JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/182921 182921 Research Article Two-Weight Extrapolation on Lorentz Spaces Li Wenming 1 Zhang Tingting 1 Xue Limei 2 Hencl Stanislav 1 College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei 050024 China 2 School of Mathematics and Science Shijiazhuang University of Economics Shijiazhuang 050031 China sjzue.edu.cn 2015 112015 2015 24 10 2014 17 12 2014 1 1 2015 2015 Copyright © 2015 Wenming Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give several two-weight extrapolation theorems on Lorentz spaces which extend the results of Cruz-Uribe and Pérez (2000) and some applications for the singular integral operators, the potential type operators, and commutators.

1. Introduction and Main Results

The classical extrapolation theorem is due to Rubio de Francia , who showed that if sublinear operator T is bounded on L p 0 ( w ) for some p 0 , 1 p 0 < , and every w A p 0 , then T is bounded on L p ( w ) for every p , 1 < p < and every w A p . Cruz-Uribe et al.  and Curbera et al.  generalized the extrapolation theorem of Rubio de Francia and got a lot of new extrapolation theorems associated with A weights. These theorems have proved to be the key to solving many problems in harmonic analysis.

Cruz-Uribe and Pérez  gave several other extrapolation theorems for pairs of weights of the forms ( w , M k w ) and ( w , ( M w / w ) r w ) , where w is any nonnegative function, r > 1 , and M k is the k th iteration of the Hardy-Littlewood maximal operator. The purpose of this paper is to extend the extrapolation theorems in  and derive some extrapolation results on Lorentz spaces.

Let w ( x ) , x R n , be a nonnegative, locally integrable weight function. For a measurable function f , (1) λ f w ( t ) = w x R n : f x > t , t 0 , is the distribution function of f with respect to the measure w ( x ) d x . For 0 < p , q , the weighted Lorentz spaces L p , q ( w ) are the collection of all functions f , such that f L p , q ( w ) < , where (2) f L p , q ( w ) = q 0 λ f w t q / p t q - 1 d t 1 / q , 0 < q < , sup t > 0 t λ f w t 1 / p , q = .

For 1 p < , p denote the conjugate exponent of p : 1 / p + 1 / p = 1 .

In the case when either 1 < p < and 1 q , or p = q = 1 , or p = q = , L p , q ( w ) is a Banach space. Furthermore, we have the relationship (3) C 1 f L p , q ( w ) sup g L p , q w 1 R n f x g x w x d x C 2 f L p , q ( w ) .

Our main results are the following.

Theorem 1.

Let S and T be operators (not necessarily linear), and let f be a function in a suitable test class for both S and T . Suppose that there exist positive constants p 0 and C 0 and a positive integer k such that for all weights w (4) T f L p 0 ( w ) C 0 S f L p 0 ( M k w ) . Then, for all p , p 0 < p < , p 0 q , there exists a constant C p depending only on C 0 , p , p 0 , k , and n , such that, for all weights w , (5) T f L p , q ( w ) C p S f L p , q ( M [ k p / p 0 ] + 1 w ) , where [ k p / p 0 ] is the largest integer less than or equal to k p / p 0 .

Theorem 2.

Let S and T be operators (not necessarily linear), and let f be a function in a suitable test class for both S and T . Suppose that there exist positive constants p 0 and C 0 and a positive integer k such that for all weights w (6) T f L p 0 ( w ) C 0 S f L p 0 ( M w ) . Then, for all p , p 0 < p < , p 0 q , there exists a constant C p depending only on C 0 , p , p 0 , k , and n , such that, for all weights w , (7) T f L p , q ( w ) C p S f L p , q ( ( M w / w ) [ p / p 0 ] M w ) .

2. Applications

Let M be the Hardy-Littlewood maximal operator; Fefferman and Stein  showed that, for every p , 1 < p < , every nonnegative function w , and every function f , (8) R n M f x p w x d x C R n f x p M w x d x .

Using Theorem 1, we can get the following result.

Proposition 3.

For p , 1 < p < , 1 q , there exists a constant C > 0 , such that, for all weights w and function f , (9) M f L p , q ( w ) C f L p , q ( M w ) .

Let T be a Calderón-Zygmund operator. Wilson  and Pérez  showed that if 1 < p < , then, for any weight w and every f , (10) R n T f x p w x d x C R n f x p M [ p ] + 1 w ( x ) d x , where the exponent [ p ] + 1 is sharp. Using Theorem 1 we can get the following result.

Proposition 4.

Let T be a Calderón-Zygmund operator, 1 < p < , 1 q ; then, for any weight w and every f , (11) T f L p , q ( w ) C f L p , q ( M [ p ] + 1 w ) .

For the commutators of singular integral operators with a BMO function [ T , b ] , Pérez  showed the analogous weighted inequalities as in (10). By Theorem 1, we can give similar results as in Proposition 4; details are omitted.

Lerner  establishes (12) R n T f x w x d x C R n M f x M w x d x , for any Calderón-Zygmund operator T and any arbitrary weight w . Using Theorems 1 and 2, we can get the following results.

Proposition 5.

Let T be a Calderón-Zygmund operator, 1 < p < , 1 q ; then, for any weight w and all f , (13) T f L p , q ( w ) C M f L p , q ( M [ p ] w ) , T f L p , q ( w ) C M f L p , q ( ( M w / w ) [ p ] M w ) .

Now, we consider the potential type operators and commutators. For a nonnegative, locally integrable function Φ on R n , assume that Φ satisfies the following weak growth condition: there are constants δ , c > 0 , 0 ɛ < 1 with the property that, for all k Z , (14) sup 2 k < | x | 2 k + 1 Φ x c 2 k n δ ( 1 - ɛ ) 2 k < | y | 2 δ ( 1 + ɛ ) 2 k Φ ( y ) d y . The basic examples are provided by the Riesz potential of order α , I α , defined by the kernel Φ ( x ) = | x | α - n , 0 < α < n .

Define the potential type operator T Φ and the maximal operator M Φ ~ by (15) T Φ f ( x ) = R n Φ x - y f y d y , M Φ ~ f x = sup x Q Φ ~ l Q Q Q f x d x , where (16) Φ ~ ( t ) = | z | < t Φ ( z ) d z , t 0 . For the Riesz potential of order α , Φ ~ ( t ) = t α . Pérez  studies the two-weight strong type ( p , q ) inequalities for T Φ , 1 < p q < .

Lemma 6 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

Let T Φ be the potential type operator with Φ satisfying (14), let f and g be bounded functions with compact support, and let a > 2 n ; then, there exist a family of cubes Q k , j and a family of pairwise disjoint subsets E k , j , E k , j Q k , j , with Q k , j < ( 1 - 2 n / a ) - 1 E k , j for all k , j , such that (17) R n T Φ f x g x d x C k , j 1 3 Q k , j 3 Q k , j f x d x Φ ~ l ρ Q k , j ρ Q k , j × ρ Q k , j g x d x E k , j , where ρ = δ ( 1 + ɛ ) .

By Lemma 6, we can easily prove the following result.

Lemma 7.

Let T Φ be the potential type operator with Φ satisfying (14); then, there is a constant C > 0 such that, for any weight w and all f , (18) R n T Φ f x w x d x C R n M Φ ~ f x M w x d x .

Using (18) and Theorems 1 and 2, we can get the following results.

Proposition 8.

Let T Φ be the potential type operator with Φ satisfying (14), 1 < p < , 1 q ; then, for any weight w and every f , (19) T Φ f L p , q ( w ) C M Φ ~ f L p , q ( M [ p ] w ) , T Φ f L p , q ( w ) C M Φ ~ f L p , q ( ( M w / w ) [ p ] M w ) .

For k 1 and b BMO , the commutators of potential type integral operator with a BMO function are defined by (20) T Φ , b k f ( x ) = R n b x - b y k Φ x - y f y d y . Li  gave the two-weight strong type ( p , q ) inequalities for T Φ , b k , 1 < p q < . We can easily get similar results as in Lemmas 6 and 7 from ; by Theorem 1, we can obtain some weighted inequalities for the commutators of potential type integral operators; details are omitted.

3. Proofs of Theorems

We need some notations and facts. Let B ( t ) : [ 0 , ) [ 0 , ) be a Young function, that is, a continuous, convex, and increasing function with B ( 0 ) = 0 such that B ( t ) as t . Given a Young function B , we define the B -average of a function f over a cube Q by (21) f B , Q = inf λ > 0 : 1 Q Q B f x λ d x 1 . In particular, for the Young function B k ( t ) = t ( 1 + log + t ) k , k = 1,2 , , the B k -average of a function f over a cube Q is denoted by f L ( log L ) k , Q . Define the maximal function associated with the Young function B k as (22) M L ( log L ) k f x = sup Q x f L ( log L ) k , Q . Pérez  obtained that M L ( log L ) k f ( y ) M k + 1 f ( y ) .

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Fix p , p 0 < p < , p 0 q , and let r = p / p 0 , s = q / p 0 . Then, by duality, (23) T f L p , q ( w ) p 0 = T f p 0 L r , s ( w ) sup g L r , s ( w ) , g L r , s ( w ) 1 R n T f x p 0 g x w x d x . The last inequality follows since L r , s ( w ) and L r , s ( w ) are associate spaces. Fixing one of these g ’s, we use (4) to continue with (24) R n T f x p 0 g x w x d x C R n S f x p 0 M k g w x d x C R n S f x p 0 M L ( log L ) k - 1 g w x M ~ w ( x ) M ~ ( w ) ( x ) d x C S f p 0 L r , s ( M ~ ( w ) ) M L ( log L ) k - 1 g w x M ~ w x L r , s ( M ~ ( w ) ) = C S f L p , q ( M ~ ( w ) ) p 0 M L ( log L ) k - 1 g w x M ~ w x L r , s ( M ~ ( w ) ) , where M ~ is an appropriate maximal operator to be chosen soon. To conclude, we just need to show that (25) M L log L k - 1 g w x M ~ w x L r , s ( M ~ ( w ) ) C g L r , s ( w ) , or equivalently (26) F : L r , s w L r , s M ~ w , where (27) F f = M L ( log L ) k - 1 f w M ~ w . To do this, we choose M ~ pointwise bigger than M L ( log L ) k - 1 ; then, we trivially have (28) F : L w L M ~ w . Therefore, by Marcinkiewicz’s interpolation theorem for Lorentz spaces in , it will be enough to show that for some ɛ > 0 (29) F : L r + ɛ w L r + ɛ M ~ w , which amounts to proving (30) R n M L log L k - 1 ( f w ) x r + ɛ M ~ w x 1 - r + ɛ d x C R n f x r + ɛ w ( x ) d x . But this result follows from : it is shown there that, for ν > 1 and η > 0 , (31) R n M L log L k - 1 f x ν M L log L k ν - 1 + η w x 1 - ν d x C R n f x ν w x 1 - ν d x . We finally choose the appropriate parameters and weight. Let ν = r + ɛ , η = ɛ = [ k r ] + 1 - k r / k + 1 > 0 , and pick the weight (32) M ~ w = M L ( log L ) [ k r ] w . This shows that (33) T f L p , q ( w ) C S f L p , q ( M L ( log L ) [ k p / p 0 ] ( w ) ) . We conclude the proof of (5).

Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>.

The proof of this theorem proceeds exactly as that of (5) with minor changes. At inequality (23), fixing one of these g ’s, we use (6) to continue with (34) R n T f x p 0 g x w x d x C R n S f x p 0 M g w x d x . For g and w , we have M ( g w ) C M c ( g w ) , where C is a constant depending only on the dimension n and M c is the unweighted centered Hardy-Littlewood maximal operator: (35) M c ( f ) ( x ) = sup r > 0 1 B r x B r ( x ) f x d x , in which B r ( x ) is the ball of radius r centered at x . Furthermore, (36) M c ( g w ) ( x ) = sup r > 0 1 B r x B r ( x ) g y w y d y = sup r > 0 w B r x B r x 1 w B r x B r ( x ) g y w y d y M w , c g x M w x , where M w , c is the weighted centered maximal operator. Then, for δ > 0 , (37) R n S f p 0 M g w d x C R n S f p 0 M w , c g w M w r + δ - 1 M w w r + δ w d x C S f p 0 L r , s ( ( M w / w ) r + δ w ) × M w , c g w M w r + δ - 1 L r , s ( ( M w / w ) r + δ w ) . To conclude, we just need to show that (38) M w , c g w M w r + δ - 1 L r , s ( ( M w / w ) r + δ w ) C g L r , s ( w ) , or equivalently (39) F : L r , s w L r , s M w w r + δ w , where (40) F f = M w , c f w M w r + δ - 1 . We notice that M w ( M w / w ) r + δ w for each w . With this, we trivially have (41) F : L ( w ) L M w w r + δ w . Therefore, by Marcinkiewicz’s interpolation theorem for Lorentz spaces, it will be enough to show that for some ɛ > 0 (42) F : L r + ɛ w L r + ɛ M w w r + δ w , which amounts to proving (43) R n M w , c g w M w r + δ - 1 r + ɛ M w w r + δ w d x C R n g r + ɛ w d x . Taking ɛ = δ and using the well-known fact that M w , c is bounded on L p ( w ) , 1 < p < , with a constant that depends only on p and n , we get (43).

This shows that (44) T f L p , q ( w ) p 0 C S f L p , q ( ( M w / w ) p / p 0 + δ w ) p 0 . Taking δ = [ p / p 0 ] + 1 - p / p 0 > 0 , we get (7). This ends the proof of Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their gratitude to the referee for his very valuable comments and suggestions. This study is supported by the Natural Science Foundation of Hebei Province (A2014205069 and A2015403040) and Postdoctoral Science Foundation of Hebei Province (B2013003007).

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