Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential Type Weights

Volume 2009, Article ID 528454,22pages doi:10.1155/2009/528454

Research Article

Derivatives of Integrating Functions for

Orthonormal Polynomials with Exponential-Type

Weights

Hee Sun Jung

_{and Ryozi Sakai}

1_{Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, South Korea}

2_{Fukami 675, Ichiba-cho, Toyota-city, Aichi 470-0574, Japan}

Correspondence should be addressed to Hee Sun Jung,[email protected]

Received 22 December 2008; Revised 26 February 2009; Accepted 22 April 2009

Recommended by Jozsef Szabados

Letwρx:|x|ρ_{exp−Qx,ρ >−1/2, whereQ∈C}2_{:−∞,∞ → 0,∞is an even function. In}

2008 we have a relation of the orthonormal polynomialpnw2

ρ;xwith respect to the weightwρ2x; pnx Anxpn−1x−Bnxpnx−2ρnpnx/x, whereAnxandBnxare some integrating

functions for orthonormal polynomialspnw2

ρ;x. In this paper, we get estimates of the higher derivatives ofAnxandBnx, which are important for estimates of the higher derivatives of pnw2

ρ;x.

Copyrightq2009 H. S. Jung and R. Sakai. 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 Results

Let R −∞,∞. LetQ ∈ C2 _{: R → R 0,∞ be an even function, and letwx}

exp−Qxbe such that∞₀xn_w2_{xdx <∞for alln0,1,2, . . . .Forρ >−1/2, we set}

wρx:|x|ρwx, x∈R. 1.1

Then we can construct the orthonormal polynomials pn,ρx pnw2

ρ;xof degree nwith

respect tow2

ρx. That is,

_∞

−∞pn,ρxpm,ρxw 2

ρxdxδmn Kronecker’s delta,

pn,ρx γnxn· · ·, γnγn,ρ >0.

A functionf :R → Ris said to be quasi-increasing if there existsC >0 such that

fx ≤ Cfyfor 0 < x < y. For any two sequences {bn}∞_n1 and {cn}∞_n1 of nonzero real numbersor functions, we writebn cnif there exists a constantC >0 independent ofnor

xsuch thatbn≤Ccnfornlarge enough. We writebn ∼cnifbncnandcn bn. We denote the class of polynomials of degree at mostnbyPn.

Throughout C, C1, C2, . . . denote positive constants independent of n, x, t, and polynomials of degree at mostn. The same symbol does not necessarily denote the same constant in different occurrences.

We will be interested in the following subclass of weights from1.

Definition 1.1. LetQ:R → Rbe even and satisfy the following properties. aQxis continuous inR, withQ0 0.

bQxexists and is positive inR\ {0}. c

lim

x→ ∞Qx ∞. 1.3

dThe function

Tx: xQ

x

Qx , x /0 1.4

is quasi-increasing in0,∞with

Tx≥Λ>1, x∈R\ {0}. 1.5

eThere existsC1 >0 such that

Qx |Qx| ≤C1

Qx

Qx , a.e. x∈R\ {0}. 1.6

Then we writew ∈ FC2. If there also exist a compact subintervalJ 0ofRandC2 > 0 such that

Qx |Qx|≥C2

Qx

Qx , a.e. x∈R\J, 1.7

then we writew∈ FC2_.

In the following we introduce useful notations.

aMhaskar-Rahmanov-Saff MRSnumbersaxare defined as the positive roots of the following equations:

x 2

π

₁

0

axuQaxu

bLet

ηx xTax−2/3, x >0. 1.9

cThe functionϕuxis defined as follows:

ϕux

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

a2 2u−x2

uauxauηuau−xauηu1/2 , |x| ≤au,

ϕuau, au<|x|.

1.10

In the rest of this paper we often denotepn,ρxsimply bypnx. Letρnρifnis odd,

ρn 0 otherwise and define the integrating functionsAnxandBnxwith respect topnx

as follows:

Anx:2bn

_∞

−∞p 2

nuQx, uw2ρudu,

Bnx:2bn

_∞

−∞pnupn−1uQx, uw 2

ρudu,

1.11

whereQx, u Qx−Qu/x−uandbn γn−1/γn. Then in2, Theorem 4.1we have a relation of the orthonormal polynomialpnxwith respect to the weightw2

ρx:

pnx Anxpn−1x−Bnxpnx−2ρn pnx

x , ρn

⎧ ⎨ ⎩

ρ, nis odd,

0, nis even, 1.12

and in 2, Theorem 4.2we already have the estimates of the integrating functions Anx

andBnxwith respect topnx. So, in this paper we will estimate the higher derivatives of

AnxandBnxfor the estimates of the higher derivatives ofpnwρ2;x, because the higher

derivatives ofpn,ρxplay an important role in approximation theory such as investigating convergence of Hermite-Fej´er and Hermite interpolation based on the zeros ofpnw2

ρ;x see

3,4.

To estimate of the higher derivatives ofAnxandBnxwe need further assumptions forQxas follows.

Definition 1.2. Letwx exp−Qx∈ FC2_{, and letνbe a positive integer. Assume that}

Qxisν−times continuously differentiable onRand satisfies the followings. aQν1_{xexists andQ}i_{x,i0,1, . . . , ν1 are nonnegative forx >0.}

bThere exist positive constantsCi>0 such that forx∈R\ {0}

Qi1x≤CiQix

Qx

cThere exist constants 0≤δ <1 andc1>0 such that on0, c1

Qν1_x≤C 1

x

δ

. 1.14

Then we writewx∈ FνC2.

Letνbe a positive integer. Define formα−ν >0,m≥0, l≥1, andα≥0,

Ql,α,mx:|x|mexp_l|x|α−α∗exp_l0, 1.15

whereα∗0 ifα0, otherwiseα∗1 and define

Qαx: 1|x||x|α−1, α >1. 1.16

Here we let exp₀x : xand for l ≥ 1, exp_lx : expexp· · ·expx· · · denotes the

lth iterated exponential. In particular, exp_lx expexp_l−1x. Then exp−Ql,α,mxand exp−Qαxare typical examples ofFνC2 see5.

In the following we improve the inequality4.3in2, Theorem 4.2.

Theorem 1.3. Letρ >−1/2andwx exp−Qx∈ FC2_{. Additionally assume thatQx}

is nondecreasing. Then for|x| ≤εanwith0< ε <1/2one has

|Bnx|< λε, nAnx, 1.17

where

lim

ε→0nlim→ ∞λε, n 0. 1.18

In this paper our main theorem is as follows.

Theorem 1.4. Letρ >−1/2andwx exp−Qx∈ FνC2for positive integerν≥2. Assume that12ρ−δ≥0forρ <0and

ann1/1ν−δ, 1.19

where0≤δ <1is defined in1.14.

a If Qx/Qxis quasi-increasing on c2,∞, then one has for |x| ≤ an1ηnand

j0, . . . , ν−1

AnjxAnx Tan an

j

, BnjxAnx Tan an

j

Moreover, for any0< ε <1/2there existsε∗ε, n>0such that for|x| ≤εanandj1, . . . , ν−1,

Anjx≤ε∗ε, nAnx n an

j

, Bnjx≤ε∗ε, nAnx n an

j

, 1.21

withε∗ε, n → 0asn → ∞.

bIfQν1_{xis non-decreasing onc}

2,∞, then one has1.20and1.21for the respective

ranges ofx.

cIf there exists a constant0 ≤δ <1such thatQν1x≤C1/xδonc2,∞, then one

has1.20and1.21for the respective ranges ofx.

The examples satisfying the conditionsa,b, orcofTheorem 1.4are given in5.

Remark 1.5. Under the assumptions ofTheorem 1.4, we have from2, Theorem 4.2that there existsC, n0>0 such that forn≥n0and|x| ≤an1Lηn,

Anx

2bn ∼ϕnx

−1_a2

n12Lηn2−x2

₋1/2

, |Bnx|Anx, 1.22

becausewx exp−Qx∈ FνC2for positive integerν≥1 and 12ρ−δ≥0 forρ <0.

In addition, for our future work we estimateatandTatusingλC1in1.6for the weight classFC2_.

Theorem 1.6. Letwx exp−Qx∈ FC2_{, and we assume}

Qx |Qx|≤λ

Qx

Qx , |x| ≥b >0, 1.23

whereb >0is large enough.

aAssume thatTxis unbounded. Then for anyη >0there existsCη> 0such that for t≥1,

at≤Cηtη. 1.24

bSuppose that there exist constantsη >0andC2>0such thatat≤C2tη. Then there exists

a constantCdepending only onλ,η, andC2such that forat≥1, ifλ >1

Tat≤Ct2ηλ−1/λ1, 1.25

and if0< λ≤1,

Remark 1.7. aLevin and Lubinsky showed the following1, Lemma 3.7: there existsC >0 such that for someε >0, and for large enought,

Tat≤Ct2−ε. 1.27

If from1.25and1.26we set for any 0< η <2

ε

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2−η, 0< λ≤1,

22−η

λ1 , λ >1,

1.28

then we have1.27in Levin and Lubinsky’s lemma.

bIfTxis unbounded, then1.19is trivially satisfied by1.24.

2. Proof of Theorems

In this section we will prove the theorems ofSection 1.

Lemma 2.1. Letρ >−1/2and letwx∈ FC2_{. Then uniformly forn≥1,} a

sup

x∈R

_{pn,ρxwx |x|}an n

ρ

x2−a2n

1/4

∼1. 2.1

b

sup

x∈R

_{pn,ρxwx |x|}an n

ρ

∼a−n1/2nTan1/6. 2.2

cMarkov inequality. Let0< p≤ ∞. For any polynomialP ∈ Pn

Pwx |x|an n

ρ

LpR

nTan1/2 an

P wx |x|an n

ρ

LpR

. 2.3

dLetβ∈R,0 < p ≤ ∞, andr >1. Then there exist positive constantsL,δ, andC2such

that for any polynomialP ∈ Pn

P w)x |x|ann

β

Lparn≤|x|

exp−C2nδP wx |x|an

n

β

LpLan/n≤|x|≤an1−Lηn .

Proof. afollows from2, Theorem 2.3.bfollows from2, Theorem 2.4.cfollows form 6, Theorem 2.1b.dfollows form6, Theorem 2.3.

Lemma 2.2. Letρ >−1/2and letwx∈ FC2_{. Then one has forc >0,}

0≤u≤c

pnwρ2udu 1

an. 2.5

Proof. Forρ ≥0, the results are immediate fromLemma 2.1a. So we assume−1/2 < ρ <0. First we see

0≤u≤an/n

pnwρ2udu

0≤u≤an/n

pnw2u |u|an n

2ρ _| u|2ρ

|u|an/n2ρdu

≤C 1 an

0≤u≤an/n

|u|2ρ

|u|an/n2ρdu

≤C 1 an

n an

2ρ

0≤u≤an/n |u|2ρdu

≤C 1 an

n an

2ρ an

n

12ρ

≤C1 n,

2.6

because we know thatanonfrom1, Lemma 3.5c. Next we see byLemma 2.1a

an/n≤u≤c

pnwρ2udu≤C 1

an. 2.7

Therefore, we have the result.

Lemma 2.3. Letρ >−1/2and letwx∈ FC2_{. Then} aone has

0≤u≤∞

pnw2u |u|an n

2ρ

Qudu∼ n

an, 2.8

bforx∈0, an/2one has

Qx≤Cn an

x an

_Λ−1

. 2.9

Proof ofTheorem 1.3. SinceBnxis an odd function, we prove only for 0≤ x≤ εan. Letθ :

εΛ−1/2Λ_{. Then we have the following two lemmas.}

Lemma 2.4. Uniformly forθandn

|u|≤θan

pnupn−1uw2ρuQx, udu

1

nθ1

θΛ−1 n

a2

n

. 2.10

Proof. For|u| ≤θan, we have byLemma 2.1a

p2nuw2ρu

1

a2

n−θan2

|u|2ρ

|u|an/n2ρ

1

an

|u|2ρ

|u|an/n2ρ. 2.11

Since Qx is nondecreasing and 1 − 1/2Λ1/2Λ ≤ θ − ε/θ ≤ 1, we have using

Lemma 2.3b:

Qx, u≤ Q

θan−Qx

θan−x

Qθan

θ−εan θ

Λ−2 n

a2

n

. 2.12

Moreover we know that forρ >−1/2,

θan

0

|u|2ρ

|u|an/n2ρdx

|u|≤an/n

an/n≤|u|≤θan an

n θan. 2.13

Therefore, we have

|u|≤θan p2

nuwρ2uQx, udu

1

nθ 1

θΛ−1 n

a2

n

. 2.14

Consequently, we have the result using Cauchy-Schwartz inequality

|u|≤θan

pnupn−1uw2ρuQx, udu

1

nθ1

θΛ−1 n a2

n

. 2.15

Lemma 2.5. Uniformly forθεΛ−1/2Λand forn

θan≤|u|≤a2n

pnupn−1uwρ2uQx, udu

ε1−1/ΛΛ−1ε1/Λn a2

n

Proof. Forθan≤ |u| ≤a2n, we have similarly to2,4.6

Qx, u−Qx,−u2uQ

x−xQu x2−_u2

an

QεanεanQu θan2

ε1−1/ΛΛ−1 n a2

n

ε1/Λ

an

Qu

2.17

seeLemma 2.3b. Therefore, we have byLemma 2.3a,

θan≤|u|≤a2n

pnupn−1uw2ρuQx, udu

≤

θan≤|u|≤a2n

pnupn−1uw2ρuQx, u−Qx,−udu

ε1−1/ΛΛ−1 n a2n

θan≤|u|≤a2n

_pnupn−1uw2

ρudu

ε1/Λ

an

θan≤|u|≤a2n

_pnupn−1uw2

ρuQ

udu

ε1−1/ΛΛ−1 n a2

n

ε1/Λ n a2

n .

2.18

Here we usedLemma 2.1b.

Since for a constantC >0

a2n≤|u|

pnupn−1uwρ2uQx, udu

O

e−nC

, 2.19

see2, page 233, there existsλn>0 such that

a2n≤|u|

pnupn−1uw2ρuQx, udu

λn_an2

n

, 2.20

and λn → 0 asn → ∞. We know from2, Lemma 4.7that bn γn−1/γn ∼ an. From 1.22we haveAnx/bn∼n/a2

nfor|x| ≤εanand from the preceding considerations and the

definition ofBnxit follows that for|x| ≤εan

|Bnx|

bn

λε, nn a2n

∼ λε, nAnx

where for some positive constantC >0

λε, n:C·max 1

nθ1

θΛ−1, ε1−1/ΛΛ−1, ε1/Λ, λn

. 2.22

Consequently,1.17is proved, and we can obtain that limε→0limn→ ∞λε, n 0. Now, we have for|x| ≤εan

Anx∼ n

an, |Bnx|< λε, n n

an. 2.23

Proof ofTheorem 1.4. First, we see that for 1≤j≤ν−1

Anjx 2bn

_∞

−∞pnwρ 2

u d j

dxjQx, udu. 2.24

We split proof of1.20into some lemmas as follows:

1Lemma 2.6is for 0≤x≤an1ηn,a4n≤u, and 1≤j ≤ν−1;

2Lemma 2.9is foran/2≤x≤an1ηn, 0≤u≤a4n, andjν−1;

3Lemma 2.10is for 0≤x≤an/2, 0≤u≤a4n, andjν−1;

4Lemma 2.11is for 0≤x≤an1ηn, 0≤u≤a4n, and 1≤j≤ν−2;

on the other hand,1.21will be proved by Lemmas2.13and2.6. For 1≤j≤ν−1 there existsηbetweenuandxsuch that

dj dxjQx, u

j! x−uj1

_j

k0 −1kQ

j1−k_x

j−k! x−u

j−k ₋₁j1

Qu

Qj1x−Qj1

η

x−u .

2.25

Then forx≥0 andu≥0, sinceQj1_{uis increasing for 1≤j≤ν−1, we have}

0≤ d

j

dxjQx, u≤

Qj1_x−Qj1_u

x−u . 2.26

Ifu <0 andx >0, then since|Qj1_{η| ≤Q}j1_{−uforη <0,}

d

j

dxjQx, u

Qj1_x−Qj1_η

x−u

≤ Qj1x Qj1−u

x −u

≤ Qj1x−Qj1−u

x−−u 2

Qj1_−u−Qj1₀

−u−0 .

So, for this case we can prove the result similarly to the casex, u > 0. For the other cases, we can prove it by the symmetry of Q, similarly. Therefore, we assume that u and xare nonnegative, and we will prove this theorem only for nonnegativexandu. Moreover, for simplicity, we letc1c2without loss of generality, because we know by1.13thatQν1u is bounded for anyubetweenc1andc2.

On the other hand, ifQj2_{uis increasing, then}

Qj1_u−Qj1_x

u−x 2.28

is also increasing forubecause there exists a pointξbetweenxandusuch that

d du

Qj1_u−Qj1_x

u−x

Qj2u−

Qj1_u−Qj1_x/u−x

u−x

Qj2u−Qj2ξ

u−x ≥0.

2.29

Moreover, ifQν1_t≤C1/tδ_for

tbetweenxandu, then we see

Qν_u−Qν_x

u−x

1

u−x

u

x

Qν1tdt≤ C u−x

u1−δ−x1−δ≤C 1 u

δ

. 2.30

To complete the proof ofTheorem 1.4we prove a series of lemmas.

Lemma 2.6. Let0≤x≤an1ηnand1≤j≤ν−1:

a4n<u

pnwρ2u d j

dxjQx, udu

Tan an

j_Anx

an . 2.31

Proof. Since

Anjx

2bn

0≤u≤a4n

a4n<u

pnwρ2u d j

dxjQx, udu, 2.32

we have to estimate

a4n<u

pnwρ2u d j

dxjQx, udu:

a4n<u

. 2.33

First, we see forx >0 large enough,

is decreasing because

Qj1_xe−Qx_Qj2_x−Qj1_xQxe−Qx_{, 2.35}

and so from our assumption,

Qj2_x−Qj1_xQx≤CQj1_xQ

x Qx −Q

j1_xQx

Qj1xQx C Qx−1

<0,

2.36

ifC < Qx. We use this fact. Let 2ρ βiwhereβ <0, and letibe a nonnegative integer, and letPu p2

nuui. Letu >0. Then since

Qj1_a

4n Qa4n ≤C

Tan an

j

, 2.37

by1.13, we have for someξbetweenxandu

a4n<u

a4n<u

pnwρ2uQj2ξdu

≤

a4n<u

pnwρ2uQj2udu

≤CQ j1_a

4nwa4n

Qa4n a

β

4n

a4n<u

PuwuQudu by2.34

≤ Tan an

j

wa4naβ4n

_∞

a4n

−Pu d

duwudu,

_∞

a4n Pu d

duwudu P wa4n−

_∞

a4n

Ptwudu.

2.38

ApplyingLemma 2.1dwithL∞,L1-norm andLemma 2.1c,

|P wa4n| ≤exp−C2nαP wxL∞Lan/n≤|x|≤an1−Lηn, _∞

a4n

Puwudu≤exp−C2nα

Pwx

L1Lan/n≤|x|≤an1−Lηn

≤exp−C2nαnTan 1/2

an P wxL1Lan/n≤|x|≤an1−Lηn.

Therefore,

_∞

a4n

PuwuQudu≤exp−C2nαP wxL∞Lan/n≤|x|≤an1−Lηn

exp−C2nαnTan 1/2

an P wxL1Lan/n≤|x|≤an1−Lηn.

2.40

Consequently we have

Tan an

j

wa4naβ_4n

a4n<u

PuwuQudu

≤ Tan an

j

exp−C2nαp2nwρ2_L∞La

n/n≤|x|≤an1−Lηn

Tan

an

j

nTan1/2

an exp−C2n α_p2

nwρ2_L

1Lan/n≤|x|≤an1−Lηn

≤Oe−nd3 Tan

an

j

Tan an

j Anx

an .

2.41

Lemma 2.7. If Qx/Qx is quasi-increasing on c1,∞ or if Qν1x is nondecreasing on c1,∞, then one has

Qν_x−Qν_u

x−u

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 Tan

an

ν−1 _n

a2

n

, 0≤u≤c1, c1≤x≤ an₂ ,

1 Tan

an

ν−1

n a2n

, c1≤u≤2c1, 0≤x≤c1,

Tan an

ν−1

n a2

n

, 2c1 ≤u≤ an₃ , 0≤x≤c1,

Tan an

ν−1

n a2n

, c1≤u≤ an

3 , c1≤x≤

an

2 .

2.42

Proof. Casea-1. 0≤u≤c1andc1≤x≤an/2. Let

Qν_u−Qν_x

u−x ≤

Qν_u−Qν_c1

u−c1

Qν_c1−Qν_x

c1−x

Then we haveQ1u1 from2.30. Then ifQx/Qxis quasi-increasing onc1,∞, there exists a pointξ∈c1, xsuch that by1.13

Q2x Q

ν1_ξ

Qξ

Q

x−Qc1 x−c1

Qξ Qξ

ν−1 Q

an/2−Qc1 an/2−c1

Qan/2

Qan/2 ν−1

Q

an/2

an

Tan an

ν−1

n a2

n .

2.44

IfQν1xis nondecreasing onc1,∞, there exists a pointξ∈c1, xsuch that by2.28and 1.13

Q2x≤ Q

ν_an/2−Qν_c1

an/2−c1

Qνan/2 Qan/2

Qan/2−Qc1 an/2−c1

Qan/2

Qan/2 ν−1

Q

an/2

an

Tan an

ν−1 _n

a2

n .

2.45

Casea-2. Forc1 ≤u≤2c1and 0≤x≤c1, we have similarly to Casea-1,

Q2x1, Q1u Tan

an

ν−1

n a2

n

. 2.46

Caseb. 2c1≤u≤an/3 and 0≤x≤c1. Using the method of Casea-1, and similarly to Casea-2,

Qνu−Qνx

u−x ∼

Qνu−Qνc1

u−c1

Tan an

ν−1

n a2n

Casec.c1 ≤u≤an/3 andc1≤x≤an/2. We can prove similarly toQ1uandQ2x of Casea-1. IfQx/Qxis quasi-increasing onc1,∞, there exists a pointξ∈c1, xsuch that by1.13

Qν_x−Qν_u

x−u

Q

ν1_ξ

Qξ

Q

x−Qu x−u

Qan/2

Qan/2 ν−1

Q an/2 an

∼ Tan an

ν−1

n a2

n .

2.48

IfQν1_{xis nondecreasing onc}

1,∞, there exists a pointξ∈c1, xsuch that by1.13

Qνx−Qνu

x−u ≤

Qνan/2−Qνu an/2−u

|Qan/2|

Qan/2 ν−1

Q an/2 an

∼ Tan an

ν−1 _n

a2

n .

2.49

Lemma 2.8. One has

Qν_x−Qν_u

x−u

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1

uδ, 0≤u≤c1, 0≤x≤c1,

Tan an

ν−1

Qx, u, 0≤u≤a4n, an

2 ≤x≤an

1ηn,

Tan an

ν−1

Qx, u an

3 ≤u≤a4n, 0≤x≤

an

2 .

2.50

Proof. Casea. 0≤u≤c1and 0≤x≤c1. From2.30and1.14

Qν_u−Qν_x

u−x ≤C

1

u

δ

. 2.51

Caseb-1. 0 ≤ u ≤ an/3 andan/2 ≤ x ≤ an1ηn. Since by1, page 64, Lemma 3.2a

Qan/2

Qan/3 ≥ 3 2

_Λ−1

we have

Qx−Qu≥Qx

1−Q

an/3

Qan/2

≥Qx

1− 2

3 Λ−1

. 2.53

Therefore, since for this case

Qx−Qu∼Qx, 2.54

we have

Qν_x−Qν_u

x−u

Qν_x−Qν_u

Qx−Qu Qx, u

Qνx Qx

Qx, u

Tan an

ν−1

Qx, u.

2.55

Caseb-2.an/3≤u≤a4nandan/2≤x≤an1ηn. There exists a pointξbetweenx

andusuch that by1.13

Qν_x−Qν_u

x−u

Qν_x−Qν_u Qx−Qu Qx, u

Qν1ξ Qξ

Qx, u

Tξ ξ

ν−1

Qx, u

Tan an

ν−1

Qx, u.

2.56

Casec.an/3≤u≤a4nand 0≤x≤an/4. By the same method as Caseb, we have

Qν_x−Qν_u

x−u

Tan an

ν−1

Qx, u. 2.57

Lemma 2.9. Letan/2≤x≤an1ηn. Then

0≤u≤a4n

pnwρ2u d ν−1

dxν−1Qx, udu

Tan an

ν−1_Anx

an . 2.58

Lemma 2.10. Let0≤x≤an/2.

aIf0≤x≤c1, then

0≤u≤c1

pnwρ2u d ν−1

dxν−1Qx, udu 1

an. 2.59

Moreover, one knows that

1

an

Tan an

ν−1_Anx

an . 2.60

bIfQx/Qxis quasi-increasing onc1,∞, or ifQν1xis nondecreasing onc1,∞,

then

0≤u≤a4n

pnwρ2u d ν−1

dxν−1Qx, udu

Tan an

ν−1

Anx

an . 2.61

cIf there exists a constant0 ≤δ < 1such thatQν1x≤ C1/xδon0,∞, then one has2.61.

Proof. aFor 0≤x≤c1we have from Lemmas2.8,2.1a, and2.2

0≤u≤c1

pnwρ2u d ν−1

dxν−1Qx, udu

0≤u≤c1

pnwρ2uu−δdu

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1

an, ρ≥0,

1

an n an

2ρ

0≤u≤c1

u2ρ−δ 1 an

n an

2ρ

, ρ <0

1 an,

2.62

because 12ρ−δ≥0 forρ <0. On the other hand, from1.19we seeaν

n≤nν/1ν−δ≤n, and

from1.22we seeAnx∼n/anfor 0≤x≤c1. So we have

1

an

n aνn1

n a2

n

Tan an

ν−1

∼ Anx an

Tan an

ν−1

bFor 0≤x≤c1, we have froma, Lemmas2.7, and2.8

0≤u≤a4n

pnwρ2u d ν−1

dxν−1Qx, udu

0≤u≤c1

c1≤u≤2c1

2c1≤u≤an/3

an/3≤u≤a4n

Tan an

ν−1_Anx

an .

2.64

Similarly, forc1≤x≤an/2 we have from Lemmas2.7and2.8

0≤u≤a4n

pnwρ2u d ν−1

dxν−1Qx, udu

0≤u≤c1

c1≤u≤an/3

an/3≤u≤a4n

Anx an

Tan an

ν−1

.

2.65

cThen by2.26andLemma 2.1a

0≤u≤a4n

pnwρ2u d ν−1

dxν−1Qx, udu

0≤u≤a4n

pnwρ2uu−δdu

0≤u≤a4n

u2ρ−δ

uan/n2ρ

u2_−a2

n du

0<u<an/n

an/n≤u≤an/2

an/2≤u≤a4n

1 aδ

n

n a2n

1

an

ν−1

Anx an

Tan an

ν−1

.

2.66

Here, we use the fact 1/aδ

n< n/aνn1from1.19for the last inequality.

Lemma 2.11. Let0≤x≤an1ηn. Then for1≤j≤ν−2,

0≤u≤a4n

pnwρ2u d j

dxjQx, udu Anx

an

Tan an

j

. 2.67

To prove1.21we need some lemmas.

Lemma 2.12. Let0< ε <1and|x| ≤εan.

aFor someC >0one has

Qεan Qεan ≤C

εΛ−1

Qεan n

an. 2.68

bFor any0< ε <1, there existsε1ε, n>0such that for2εan≤u

dj

dxjQx, u≤ε1ε, n Anx

an n an

j

Qx, u

εanj , 2.69

andε1ε, n → 0asn → ∞.

Proof. aIt follows fromLemma 2.3b.bBy2.25,Lemma 2.3b, anda, we have

dj

dxjQx, u≤C j−1

k0

Qj1−k_x

εank1

Qx, u

εanj

≤C j−1

k0

Qεan

εank1

εΛ−1

Qεan n an

j−k

Qx, u εanj

≤C j−1

k0

εΛ−1 εank1

n an

εΛ−1

Qεan n an

j−k

Qx, u εanj

≤Cn a2

n n an

j j−1

k0

εΛ−k−2

εΛ−1

Qεan

j−k

1

n

k

Qx, u εanj

≤ε1ε, nAnx an

n an

j

Qx, u εanj ,

2.70

where we let

ε1ε, n:C j−1

k0

εΛ−k−2

εΛ−1

Qεan

j−k

1

n

k

−→0 2.71

asn → ∞. Therefore, this lemma is proved.

Lemma 2.13. Suppose that the one of the three conditions (a), (b), and (c) inTheorem 1.4is satisfied. Then for any0< ε <1/2, there existsε2ε, n>0such that for|x| ≤εanandj1, . . . , ν−1,

0≤u≤a4n

pnwρ2u d j

dxjQx, uduε2ε, n Anx

an n an

j

, 2.72

Proof. First, we consider the case of which cin Theorem 1.4is satisfied. Then the lemma follows from2.66withε2ε, n: 1/nν−1. Now, we consider the other cases. If we consider only for |x| ≤ εan and |u| ≤ 2εan in proving Lemmas 2.7and 2.8, then we know that for

|x| ≤εanandj1, . . . , ν−1

dj

dxjQx, u

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1u−δ Qεan Qεan

j

Qεan

εan , 0≤u≤2c1,

Qεan Qεan

j

Qεan

εan , 2c1 ≤u≤

ε

2an,

Qεan Qεan

j_Qεan

εan ,

ε

2an≤u≤2εan.

2.73

Then we have byLemma 2.12a

0≤u≤2εan

pnwρ2u d j

dxjQx, udu

0≤u≤2c1

2c1≤u≤2εan

1 aδn

Qεan Qεan

j

Qεan εan

⎛ ⎝a2nj−δ

n1j ε

Λ−2

εΛ−1

Qεan

j⎞

⎠Anx

an n an j , 2.74

and we can see that

ε3ε, n: a 2j−δ n n1j ε

Λ−2

εΛ−1

Qεan

j

−→0 as n−→ ∞. 2.75

Finally, we estimate_2εa

n≤u≤a4n. ByLemma 2.12bwe have

2εan≤u≤a4n

pnwρ2u d j

dxjQx, udu

≤

2εan≤u≤a4n

ε1ε, nAnx an

n an

j

1

εanjQx, u

pnwρ2udu

≤

ε1ε, n 1

εnj

Therefore, if we letε2ε, n:ε3ε, n ε1ε, n 1/εnj,then

0≤u≤a4n

pnwρ2u d j

dxjQx, uduε2ε, n Anx

an n an

j

, 2.77

andε2ε, n → 0 asn → ∞by2.75and2.76.

From the proof ofLemma 2.6, we have the following. There existsε4n>0 satisfying

ε4n → 0 asn → ∞such that

u≥a4n

pnwρ2u d j

dxjQx, udu≤ε4n Anx

an n an

j

. 2.78

Therefore, from Lemmas2.6,2.9,2.10, and2.11we obtain the estimate forAnjxin1.20,

and fromLemma 2.13and2.78we have the estimate for Anjxin1.21. Using

Cauchy-Schwarz Inequality we also have the estimate forBnjxin1.20and1.21. Consequently,

we provedTheorem 1.4, completely.

Proof ofTheorem 1.6. a 1.24follows from1,3.45easily.

bSuppose that1.23is satisfied on|x| ≥Dfor someD >0 large enough. Letx > D. From1.23we have for largex > D

ln

Qx QD

≤ln Qx

QD

λ

, 2.79

and we have for largex > D

Qx QD ≤

Qx QD

λ

. 2.80

Caseλ >1. Then we can see by1, Lemma 3.43.18and2.80

Tat atQ

at Qat ≤

QD

QDλatQat λ−1_≤

Cat

t

Tat

λ−1

. 2.81

Therefore from the assumptionat≤C2tηwe have for anyη >0

Tat≤Cλ, ηt2ηλ−1/λ1. 2.82

Case 0< λ≤1. Then we have by2.80

Tx xQ

x Qx ≤x

QD QDλQx

λ−1_≤

xQ

D

Therefore, from the assumptionat≤C2tηwe have for anyη >0

Tat≤Cλ, ηtη. 2.84

Acknowledgments

The authors thank the referees for many kind suggestions and comments. The first author was supported by the Korea Research Foundation Grant funded by the Korean Government KRF-2008-C00017.

References

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