On the Boundedness of a Generalized Fractional Integral on Generalized Morrey Spaces

On the Boundedness of a Generalized Fractional

Integral on Generalized Morrey Spaces

Eridani

Department of Mathematics Airlangga University, Surabaya

dani@unair.ac.id

Abstract

In this paper we extend Nakai’s result on the boundedness of a general-ized fractional integral operator from a generalgeneral-ized Morrey space to another generalized Morrey or Campanato space.

keywords: Generalized fractional integrals, generalized Morrey spaces, ge-neralized Campanato spaces

1. Introduction and Main results

For a given function ρ : (0,∞) −→ (0,∞), let Tρ be the generalized fractional

integral operator, given by

Tρf(x) =

Z

Rn

f(y)ρ(|x−y|) |x−y|n dy, and put

e

Tρf(x) =

Z

Rn

f(y)

µ

ρ(|x−y|) |x−y|n −

ρ(|y|)(1−χBo(y))

|y|n

¶

dy,

the modified version ofTρ, whereBo is the unit ball about the origin, andχBo is

the characteristic function ofBo.

In [4], Nakai proved the boundedness of the operatorsTeρandTρfrom a

general-ized Morrey spaceM1,φto another generalized Morrey spaceM1,ψ or generalized

Campanato spaceL1,ψ.More precisely, he proved that

kTρfkM1,ψ ≤CkfkM1,φ and kTeρfkL1,ψ ≤CkfkM1,φ,

whereC >0,with some appropriate conditions onρ, φandψ.Using the techniques developed by Kurataet.al.[1], we investigate the boundedness of these operators from generalized Morrey spacesMp,φ to generalized Morrey spacesMp,ψ or

gen-eralized Campanato spacesLp,ψ for 1< p <∞.

The generalized Morrey and Campanato spaces are defined as follows. For a given functionφ: (0,∞)−→(0,∞),and 1< p <∞,let

the Morrey spaceMp,φby

Mp,φ={f ∈Lploc(R n_{) :kfk}

Mp,φ <∞},

and the Campanato spaceLp,φ by

Lp,φ={f ∈Lploc(R n_{) :kfk}

Lp,φ <∞}.

Our results are the following:

Theorem 1.2 _{If ρ, φ, ψ: (0,∞)−→(0,∞) satisfying the conditions below}

1 2 ≤

t r ≤2⇒

1

A1

≤ φ(t)

φ(r) ≤A1, and 1

A2

≤ ρ(t)

ρ(r) ≤A2 (4)

Z 1

0

ρ(t)

t dt <∞,and for allr >0,we have

Z ∞

r

φ(t)p

t dt≤A3φ(r)

p

, (5)

|ρ(r)

rn −

ρ(t)

tn | ≤A4|r−t|

ρ(r)

rn+1, f or

1 2 ≤

t

r ≤2, (6) φ(r)

Z r

0

ρ(t)

t dt+r

Z ∞

r

ρ(t)φ(t)

t2 dt≤A5ψ(r),for allr >0, (7)

where Ai > 0 are independent of t, r > 0, then for each 1 < p < ∞ there exists

Cp>0 such that

kTeρfk_Lp,ψ ≤CpkfkMp,φ.

2. Proof of the Theorems

To prove the theorems, we shall use the following result of Nakai [2] (in a slightly modified version) about the boundedness of the standard maximal functionM fon a generalized Morrey spaceMp,φ.The standard maximal functionM f is defined

by

M f(x) = sup

B∋x

1 |B|

Z

B

|f(y)|dy, x∈Rn_,

where the supremum is taken over all open ballsB containingx.

Theorem 2.1 _{(Nakai).Ifφ: (0,∞)−→(0,∞)satisfying the conditions below :}

(a). 1₂ ≤ t r ≤2⇒

1

A1 ≤

φ(t)

φ(r)≤A1,

(b). R_r∞φ(t_t)pdt≤A2φ(r)p,for all r >0,

where Ai > 0 are independent of t, r > 0, then for each 1 < p < ∞ there exists

Cp>0 such that

kM fk_Mp,φ ≤CpkfkMp,φ.

From now on, C andCp will denote positive constants, which may vary from

Proof of Theorem 1.1

Forx∈Rn,and r >0,write

Tρf(x) =

Z

|x−y|<r

f(y)ρ(|x−y|) |x−y|n dy+

Z

|x−y|≥r

f(y)ρ(|x−y|)

|x−y|n dy=I1(x) +I2(x).

Note that, fort∈[2k_r,2k+1_{r],there exist constantsC}

i>0 such that

ρ(2kr)≤C1 Z 2k+1

r

2k_r

ρ(t)

t dt

and

ρ(2kr)φ(2kr)≤C2 Z 2k+1

r

2k_r

ρ(t)φ(t)

t dt.

So, we have

|I1(x)| ≤ Z

|x−y|<r

|f(y)|ρ(|x−y|) |x−y|n dy

≤ −1 X

k=−∞ Z

2k_{r≤|x−y|<2}k+1_r

|f(y)|ρ(|x−y|) |x−y|n dy

≤C

−1 X

k=−∞

ρ(2k_r)

(2k_r)n

Z

|x−y|<2k+1_r

|f(y)|dy

≤C

−1 X

k=−∞

ρ(2kr)M f(x)

≤CM f(x) −1 X

k=−∞ Z 2k+1

r

2k_r

ρ(t)

t dy

≤CM f(x)

Z r

0

ρ(t)

t dy

Meanwhile,

Combining the two estimates, we obtain

1

ψ(r)p|B|

Z

B

|Tρf(x)|pdx≤CpkfkpMp,φ,

and the result follows. ¤

Proof of Theorem 1.2

E_B1(x) =

With the same technique as in the proof of the previous theorem, we have

|E_B1(x)| ≤

and the result follows as before. ¤

3. Remark

We also suspect thatTeρ, the modified version ofTρ, is bounded fromLp,φtoLp,ψ

References

[1] K. Kurata, S. Nishigaki, and S. Sugano,Boundedness of integral operators on generalized Morrey spaces and its application to Schr¨odinger operators,Proc. Amer. Math. Soc.128_{(1999), 1125-1134.}

[2] E. Nakai,Hardy-Littlewood maximal operator, singular integral operators, and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166_(1994),

95-103.

[3] E. Nakai,On generalized fractional integrals, Proceedings of the International Conference on Mathematical Analysis and its Applications 2000 (Kaohsiung, Taiwan), Taiwanese J. Math.5_{(2001), 587-602.}

[4] E. Nakai, On generalized fractional integrals on the weak Orlicz spaces,

BM Oφ, the Morrey spaces and the Campanato spaces, Proceedings of the