Asymptotic Behavior for a Class of Modified Potentials in a Half Space

Journal of Inequalities and Applications Volume 2010, Article ID 647627,14pages doi:10.1155/2010/647627

Research Article

Asymptotic Behavior for a Class of Modified

α

-Potentials in a Half Space

Lei Qiao

_{and Guantie Deng}

1_{Department of Mathematics and Information Science, Henan University of Finance and Economics,}

Zhengzhou 450002, China

2_{Laboratory of Mathematics and Complex Systems, School of Mathematical Science,}

Beijing Normal University, MOE, Beijing 100875, China

Correspondence should be addressed to Guantie Deng,[email protected]

Received 13 March 2010; Accepted 21 June 2010

Academic Editor: Shusen Ding

Copyrightq2010 L. Qiao and G. Deng. 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.

A class ofα-potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimatesRα,l,lx ox

β

n|x|l−2β2h|x|−1at infinity in the

upper-half space of then-dimensional Euclidean space, where the functionh|x|is a positive non-decreasing function on the interval0,∞satisfying certain conditions. This result generalizes the growth properties of analytic functions, harmonic functions, and superharmonic functions.

1. Introduction and Main Results

Let Rn_{n ≥ 2 denote the n-dimensional Euclidean space with points x}

x1, x2, . . . , xn−1, xn x, xn, where x ∈ Rn−1 and xn ∈ R. The boundary and closure of

an open Ω of Rn _{are denoted by ∂Ω and Ω, respectively. The upper half-space is the set}

H {x x, xn ∈ Rn; xn > 0}, whose boundary is∂H. We identify Rn with Rn−1×R

and Rn−1 _{with R}n−1 _{× {0}, writing typical pointsx, y ∈ R}n _{as x x, x}

n, y y, yn,

wherex x1, x2, . . . , xn−1, y y1, y2, . . . , yn−1 ∈ Rn−1 and putting x·y

_n

j1xjyj

x·yxnyn, |x|

√

x·x, |x|√x·x.

Forx∈Rn_{andr >0, letB}

nx, rdenote the open ball with center atxand radiusrin

Rn_.

It is well known that see, e.g., 1, Chapter 6 the positive powers of the Laplace operatorΔcan be defined by

whereα >0,fis a Schwarz function and

Ffξ fξ

Rn

fxe−ixξdx. 1.2

It follows that we can extend definition 1.1 to certain negative powers of −Δ,

−Δ−α/2_{for 0< α < nand define an operatorI}

αby

Iαf −Δ−α/2f F−1

|ξ|−αf, 1.3

where 0< α < nandfis a function in the Schwartz class.

IfIαis defined as the inverse Fourier transform of|ξ|−αin the sense of distributions,

one can show that

Iαx γα|x|α−n, 1.4

whereγαis a certain constantsee, e.g.,1, page 414for the exact value ofγα.

The functionIα is known as the Riesz kernel. It follows immediately from the rules

for manipulating Fourier transforms that any Schwartz functionf can be written as a Riesz potential,

fx Iαgx

Iα∗g

x γα

Rn

gy

x−y n−αdy, 1.5

where 0< α < nandg −Δα/2f.

This Riesz kernelIαinRninspired us to introduce the modified Riesz kernel forH. To

do this, we first set

Eαx

⎧ ⎨ ⎩

−log|x| ifαn2,

|x|α−n if 0< α < n.

1.6

LetGαx, ybe the modified Riesz kernel forH, that is,

Gα

x, yEα

x−y−Eα

x−y∗, x, y∈H, x /y, 0< α≤n, 1.7

where∗denotes reflection in the boundary plane∂Hjust asy∗ y1, y2, . . . , yn−1,−yn.

We define the kernel functionPαx, ywhenx∈Handy∈∂Hby

Pα

x, y ∂Gαx, y

∂yn

yn0 Cα

xn

x−y n−α2, 1.8

We remark thatG2x, yand P2x, yare the classical Green function and classical Poisson kernel forHrespectivelysee, e.g.,2, page 127.

Next we use the following modified kernel functionPα,mx, ydefined by

Pα,m x, y ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Pα

x, y if y <1,

Pα

x, y−

m−1

k0

Cαxn|x|k

y n−α2kC n−α2/2

k

x·y |x| y

if y ≥1, 1.9

wheremis a nonnegative integer;Cω_kt ω n−α/2 is the ultrasphericalor Gegenbauer polynomialssee3. The Gegenbauer polynomials come from the generating function

1−2 trr2−ω

∞

k0

Cω_ktrk, 1.10

where |r| < 1, |t| ≤ 1, and ω > 0. The coefficients Cω

kt are called the ultraspherical or

Gegenbauerpolynomials of degreekassociated withω, each functionCω_ktis a polynomial of degreekint. Here note thatP2,mx, yis the modified Poisson kernel inH, which has been

used by several authorssee, e.g.,4–8.

Motivated by this modified kernel function Pα,mx, y, it is natural to ask if the

functionGαx, ycan also be modified? In this paper, we give an affirmative answer to this

question.

First we consider the modified kernel function in caseαn2, which is defined by

En,l

x−y

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ En

x−y if y <1,

En

x−yR

logy−

l−1

k1

xk

kyk

if y ≥1. 1.11

In case 0< α < n, we define

Eα,l

x−y

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Eα

x−y if y <1,

Eα

x−y−

l−1

k0

|x|k y n−αkC

n−α/2

k

x·y |x| y

if y ≥1, 1.12

wherelis a nonnegative integer,x, y∈H, andx /y.

Then we define the modified kernel functionGα,lx, yby

Gα,l x, y ⎧ ⎨ ⎩

En,l1

x−y−En,l1

x−y∗ if αn2,

Eα,l1

x−y−Eα,l1

x−y∗ if 0< α < n.

Write

Gα,l

x, μ

H

Gα,l

x, ydμy,

Uα,mx, ν

∂H

Pα,m

x, ydνy,

1.14

where μresp.,ν is a nonnegative measure onHresp., ∂H. Here note thatGα,0x, μ is

nothing but the Green potential of general ordersee9–11. Following Fugledesee6, we set

ky, μ

E

ky, xdμx, kμ, x

E

ky, xdμy, 1.15

for a nonnegative Borel measurable functionkonRn_×Rn_{and a nonnegative measureμon a}

Borel setE⊂Rn_{. We define a capacityC} kby

CkE sup μRn, E⊂H, 1.16

where the supremum is taken over all nonnegative measuresμsuch thatSμ the support of

μis contained inEandky, μ≤1 for everyy∈H.

Forβ≤1 andδ≤1, we consider the functionkα,β,δdefined by

kα,β,δ

y, xx−nβyn−δGα

x, y forx, y∈H. 1.17

Ifβ δ 1, thenkα kα,1,1 is extended to be continuous onH×Hin the extended sense,

whereHH∪∂H.

Now we will discuss the behavior at infinity of the modified Green potential and modified Poisson integral in the upper-half space, respectively. For related results, we refer the readers to the papers by Mizutasee9, Siegel and Talvilasee8, and Mizuta and Shimomurasee7.

Theorem 1.1. Lethrbe a positive nondecreasing function on the interval0,∞such that

arβ−1_{hris nondecreasing on0,∞,} brβ−2_{hris nonincreasing on0,∞andlim}

r→ ∞rβ−2hr 0,

cthere exists a positive constantMsuch thath2r≤Mhrfor anyr >0.

Letμbe a nonnegative measure onHsatisfying

H

yδ nh y

1 y nl−α−βδ2

dμy<∞. 1.18

ilim|x| → ∞,x∈H−Exn−β|x|−l2β−2h|x|Gα,lx, μ 0;

ii∞_i12−in−αβδ_C

kα,β,δEi<∞,

whereE_i{x∈E: 2i_{≤ |x|<2}i1_}.

Corollary 1.2. Letμbe a nonnegative measure onHsatisfying

H

yn

1 y nl−α2dμ

y<∞. _1.19

Then there exists a Borel setE⊂Hwith properties

ilim|x| → ∞,x∈H−Exn−β|x|β−l−1Gα,lx, μ 0;

ii∞_i12−in−αβ1_C

kα,β,1Ei<∞, whereEi{x∈E: 2i≤ |x|<2i1}.

Theorem 1.3. Lethbe defined as inTheorem 1.1andνa nonnegative measure on∂Hsatisfying

∂H

h y

1 y nm−α−β3dν

y<∞. 1.20

Then there exists a Borel setF⊂Hwith properties

ilim|x| → ∞,x∈H−Fx−nβ|x|−m2β−2h|x|Uα,mx, ν 0;

i∞_i12−in−αβ1_C

kα,β,1Fi<∞, whereFi{x∈F: 2i≤ |x|<2i1}.

Remark 1.4. In the caseml,FE.

Corollary 1.5. Letνbe a nonnegative measure on∂Hsatisfying

∂H

1

1 y nl−α2dν

y<∞. _1.21

Then there exists a Borel setE⊂HsatisfyingCorollary 1.2(ii) such that

lim

|x| → ∞,x∈H−Ex

−β

n |x|β−l−1Uα,lx, ν 0. 1.22

We define the modifiedα-potentials onHby

Rα,l,mx Gα,l

x, μUα,mx, ν, 1.23

The following theorem follows readily from Theorems1.1and1.3.

Theorem 1.6. Lethbe defined as inTheorem 1.1andRα,l,lxdefined by1.23. Then there exists a

Borel setE⊂HsatisfyingCorollary 1.2(ii) such that

lim

|x| → ∞,x∈H−Ex

−β

n |x|−l2β−2h|x|Rα,l,lx 0. 1.24

Remark 1.7. In the caseh|x| |x|1−β_{0 ≤β ≤ 1, by usingLemma 2.5below, we can easily}

show thatCorollary 1.2iiwithα 2 means thatEisβ-rarefied at infinity in the sense of

12. In particular, This condition withα 2,β 1, andh|x| ≡ 1resp.,α 2,β 0, and

h|x| |x|means thatEis minimally thin at infinityresp., rarefied at infinityin the sense of13.

Theorem 1.6is the best possibility as to the size of the exceptional set. In fact we have the following result. The proof of it is essentially due to Mizutasee9, Theorem 2, so we omit the proof here.

Theorem 1.8. LetE⊂Hbe a Borel set satisfyingCorollary 1.2(ii),hdefined as inTheorem 1.1, and

Rα,l,lxdefined by1.23. Then we can find a nonnegative measureλdefined onHsatisfying

H

h y

1 y nl−α−β3dλ

y<∞, 1.25

such that

lim sup

|x| → ∞,x∈E

xn−β|x|−l2β−2h|x|Rα,l,lx ∞, _1.26

wheredλy yndμyy∈Handdλy dνyy∈∂H.

2. Some Lemmas

Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 2.1. There exists a positive constantMsuch thatGαx, y≤Mxnyn/|x−y|n−α2,where

0< α≤n, x x1, x2, . . . , xn, andy y1, y2, . . . , yninH.

This can be proved by simple calculation.

Lemma 2.2. Gegenbauer polynomials have the following properties: i|Cω

kt| ≤Ckω1 Γ2ωk/Γ2ωΓk1, |t| ≤1;

ii d/dtCω_kt 2ωC_kω₋₁1t, k≥1;

iii∞_k0Cω_k1rk _1−r−2ω_;

Proof. iandiican be derived from3.iiifollows by takingt 1 in1.10;ivfollows byi,iiand the Mean Value Theorem for Derivatives.

Lemma 2.3. Letlbe a nonnegative integer andx,y ∈Rn_{α n 2, then one has the following}

properties:

i|Il_k0xk_/yk1_{| ≤}l−1

k02kxn|x|k/|y|k2; ii|I∞_k0xkl1_/yk_{| ≤2}l1_x

n|x|l;

iii|Gn,lx, y−Gnx, y| ≤Mlk1kxnyn|x|k−1/|y|k1;

iv|Gn,lx, y| ≤M∞kl1kxnyn|x|k−1/|y|k1.

Lemma 2.4see14. Letmbe a nonnegative integer andM >0.

iIf1≤ |y| ≤ |x|/2, then|Pα,mx, y| ≤Mxn|x|m−1/|y|nm−α1.

iiIf|y| ≥2|x|and|y| ≥1, then|Pα,mx, y| ≤Mxn|x|m/|y|nm−α2.

The following lemma can be proved by using Fuglede6, Th´eor`em 7.8.

Lemma 2.5. For any Borel setEinH, we haveCkα,β,1E Ckα,β,1Eand

Ckα,β,δE infλH

resp. infλH ifδ <1 resp., δ1, 2.1

where the infimum is taken over all nonnegative measuresλonH (resp.,H) such thatkα,β,δλ, x≥1

for everyx∈E.

3. Proof of

Theorem 1.1

For any1 >0, there existsR1>2 such that

{y∈H,|y|≥R₁}

yδ nh y

1 y nl−α−βδ2dμ

y< 2. 3.1

For fixedx∈Hand|x| ≥2R1, we write

Gα,l

x, μ

H1

Gα

x, ydμy

H2

Gα

x, ydμy

H3

Gα,l

x, y−Gα

x, ydμy

H4

Gα,l

x, ydμy

H5

Gα

x, ydμy

H6

Gα,l

x, y−Gα

x, ydμy

H7

Gα,l

x, ydμy

V1x V2x V3x V4x V5x V6x V7x,

where

H1

y∈H: y ≥R1, x−y ≤

|x|

2

, H2

y∈H: y ≥R1,

|x|

2 < x−y ≤3|x|

,

H3

y∈H: y ≥R1, x−y ≤3|x|

, H4

y∈H: y ≥R1, x−y >3|x|

,

H5 H6

y∈H: 1≤ y < R1

, H7

y∈H: y <1.

3.3

We distinguish the following two cases.

Case 10< α < n. Note thatV1x xβn

H1kα,β,δy, xy

δ

ndμy.In view of1.18, we can find

a sequence{ai}of positive numbers such that limi→ ∞ai∞and∞i1aibi<∞, where

bi

{y∈H:2i−1_<|y|<2i2_}

yδ nh y

y nl−α−βδ2dμ

y. 3.4

Consider the sets

E_i

x∈H: 2i≤ |x|<2i1, x−nβV1x≥a−i12il−2β2h

2i1−1

, 3.5

fori1,2, . . . .Ifωis a nonnegative measure onHsuch thatSω ⊂E_iandkα,β,δy, ω≤1 for

y∈H, then we have

H

dω

≤ai2−il−2β2h

2i1 _x−β

n V1xdωx

ai2−il−2β2h

2i1

{y∈H:2i−1_<|y|<2i2_}

kα,β,δ

y, ωy_nδdμy

≤Mai2−il−2β2h

2i1

{y∈H:2i−1_<|y|<2i2_}

yδ_ndμy

Mai2−il−2β2h

2i1

{y∈H:2i−1_<|y|<2i2_}

y 2−β h y

y nl−α2

1 y 2−δ

yδ nh y

1 y nl−α−βδ2dμ

y

≤M2−il−2β22i22−β2i2nl−α22−i2−δ

{y∈H:2i−1_<|y|<2i2_}

h y _y nm−α−β3dμ

y

≤M2in−αβδaibi.

So that

Ckα,β,δ

E_i≤M2in−αβδaibi, 3.7

which yields

∞

i1

2−in−αβδCkα,β,δ

E_i<∞. 3.8

SettingE∞_i1E_i, we see thatTheorem 1.1iiis satisfied and

lim sup

|x| → ∞,x∈H−Ex

−β

n |x|−l2β−2h|x|V1x≤lim sup i→ ∞ a

−1

i 0. 3.9

Moreover byLemma 2.1,

|V2x| ≤Mxn

H2

yn

_x−y n−α2dμ

y

≤Mxn|x|α−n−2

H2

_y 2−β

h y y

nl−α1 yδnh y

1 y nl−α−βδ2dμ

y

≤M2xn|x|l−β1h4|x|−1.

3.10

Note thatCω

0t ≡ 1. By iiiandivin Lemma 2.2, we taket x·y/|x||y|,t∗ x·y∗/|x||y∗|inLemma 2.2ivand obtain

|V3x| ≤

H3

l

k1

|x|k

_y n−αk2n−αC

n−α2/2

k−1 1

xnyn

|x| y dμ

y

≤Mxn|x|l−1 l−1

k1

1 4k−1C

n−α2/2

k−1 1

H3

_y 2−β

h y

yδ nh y

1 y nl−α−βδ2dμ

y

≤M1xn|x|l−β1h4|x|−1.

3.11

Similarly, we have byiiiandivinLemma 2.2

|V4x| ≤

H4 ∞

kl1

|x|k

_y n−αk2n−αC

n−α2/2

k−1 1

xnyn

|x| y dμ

y

≤Mxn|x|l

∞

kl1

1 2k−1C

n−α2/2

k−1 1

H4

_y 1−β

h y yδ

nh y

1 y nl−α−βδ2dμ

y

≤M1xn|x|l−β1h2|x|−1.

ByLemma 2.1, we have

|V5x| ≤Mxn

H5

yn

x−y n−α2dμ

y

≤Mxn|x|α−n−2

H5

y 2−β h y y

nl−α1 yδnh y

1 y nl−α−βδ2dμ

y

≤Mxn|x|l−1R2−1βhR2 −1_.

3.13

Similarly asV3x, we obtain

|V6x| ≤

H6

l

k1

|x|k

y n−αk2n−αC

n−α2/2

k−1 1

xnyn

|x| y dμ

y

≤Mxn l

k1

C_kn₋−₁α2/21|x|k−1Rl−k1

2

H6

y 1−β h y

yδ nh y

1 y nl−α−βδ2

dμy

≤MRl₁xn|x|l−1h1−1.

3.14

Finally, byLemma 2.1, we have

|V7x| ≤Mxn|x|l−1h1−1. 3.15

Combining3.9–3.15, we prove Case1.

Case 2αn2. In this case, the growth estimates ofV1x,V2x,V5xandV7xcan be proved similarly as in Case1. Inequations3.9,3.10,3.13and3.15still hold.

Moreover we have byLemma 2.3iii

|V3x| ≤M

H3

l

k1

kxnyn|x|k−1

_y k1 y

l1 y 2−β

h y yδ

nh y

1 y l−βδ2dμ

y

≤Mxn|x|l−1 l

k1

k

4k−1

H3

_y 2−β

h y yδ

nh y

1 y l−βδ2dμ

y

≤M1xn|x|l−β1h4|x|−1.

ByLemma 2.3iv, we have

|V4x| ≤M

H4 ∞

kl1

kxnyn|x|k−1

_y k1 y

l2 y 1−β

h y yδ

nh y

1 y l−βδ2dμ

y

≤Mxn|x|l

∞

kl1

k

2k−1

H4

_y 1−β

h y yδ

nh y

1 y l−βδ2dμ

y

≤M1xn|x|l−β1h2|x|−1.

3.17

Similarly asV3x, we have

|V6x| ≤MRl₁xn|x|l−1h1−1. 3.18

Combining3.9,3.10,3.13,3.15, and3.16–3.18, we prove Case2.

Hence we complete the proof ofTheorem 1.1.

4. Proof of

Theorem 1.3

For any2 >0, there existsR2>2 such that

{y∈∂H,|y|≥R₂}

h y

1 y nm−α−β3dν

y< 2. 4.1

For fixedx∈Hand|x| ≥2R2, we write

Uα,mx, ν

G1

Pα,m

x, ydνy

G2

Pα,m

x, ydνy

G3

Pα,m

x, y−Pα

x, ydνy

G4

Pα

x, ydνy

G5

Pα,m

x, ydνy

U1x U2x U3x U4x U5x,

4.2

where

G1

y∈∂H: y <1, G2

y∈∂H: 1≤ y < |x|

2

,

G3G4

y∈∂H: |x| 2 ≤ y

_{<2|x|, G}

5

y∈∂H: y ≥2|x|.

First note that

|U1x| ≤Mxn

_| x|

2

α−n−2

G1

dνy

≤Mxn|x|α−n−2

G1

_y 2−β

h y y

nm−α1 h y

1 y nm−α−β3ν

y

≤Mxn|x|m−1h1−1.

4.4

Write

U2x U21x U22x, 4.5

where

U21x

G2Bn−10,R2

Pα,m

x, ydνy,

U22x

G2−Bn−10,R2

Pα,m

x, ydνy.

4.6

We obtain byLemma 2.4i

|U2x| ≤Mxn|x|m−1

G2

1

_y nm−α1dν

y

≤Mxn|x|m−1

G2

_y 2−β

h y

h y

1 y nm−α−β3ν

y.

4.7

For|x|>2R2, by4.7we have

|U21x| ≤Mxn|x|m−1R22−βhR2

−1_{. 4.8}

On the other hand,4.7yields that

|U22x| ≤M2xn|x|m−β1h

_| x|

2

−1

. 4.9

Combining4.8and4.9, we have

lim

|x| → ∞,x∈Hx

−β

We have byLemma 2.2iii

|U3x| ≤M

G3

m−1

k0

xn|x|k

y n−α2kC

n−α2/2

k 1dν

y

≤Mxn|x|m m−1

k0

1 2kC

n−α2/2

k 1

G3

y 1−β h y

h y y nm−α−β3dν

y

≤Mεxn|x|m−β1h

_| x|

2

−1

.

4.11

ByLemma 2.4ii, we obtain

|U5x| ≤Mxn|x|m

G5

1

y nm−α2dν

y

≤Mxn|x|m

G5

y 1−β h y

h y

1 y nm−α−β3ν

y

≤M2xn|x|m−β1h2|x|−1.

4.12

Note thatU4x xβn

G4kα,β,1y

_{, xdνy. By the lower semicontinuity ofk}

α,β,1y, x,

we can prove the following fact in the same way asV1xin the proof ofTheorem 1.1:

lim sup

|x| → ∞,x∈H−F

x−nβ|x|−m2β−2h|x|U4x 0, 4.13

whereF∞_i1Fi, Fi{x∈F: 2i≤ |x|<2i1}, and∞i12−in−αβ1Ckα,β,1Fi<∞.

Combining4.4and4.10–4.13, we complete the proof ofTheorem 1.1.

Acknowledgments

This work is supported by The National Natural Science Foundation of China under Grant 10671022 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060027023.

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