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Volume 2007, Article ID 36503, 8 pagesdoi:10.1155/2007/36503 Research Article On the p,q-Boundedness of Nonisotropic Spherical Riesz Potentials Mehmet Zeki Sarikaya and H¨useyin Yildiri

Volume 2007, Article ID 36503, 8 pages

doi:10.1155/2007/36503

Research Article

On the ( p,q)-Boundedness of Nonisotropic Spherical

Riesz Potentials

Mehmet Zeki Sarikaya and H¨useyin Yildirim

Received 20 November 2006; Accepted 1 March 2007

Recommended by Shusen Ding

We introduced the concept of nonisotropic spherical Riesz potential operators generated

investi-gated

Copyright © 2007 M Z Sarikaya and H Yildirim This is an open access article distrib-uted 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

Let

Rn =x =x1,x2, ,x n

 :x i ∈ R, 1≤ i ≤ n

L p = L p



Ωn,λ



=



f (x) :  f  p =



Ωn,λ

f (x) p

dx

1/ p

< ∞

 , 1≤ p < ∞

L ∞ = L ∞

Ωn,λ



=f (x) :  f  ∞ =ess sup

x ∈Ωn,λ

f (x) < ∞, (1.2)

| x − y | λ:=

x1− y1 1/λ1

+ x2− y2 1/λ2

+···+ x n − y n 1/λ n | λ | /n

wherex, y ∈Ωn,λ,λ =(λ1,λ2, ,λ n),λ k > 0, k =1, 2, ,n, | λ | = λ1+λ2+···+λ n Note

t λ1 x1 1/λ1+···+ t λ n x n 1/λ n | λ | /n

| λ | /n So the nonisotropic λ-distance has the following properties:

(2)| t λ x | λ = | t | | λ | /n | x | λ,

(3)| x + y | λ ≤2(1+1/λmin )| λ | /n(| x | λ+| y | λ)

x1=ρ cosθ1

 2λ1

, ,x n =ρ sinθ1sinθ2···sinθ n −1

 2λ n

on anglesθ1,θ2, ,θ n −1 It is clear that ifλ1= λ2= ··· = λ n =1/2, then the λ-distance is

the Euclidean distance

We define angle

For f ∈ L(Ω n,λ), 0< α(x) < n, we will consider the following nonisotropic spherical

I λ α(x) f (x) =



Ωn,λ

| x − y | α(x) − n

The aim of this paper to show that the well-known properties of classical Riesz

Riesz potentials and theirs generalizations were studied by many authors We refer to

impor-tance of the nonisotropic kernel is that it does not have the classical triangle inequality

oper-ators fromL p(R n) toL q(R n) for 1/q =1/ p − α/n, 0 < α < n, 1 ≤ p < q < ∞[10]

n,λ f (x)K(x, y)dy, x ∈Ω2

n,λ ,

k1= sup

y ∈Ω 1

n,λ



Ω 2

n,λ

| K(x, y) | q dx

1/r

< ∞, k2= sup

x ∈Ω 2

n,λ



Ω 1

n,λ

| K(x, y) | q dy

1/q −1/r

< ∞

(1.8)

and the following conditions are carried out: 1 ≤ p ≤ r ≤ ∞ , 1 −1/ p + 1/r =1/q, f ∈

L p(Ω1

J λ

L p(Ω 1

n,λ)≤f

L r(Ω 2

Proof Let λ, μ, ν be positive numbers such that 1/λ + 1/μ + 1/ν =1 We write

J λ(x) =



Ω 1

n,λ

f p(1/ p −1/μ)(x) f p/μ(x)K q(1/q −1/ν)(x, y)K q/ν(x, y)dy. (1.10)

J λ(x) ≤



Ω 1

n,λ

f (y) pλ(1/ p −1/μ) K(x, y) λq(1/q −1/ν) dy

1/λ

Ω 1

n,λ

f (y) p dy

1/μ

Ω 1

n,λ

K(x, y) q dy

1/ν

.

(1.11)

μ, ν in such a way

1

λ =1

p −1

μ ,

1

λ =1

q −1

ν ,

1

λ =1

J λ(x) ≤  f  L p/μ p(Ω1

n,λ)



Ω 1

n,λ

K(x, y) q dy

1/ν

Ω 1

n,λ

f (y) p K(x, y) q dy

1/λ



Ω 2

n,λ

J λ(x) r

dx

≤  f  r p/μ L p(Ω1

n,λ)



Ω 2

n,λ



Ω 1

n,λ

K(x, y) q dy

r/ν

Ω 1

n,λ

f (y) p K(x, y) q dy

r/λ

dx

≤  f  r p/μ L p(Ω1

n,λ) sup

x ∈Ω 2

n,λ



Ω 1

n,λ

K(x, y) q dy

r/ν

Ω 2

n,λ

sup

y ∈Ω 1

n,λ

K(x, y) q



Ω 1

n,λ

f (y) p dy dx

≤  f  r p/μ L p(Ω1

n,λ) f  L p p(Ω1

n,λ) sup

y ∈Ω 1

n,λ



Ω 2

n,λ

K(x, y) q dx sup

x ∈Ω 2

n,λ



Ω 1

n,λ

K(x, y) q dy

r/ν

.

(1.14) Hence

J λ(x)r

L r( Ω 2

n,λ)≤  f  p(r/μ+1) L p(Ω1

n,λ) sup

y ∈Ω 1

n,λ



Ω 2

n,λ

K(x, y) q dx sup

x ∈Ω 2

n,λ



Ω 1

n,λ

K(x, y) q dy

r/ν

.

(1.15)

Takingrth roots, we have the following inequality:

J λ(x)

L r( Ω 2

n,λ)

≤  f  L p(1/μ+1/r)

p( Ω 1

n,λ) sup

y ∈Ω 1

n,λ



Ω 2

n,λ

K(x, y) q dx 1/r sup

x ∈Ω 2

n,λ



Ω 1

n,λ

K(x, y) q dy

1/ν

≤  f  L p(Ω1

n,λ) sup

y ∈Ω 1

n,λ



Ω 2

n,λ

K(x, y) q dx

1/r

sup

x ∈Ω 2

n,λ



Ω 1

n,λ

K(x, y) q dy

1/q −1/r

≤  f  L p( Ω 1

n,λ)k1k2.

(1.16)



weak types ( p0,q0) and ( p1,q1), 1 ≤ p i,q i ≤ ∞ If 0 < t < 1 and 1/ p t =(1− t)/ p0+t/ p1,

1/q t =(1− t)/q0+t/q1, then T is of type (p t,q t ), and

 T (p t,q t)≤  T 1− t

(p0 ,q0 ) T  t

The following theorem gives the condition of absolute convergence of the potential

I λ α(x) f

convergent for almost every x.

Proof Let L y,θ = { x ∈Ωn,λ:y · x =cosθ },| L y,θ | = |Ωn −1,λ |sin2| λ |−1θ Hence we have



Ωn,λ

I α(x)

λ f (x) dx

≤



Ωn,λ

f (y)

| x − y | n − α(x)

λ

dy dx

=



Ωn,λ

f (y) 

Ωn,λ

1

| x − y | n λ − α(x) dx dy

=



Ωn,λ

f (y) π

0



L y,θ

1

θ(2| λ | /n)(n − α(x)) dL y,θ(x) dθ



dy

=



Ωn,λ

f (y) 1

0+

π

1 L y,θ

1

θ(2| λ | /n)(n − α(x)) dL y,θ(x) dθ



dy

≤



Ωn,λ

f (y) 1

0



L y,θ

1

θ(2| λ | /n)(n − m) dL y,θ(x) dθ +

π

1



L y,θ

dL y,θ(x) dθ



dy

≤



Ωn,λ

f (y) 1

0

Ω

n −1,λ sin2| λ |−1

θ

θ(2| λ | /n)(n − m) dθ +

π

1

Ω

n −1,λ sin2| λ |−1

θdθ



dy

≤ Ωn −1,λ 

Ωn,λ

f (y) 1

0

1

θ1−(2| λ | /n)m dθ+

π

1 dθ



dy ≤ M  f 1< ∞

(1.18)

Theorem 1.4 Let 0 < m ≤ α(x) < n, 1 ≤ p < ∞ Then I λ α(x) f is of type (p, p), that is,

I α(x)

λ f

where the constant M is dependent on λ, m, and n.

Proof Let

S θ f (x) = L1x,θ L

x,θ

Thus we have

S θ f

By the Minkowsky inequality for integrals, we have the following inequality:



Ωn,λ

I α(x)

λ f (x) p

dx

1/ p

=



Ωn,λ

Ω

n,λ

f (y)

| x − y | n − α(x) λ

dy

p dx

1/ p

≤



Ωn,λ

π

0

Ωn −1,λ sin2 λ |−1

θ

θ(2| λ/n)(n − α(x))

1

L x,θ L

x,θ

f (y)dL x,θ dθ

p dx

1/ p

≤ M



Ωn,λ

π

0

1

θ1−(2| λ | /n)α(x) S θ(f ) dθ p

dx

1/ p

≤ M

 1

0+

π

1



Ωn,λ

1

θ(1−(2| λ | /n)α(x))p S θ(f ) p

dx

1/ p

dθ

≤ M

 1

0

1

θ1−(2| λ | /n)α(x)



Ωn,λ

S θ(f ) p

dx

1/ p

dθ + M

π

1



Ωn,λ

S θ(f ) p

dx

1/ p

dθ

≤ M

1

0

1

θ1−(2| λ | /n)α(x)S θ(f )

p dθ + M

π

1

S θ(f )

p dθ

≤ M  f  p

 1 0

dθ

θ1−(2| λ | /n)α(x)+

π

1 dθ ≤ M  f  p

(1.22)

Theorem 1.5 Let 0 < m ≤ α(x) < n, 1 < p ≤ r, n/ p − n/r < m Then I λ α(x) f is of type (p,r), that is,

I α(x)

λ f

where the constant M is dependent on λ, m, and n.

Proof Let q = pr/(pr + p − r), 1/q + 1/q  =1 We show thatI λ α(x) f is of type (1,q) and

I α(x)

λ f

q

≤



Ωn,λ



Ωn,λ

f (y)

| x − y | n λ − α(x) dy

q dx

1/q

≤



Ωn,λ



Ωn,λ

f (y) q

| x − y |(λ n − α(x))q dx

1/q

dy

=



Ωn,λ

f (y) 

Ωn,λ

1

| x − y |(λ n − α(x))q dx

1/q

dy

≤



Ωn,λ

f (y) π

0



L y,θ

1

θ(2| λ | /n)(n − α(x))q dL y,θ(x) dθ



dy

≤



Ωn,λ

f (y) 1

0



L y,θ

1

θ(2| λ | /n)(n − m)q dL y,θ(x) dθ +

π

1

Ω

n −1,λ sin2| λ |−1

θdθ



dy

≤



Ωn,λ

f (y) 1

0

Ωn −1,λ sin2| λ |−1

θ

θ(2| λ | /n)(n − m)q dθ + M



dy

≤



Ωn,λ

f (y)  Ωn −1,λ 1

0

1

θ(2| λ | /n)(n − m)q −2| λ |+1dθ + M



dy ≤ M  f 1.

(1.24) Thus the last integral is convergence where

pr

pr + p − r <

n

n − m for

n

p − n

r < m,

q < n

n − m =⇒2| λ |

n (n − m)q + 1 −2| λ | < 1.

(1.25)

have

I α(x)

λ f <

Ωn,λ

f (y)

| x − y | n − α(x) λ

dy

≤



Ωn,λ

f (y) q 

dy

1/q 

Ωn,λ

1

| x − y |(λ n − α(x))q dy

1/q

≤ M  f  q 

(1.26)

Therefore we have

I α(x)

λ f

This shows thatI λ α(x) f is of type (q ,∞)

Lett = q(1 −1/ p), then fromTheorem 1.2,I λ α(x) f is of type (p,r) where 1/ p =(1− t)/1+

Theorem 1.6 Let 0 < m ≤ α(x) < n, 1 < p < r, n/ p − n/r = m Then I λ α(x) f is of type (p,r) Proof Firstly, for a constant m we will consider the α(x) = m Thus, by usingLemma 1.1

forK(x, y) = | x − y | m − n

I m

λ f

Let

Ωn,λ,x =y ∈Ωn,λ:| x − y | λ ≥1

, Ωn,λ,x =Ωn,λ \Ωn,λ,x (1.29) Then

I α(x)

λ f ≤

Ωn,λ

f (y)

| x − y | n − α(x) λ

dy ≤



Ωn,λ,x

f (y) dy +

Ωn,λ,x

f (y)

| x − y | n − m λ

dy

≤



Ωn,λ,x

f (y) dy + I m

λ f (x) ≤ M   f  p+I m

λ f (x).

(1.30)

Therefore we have

I α(x)

λ f

r ≤M   f  p+I m

λ f (x)

r ≤ M   f  p+I m

λ f (x)

r

ThusI λ α(x) f is of type (p,r).

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Mehmet Zeki Sarikaya: Department of Mathematics, Faculty of Science and Arts,

Kocatepe University, 03200 Afyon, Turkey

Email address:sarikaya@aku.edu.tr

H¨useyin Yildirim: Department of Mathematics, Faculty of Science and Arts,

Kocatepe University, 03200 Afyon, Turkey

Email address:hyildir@aku.edu.tr

The following theorem gives the condition of absolute convergence of the potential

I λ α(x) f

convergent for almost every x.

Proof... f

where the constant M is dependent on λ, m, and n.

Proof Let q =... class="text_page_counter">Trang 8

[11] Z Zhou, Y Hong, and C Z Zhou, ? ?The (p ,q)-boundedness of Riesz potential operators of< /small>

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