem 1.1, and then using a Yano-type extrapolation technique [27].

By another suitable application ofTheorem 1.1, we will prove the following extension of [3, Theorem 1.2].

Theorem 1.3. Suppose thatΩ∈B^0,_q^−1/2(Sⁿ⁻¹),q >1, satisfying (1.1). LetP:Rⁿ→Rbe a real-valued polynomial. ThenᏹΩ,P is bounded onL^p(Rⁿ) for all p≥2 withL^pbounds independent of the coefficients of the polynomialP.

As an immediate consequence ofTheorem 1.1and the observation that

T_Ω,P,h(f)(x)≤ hL²(_R+,r⁻¹dr)ᏹΩ,P(f)(x), (1.5) we obtain the following result concerning oscillatory singular integrals.

Theorem 1.4. LetΩ∈L^q(Sⁿ⁻¹),q >1, be a homogeneous function of degree zero onRⁿ with Ω1≤1. Let P : Rⁿ→R be a real-valued polynomial of degree d and let h∈ L²(R+,r⁻¹dr). Then the oscillatory singular integral operatorᏹΩ,P;

T_Ω,P,h(f)(x)=p·v

Rⁿe^iP(y)f(x−y)|y|⁻ⁿh|y|

Ωy)d y (1.6) satisfies

T_Ω,P,h(f)_p≤

1 + log^1/2e+Ωq

hL²(_R+,r⁻¹dr)Cp,qfp (1.7) for allp≥2, whereC_p,q=(2^1/q/(2^1/q−1))C_p. Here 1/q=1−1/qandC_pis a constant that may depend on the degree of the polynomialPbut it is independent of the functionΩ, the indexq, and the coefficients of the polynomialP.

ByTheorem 1.4, we obtain the following two results.

Corollary 1.5. Let Ω∈L(logL)^1/2(Sⁿ⁻¹) be a homogeneous function of degree zero on Rⁿand satisfies (1.1). LetP:Rⁿ→Rbe a real-valued polynomial of degreedand leth∈ L²(R+,r⁻¹dr). Then the oscillatory singular integral operatorᏹΩ,P;

T_Ω,P,h(f)(x)=p·v

Rⁿe^iP(y)f(x−y)|y|⁻ⁿh|y|

Ω(y)d y (1.8) is bounded onL^pfor allp≥2 withL^pbounds that may depend on the degree of the polyno- mialPbut they are independent of the coefficients of the polynomialP.

Corollary 1.6. LetΩ∈B^0,_q^−1/2(Sⁿ⁻¹), s >1, be a homogeneous function of degree zero onRⁿand satisfies (1.1). LetP:Rⁿ→Rbe a real-valued polynomial of degreedand let

h∈L²(R+,r⁻¹dr). Then the oscillatory singular integral operatorᏹΩ,P; T_Ω,P,h(f)(x)=p·v

Rⁿe^iP(y)f(x−y)|y|⁻ⁿh|y|

Ω(y)d y (1.9) is bounded onL^pfor allp≥2 withL^pbounds that may depend on the degree of the polyno- mialPbut they are independent of the coefficients of the polynomialP.

Further applications of the results stated above will be presented inSection 6.

Throughout this paper, the letterCwill stand for a constant that may vary at each occurrence, but it is independent of the essential variables.

2. Preliminary estimates

We start by recalling the following result in [10].

Lemma 2.1 (see [10]). Letᏼ=(P1,. . .,Pd) be a polynomial mapping from Rⁿ intoR^d. Suppose thatΩ∈L¹(Sⁿ⁻¹) and

M_Ω,ᏼf(x)=sup

j∈Z

2^j≤|y|<2^j+1

fx−ᏼ(y)|y|⁻ⁿΩy)d y. (2.1)

Then for 1< p≤ ∞, there exist a constantCp>0 independent ofΩand the coefficients of P1,. . .,Pdsuch that

M_Ω,ᏼf_p≤CpΩL¹(Sⁿ⁻¹)fp (2.2) for every f ∈L^p(R^d).

Lemma 2.2 (van der Corput [26]). Supposeφis real valued and smooth in (a,b), and that

|φ^(k)(t)| ≥1 for allt∈(a,b). Then the inequality _b

ae^−iλφ(t)ψ(t)dt≤Ck|λ|^−1/k (2.3)

holds when (i)k≥2, or

(ii)k=1 andφis monotonic.

The boundCkis independent ofa,b,φ, andλ.

Lemma 2.3. LetΩ∈L^q(Sⁿ⁻¹),q >1, be a homogeneous function of degree zero onRⁿwith Ω1≤1. LetP(x)= _|α|≤da_αx^α be a real-valued polynomial of degreed >1 such that

|x|^d is not one of its terms. Fork∈Z, letEk,Ω: [1, log(e+Ωq)]×P(Sⁿ⁻¹)×R→Cand let Jk,Ω:Rⁿ→Rbe given by

E_k,Ωr,P(y),s=e^−i[P(2−(k+1) log(e+Ωq)r y)+2⁻(k+1) log(e+Ωq)sr], Jk,Ω(ξ)=

₂^{2 log(e+Ω}q)

1

Sⁿ⁻¹Ω(y)Ek,Ω

r,P(y),ξ·ydσ(y)

2d rr.

(2.4)

Then, Jk,Ωsatisfies

sup

ξ∈Rⁿ

Jk,Ω(ξ)≤2^(k+1)/4qloge+Ωq

|α|=d

aα−ε/q

C (2.5)

for some 0< ε <1, whereCis a constant that may depend on the degree of the polynomialP but it is independent of the functionΩ, the indexq, and the coefficients of the polynomialP.

Proof ofLemma 2.3. First, we notice the following:

Jk,Ω(ξ)≤loge+Ωq

, (2.6)

Jk,Ω(ξ)^q≤ Ω^2qq

Sⁿ⁻¹

2^{2 log(e+Ωq)}

1 Ek,Ω

r,P(y),ξ·y

×E_k,Ωr,P(z),ξ·zdr r

^qdσ(y)dσ(z).

(2.7)

Next, notice that

P2^−γk,Ωr y+ 2^−γk,Ω(ξ·y)r−P2^−γk,Ωrz+ 2^−γk,Ω(ξ·z)r

=2^−γk,Ωdr^d

|α|=d

aαy^α−

|α|=d

aαz^α

+ 2^−γk,Ωξ·(y−z)r+Hk(r,y,z,ξ) (2.8) with (d^d/dr^d)Hk=0, whereγk,Ω=(k+ 1) log(e+Ωq). Thus, byLemma 2.2, we have

₂^{2 log(e+Ω}q)

1 Ek,Ω

r,P(y),ξ·yEk,Ω

r,P(z),ξ·zdr r

≤2^−dγk,ΩP(y)−P(z)^−1/d. (2.9) Now, by (2.9) and the inequality

2^{2 log(e+Ωq)}

1 E_k,Ωr,P(y),ξ·yE_k,Ωr,P(z),ξ·zdr r

≤Cloge+Ωq

, (2.10)

we obtain

2^{2 log(e+Ωq)}

1 E_k,Ωr,P(y),ξ·yE_k,Ωr,P(z),ξ·zdr r

≤2^−dγk,ΩP(y)−P(z)^−1/4dqCloge+Ωq

1₋1/4q

.

(2.11)

Therefore, by (2.7), (2.11), and [12, (3.11)], we obtain J_k,Ω(ξ)≤2^γk,Ω/4qΩ^2qqCloge+Ωq

1₋1/4q

. (2.12)

Hence, by (2.6) and (2.12), we get

J_k,Ω(ξ)≤2^{γk,Ω/4 log(e+Ωq)q}Ω^2/q^log(e+Ωq)loge+Ωq

≤2^(k+1)/4qloge+Ωq C.

(2.13)

This completes the proof.

Now, we will need the following lemma.

Lemma 2.4. LetΩ∈L^q(Sⁿ⁻¹),q >1, be a homogeneous function of degree zero onRⁿwith Ω1≤1. Then

ᏹΩ(f)_p≤log^1/2e+Ωq

2^1/q 2^1/q−1

C_pfp (2.14)

for allp≥2 with constantsCpindependent of the functionΩand the indexq.

We remark here that sinceL^q(Sⁿ⁻¹)⊂Llog^1/2L, it follows from [2, Theorem B(a)] that ᏹΩp≤ ΩqC_pfor allp_≥2. But, clearly the constant{1 + log^1/2(e+Ωq)}in (2.14) is sharper than the constantΩqthat can be deduced from [2, Theorem B(a)]. However, the former constant can be obtained by following a similar argument as in the proof of Theorem B(a) in [2] and keeping track of certain constants. For completeness, we, below, present the main ideas of the proof.

Proof ofLemma 2.4. Choose a collection of Ꮿ^∞ functions {ω_k}k∈Z on (0,∞) with the properties sup(ωk)⊆[2^{−log(e+Ωq)(k+1)}, 2^{−log(e+Ωq)(k−1)}], 0≤ωk≤1, _k∈Zωk(u)=1,

|(d^sωk/du^s)(u)| ≤Csu^−s, whereCsis a constant independent of log(e+Ωq). Fork∈Z, letG_kbe the operator defined by (G_k(f))(ξ)=ω_k(|ξ|)f(ξ). Let

E_j(f)(x)=

k∈Z

2^{2 log(e+Ωq)} 1

Sⁿ⁻¹Ω(y)G_k+j(f)x−2^klog(e+Ωq)r ydσ(y)

2

r⁻¹dr 1/2

. (2.15) Then

ᏹΩ(f)(x)≤

j∈Z

Ej(f)(x). (2.16)

By exactly the same argument in [2], we obtain

Ej(f)₂≤C2^−β|j|/qlog^1/2e+Ωq

f2. (2.17)

On the other hand, by a duality argument; see (3.24)-(3.25) for similar argument, we get Ej(f)_p≤Clog^1/2e+Ωq

fp (2.18)

for all 2< p <∞. Thus, by interpolation between (2.17) and (2.18), we have Ej(f)_p≤C2^−ε(|j|/q)log^1/2e+Ωq

fp (2.19)

for someε >0 and for all 2≤p <∞, andj∈Zwith constantCindependent ofΩ,k, and j. Hence, (2.14) follows by (2.16) and (2.19). This completes the proof.

3. Proof ofTheorem 1.1

Proof ofTheorem 1.1. We will argue by induction on the degree of the polynomialP. If d₌deg(P)₌0, then (1.4) follows easily fromLemma 2.4. In fact, ifd₌0, then by duality it can be easily seen that

ᏹΩ,P(f)(x)≤CᏹΩ(f)(x). (3.1) Thus, byLemma 2.4, we have

ᏹΩ,P(f)_p≤ 2^1/q

2^1/q−1

log^1/2e+Ωq

C_pfp

≤ 2^1/q

2^1/q−1

1 + log^1/2e+Ωq

Cpfp

(3.2)

for allp≥2.

Now, ifd=1, that is,P(y)= −→a·yfor some^−→a∈Rⁿ, then by (3.2), we have ᏹΩ,P(f)_p≤

2^1/q 2^1/q−1

log^1/2Ωq Cpgp

= 2^1/q

2^1/q−1

1 + log^1/2e+Ωq

C_pfp,

(3.3)

whereg(y)=e^−iP(y)f(y).

Next, assume that (1.4) holds for all polynomialsQ of degree less than or equal to d >1. Let

P(x)=

|α|≤d+1

a_αx^α (3.4)

be a polynomial of degreed+ 1. Then by duality, we have ᏹΩ,P(f)(x)=

_∞

0

Sⁿ⁻¹e^{iP(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

. (3.5) We may assume thatPdoes not contain|x|^d+1as one of its terms. By dilation invari- ance, we may also assume that

|α|=d+1

a_α=1. (3.6)

We now choose a collection{ωk}k∈ZofᏯ^∞functions defined on (0,∞) that satisfy the following properties:

suppψk

⊆

2^{−log(e+Ωq)(k+1)}, 2^{−log(e+Ωq)(k−1)}, 0≤ψk≤1,

k∈Z

ψk(u)=1. (3.7)

Set

η_∞(u)= 0 k=−∞

ψk(u), η0(u)= ∞ k=1

ψk(u). (3.8)

Then,

η_∞(u) +η0(u)=1, suppη_∞(u)⊂

2^{−log(e+Ωq)},∞

, suppη0(u)⊂(0, 1]. (3.9) Define the operators᏿Ω,P,∞and᏿Ω,P,0by

᏿Ω,P,∞(f)(x)= _∞

2^{−log(e+Ωq)}

η_∞(r)

Sⁿ⁻¹e^{iP(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

,

᏿Ω,P,0(f)(x)= ₁

0

η0(r)

Sⁿ⁻¹e^{iP(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

. (3.10) Thus, by (3.9), we have

᏿Ω,P(f)(x)≤᏿Ω,P,0(f)(x) +᏿Ω,P,∞(f)(x). (3.11) Now, we estimate᏿Ω,P,0p.

Let

Q(x)=

|α|≤d

aαx^α. (3.12)

Assume that deg(Q)=l, where 0≤l≤d. Define the operators᏿⁽¹⁾Ω,P,0and᏿⁽²⁾Ω,Q,0by

᏿⁽¹⁾_Ω,P,0(f)(x)= 1

0

Sⁿ⁻¹

e^{iP(r y)}−e^{iQ(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

,

᏿⁽²⁾_Ω,Q,0(f)(x)= ₁

0

Sⁿ⁻¹e^{iQ(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

.

(3.13)

Now, by the observation thatη0(r)≤1 and by Minkowski’s inequality, we obtain

᏿Ω,P,0(f)(x)≤᏿⁽¹⁾_Ω,P,0(f)(x) +᏿_Ω⁽²⁾,Q,0(f)(x). (3.14)

By induction assumption, it follows that ᏿⁽²⁾_Ω,Q,0(f)_p≤

1 + log^1/2(e+Ωq) 2^1/q 2^1/q−1

C_pfp (3.15) for allp≥2.

On the other hand, by Cauchy-Schwarz inequality, by the fact thatΩ1≤1, and the inequality

e^{iP(r y)}−e^{iQ(r y)}≤r^d+1

|α|=d+1

a_αy^α

≤r^d+1,

(3.16)

we get

᏿Ω,P,0⁽¹⁾ (f)(x)≤ 1

0

Sⁿ⁻¹

e^{iP(r y)}−e^{iQ(r y)2}Ω(y)f(x−r y)²dσ(y)r⁻¹dr 1/2

≤ 1

0

Sⁿ⁻¹

Ω(y)f(x−r y)²dσ(y)r^2d+1dr 1/2

= ₋₁

j=−∞

2^j+1 2^j

Sⁿ⁻¹

Ω(y)f(x−r y)²dσ(y)r^2d+1dr 1/2

≤ ₋₁

j=−∞

2^(2d+2)j 2^j+1

2^j

Sⁿ⁻¹

Ω(y)f(x−r y)²dσ(y)r⁻¹dr 1/2

≤CM_Ω|f|²1/2

(x),

(3.17) whereM_Ωis the operator given by (2.1) withᏼ(y)=y. Thus, by (3.17), by the fact that Ω1≤1, andLemma 2.1, we obtain

᏿⁽¹⁾_Ω,P,0(f)_p≤C_pfp (3.18)

for allp≥2 with constantC_pindependent of the functionΩand the coefficients of the polynomialP. Therefore, by (3.14), by Minkowski’s inequality, by (3.15), and (3.18), we obtain

᏿Ω,P,0(f)_p≤

1 + log^1/2e+Ωq

2^1/q 2^1/q−1

Cpfp (3.19) for allp≥2.

Finally, we prove theL^pboundedness of᏿Ω,P,∞. By generalized Minkowski’s inequal- ity, we can write᏿Ω,P,∞as

᏿Ω,P,∞(f)(x)= _∞

2^{−log(e+Ωq)}

η_∞(r)

Sⁿ⁻¹e^{iP(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

= _∞

0

0 k=−∞

ψk(r)

Sⁿ⁻¹e^{iP(r y)}Ω(y)f(x−r y)dσ(y)

21 rdr

1/2

≤ 0 k=−∞

᏿Ω,P,∞,k(f)(x),

(3.20) where

᏿Ω,P,∞,k(f)(x)=

₂^−log(e+Ωq)(k−1)

2^{−log(e+Ωq)(k+1)}

Sⁿ⁻¹e^{iP(r y)}Ω(y)f(x−r y)dσ(y)

2

r⁻¹dr 1/2

. (3.21) By Plancherel’s theorem, Fubini’s theorem, andLemma 2.3, we have

᏿Ω,P,∞,k(f)²₂=

Rⁿ f(ξ)²J_k,Ω(ξ)dξ≤2^(k+1)/4qloge+Ωq

f²2. (3.22) Thus,

᏿Ω,P,∞,k(f)₂≤2^(k+1)/8qlog^1/2e+Ωq

f2. (3.23)

Now, forp >2, chooseg∈L^(p/2)withg(p/2)=1 such that ᏿Ω,P,∞,k(f)²_p

=

Rⁿ

2^{2 log(e+Ωq)} 1

Sⁿ⁻¹E_k,Ωr,P(y), 0Ω(y)fx₋2^−γk,Ωr ydσ(y)

2

r⁻¹drg(x)dx

≤

Rⁿ

f(z)²

2^{2 log(e+Ωq)} 1

Sⁿ⁻¹

Ω(y)gz+ 2^−γk,Ωr ydσ(y)dr

r dz

≤Cloge+Ωq

f²_pM_Ωg(z) _(p/2),

(3.24) whereM_Ω is the operator given by (2.1) withᏼ(y)=y. Thus,Lemma 2.1 and (3.24) imply that

᏿Ω,P,∞,k(f)_p≤log^1/2e+Ωq

Cfp, (3.25)

which when combined with (3.23) implies

᏿Ω,P,∞,k(f)_p≤2^(k+1)δ/8qlog^1/2e+Ωq

Cfp, (3.26)