Asymptotic Behavior for a Class of Modified α-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

1Department of Mathematics and Information Science, Henan University of Finance and Economics, Zhengzhou 450002, China

2Laboratory 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^βn|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 Rⁿn ≥ 2 denote the n-dimensional Euclidean space with points x x1, x2, . . . , x_n−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 Rⁿ⁻¹ × {0}, writing typical pointsx, y ∈ Rⁿ as x x, x_n, y y, y_n, wherex x1, x₂, . . . , x_n−1, y y1, y₂, . . . , y_n−1 ∈ Rⁿ⁻¹ and putting x·y _n

j1x_jy_j x·yx_ny_n, |x|√

x·x, |x|√ x·x.

Forx∈Rⁿandr >0, letBnx, 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

−Δ^α/2fx F⁻¹

|ξ|^αfξ

, 1.1

whereα >0,fis a Schwarz function and

Ffξ fξ

Rⁿfxe^−ixξdx. 1.2

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

−Δ^−α/2for 0< α < nand define an operatorIαby Iαf −Δ^−α/2f F⁻¹

|ξ|^−αf

, 1.3

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

IfI_αis defined as the inverse Fourier transform of|ξ|^−αin the sense of distributions, one can show that

I_αx γ_α|x|^α−n, 1.4

whereγ_αis a certain constantsee, e.g.,1, page 414for the exact value ofγ_α.

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 I_αgx I_α∗g

x γ_α

Rⁿ

g y

x−y ^n−αdy, 1.5 where 0< α < nandg −Δ^α/2f.

This Riesz kernelI_αinRⁿinspired us to introduce the modified Riesz kernel forH. To do this, we first set

E_αx

⎧⎨

⎩

−log|x| ifαn2,

|x|^α−n if 0< α < n. 1.6

LetG_αx, ybe the modified Riesz kernel forH, that is, G_α

x, y E_α

x−y

−E_α x−y^∗

, x, y∈H, x /y, 0< α≤n, 1.7 where∗denotes reflection in the boundary plane∂Hjust asy^∗ y1, y2, . . . , y_n−1,−yn.

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

Pα

x, y

∂Gαx, y

∂yn

yn0

Cα

x_n

x−y ^n−α2, 1.8 whereC_α2n−αif 0< α < nand2 ifαn2.

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 functionP_α,mx, ydefined by

P_α,m x, y

⎧⎪

⎪⎨

⎪⎪

⎩ Pα

x, y

if y <1, P_α

x, y

−^m−1

k0

C_αx_n|x|^k

y ^n−α2kC^n−α2/2_k

x·y

|x| y

if y ≥1, 1.9

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

1−2 trr²_−ω ^∞

k0

C^ω_ktr^k, 1.10

where |r| < 1, |t| ≤ 1, and ω > 0. The coefficients C^ω_kt are called the ultraspherical or Gegenbauerpolynomials of degreekassociated withω, each functionC^ω_ktis a polynomial of degreekint. Here note thatP_2,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 functionG_αx, 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

E_n,l x−y

⎧⎪

⎪⎨

⎪⎪

⎩ E_n

x−y

if y <1, E_n

x−y R

logy−^l−1

k1

x^k ky^k

if y ≥1. 1.11 In case 0< α < n, we define

E_α,l x−y

⎧⎪

⎪⎨

⎪⎪

⎩ E_α

x−y

if y <1, E_α

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 functionG_α,lx, yby

G_α,l x, y

⎧⎨

⎩ E_n,l1

x−y

−E_n,l1 x−y^∗

if αn2, E_α,l1

x−y

−E_α,l1 x−y^∗

if 0< α < n. 1.13

Write

G_α,l x, μ

H

G_α,l x, y

dμ y

, U_α,mx, ν

∂H

P_α,m x, y

dν y

,

1.14

where μresp.,ν is a nonnegative measure onHresp., ∂H. Here note thatG_α,0x, μ is nothing but the Green potential of general ordersee9–11.

Following Fugledesee6, we set

k y, μ

E

k y, x

dμx, k

μ, x

E

k y, x

dμ y

, 1.15

for a nonnegative Borel measurable functionkonRⁿ×Rⁿand a nonnegative measureμon a Borel setE⊂Rⁿ. We define a capacityCkby

C_kE sup μRⁿ, E⊂H, 1.16

where the supremum is taken over all nonnegative measuresμsuch thatS_μ 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, x

x^−β_n y_n^−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 ar^β−1hris nondecreasing on0,∞,

br^β−2hris nonincreasing on0,∞and limr→ ∞r^β−2hr 0,

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

Letμbe a nonnegative measure onHsatisfying

H

y_n^δh y

1 y ^{nl−α−βδ2}dμ y

<∞. 1.18

Then there exists a Borel setE⊂Hwith properties

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

ii_∞

i12^{−in−αβδ}C_kα,β,δE_i<∞, whereE_i{x∈E: 2ⁱ≤ |x|<2ⁱ¹}.

Corollary 1.2. Letμbe a nonnegative measure onHsatisfying

H

y_n

1 y ^nl−α2dμ y

<∞. 1.19

Then there exists a Borel setE⊂Hwith properties ilim|x| → ∞,x∈H−Ex^−βn |x|^β−l−1Gα,lx, μ 0;

ii_∞

i12^{−in−αβ1}Ckα,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−β3}dν y

<∞. 1.20

Then there exists a Borel setF⊂Hwith properties

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

i_∞

i12^{−in−αβ1}C_kα,β,1Fi<∞, whereF_i{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

|x| → ∞,x∈H−Elim x^−β_n |x|^β−l−1U_α,lx, ν 0. 1.22

We define the modifiedα-potentials onHby R_α,l,mx G_α,l

x, μ

U_α,mx, ν, 1.23 where 0< α≤nandμ(resp.,ν) is a nonnegative measure onH(resp.,∂H) satisfying1.18(δ1) (resp.,1.20). Clearly,R_2,0,mxis a superharmonic function onH.

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

Theorem 1.6. Lethbe defined as inTheorem 1.1andR_α,l,lxdefined by1.23. Then there exists a Borel setE⊂HsatisfyingCorollary 1.2(ii) such that

|x| → ∞,x∈H−Elim x^−β_n |x|^−l2β−2h|x|Rα,l,lx 0. 1.24

Remark 1.7. In the caseh|x| |x|^1−β0 ≤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 R_α,l,lxdefined by1.23. Then we can find a nonnegative measureλdefined onHsatisfying

H

h y

1 y ^{nl−α−β3}dλ y

<∞, 1.25

such that

lim sup

|x| → ∞,x∈E

x_n^−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 thatG_αx, y≤Mxny_n/|x−y|^n−α2,where 0< α≤n, x x1, x₂, . . . , x_n, andy y1, y₂, . . . , y_ninH.

This can be proved by simple calculation.

Lemma 2.2. Gegenbauer polynomials have the following properties:

i|C^ω_kt| ≤C_k^ω1 Γ2ωk/Γ2ωΓk1, |t| ≤1;

ii d/dtC^ω_kt 2ωC^ω1_k−1t, k≥1;

iii_∞

k0C^ω_k1r^k 1−r^−2ω;

iv|C^n−α/2_k t−C^n−α/2_k t^∗| ≤n−αC^n−α2/2_k−1 1|t−t^∗|, |t| ≤1, |t^∗| ≤1.

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 ∈Rⁿα n 2, then one has the following properties:

i|I_l

k0x^k/y^k1| ≤_l−1

k02^kxn|x|^k/|y|^k2; ii|I_∞

k0x^kl1/y^k| ≤2^l1x_n|x|^l; iii|Gn,lx, y−Gnx, y| ≤M_l

k1kxnyn|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.

iIf 1≤ |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 haveC_kα,β,1E C_kα,β,1Eand

C_kα,β,δE infλH

resp. infλ

H

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}

y_n^δh y

1 y ^{nl−α−βδ2}dμ y

< ₂. 3.1

For fixedx∈Hand|x| ≥2R₁, we write

G_α,l x, μ

H1

G_α x, y

dμ y

H2

G_α x, y

dμ y

H3

G_α,l x, y

−G_α x, y

dμ y

H4

Gα,l

x, y dμ

y

H5

Gα

x, y dμ

y

H6

Gα,l

x, y

−Gα

x, y dμ

y

H7

Gα,l

x, y dμ

y

V1x V2x V3x V4x V5x V6x V7x,

3.2

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 x^β_n

H1k_α,β,δy, xy_n^δdμy.In view of1.18, we can find a sequence{ai}of positive numbers such that lim_{i→ ∞}ai∞and_∞

i1aibi<∞, where

b_i

{y∈H:2ⁱ⁻¹<|y|<2ⁱ²}

y^δ_nh y y ^{nl−α−βδ2}dμ

y

. 3.4

Consider the sets

E_i

x∈H: 2ⁱ≤ |x|<2ⁱ¹, x^−β_n V1x≥a⁻¹_i 2^il−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

≤a_i2^−il−2β2h

2ⁱ¹ x_n^−βV₁xdωx a_i2^−il−2β2h

2ⁱ¹

{y∈H:2ⁱ⁻¹<|y|<2ⁱ²}k_α,β,δ y, ω

y_n^δdμ y

≤Ma_i2^−il−2β2h 2ⁱ¹

{y∈H:2ⁱ⁻¹<|y|<2ⁱ²}y^δ_ndμ y

Mai2^−il−2β2h 2ⁱ¹

{y∈H:2ⁱ⁻¹<|y|<2ⁱ²}

y ^2−β

h y y ^nl−α2

1 y ^2−δ y^δ_nh y

1 y ^{nl−α−βδ2}dμ y

≤M2^−il−2β22^i22−β2^i2nl−α22^−i2−δ

{y∈H:2ⁱ⁻¹<|y|<2ⁱ²}

h y y ^{nm−α−β3}dμ

y

≤M2^in−αβδaibi.

3.6

So that

C_kα,β,δ E_i

≤M2^in−αβδa_ib_i, 3.7

which yields

∞ i1

2^{−in−αβδ}Ckα,β,δ

E_i

<∞. 3.8

SettingE_∞

i1E_i, we see thatTheorem 1.1iiis satisfied and lim sup

|x| → ∞,x∈H−Ex_n^−β|x|^−l2β−2h|x|V1x≤lim sup

i→ ∞ a⁻¹_i 0. 3.9

Moreover byLemma 2.1,

|V2x| ≤Mx_n

H2

y_n

x−y ^n−α2dμ y

≤Mx_n|x|^α−n−2

H2

y ^2−β

h y y ^nl−α1 y^δ_nh y

1 y ^{nl−α−βδ2}dμ y

≤M2xn|x|^l−β1h4|x|⁻¹.

3.10

Note thatC₀^ωt ≡ 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 1xnyn

|x| y dμ y

≤Mx_n|x|^l−1l−1

k1

1

4^k−1C_k−1^n−α2/21

H3

y ^2−β

h y y_n^δh y

1 y ^{nl−α−βδ2}dμ 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 1xnyn

|x| y dμ y

≤Mx_n|x|^l∞

kl1

1

2^k−1C^n−α2/2_k−1 1

H4

y ^1−β

h y y^δ_nh y

1 y ^{nl−α−βδ2}dμ y

≤M1xn|x|^l−β1h2|x|⁻¹.

3.12

ByLemma 2.1, we have

|V5x| ≤Mx_n

H5

yn

x−y ^n−α2dμ y

≤Mx_n|x|^α−n−2

H5

y ^2−β

h y y ^nl−α1 y^δ_nh y

1 y ^{nl−α−βδ2}dμ y

≤Mxn|x|^l−1R^2−β₁ hR2⁻¹.

3.13

Similarly asV₃x, we obtain

|V6x| ≤

H6

l k1

|x|^k

y ^n−αk2n−αC_k−1^n−α2/21x_ny_n

|x| y dμ y

≤Mx_n l k1

C_k−1^n−α2/21|x|^k−1R^l−k1₂

H6

y ^1−β

h y y^δ_nh y

1 y ^{nl−α−βδ2}dμ y

≤MR^l₁x_n|x|^l−1h1⁻¹.

3.14

Finally, byLemma 2.1, we have

|V7x| ≤Mx_n|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 y^δ_nh y 1 y ^l−βδ2dμ

y

≤Mx_n|x|^l−1l

k1

k 4^k−1

H3

y ^2−β

h y y^δ_nh y 1 y ^l−βδ2dμ

y

≤M1xn|x|^l−β1h4|x|⁻¹.

3.16

ByLemma 2.3iv, we have

|V4x| ≤M

H4

∞ kl1

kxnyn|x|^k−1

y ^k1 y ^l2 y ^1−β

h y y^δ_nh y 1 y ^l−βδ2dμ

y

≤Mx_n|x|^l∞

kl1

k 2^k−1

H4

y ^1−β

h y y_n^δh y 1 y ^l−βδ2dμ

y

≤M₁x_n|x|^l−β1h2|x|⁻¹.

3.17

Similarly asV3x, we have

|V6x| ≤MR^l₁xn|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−β3}dν y

< 2. 4.1

For fixedx∈Hand|x| ≥2R₂, we write

U_α,mx, ν

G1

P_α,m x, y

dν y

G2

P_α,m x, y

dν y

G3

P_α,m x, y

−P_α x, y

dν y

G4

P_α x, y

dν y

G5

P_α,m x, y

dν y

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|

, G₅

y∈∂H: y ≥2|x|

.

4.3

First note that

|U1x| ≤Mx_n |x|

2

_α−n−2

G1

dν y

≤Mx_n|x|^α−n−2

G1

y ^2−β

h y y ^nm−α1 h y

1 y ^{nm−α−β3}ν y

≤Mxn|x|^m−1h1⁻¹.

4.4

Write

U2x U21x U22x, 4.5

where

U₂₁x

G2B_n−10,R2P_α,m x, y

dν y

,

U₂₂x

G2−Bn−10,R2P_α,m x, y

dν y

.

4.6

We obtain byLemma 2.4i

|U2x| ≤Mx_n|x|^m−1

G2

y ^nm−α11 dν y

≤Mx_n|x|^m−1

G2

y ^2−β

h y h y

1 y ^{nm−α−β3}ν y

.

4.7

For|x|>2R₂, by4.7we have

|U21x| ≤Mxn|x|^m−1R^2−β₂ hR2⁻¹. 4.8 On the other hand,4.7yields that

|U22x| ≤M2xn|x|^m−β1h |x|

2 ₋₁

. 4.9

Combining4.8and4.9, we have

|x| → ∞,x∈Hlim x_n^−β|x|^−m2β−2h|x|U2x 0. 4.10

We have byLemma 2.2iii

|U3x| ≤M

G3

m−1

k0

xn|x|^k

y ^n−α2kC^n−α2/2_k 1dν y

≤Mx_n|x|^mm−1

k0

1

2^kC^n−α2/2_k 1

G3

y ^1−β

h y h y y ^{nm−α−β3}dν

y

≤Mεxn|x|^m−β1h |x|

2 ₋₁

.

4.11

ByLemma 2.4ii, we obtain

|U5x| ≤Mxn|x|^m

G5

y ^nm−α21 dν y

≤Mxn|x|^m

G5

y ^1−β

h y h y

1 y ^{nm−α−β3}ν y

≤M₂x_n|x|^m−β1h2|x|⁻¹.

4.12

Note thatU₄x x^β_n

G4k_α,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−Fx^−β_n |x|^−m2β−2h|x|U4x 0, 4.13 whereF_∞

i1Fi, Fi{x∈F: 2ⁱ≤ |x|<2ⁱ¹}, and_∞

i12^{−in−αβ1}Ckα,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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