OF HOMOGENEOUS TYPE

VOL. LXIV 1993 FASC. 1

ON THE EXPONENTIAL INTEGRABILITY OF FRACTIONAL INTEGRALS ON SPACES

OF HOMOGENEOUS TYPE

BY

A. EDUARDO G A T T O

STEPHEN V ´ A G I (CHICAGO, ILLINOIS)

In this paper we show that the fractional integral of order α on spaces of homogeneous type embeds L ^1/α into a certain Orlicz space. This extends results of Trudinger [T], Hedberg [H], and Adams–Bagby [AB].

1. Definitions and statement of results. We will state the main definitions needed in this paper and will refer to [GV] for other definitions and properties. In this paper (X, δ, µ) will denote a space of homogeneous type that is normal and will be referred to as a normal space. The property of normality is defined as follows: Let B r (x) be the ball of center x and radius r; then there are positive constants A 1 and A 2 such that for all x in X

A 1 r ≤ µ(B r (x)) if 0 < r < R x

and

µ(B r (x)) ≤ A 2 r if r ≥ r x ,

where r x = 0 if µ({x}) = 0, r x = sup{r > 0 : B r (x) = {x}} if µ({x}) 6= 0 and R x = ∞ if µ(X) = ∞, R x = inf{r > 0 : B r (x) = X} if µ(X) < ∞. For 1 ≤ p ≤ ∞, L ^p = L ^p (X, δ, µ) has its usual meaning. The space (X, δ, µ) is said to be of order γ , 0 < γ ≤ 1, if there exists a positive constant M such that for every x, y, and z in X,

|δ(x, z) − δ(y, z)| ≤ M δ(x, y) ^γ (max {δ(x, z), δ(y, z)}) ^1−γ .

In order to define the kernel of the fractional integral without having to distinguish the case when the measure µ has atoms we shall adopt the following abuse of notation: for 0 < α < 1 we define

1

δ(x, y) ^1−α =  1/δ(x, y) ^1−α if x 6= y ,

0 if x = y .

1991 Mathematics Subject Classification: Primary 42B99; Secondary 44A99, 43A99.

Supported in part by a grant from the Faculty Research and Development Fund of

the College of Liberal Arts and Sciences of DePaul University.

The fractional integral of order α, 0 < α < 1, in L ^1/α is defined by I α f (x) = R f (y)

δ(x, y) ^1−α dµ(y) if f has bounded support, and otherwise by

I e α f (x) = R ^ 1

δ(x, y) ^1−α − ψ z (y) δ(z, y) ^1−α



f (y) dµ(y)

where ψ z is the characteristic function of the complement of the ball B 1 (z), and z is any fixed point in X.

R e m a r k. The convergence a.e. of both integrals and the fact that they are elements of BMO was shown in [GV]. Note that the class of e I α f in BMO is independent of the choice of z. If f has bounded support then I α f and I e α f define the same class in BMO.

Theorem 1. Let (X, δ, µ) be a normal space, 0 < α < 1, and let f be in L ^1/α with support in a ball B. Then there are constants C 1 and c independent of B and f such that

R

B

exp  |I α f (x)|

C 1 kf k _1/α

 1/(1−α) 

dµ(x) ≤ cµ(B) .

Theorem 2. Let (X, δ, µ) be a normal space of order γ, 0 < γ ≤ 1. Let 0 < α < 1 and let f belong to L ^1/α . Then there is a constant C 2 independent of f such that for every ball B we have

R

B



exp  |e I α f (x) − m B ( e I α f )|

C 2 kf k _1/α

 1/(1−α) 

− 1



dµ(x) ≤ µ(B) . where m B ( e I α f ) = µ(B) ⁻¹ R

B I e α f dµ.

R e m a r k. The expression I α f − m B (I α f ) coincides a.e. with e I α f − m B ( e I α f ) if f has bounded support. Therefore it suffices to state the theorem for e I α .

As mentioned above it was shown in [GV] that for f in L ^1/α , e I α f is in BMO and k e I α f k BMO ≤ ckf k _1/α . This result and the John–Nirenberg theorem [JN], [CW] imply that there are constants K 1 and K 2 such that

R

B

exp  |e I α f − m B ( e I α f )|

K 1 kf k _1/α



dµ ≤ K 2 µ(B)

for every ball B. But a stronger result is true as stated in Theorem 2. To prove Theorem 2 it is convenient to introduce the related Orlicz space norms.

Let φ be a convex increasing continuous function on [0, ∞) with φ(0) = 0,

and φ(t)/t → ∞ as t → ∞. Let B be a ball in (X, δ, µ). We say that a

measurable function g on B is in L φ (B) if there exists a λ > 0 such that R

B φ(|g(x)|/λ) dµ(x) < ∞. For c > 0 we define the norm N B,c (g) = inf n

λ > 0 : R

B

φ(|g|/λ) dµ ≤ cµ(B) o .

Then L φ (B) is a Banach space with respect to the norm N B,c and these norms are equivalent for different choices of c as shown in Lemma 2.

2. Lemmata and proofs of the theorems

Lemma 1. Let (X, δ, µ) be a normal space and 0 < r ≤ R < ∞. Then there is a constant B 1 independent of x, r and R such that

R

r≤δ(x,y)≤R

dµ(y)

δ(x, y) ≤ B ₁ log 2R r .

P r o o f. Without loss of generality we can assume that r x ≤ r. Let K be the smallest positive integer such that 2 ^K+1 r > R. Then using normality we have

R

r≤δ(x,y)≤R

dµ(y) δ(x, y) ≤

K

X

k=0

R

2

r≤δ(x,y)<2

r

dµ(y) δ(x, y)

≤

K

X

k=0

1 2 ^k r

R

δ(x,y)<2

r

dµ(y) ≤ 2A 2 (K + 1) ≤ 4A 2 K .

Now observe that 2 ^K−1 r ≤ R, and that therefore K ≤ (1/ log 2) log(2R/r).

This proves the lemma with B 1 = 4A 2 / log 2 . Lemma 2. If 0 < c 1 < c 2 , then

N B,c

≤ N _B,c

≤ c 2

c 1

N B,c

.

P r o o f. The first inequality is immediate from the definition of N _B,c . To prove the second inequality let λ > N B,c

. Then

R

B

φ(|f |/λ) dµ ≤ c 2 µ(B) .

Multiplying this by c 1 /c 2 and using the fact that for 0 < ν < 1, φ(νt) ≤ νφ(t), we get

R

B

φ

 |f | (c 2 /c 1 )λ



dµ ≤ c 1 µ(B) .

This implies the second inequality.

P r o o f o f T h e o r e m 1. Let B = B r (x 0 ). If x 0 is an atom and r ≤ r x

then I α f (x 0 ) = 0 and the estimate is trivial. Let, then, r > r x

, let x ∈ B and let 0 < % < 2κr where κ is the constant in the “triangle inequality”

δ(x, y) ≤ κ(δ(x, z) + δ(z, y)). Then

|I _α f (x)| ≤ R

B

|f (y)|

δ(x, y) ^1−α dµ(y) ≤ R

δ(x,y)≤2κr

|f (y)|

δ(x, y) ^1−α dµ(y)

≤ R

δ(x,y)<%

+ R

%≤δ(x,y)≤2κr

= I 1 + I 2 .

We first estimate I 1 . If x is an atom and % ≤ r x then I 1 = 0. Let % > r x

and let K be the set of nonnegative integers k such that 2 ^−k % > r x . Denote by M f the Hardy–Littlewood maximal function of f . Then

I 1 = X

k∈K

R

2

%≤δ(x,y)<2

%

|f (y)|

δ(x, y) ^1−α dµ(y)

≤ X

k∈K

µ(B ₂

_% (x))

(2 ^−k−1 %) ^1−α M f (x)

≤ M f (x)

∞

X

k=0

A 2 2 ^−k %

(2 ^−k−1 ) ^1−α % ^1−α = A α % ^α M f (x) , with A α = A 2 · 2/(2 ^α − 1).

We now estimate I 2 . Using H¨ older’s inequality with p = 1/α and Lemma 1 we have

I 2 ≤ kf k _1/α

 R

%≤δ(x,y)≤2κr

dµ(y) δ(x, y)

 1−α

≤ kf k _1/α



B 1 log 4κr

%

 1−α

. If A α (2κr) ^α M f (x) ≤ kf k _1/α we set % = 2κr, since supp(f ) is contained in B, I ₂ = 0 and hence

|I _α f (x)| ≤ I 1 ≤ kf k _1/α .

If, on the other hand, A α (2κr) ^α M f (x) > kf k _1/α then there is a unique % in (0, 2κr) for which A α % ^α M f (x) = kf k _1/α , i.e. % = [kf k _1/α /(A α M f (x))] ^1/α . With this value of % we have

|I _α f (x)| ≤ I 1 + I 2 ≤ kf k _1/α

 1 +



B 1 log 4κrA ^1/α α M f (x) ^1/α kf k ^1/α _1/α

 1−α 

and hence in both cases

 I α f (x) C 1 kf k _1/α

 1/(1−α)

≤ 1 + log ⁺ 4κrA ^1/α α M f (x) ^1/α

kf k ^1/α _1/α

where C 1 = 2 ^α max(1, B ₁ ^1−α ).

Finally, using kM f k _1/α ≤ c ₁ kf k _1/α and normality we have

R

B

exp

   

I α f (x) C 1 kf k _1/α

1/(1−α)  dµ(x)

≤ e



µ(B) + A ^1/α α 4κr kf k ^1/α _1/α

R

X

M f (x) ^1/α dµ(x)



≤ e



1 + A ^1/α α 4κc ^1/α ₁ A 1



µ(B) = cµ(B) .

This concludes the proof of the theorem with C 1 = 2 ^α max(1, B ₁ ^1−α ) and c = e(1 + A ^1/α α 4κc ^1/α ₁ /A 1 ).

P r o o f o f T h e o r e m 2. We consider a ball B = B r (x 0 ) and the Orlicz norm N B,1 defined with φ(t) = e ^t

− 1. For f ∈ L ^1/α (X) we write

I e α f (x) − m B ( e I α f )

= R

X

 1

δ(x, y) ^1−α − ψ z (y) δ(z, y) ^1−α



f (y) dµ(y)

− 1

µ(B)

R

B

R

X

 1

δ(t, y) ^1−α − ψ z (y) δ(z, y) ^1−α



f (y) dµ(y) dµ(t)

= 1 µ(B)

R

B

R

X

 1

δ(x, y) ^1−α − 1 δ(t, y) ^1−α



f (y) dµ(y) dµ(t) . Decompose X = e B ∪ e B ^c where e B = B _4κ

r (x 0 ). The last expression can be written as

R

B e

1

δ(x, y) ^1−α f (y) dµ(y) − 1 µ(B)

R

B

R

B e

1

δ(t, y) ^1−α f (y) dµ(y) dµ(t) + 1

µ(B)

R

B

R

B e

 1

δ(x, y) ^1−α − 1 δ(t, y) ^1−α



f (y) dµ(y) dµ(t)

= J 1 − J ₂ + J 3 . Since k e I α f − m _B ( e I α f )k _B,1 ≤ kJ 1 k _B,1 + kJ 2 k _B,1 + kJ 3 k _B,1 it is enough to show that kJ i k _B,1 ≤ M _i kf k _1/α , 1 ≤ i ≤ 3, with M i independent of f . Since J 1 (x) = I α (f χ

B e ) we can use Theorem 1 and normality to obtain

R

B

φ

 |J 1 | c 1 kf k _1/α

 1/(1−α)

dµ ≤ R

B

φ

 |I _α (f χ

B e )|

c 1 kf χ

B e k _1/α

 1/(1−α)

dµ

≤ cµ( e B) ≤ cµ(B) .

From the definition of k k B,c and Lemma 2 it follows that kJ ₁ k _B,1 ≤ M ₁ kf k _1/α .

To estimate J 2 we use Jensen’s inequality and the estimate above to obtain

R

B

φ

 J 2

c 1 kf k _1/α



dµ ≤ 1 µ(B)

R

B

R

B

φ

 |I _α (f χ

B e )|

c 1 kf k _1/α



dµ(x) dµ(t) ≤ cµ(B) . As before, from the definition of k k B,c and Lemma 2 it follows that kJ 2 k _B,1 ≤ M 2 kF k _1/α .

Finally, for J 3 we will first show that H f (x, t) = R

B e

 1

δ(x, y) ^1−α − 1 δ(t, y) ^1−α



f (y) dµ(y)

is bounded and kH f k _∞ ≤ ckf k _1/α .

Since x and t are in B, and y in e B ^c , and the space has order γ, Lemma II.3 of [GV] states that

1

δ(x, y) ^1−α − 1 δ(t, y) ^1−α

≤ B ₂ δ(x, t) ^γ δ(x, y) ^α−γ−1 . Using this lemma and H¨ older’s inequality with p = 1/α we obtain

|H _f (x, t)| ≤ B 2 δ(x, t) ^γ  R

B e

δ(x, y) ^{−1−γ/(1−α)} dµ(y)  1−α  R

|f | ^1/α dµ  α

.

Using inequality II.2 of [GV]:

R

B e

δ(x, y) ^{−1−γ/(1−α)} dµ(y) ≤ cr ^{−γ/(1−α)} ,

and δ(x, t) ≤ r we get the desired estimate for kH f k _∞ .

Therefore kJ 3 k _∞ ≤ ckf k _1/α . On the other hand, it is easy to show that kJ ₃ k _B,1 ≤ ckJ ₃ k _∞ , and hence kJ 3 k _B,1 ≤ M ₃ kf k _1/α . This concludes the proof of Theorem 2.

REFERENCES

[AB] D. R. A d a m s and R. J. B a g b y, Translation-dilation invariant estimates for Riesz potentials, Indiana Univ. Math. J. 23 (1974), 1051–1067.

[CW] R. R. C o i f m a n and G. W e i s s, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.

[GV] A. E. G a t t o and S. V ´ a g i, Fractional integrals on spaces of homogeneous type, in:

Analysis and Partial Differential Equations, C. Sadosky (ed.), Dekker, New York

1990, 171–216.

[H] L. I. H e d b e r g, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.

[JN] F. J o h n and L. N i r e n b e r g, On functions of bounded mean oscillation, Comm.

Pure Appl. Math. 14 (1964), 415–426.

[T] N. S. T r u d i n g e r, On imbeddings into Orlicz spaces and some applications, J.

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[Z] A. Z y g m u n d, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge 1959.

DEPARTMENT OF MATHEMATICAL SCIENCES DePAUL UNIVERSITY

2219 NORTH KENMORE AVE.

CHICAGO, ILLINOIS 60614 U.S.A.

Re¸ cu par la R´ edaction le 25.1.1991;

en version modifi´ ee le 20.1.1992