WEAK TYPE BOUNDS FOR A CLASS OF ROUGH OPERATORS

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 10, October 1997, Pages 2939–2942 S 0002-9939(97)03914-2

WEAK TYPE BOUNDS FOR A CLASS OF ROUGH OPERATORS WITH POWER WEIGHTS YONG DING (Communicated by J. Marshall Ash) Abstract. In this note we show that TΩ,α and MΩ,α , the fractional integral and maximal operators with rough kernel respectively, are bounded operators from L1 (|x|β(n−α)/n , Rn ) to Ln/(n−α),∞ (|x|β , Rn ), where 0 < α < n and −1 < β < 0.

§1. Introduction Suppose that 0 < α < n, and Ω ∈ Ls (S n−1 ) (s ≥ 1), where S n−1 denotes the unit sphere of Rn . Moreover, Ω is homogeneous of degree zero. We define the fractional maximal operator by Z 1 |Ω(x − y)||f (y)| dy MΩ,α f (x) = sup n−α r>0 r |x−y| 0 and any f ∈ L1 (|x|β(n−α)/n , Rn ),  Z n/(n−α) Z 1 β β(n−α)/n |x| dx ≤ C |f (x)||x| dx , λ Rn {x:|TΩ,α f (x)|>λ} where C is independent of λ and f . Received by the editors January 24, 1996 and, in revised form, May 3, 1996. 1991 Mathematics Subject Classification. Primary 42B20. Key words and phrases. Fractional integral and maximal operators, power weights. The author was supported by NSF of Jiangxi in China. c

1997 American Mathematical Society

2939

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

2940

YONG DING

Theorem 2. Let 0 < α < n, −1 < β < 0, n/(n − α) ≤ s ≤ ∞ and Ω ∈ Ls (S n−1 ). Then MΩ,α is a bounded operator from L1 (|x|β(n−α)/n , Rn ) to Ln/(n−α),∞ (|x|β , Rn ). That is, for any λ > 0 and any f ∈ L1 (|x|β(n−α)/n , Rn ),  Z n/(n−α) Z 1 β β(n−α)/n |x| dx ≤ C |f (x)||x| dx , λ Rn {x:|MΩ,α f (x)|>λ} where C is independent of λ and f . §2. Proof of the theorems Let us begin by giving some lemmas. Lemma 1. Let q > 1 and T be a sublinear operator satisfying for each a > 0 the estimate (2.1)   a 2 ≤ |x| ≤ a : |T (f χ{|x|>2a})(x)| > λ ≤ C

1 λ

Z

a |f (y)|( )1/q dy |y| |y|>2a

!q .


Then , if T is of weak type (1, q), it is also of weak type L1 (|x|β/q ), Lq,∞ (|x|β ) for −1 < β < 0. Proof. Given f , we now define, for each k ∈ Z, fk,0 = f χ{|x|≤2k+1 } and fk,1 = f − fk,0 . Then we can write, as usual, X X |T f (x)| ≤ |T fk,0 |χIk + |T fk,1 |χIk = T0 f (x) + T1 f (x), k

k

where Ik = {x ∈ Rn : 2k−1 ≤ |x| < 2k } for each k ∈ Z. If we call ωβ (x) = |x|β , we have Z ωβ (x) dx ωβ {x : T0 f > λ} = ≤C

X

{x:T0 f >λ}

k

ωβ (2 )|{x ∈ Ik : T0 fk,0 > λ}|

k

≤ = ≤

= ≤

Z q C X k ω (2 ) |f (y)| dy β λq |y|≤2k+1 k  q X Z C X k ω (2 ) |f (y)| dy β λq k j≤k+1 Ij X X Z q 1/q q C k |f (y)| dy ω (2 ) β λq Ij j k≥j−1   X 1/q q Z C X k |f (y)| dy ω (2 ) β λq Ij j k≥j−1  q  Z q Z C X 1 j 1/q 1/q |f (y)|ωβ (2 ) dy ≤ C |f (y)|ωβ (y) dy . λq λ Rn Ij j

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

WEAK TYPE BOUNDS FOR A CLASS OF ROUGH OPERATORS

2941

Here we have used that T is a weak type (1, q) bounded operator and β < 0. In order to estimate T1 , we make use of (2.1): Z ωβ {x : T1 f > λ} =

{x:T1 f >λ}

≤C

X

ωβ (x) dx

ωβ (2k )|{x ∈ Ik : T1 fk,1 > λ}|

k

Z  k 1/q q C X 2 ωβ (2k ) |f (y)| dy = q λ |y| k+1 |y|>2 k XZ  1/q q C X 1 k k = q ωβ (2 ) 2 |f (y)| dy λ |y| k j≥k Ij   Z q 1/q q C X X 1 1/q k k ≤ q ωβ (2 )2 |f (y)|( ) dy λ |y| Ij j k≤j  1/q q X Z C X 1 1/q k k = q |f (y)|( ) dy ωβ (2 )2 λ |y| Ij j k≤j  q Z C X 1 j 1/q j/q ≤ q |f (y)| dy j/q ωβ (2 ) 2 λ 2 Ij j  Z q 1 ≤C |f (y)|ωβ (y)1/q dy , λ Rn where we have used that β > −1. This finishes the proof of Lemma 1. Lemma 2. Let 0 < α < n, Ω ∈ Ls (S n−1 ) and s ≥ 1. Then there is a C > 0 depending only on n and α, such that (2.2)

MΩ,α f (x) ≤ CT|Ω|,α (|f |)(x).

Proof. Fix r > 0; then we have Z T|Ω|,α (|f |)(x) ≥ (2.3)

≥

|x−y|0

1 rn−α

Z |x−y| 0. In fact, a |{ ≤ |x| ≤ a : |TΩ,α (f χ{|x|>2a} )(x)| > λ}| 2Z 1 ≤ q |TΩ,α (f χ{|x|>2a} )(x)|q dx λ |x|≤a Z q Z 1 Ω(x − y) = q f (y) dy dx λ |x|≤a |y|>2a |x − y|n−α Z Z 1/q q 1 |Ω(x − y)|q |f (y)| dx dy ≤ q (n−α)q λ |y|>2a |x|≤a |x − y| Z Z 1/q q 1 |Ω(x)|q ≤ q |f (y)| dx dy (n−α)q λ |y|>2a |x−y|≤a |x| Z Z 1/q q C 1 q ≤ q |f (y)| n−α |Ω(x)| dx dy λ |y| |y|>2a |x−y|≤a Z  Z |y|+a Z 1/q q C 1 q n−1 ≤ q |f (y)| n−α |Ω(θ)| dθr dr dy λ |y| |y|>2a S n−1 |y|−a Z q C 1 ≤ q |f (y)| n−α kΩkq (a|y|n−1 )1/q dy λ |y| |y|>2a  Z q a 1/q q 1 = CkΩkq |f (y)|( ) dy , λ |y|>2a |y| R where kΩkqq = S n−1 |Ω(θ)|q dθ. Thus, the conclusion of Theorem 1 immediately follows from Lemma 1. It is easy to see that the conclusion of Theorem 2 is a direct consequence of Theorem 1 and Lemma 2. Acknowledgement The author would like to thank the referee for his very valuable comments. References 1. S. Chanillo, D. Watson and R. L. Wheeden, Some integral and maximal operator related to starlike sets, Studia Math. 107 (1993), 223–255. MR 94j:42027 2. B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249–258. MR 44:3155 3. F. Soria and G. Weiss, A remark on singular integrals and power weights, Indiana Univ. Math. Jour. 43 (1994), 187–204. MR 95g:42028 Department of Mathematics, Nanchang Vocational and Technical Teacher’s College, Nanchang, Jiangxi, 330013, People’s Republic of China Current address: No. 35, Xianshi One Road, Nanchang, Jiangxi, 330006, People’s Republic of China

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use