Ostrowski type inequalities related to the generalized Baouendi-Grushin vector fields

Ostrowski Type Inequalities Related to the Generalized Baouendi-Grushin Vector Fields

JINGBODOU(*) - YAZHOUHAN(**)

ABSTRACT- In this paper, we employ a new method to prove a representation for- mula related to the generalized Baouendi-Grushin vector fields, and then the Ostrowski type inequalities is established in the ball and bounded domain, re- spectively, via the representation formula andL¹norm of the horizontal gra- dient. In addition, in the same spirit, we show the Hardy inequalities with boundary term related to the generalized Baouendi-Grushin vector fields.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 26D10, 35J70.

KEYWORDS. Generalized Baouendi-Grushin vector fields; representation formula;

Ostrowski type inequality; Hardy inequality.

1. Introduction

In 1938, A. Ostrowski [21] established the following sharp integral in- equality

1 b a

Z^b

a

f(y)dy f(x)

1

4‡(x ^a‡b₂ )² (b a)² (b a)

kf⁰k₁ (1:1)

for f 2C¹[a;b] and x2[a;b], which is known as the Ostrowski type in- equality. Because the Ostrowski type inequality is useful in some fields, it

(*) Indirizzo dell'A.: School of Statistics, XI'AN University of Finance and Economics, XI'AN, Shaanxi, 710100, P. R. China.

E-mail: djbmn@126.com

(**) Indirizzo dell'A.: Department of Mathematics, College of Science, China Jiliang University, Hangzhou, 310018, P. R. China.

E-mail: yazhou.han@gmail.com

was extended from intervals to rectangles and general domains inR^N(see [3] and [4]). Moreover, many authors improved inequality (1.1) to the higher-order derivatives case (see [5, 17, 22] and the references therein).

Recently, Lian and Yang [16] obtained the Ostrowski type inequality on the H-type group by the representation formula.

Our main interesting in this paper is to prove the Ostrowski type in- equality related to the generalized Baouendi-Grushin (B-G) vector fields by an improved method. To do this, we first introduce the generalized B-G vector fields.

Let xˆ(x₁;x₂; ;x_n)2Rⁿ;yˆ(y₁;y₂; ;y_m)2R^m;a>0: The generalized B-G vector fields is defined as

Z_iˆ @

@x_i; Z_n‡jˆ jxj^a @

@y_j (iˆ1;2;. . .;n;jˆ1;2;. . .;m):

(1:2)

Denote the horizontal gradient

r_Lˆ(Z1;. . .;Zn;Zn‡1. . .;Zn‡m);

and div_L(u₁;u₂;. . .;u_n;u_n‡1 . . .;u_n‡m)ˆ^n‡mP

iˆ1 Z_iu_i denotes the general- ized divergence. Then the second order degenerate elliptic operator andp- degenerate sub-elliptic operator are defined as

L_aˆD_x‡ jxj^2aD_yˆ^n‡mX

iˆ1

Z_iˆ r_L r_L; and L_p;auˆdiv_L jr_Luj^{p 2}r_Lu

ˆ r_L jr_Luj^{p 2}r_Lu

; p>1;

respectively, whereD_x andD_y are Laplace operators onRⁿ andR^m, re- spectively.

La is usually called the B-G operator. This second order operator be- longs to the wide class of sub-elliptic operators introduced and studied by Franchi and Lanconelli in [13, 15]. Whenx6ˆ0,L_ais elliptic and becomes degenerate on the manifoldf0g R^m. Moreover, Foraˆ1, the B-G op- erator can be viewed as Tricomi operator for transonic flow restricted to subsonic regions.

In particular, Caffarelli and Silvestre [6] established an equivalent defi- nition of the fractional LaplacianD^s;s2(0;1), which can be written to a nondivergence form asL_awithmˆ1 by the suitable change of variables.

Thus it seems conveniently to study fractional Laplacian equation in some sense. Furthermore, a series of results obtained for B-G operator can be carried over to fractional Laplacian. For example, the fundamental solution of equationLauˆ0 can apply to the fractional Laplacian problem, see [6, 19].

In recent years, the integral inequalities and partial differential equa- tions related to the generalized B-G vector fields have been paid much attention by many scholars (see, e.g., [8, 9, 11, 10, 19, 20]).

To state our main results, we need also some notations. There exists a natural family of anisotropic dilations attached toL_a;i.e.,

dl(x;y)ˆ(lx;l^a‡1y); l>0; (x;y)2R^n‡m: (1:3)

The homogeneous dimension with respect to the dilations (1.3) is Qˆn‡(a‡1)m:

The distance function is written as

d:ˆd(x;y)ˆ j(x;y)j ˆ(jxj^2(a‡1)‡(a‡1)²jyj²)^2(a‡1)1 : Takejˆ(x;y)2R^n‡m;jjj ˆd(j);xˆ x

jjj;yˆ y

jjj^a‡1;jˆ(x;y);and write rjˆdr(j)ˆdr(x;y). The open ball of radius R and centered at (0;0)2 R^n‡m is denoted by BL(R)ˆBL((0;0);R)ˆ f(x;y)2R^n‡mjd(x;y)5Rg. Let

S:ˆ f(x;y)2R^n‡mjd(x;y)ˆ1gbe the B-G unit sphere andjSjbe the vo-

lume of theS. Obviously,j2S. Forf 2C(BL(R));let

~f(r)ˆ 1 jSj

Z

S

f(rj)dm(j)

be the avenge of f over the B-G sphere, here 05r5R;dm is the surface measureonS.DenoteN(f)ˆ kf ~fk₁ˆ kf ~fk_L1(BL(R))ˆ sup

j2BL(R)jf ~f(r)j.

One of the main results in this paper is

THEOREM 1.1. Assume that f 2C¹(B_L(R)) and 05a5n. Then for jˆrj, there holds

f(j) 1 jB_L(R)j

Z

BL(R)

f(h)dh

N(f)‡G _2(a‡1)^{n a}

G _2(a‡1)^Q G _2(a‡1)ⁿ

G _2(a‡1)^{Q a}

Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kr_Lfk₁: (1:4)

It seems that the estimate (1.4) is not sharp since jr_Ldj 6ˆ1 with (x;y)6(x;0) (see next section).

Suppose thatf(j)ˆf(d(j)) is the radial function. Then, we can improve the result of Theorem 1.1 as follows.

COROLLARY 1.2. Let f 2C¹(BL(R)) be a radial function. Then for jˆrj, it holds

(1:5)

f(j) 1 jB_L(R)j

Z

BL(R)

f(h)dh

N(f)‡

Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kf⁰k₁:

LetVR^n‡mbe a bounded domain, and let Lip(V)ˆ fu2C(V): ju(x) u(y)j Kjx yj

for someK>0 and for anyx;y2Vg:

We define

C¹₀(V)ˆ fu2C¹(V);rLuˆ0 on@Vg;

and

W^1;1(V):ˆ fu2C¹(V);r_Lu2Lip(V)g:

Iff 2W^1;1(V)\C¹₀(V), we can extend the functionf by zero toFonBL(R), the smallest ball centered at the origin and containing V. We state the following theorem.

THEOREM 1.3. Assume that 05a5n and f 2W^1;1(V)\C¹₀(V).

Let RˆinffR₀>0jVB_L(R₀)g. Then for all jˆrj2V and F2W^1;1(BL(R)),

f(j) 1 jVj

Z

V

f(h)dh

N(F)‡

1 jVj

jBL(R)j 1

jVj Z

V

f(h)dh

‡G _2(a‡1)^{n a}

G _2(a‡1)^Q G _2(a‡1)ⁿ

G _2(a‡1)^{Q a} Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kr_Lfk_L1(V): (1:6)

Note that Liu and Luan [18] established the Ostrowski type inequality associated with Carnot-CaratheÂodory distance dcc on the Grushin plane (aˆ2;nˆmˆ1). Since jrLdccj ˆ1, the estimate (1.4) is sharp for Grushin plane.

A key process in the proof of our results is to establish the re- presentation formula related to the generalized B-G vector fields. We employ a new technique to prove the representation formula which is different from Lianet al.[16] and Donget al.[12], bases on the appropriate use of a suitable vector field and elementary calculations.

We also can obtain the following Hardy inequalities with boundary term by the above mentioned technique. It has the advantage that it

allows us to compute explicit constants for the remainder term. We point out that our method is different from M. Zhu and Z. Wang [23] and Adimurthiet al. [1, 2].

Let D^1;p₀ (V) be the completion of C¹₀ (V) with respect to the norm k ku _D1;pˆ R

Vjr_Luj^pdj_p¹ .

THEOREM 1.4. Let R>0;15p51 and p6ˆQ. Then for all u2D^1;p(R^n‡m),

Z

BL(R)

jr_Luj^pdj Q p

p ^p Z

BL(R)

c_pajuj^p d^p dj

Q p

p

Q p p

^{p 2} 1 R^{p 1}

Z

@BL(R)

c_pa juj^p

jrdjdH_{Q 1}; (1:7)

and

Z

B^c_L(R)

jr_Luj^pdj Q p

p ^p Z

B^c_L(R)

c_pajuj^p d^p dj

Q p p

Q p p

^{p 2} 1 R^{p 1}

Z

@BL(R)

c_pa juj^p

jrdjdH_{Q 1}; (1:8)

whereB^c_L(R):ˆR^n‡mnBL(R);dH_{Q 1}is the (Q 1)-dimensional Hausdorff measure and r is usual gradient onR^n‡m. For further improvement of these inequalities see Sec. 4.

REMARK 1.5. The Hardy inequalities related to the generalized B-G vector fields have been established in space C₀¹(V) or C¹₀ (R^n‡m), see [8, 9, 11]. In this work, we extend these inequalities to D^1;p(V) or D^1;p(R^n‡m).

This paper is organized as follows. In section 2, we give some elemen- tary facts and provide a representation formula related to the generalized B-G vector fields for function without compact support. In section 3, we prove Theorem 1.1 and 1.3. In section 4, we derive a family of Hardy in- equalities with boundary term.

2. Arepresentation formula related to the generalized B-G vector fields Let us note that some essential differences appearing unavoidably. We recall some known facts about the generalized B-G vector fields (see, e.g., [15, 19, 8]).

We need the following concepts. A functionu:V R^n‡m!Ris said cylindrical, ifu(x;y)ˆu(r;s) (i.e.,udepends only onrˆ jxjandsˆ jyj), and in particular, if u(x;y)ˆu(d(x;y)), we call u is radial. Let VˆB_L(R₂)nB_L(R₁) with 0R₁R₂ ‡1;u2L¹(V). Then, the change of polar coordinates defined in [19, 10]

(x1;. . .;xn;y1;. . .;ym)!(r;u;u1;. . .;un 1;g₁;. . .;g_{m 1}) allows the following formula to hold

Z

V

u(x;y)dxdyˆ Z^R2

R1

Z

S

u(dr(x;y))r^{Q 1}dmdr

ˆsn;m

Z^R2

R1

r^{Q 1}u rjsinuj^a‡11; r^a‡1 a‡1jcos uj

dr;

(2:1)

where sn;mˆ 1 a‡1 _m

vnvmR^a2

a1

jsinuj^{a‡1n 1}jcos uj^{m 1}du, vn;vm are the Lebesgue measure of the unitary Euclidean sphere in Rⁿ and R^m; re- spectively,a₁;a₂depend onnandm, see [19].

Let us recall some elementary facts about distance functiondˆd(x;y).

r_Ldˆ jxj^a

d^2a‡1…jxj^ax1;jxj^ax2;. . .;jxj^axn;(a‡1)y1;. . .;(a‡1)ym†;

jrLdj^pˆc_pa ˆjxj^pa

d^pa 1; Ladˆc_2aQ 1

d ; Lp;adˆQ 1

d c_pa; p1:

It is easy to seedandrLdare homogeneous of degree one and zero with respect to the dilations (1.3), respectively. From the formula ofrLd, it holds

rL(d^2(a‡1))ˆ2(a‡1)d^2a‡1rLd

ˆ2(a‡1)jxj^a jxj^ax₁;jxj^ax₂; ;jxj^ax_n;(a‡1)y₁; ;(a‡1)y_m (2:2) :

Noticingr_L(d^2(a‡1))ˆ2(a‡1)d^2a‡1r_Ld, it is not difficult to obtain jrL(d^2(a‡1))j²ˆ[2(a‡1)d^2a‡1]²jrLdj²ˆ[2(a‡1)]²jxj^2ad^2(a‡1): (2:3)

Introduce the following two matrix A_a ˆ I_n O

O jxj^2aI_m

; s_aˆ I_n O

O jxj^aI_m

;

whereInandImare thenandmorder unit matrix, respectively. Thus, we can denote

LauˆdivL(ru)ˆdiv(Aaru)ˆdiv(s^T_asaru); and rLˆsar:

Letjˆ(x;y)2R^n‡m;xˆ x

jjj;yˆ y

jjj^a‡1;jˆ(x;y) andrjˆd_r(j).

We havej2Sˆ f(x;y)2R^n‡mjd(x;y)ˆ1g. Arguing as in [14], we can obtain the following result from the polar coordinates and dilations (1.3).

LEMMA2.1. Iff 2L¹(R^n‡m), then Z

R^n‡m

f(j)djˆ Z¹

0

Z

S

f(rj)r^{Q 1}dm(j)dr:

(2:4)

On the other hand, by Federer's coarea formula Z

R^N

f(x)dxˆ Z¹

1

ds Z

gˆs

f(x)

jrg(x)jdHN 1;

where g2Lip(R^N), dHN 1 denotes the (N 1)-dimensional Hausdorff measure andris usual gradient. It follows

(2:5) Z

BL(R)

f(j)djˆ Z^R

0

dr Z

@BL(r)

f(j)

jrdjdH_{Q 1}ˆ Z^R

0

dr Z

S

f(j)

jrdjr^{Q 1}dH_{Q 1}d_r(j):

Combining (2.4) with (2.5), it yields Z

S

f(rj)dmˆ Z

S

f(j)

jrdjdH_{Q 1}d_r(j)ˆ Z

@BL(r)

f(j)

d^{Q 1}jrdjdH_{Q 1}: (2:6)

Using the polar coordinates and (2.1), as in [7], we have LEMMA2.2. Suppose thatg> n, and

C_gˆ Z

S

jxj^gdm:

Then,

C_gˆ 1 a‡1

_m4p^n‡m2 G_2(a‡1)^n‡g G(ⁿ₂)G_2(a‡1)^Q‡g: In particular,

jSj ˆC0 ˆ Z

S

dmˆ 1 a‡1

_m4p^n‡m2 G_2(a‡1)ⁿ G(ⁿ₂)G_2(a‡1)^Q : PROOF. Forg> Q, it follows from (2.4) that

C_gˆ Z

S

jxj^gdmˆ(Q‡g) Z¹

0

r^{Q‡g 1}dr Z

S

jxj^gdm

ˆ(Q‡g) Z¹

0

Z

S

r^{Q 1}jrxj^gdmdr

ˆ(Q‡g) Z

BL(1)

jxj^gdxdy:

Now, invoking (2.1), as in [19], it gets forg> n C_g ˆ(Q‡g)

Z

BL(1)

jxj^gdxdy

ˆ(Q‡g) 1 a‡1 _m

v_nv_m Z¹

0

r^{Q‡g 1}dr Z^a2

a1

jsinuj^{n‡ga‡1 1}jcos uj^{m 1}du

ˆ 1

a‡1 _m

v_nv_m Z^a2

a1

jsinuj^{n‡ga‡1 1}jcos uj^{m 1}du

ˆ 1

a‡1

_m 4p^n‡m2 G_2(a‡1)^n‡g G(ⁿ₂)G_2(a‡1)^Q‡g: The proof of Lemma 2.2 is finished.

LEMMA2.3. Assume05R15R251and f 2C¹(BL(R2)nBL(R1)). Then (2:7)

Z

S

f(R₂j)dm Z

S

f(R₁j)dm

ˆ Z

BL(R2)nBL(R1)

1 d^Q(j)jxj^2a

rLf;rL

d^2(a‡1)(j) 2(a‡1)

dj:

PROOF. Let T be a C¹ vector field on U:ˆB_L(R2)nB_L(R1) and be specified later. For anyf 2C¹(U), integrating by parts, we have

(2:8) Z

U

div_LTfdjˆ Z

U

divs^T_aTfdjˆ Z

U

hT;r_Lfidj Z

@U

fs^T_ahT;nidH_{Q 1};

wherenis the outer unit normal vector to the boundary.

Choose nowTˆ 1 d^Q(j)jxj^2ar_L

d^2(a‡1)(j) 2(a‡1)

. From (2.2), it is easy to see Tˆ rLd

d^{Q 2a 1}jxj^2aˆ sard

d^{Q 2a 1}jxj^2a. We compute

divs^T_aTˆdivs^T_a

rLd d^{Q 2a 1}jxj^2a

ˆ Lad

d^{Q 2a 1}jxj^2a‡ hrLd;rL(d^{2a‡1 Q}jxj ^2a)i

ˆjxj^2a d^2a

Q 1

dd^{Q 2a 1}jxj^2a‡(2a‡1 Q)jxj ^2ad^{2a Q}hr_Ld;r_Ldi

‡d^{2a‡1 Q}hr_Ld;r_L(jxj ^2a)i

ˆQ 1

d^Q ‡2a‡1 Q

d^Q 2ad^{2a‡1 Q}jxj^a

d^2a‡1 jxj ^{2a 2}jxj^a‡2

ˆ2a d^Q

2a d^Qˆ0:

(2:9)

Note that nˆ rd

jrdj and jr_Ldj²ˆA_ahrd;rdi. Combining (2.8) and (2.9), we obtain

(2:10) Z

U

1 d^Q(j)jxj^2a

rLf;rL

d^2(a‡1)(j) 2(a‡1)

djˆ

Z

@U

fs^T_a

rLd d^{Q 2a 1}jxj^2a;n

dHQ 1

ˆ Z

@U

fs^T_a

rLd

d^{Q 2a 1}jxj^2a; rd jrdj

dHQ 1

ˆ Z

@U

f jrdj

jrLdj²

d^{Q 2a 1}jxj^2adHQ 1

ˆ Z

@BL(R2)

f

d^{Q 1}jrdjdHQ 1

Z

@BL(R1)

f

d^{Q 1}jrdjdHQ 1:

Invoking (2.6), we deduce the formula (2.7). Hence, this completes the proof

of Lemma 2.3. p

3. Ostrowski type inequalities

In this section, we prove Theorem 1.1 and 1.3. To begin with, we need the following estimates.

LEMMA3.1. Let05R₁5R₂51;05a5n and f 2C¹(B_L(R₂)nB_L(R₁)).

Then Z

S

f(R₂j)dm Z

S

f(R₁j)dm

Z

BL(R2)nBL(R1)

1

d^{Q (a‡1)}jxj^ajr_Lfjdj C ajR2 R1j krLfk₁:

(3:1)

PROOF. Using the pointwise Schwartz inequality, it follows from Lemma 2.3 and (2.3) that

Z

S

f(R2j)dm Z

S

f(R1j)dm

Z

B_L(R₂)nB_L(R₁)

1

d^Qjxj^2ajrLfj rL

d^2(a‡1) 2(a‡1)

dj

Z

BL(R2)nBL(R1)

1

d^{Q (a‡1)}jxj^ajrLfjdj

krLfk₁ Z

BL(R2)nBL(R1)

1

d^{Q (a‡1)}jxj^adj :

Therefore, invoking Lemma 2.2, we get for 05a5n Z

S

f(R₂j)dm Z

S

f(R₁j)dm

kr_Lfk₁ Z

BL(R2)nBL(R1)

1 d^{Q 1}jxj^adj

ˆ kr_Lfk₁ Z^R2

R1

dr Z

S

jxj ^adm

ˆC _ajR₂ R₁jkr_Lfk₁:

The lemma is proved. p

PROOF OFTHEOREM1.1. Since

jBL(R)j ˆ jSjR^Q Q ; it follows from this and Lemma 2.1 that

f(j) 1 jB_L(R)j

Z

BL(R)

f(h)dh

jf ~f(r)j

‡ 1

jSj Z

S

f(rj)dm Q R^QjSj

Z^R

0

Z

S

f(rj)r^{Q 1}dmdr

ˆ:N(f)‡I:

(3:2)

By Lemma 3.1 and 2.2, we have Iˆ

1 jSj

Z

S

f(rj)dm Q R^QjSj

Z^R

0

Z

S

f(rj)r^{Q 1}dmdr

Q

R^QjSj Z^R

0

Z

S

f(rj)r^{Q 1}dmdr Z^R

0

Z

S

f(rj)r^{Q 1}dmdr QC _a

R^QjSj Z^R

0

jr rjr^{Q 1}drkr_Lfk₁

ˆQC a

R^QjSj R^Q‡1

Q‡1 rR^Q

Q ‡ 2r^Q‡1 (Q‡1)Q

kr_Lfk₁

ˆC _a jSj

Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kr_Lfk₁

ˆG _2(a‡1)^{n a}

G _2(a‡1)^Q G _2(a‡1)ⁿ

G _2(a‡1)^{Q a} Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kr_Lfk₁: (3:3)

Combining (3.3) and (3.2), we obtain the conclusion. p

PROOF OF COROLLARY 1.2. Let f 2C¹(BL(R)) be a radial function.

From Lemma 2.3 and 2.2, the following estimate holds Z

S

f(R2j)dm Z

S

f(R1j)dm

Z

BL(R2)nBL(R1)

1

d^Qjxj^2ajr_Lfkr_L( d^2(a‡1) 2(a‡1))jdj

Z

BL(R2)nBL(R1)

1

d^{Q (a‡1)}jxj^ajf⁰krLdjdj kf⁰k₁

Z

BL(R2)nBL(R1)

1 d^{Q 1}dj

ˆC0jR2 R1jkf⁰k₁: Similar to (3.3), we can get

Iˆ 1

jSj Z

S

f(rj)dm Q R^QjSj

Z^R

0

Z

S

f(rj)r^{Q 1}dmdr C0

jSj Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kf⁰k₁

ˆ Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

kf⁰k₁:

Hence, we obtain (1.5) from the proof of Theorem 1.1. p PROOF OFTHEOREM 1.3. Assume thatRˆinffR0>0jVB_L(R0)g.

Then

F(j):ˆ f(j); if j2V;

0; if j2B_L(R)nV;

(

and satisfiesF2W^1;1(BL(R))\C₀¹(BL(R)):We can splitf(j) 1 jVj

R

Vf(h)dh as

(3:4)

f(j) 1 jVj

Z

V

f(h)dh

F(j) 1 jBL(R)j

Z

BL(R)

F(h)dh ‡

1 jBL(R)j

Z

BL(R)

F(h)dh 1 jVj

Z

V

f(h)dh

ˆ:I₁‡I₂:

By the definition ofR, we can obtain I₂ˆ

1 jB_L(R)j

Z

BL(R)

F(h)dh 1 jVj

Z

V

f(h)dh

ˆ 1

jB_L(R)j Z

V

f(h)dh 1 jVj

Z

V

f(h)dh

ˆ 1

jB_L(R)j 1 jVj

Z

V

f(h)dh

ˆ

1 jVj

jBL(R)j 1

jVj Z

V

f(h)dh : (3:5)

Further, replacingBL(R) byVin Theorem 1.1, it follows

(3:6) I1ˆ

F(j) 1 jBL(R)j

Z

B_L(R)

F(h)dh kF Fk~ ₁‡G _2(a‡1)^{n a}

G _2(a‡1)^Q G _2(a‡1)ⁿ

G _2(a‡1)^{Q a} Q

Q‡1R r‡ 2r^Q‡1 (Q‡1)R^Q

krLfk_L1(V):

Substituting (3.5) and (3.6) into (3.4), we conclude (1.6). Theorem 1.3 is

established. p

4. Hardy inequalities with boundary term

In this section, we present some Hardy inequalities with boundary term which improves the Hardy inequality established in [8, 9, 11]. Here we point out that our interests is to establish inequalities, the best constants of inequalities are not discussed (due to the optimality of the constants in the inequalities is of its independent interest).

PROOF OFTHEOREM1.4. LetTbe aC¹ vector field onBL(R) and let it be specified later. For any u2D^1;p(R^n‡m), integrating by parts, we have

(4:1) Z

BL(R)

divLTjuj^pdjˆ p Z

BL(R)

hT;juj^{p 2}urLuidj‡ Z

@BL(R)

juj^ps^T_ahT;nidH_{Q 1}:

Using HoÈlder's inequality and Young's inequality we obtain

(4:2) p

Z

V

hT;r_Luijuj^{p 2}udjp Z

BL(R)

jr_Luj^pdj ¹_p Z

BL(R)

jTj^pp1juj^pdj ^p_p¹

Z

BL(R)

jr_Luj^pdj‡(p 1) Z

BL(R)

jTj^pp1juj^pdj:

Thereby, substituting (4.2) into (4.1), it follows (4:3)

Z

BL(R)

div_LT (p 1)jTj^pp1

juj^pdj Z

BL(R)

jr_Luj^pdj‡ Z

@BL(R)

juj^ps^T_ahT;nidH_{Q 1}:

Choose nowTˆAjAj^{p 2}jr_Ldj^{p 2}r_Ld

d^{p 1} , where AˆQ p

p forp6ˆQ:A di- rect computation gets

divLT (p 1)jTj^pp1juj^pˆ jAj^{p 1}jrLdj^p d^p ; and note thatnˆ rd

jrdj:Combining these with (4.3) yields jAj^p

Z

BL(R)

c_pajuj^p d^p dj

Z

BL(R)

jrLuj^pdj‡AjAj^{p 2} 1 R^{p 1}

Z

@BL(R)

c_pa juj^p

jrdjdHQ 1: Thus, inequality (1.7) is proved.

A similar argument can prove inequality (1.8), so we omit the details

here. p

In the same spirit with Theorem 1.4, we can obtain the following Hardy inequality with boundary term on bound domain.

THEOREM4.1. Let15p5Q. IfVR^n‡mis a smooth domain with@V being bounded and062@V, then there exists R₀ˆinffR>0jVB_L(R)g such that for all u2D^1;p(R^n‡m)

Z

V

jr_Luj^pdj Q p

p ^pZ

V

c_pajuj^p d^p dj

Q p p

^{p 1} 1 R^p₀ ¹

Z

@V

c_pa juj^p

jrdjdH_{Q 1}; (4:4)

In addition, ifVR^n‡m is a bounded and star-sharped with respect to the origin, there exists R₁ˆinffR>0jV B_L(R)g such that for all u2D^1;p(R^n‡m)

Z

V^c

jr_Luj^pdj Q p

p ^pZ

V^c

c_pajuj^p d^p dj

Q p p

^{p 1} 1 R^p₁ ¹

Z

@V

c_pa juj^p

jrdjdH_{Q 1}; (4:5)

whereV^c:ˆR^n‡mnV:

In addition, J. Dou, Q. Guo and P. Niu [11] had established the following Hardy inequalities with remainder term.

PROPOSITION 4.2. Let 02V be a bounded domain in R^n‡m and 15p51.

(1): If p6ˆQ, then there exists a positive constant R0sup

j2Vd(j)such that for any R>R₀ and all u2D^1;p₀ (Vnf0g),

(4:6) Z

V

jr_Luj^pdj Q p

p ^pZ

V

c_pajuj^p d^p dj

‡p 1 2p

Q p p

^{p 2}Z

V

c_pajuj^p d^p

lnR

d ²

dj:

In particular, if 2p5Q, one can takesup

j2Vd(j)ˆR0. (2): If pˆQ, then there exists R>sup

j2Vd(j) such that for all u2D^1;p₀ (Vnf0g)

Z

V

jr_Luj^pdj

p 1 p

_pZ

V

c_pa juj^p dln ^R_dpdj:

(4:7)

We note that a simple density argument can show thatD^1;p₀ (Vnf0g)ˆ D^1;p₀ (V). Hence, inequalities (4.6) and (4.7) can be extended to the space D^1;p(V).

THEOREM4.3. Let02V be a bounded domain inR^n‡mwith smooth boundary and15p51.

(1): If p6ˆQ, then there exists a small positive constant M₀such that for any R>R₀:ˆe^M10sup

j2Vd(j)and all u2D^1;p(V), Z

V

jr_Luj^pdj Q p

p ^pZ

V

c_pajuj^p d^p dj p 1

2p

Q p p

^{p 2}Z

V

c_pajuj^p d^p

lnR

d ²

dj

C_p;Q;R0 Z

@V

c_pa juj^p

jrdjdH_{Q 1}; (4:8)

where C_p;Q;R0ˆ

ˆ p Q

p

Q p p

^{p 2} 1‡_{Q p}^{p 1}M0‡aM²₀

R^p₀ ¹ for 15p52;05a5(2 p)(p 1) 6p²A² ; p Q

p

Q p p

^{p 2} 1‡_{Q p}^{p 1}M0

R^p₀ ¹ for 2p5Q;

Q p p

Q p p

^{p 2 Q pp 1}M0‡aM₀²

R^p₀ ¹ for p>Q;0>a>(2 p)(p 1) 6p²A² : 8>

>>

>>

>>

>>

>>

><

>>

>>

>>

>>

>>

>>

:

(2): If pˆQ, then there exists R>sup

j2Vd(j)and the definition of M0

as above such that for all u2D^1;p(V) (4:9)

Z

V

jr_Luj^pdj

p 1 p

_pZ

V

c_pa juj^p dln ^R_dpdj

p 1 p

_{p 1}M₀^{p 1} R^p₀ ¹

Z

@V

c_pa juj^p

jrdjdH_{Q 1}:

PROOF. LetB(s)ˆ 1

ln (s);s2(0;1) andAˆQ p

p . ForR>sup

j2Vd(j), there exists a constantM>0 such that

0 B d(j)

R

M; j2V:

Furthermore, rLB^g

d R

ˆgB^g‡1 _R^d rLd

d ; dB^{g r}_R

dr ˆgB^g‡1 _R^r

r ; for allg2R;

and Z^b

a

B^g‡1(s) s dsˆ1

g

B^g(b) B^g(a) :

(1) Forp6ˆQ:Let Tbe aC¹ vector field onV and let it be specified later. For anyu2D^1;p(V), similar to (4.3), it holds

(4:10) Z

V

[divLT (p 1)jTj^pp1]juj^pdj Z

V

jrLuj^pdj‡

Z

@V

juj^ps^T_ahT;nidHQ 1:

We choose

T(d)ˆAjAj^{p 2}jr_Ldj^{p 2}r_Ld

d^{p 1} 1‡p 1 pA B

d R

‡aB² d

R

!

; whereais a parameter to be chosen later.

As in [11], take (i) a>(2 p)(p 1)

6p²A² >0 for 15p525Q, (ii) aˆ0 for 2p5Q,

(iii) a5(2 p)(p 1)

6p²A² 50 forp>Q.

Thus, it is not difficult to chooseM0(small enough) in all cases such that for 05B M₀, and then

div_LT (p 1)jTj^pp1 jAj^pjr_Ldj^p d^p

1‡p 1 2pA²B²

d R

: (4:11)

Noting that the conditionB M0 is equivalent toRR0:ˆe^M10sup

j2Vd(j) andnˆ rd

jrdj;ifp5Q, then

Z

@V

juj^ps^T_ahT;nidH_{Q 1}

ˆ Z

@V

AjAj^{p 2}jrLdj^p d^{p 1}

1‡p 1 pA B

d R

‡aB² d

R juj^p

jrdjdH_{Q 1}

AjAj^{p 2} 1‡^p_pA¹M0‡aM²₀ R^p₀ ¹

Z

@V

c_pa juj^p

jrdjdH_{Q 1}; (4:12)

and substituting (4.11) and (4.12) into (4.10), we conclude inequality (4.8). If p>Q, we have

(4:13) Z

@V

juj^ps^T_ahT;nidH_{Q 1}AjAj^{p 2}

p 1

pA M₀‡aM²₀ R^p₀ ¹

Z

@V

c_pa juj^p

jrdjdH_{Q 1};

where the factsA50;a50 is used.

(2) ForpˆQ:ChooseT(d)ˆ p 1

p _{p 1}

jr_Ldj^{p 2}r_Ld d^{p 1} B^{p 1}

d R

, as in [11], we can obtain

div_LT (p 1)jTj^pp1ˆ

p 1 p

_p B^p

d R

jr_Ldj^p d^p : (4:14)

Notice

(4:15) Z

@V

juj^ps^T_ahT;nidH_{Q 1}ˆ Z

@V

p 1 p

_{p 1} jrLdj^p

d^{p 1} B^{p 1} d

R juj^p

jrdjdH_{Q 1}

p 1

p _{p 1}

M^p₀ ¹ R^p₀ ¹

Z

@V

c_pa juj^p

jrdjdH_{Q 1}:

Combining (4.14) and (4.15) yields inequality (4.9). p Acknowledgements. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11101319), the Foundation of Shaanxi Province Education Department (Grant No. 2010JK549) and Zhejiang Provincial Natural Science Foundation of China (Grant No.

Y6110118).

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Manoscritto pervenuto in redazione il 3 Gennaio 2012.