Indian J. Pure Appl. Math., 51(3): 1105-1132, September 2020 °c Indian National Science Academy DOI: 10.1007/s13226-020-0453-1

HARDY-TYPE OPERATORS IN LORENTZ-TYPE SPACES DEFINED ON SPACES1 Qinxiu Sun∗, Xiao Yu∗∗ and Hongliang Li∗∗∗

∗Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China ∗∗Department of Mathematics, Shangrao Normal University, Shangrao 334001, China ∗∗∗Department of Mathematics, Zhejiang International Studies University, Hangzhou 310012, China e-mail: [email protected] (Received 2 June 2018; accepted 12 June 2019)

Weight criteria for the boundedness and compactness of generalized Hardy-type operators Z

T f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X, (0.1) {φ(y)≤ψ(x)}

in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions φ, ψ, u1,

u2, v0 are positive measurable functions. Some sufficient conditions of boundedness of

G0 G1 G0 G1,∞ T :Λv0 (w0) → Λv1 (w1) and T :Λv0 (w0) → Λv1 (w1) are obtained on Orlicz-Lorentz spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and

p0 p1,q1 compact from a weighted Lorentz Λv0 (w0) to another Λv1 (w1). It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces.

Key words : Hardy operator; Orlicz-Lorentz spaces; weighted Lorentz spaces; boundedness; compactness.

2010 Mathematics Subject Classification : 46E30, 46B42.

1. INTRODUCTION

R x For the Hardy operator S defined by Sf(x) = 0 f(t)dt, the weighted Lebesgue- inequalities have been characterized by many authors (e.g. [3, 12, 30, 32]). Sawyer [36] characterized the weights

1Supported by Natural Science Foundation of Zhejiang Province of China (LY19A010001), National Natural Science Foundation of China (11961056), Natural Science Foundation of Jiangxi Province of China (20151BAB211002). 1106 QINXIU SUN, XIAO YU AND HONGLIANG LI u, v such that S : Lp,q(u) → Lr,s(v) is bounded, under certain restriction on exponents p, q, r and s. Later on Carro and Soria [5] described the exponents p0, p1, the weights u0, u1, w0, w1 such that p0 p1,∞ p0 p1 S :Λv0 (w0) → Λu1 (w1) or S :Λu0 (w0) → Λu1 (w1) is bounded.

1 R x For the Hardy operator A by Af(x) = x 0 f(t)dt, Sawyer [36] analyzed the weights v and w such that A : Lr(v) → Lp,q(w) is bounded under some assumptions on exponents p, q, r. Given non-negative measurable functions ψ and φ on R+ define the operator Z x H1f(x) = ψ(x) φ(t)f(t)dt, x > 0. 0 r p ,q Ferreyra [10] gave a characterization of boundedness of H1 : L 1 (u1) → L 1 1 (w1) under the p ,q assumptions 1 ≤ r1 ≤ min(p1, q1) and normability of L 1 1 (w1). Edmund, Gurka and Pick in [7, Theorems 3-4] obtained characterization of boundedness and compactness of

r0,s0 p0,q0 H1 : L (v0) → L (w0),

r ,s p ,q when max(r0, s0) ≤ min(p0, q0) and the Lorentz spaces L 0 0 (v0) and L 0 0 (w0) are normable. The result in [7] can also be used to the description of boundedness and compactness of the high dimensional Hardy operator Z Hf(x) = ψ(x) φ(y)f(y)dy, x ∈ Rn, (1.1) B(0,|x|) from Lr,s(u) to Lp,q(w) where ψ, φ are non-negative measurable functions on Rn, |x| is the Eu- clidean norm of x ∈ Rn, B(0, t) is the ball of radius t in Rn centered at 0 when max(r, s) ≤ min(p, q) and the Lorentz spaces Lr,s(u) and Lp,q(w) are normable. Mart´ın-Reyes, Ortega Salvador and Sarrion´ Gavilan´ [28] discovered the conditions for boundedness of

p0 p1,q1 H :Λv0 (w0) → Λu1 (w1) (1.2) when 0 < p0 ≤ p1 ≤ q1 ≤ ∞, w1 is a non-increasing. Li and Kaminska [24] study boundedness and compactness of

G0 G1 G0 G1,∞ H :Λv0 (w0) → Λu1 (w1) and H :Λv0 (w0) → Λu1 (w1) on the Orlicz-Lorentz spaces and (1.2) on the weighted Lorentz spaces, improving the results in [28] through enlarging the range of weights and indices.

Edmunds, Kokilashvili, and Meskhi [8] (see also [9]) considered the Hardy-type operator T on a σ−finite measure space (X, µ) defined as Z T f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X. (1.3) {φ(y)≤ψ(x)} LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1107

where the functions φ, ψ, u1, u2, , v0 are positive measurable functions on (X, µ) and for every t1, t2 with 0 < t1 < t2 < ∞ the conditions:

0 < µ{y ∈ X : t1 < φ(y) < t2} < ∞,

0 < µ{x ∈ X : t1 < ψ(x) < t2} < ∞, are fulfilled. Obviously, the operator T is a generalization of the operators H1 and H. The au- thors found a characterization of boundedness and compactness of T from a Banach function space

(X1, µ, v0) to another (X2, µ, v1) on which some special conditions are required.

The first main result in the present paper is Proposition 3.1 which gives necessary and sufficient conditions of the modular inequalities,

kG(T f)k p1 ≤ CkG(f)k p0 and kG(T f)k p1,∞ ≤ CkG(f)k p0 , Λv1 (w1) Λv0 (w0) Λv1 (w1) Λv0 (w0) when 0 < p0 ≤ p1 < ∞. It leads to Corollary 3.1 which states sufficient conditions of the bounded- ness of the operators

Gp0 Gp1 Gp0 Gp1 ,∞ T :Λv0 (w0) → Λv1 (w1) and T :Λv0 (w0) → Λv1 (w1) between Orlicz-Lorentz spaces. These results generalize the result of [24] since the operator H is a special case of T and also improves [8, 9] due to no restriction of the spaces to be Banach spaces. When discussing the sufficiency of Proposition 3.1 we exploit a method quite different from [8, 9] since the spaces studied here are not Banach function spaces and thus we can not use the principle of duality. Specifically, the principle of duality says that if X and Y are Banach function spaces on a measure space (Ω, µ), then the boundedness of

T : X → Y (1.4) is equivalent to the boundedness of T 0 : Y 0 → X0 or the establishment of the inequality ¯Z ¯ ¯ ¯ ¯ ¯ ¯ (T f)g¯ ≤ CkfkX kgkY 0 , (1.5) R R where T 0 is the conjugate operator of T defined by the formula (T f)g = fT 0g and X0 represents the Kothe¨ dual of X, ½ Z ¾ X0 = g : |fg| < ∞ for all f ∈ X , and the associate norm of g for g ∈ X0 is endowed by Z

kgkX0 = sup |fg|. kfkX ≤1 1108 QINXIU SUN, XIAO YU AND HONGLIANG LI

[8, 9] transfered the question (1.4) to the proof of (1.5). But we indicate that in the background of quasi-Banach function spaces this method is false. For example, if Y = L1,q, 1 < q ≤ ∞, then Y 0 = {0} and thus we can not change the question (1.4) to (1.5).

p1 p1,∞ Particularly, for Y = Λv1 (w1) or Λv1 (w1) under some assumptions on exponents and weights, the present Corollary 3.2 gives a sufficient and necessary condition of the boundedness of

p0 T :Λv0 (w0) → Y. (1.6) Under the conditions of Corollary 3.2, Theorem 3.1 presents the characterization of the boundedness of (1.6) which is analogous to [9, Theorem 1.6] but due to weaker assumptions it is also an improve- ment. At the same time, Theorem 3.1 extends [24, Theorem 3.9] since the operator T is more general than H. Meanwhile, we get a sufficient condition of the boundedness of

p0 P :Λv0 (w0) → Y where Z P f(x) = u1(x) f(y)u2(y)v0(y)dµ(y) {aψ(x)<φ(y)

2. PRELIMINARIES

Let (X, µ) be a σ-finite measure space and M(X, µ) be the space of all µ-measurable real valued ∗ functions on X. The decreasing rearrangement fµ of f ∈ M(X, µ) is defined in [2] by

∗ µ fµ(t) = inf{s : λf (s) ≤ t}, t ≥ 0, where µ λf (s) = µ{x ∈ X : |f(x)| > s}, s ≥ 0, is a distribution function of f. The function w : X → R+ is called a weight function, or simply a weight, whenever w is measurable, not identically equal to zero and integrable on sets of finite LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1109

R t measure. If w is a weight on R+, then we denote W (t) = 0 w(s) ds, and we always have that W (t) < ∞, t > 0. Letting 0 < p, q < ∞, we say that f ∈ M(X, µ) belongs to the Lorentz ¡R ¢ p,q ∞ 1/p ∗ q dt 1/q space L (X) [2, 14] if kfkLp,q(X) = 0 (t fµ(t)) t < ∞, and for 0 < p ≤ ∞, the p,∞ 1/p ∗ space L (X) is defined as a class of M(X, µ) such that kfkLp,∞(X) = supt>0 t fµ(t) < ∞. If p,q p,q (X, µ) = (R+, wdx), we use the notation L (X) = L (w) and µ(E) = w(E) for every Lebesgue measurable subset E of R+.

p,q Let w be a weight on R+. Define for 0 < p, q < ∞ the weighted Lorentz space ΛX (w) (see [4] or [5]) as a class of f ∈ M(X, µ) such that

° Ã ! 1 ° ° Z µ p ° ° λf (y) ° ∗ 1/q ° ° kfkΛp,q(w) = kfµkLp,q(w) = p °y w(t)dt ° < ∞, X ° 0 ° q dy L ( y ) R ∞ q dy 1/q p,∞ where kgk q dy = ( |g(y)| y ) , and the weighted Lorentz space ΛX (w) consisting of f ∈ L ( y ) 0 M(X, µ) with

à ! 1 Z λµ(y) p ∗ f kfkΛp,∞(w) = kfµkLp,∞(w) = sup y w(t)dt < ∞. X y>0 0

p p,p p,q We agree on the convention ΛX (w) = ΛX (w) and note that for 0 < p, q < ∞, ΛX (w) = q q p −1 ΛX (w ¯) where w¯ = W w.

p p Let Ldec(w) be the cone of all non-increasing functions in L (w) where w is a weight on [0, ∞), 1 R t 0 < p < ∞ and the operator A is defined by Af(t) = t 0 f(s)ds for all nonnegative measur- able functions f on R+. Arino˜ and Muckenhoupt [1] gave a characterization of the boundedness of p p A : Ldec(w) → L (w) in terms of the inequality on w called condition Bp, that is, w satisfies the following condition: Z ∞ Z r p w(x) r p dx ≤ C w(x)dx, r > 0, r x 0 p for some C > 0. Carro and Soria [6] obtained a characterization of boundedness of A : Ldec(w) → p,∞ L (w) showing that A is bounded whenever w ∈ Bp,∞ that is there exists C > 0 such that if p > 1 then à ! 0 Z µ Z ¶ 0 1/p µZ ¶ r 1 x −p r 1/p w(t)dt w(x)dx w(x)dx ≤ Cr, r > 0, 0 x 0 0 and if p ≤ 1 then Z Z 1 r C s p w(x)dx ≤ p w(x)dx, 0 < s < r. r 0 s 0 For other characterizations of Bp,Bp,∞, we refer to [4, 18, 37, 40]. We know [4, Theorem 2.2.5] that p p,∞ if p ≥ 1 and w ∈ Bp,∞ then ΛX (w) is normable and if w ∈ Bp then ΛX (w) is normable. 1110 QINXIU SUN, XIAO YU AND HONGLIANG LI

Let G : [0, ∞) → [0, ∞) be an Orlicz function [31], in symbol G ∈ F, that is G is continuous and strictly increasing on R+, such that lim G(t) = ∞ and G(0) = 0. Given G ∈ F and a weight t→∞ G G,∞ w on R+, the Orlicz-Lorentz space ΛX (w) (resp. ΛX (w)) [15, 16, 23, 27, 29, 31] is the set of G G,∞ f ∈ M(X, µ) such that for some λ > 0, we have IX,w(λf) < ∞ (resp. IX,w (λf) < ∞), where Z ∞ µ ¶ G ∗ G,∞ ∗ IX,w(f) = G(fµ(t))w(t)dt, resp. IX,w (f) = sup G(fµ(t))W (t) , 0 t>0 and we let ½ ³ ´ ¾ µ ½ ³ ´ ¾¶ G f G,∞ f kfkΛG (w) = inf ² > 0 : IX,w ≤ 1 resp. kfkΛG,∞(w) = inf ² > 0 : IX,w ≤ 1 . X ² X ² We will assume further, without loss of generality, that the weight w vanishes on the interval [µ(X), ∞) if µ(X) < ∞.

For G ∈ F, define its lower and upper Matuszewska-Orlicz indices [27] as follows: G(at) G(at) αG = sup{r > 0 : sup r < ∞}, βG = inf{r > 0 : inf r > 0}. 00 G(t)a 00 G(t)a

We say that a function G : [0, ∞) → [0, ∞) satisfies condition ∆2, in symbol G ∈ ∆2, whenever supt>0 G(2t)/G(t) < ∞. It is well known that βG < ∞ if and only if G ∈ ∆2. Kaminska´ and

Raynaud showed in [20, Proposition 4.5] that if αG > 0 and W ∈ ∆2, then k·kΛG (w) and k·k G,∞ X ΛX (w) are quasi-norms.

p G p G,∞ p,∞ If G(t) = t , 0 < p < ∞, then ΛX (w) = ΛX (w) and ΛX (w) = ΛX (w) are weighted G G Lorentz spaces (see [4, 26]). If a measure vdµ(y) is given on X, we denote ΛX (w) = Λv (w), G,∞ G,∞ p p p,∞ p,∞ p,q p,q ΛX (w) = Λv (w), ΛX (w) = Λv(w), ΛX (w) = Λv (w) and if v = 1 then L (X) = L .

∗∗ ∗ Letting 0 < p < ∞, fµ (t) = A(fµ)(t), t > 0, for f ∈ M(X, µ), define the space [4, Section 2.2.4] Z ∞ p p ∗∗p ΓX (w) = {f ∈ M(X, µ): kfkΓ (w) = fµ (t)w(t)dt < ∞}, X 0 and if Φ is a non-negative function on R+, define

p,∞ p,∞ ∗∗ 1/p ΓX (dΦ) = {f ∈ M(X, µ): kfkΓ (dΦ) = sup fµ (t)Φ (t) < ∞}. X t>0 p p p,∞ p,∞ If a measure vdµ(y) is given on X, denote ΓX (w) = Γv(w) and ΓX (dΦ) = Γv (dΦ). A function G ∈ F is said to satisfy ∆0 (resp. ∇0) condition [22, 35], in symbol G ∈ ∆0 (resp. ∇0) if there exists C > 0 such that

G(xy) ≤ CG(x)G(y), x, y ≥ 0 (resp. G(Cxy) ≥ G(x)G(y), x, y ≥ 0). LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1111

We have that G ∈ ∇0 if and only if there exists C > 0 such that G(xy) ≥ CG(x)G(y), x, y ≥ 0. The 0 explanation can be found in [24]. It is also easy to see that G ∈ ∆ yields G ∈ ∆2.

A strictly increasing positive sequence {xj}j∈Z is called a covering sequence [13] if the sequence j=∞ j=M is of the form {xj}j=−∞ or of the form {xj}j=N , where M and/or N is finite. In the latter case we define xN−1 = 0 and/or xM+1 = ∞.

0 ∞ Throughout the paper, we assume that the expressions of the form 0 · ∞, 0 , ∞ are equal to 0 1 1 zero. Given 1 ≤ p < ∞ denote by p its conjugate index that is p + p0 = 1. The notation f ≈ g indicates the existence of a universal constant C > 0 independent of all parameters involved, so that (1/C)f ≤ g ≤ Cf. The symbol f ↓ indicates that f is a non-negative non-increasing function in (0, b) for given b > 0 with 0 < b ≤ ∞. Note that the constant C, unless specifically stated otherwise, may differ from one occurrence to another.

3. BOUNDEDNESSOF HARDY OPERATORS

Let for the rest of the paper G ∈ F, w0, w1 be the weights on R+ and v0, v1 the weights on Gp0 (X, µ). The operator T is defined by (1.3). Let us consider first when the operator T :Λv0 (w0) → p Gp1 G 1 ,∞ Λv1 (w1) (resp. Λv1 (w1)) is bounded. We begin from looking for necessary and sufficient con- ditions of the modular inequality

Gp0 kG(T f)kY ≤ CkG(|f|)k p0 , f ∈ Λv (w0), Λv0 (w0) 0

p1 p1,∞ where Y = Λv1 (w1) or Λv1 (w1). The following lemma is [24, Lemma 3.1] essentially however they have different pattern for representation and we omit the proof.

Lemma 3.1 — (i) Let 0 < p0 ≤ σ. There exists C > 0 such that for every {tk}k ⊂ R+ with P k tk ≤ v0(X), σ Ã P ! σ µZ ¶ Z p X tk p0 k tk 0 w0(t)dt ≤ C w0(t)dt (3.1) k 0 0 if and only if there exists a constant C > 0 such that for every collection of measurable sets Ek with P p0 k χEk ≤ c and a function f ∈ Λv0 (w0), ° °σ X ° ° σ °fχ∪ E ° ≥ C kfχE k p0 . (3.2) k k p0 k Λv (w0) Λ (w0) 0 v0 k P (ii) Let p1 ≥ σ > 0. There exists C > 0 such that for all {tk}k ⊂ R+ satisfying k tk < v1(X),

σ Ã P ! σ µZ ¶ Z p X tk p1 k tk 1 w1(t)dt ≥ C w1(t)dt , (3.3) k 0 0 1112 QINXIU SUN, XIAO YU AND HONGLIANG LI

if and only if there exists a constant C > 0 such that for every collection of measurable sets Ek with P p1 k χEk ≤ c and a function f ∈ Λv1 (w1), ° °σ X ° ° σ °fχ∪ E ° ≤ C kfχE k p1 . (3.4) k k p1 k Λv (w1) Λ (w1) 1 v1 k

(iii) Let p1, σ > 0. Then w1 satisfies (3.3) if and only if there exists a constant C > 0 such that P p1,∞ for every collection of measurable sets Ek with k χEk ≤ c and a function f ∈ Λu1 (w1), ° °σ X ° ° σ °fχ∪ E ° ≤ C kfχE k p1,∞ . k k p1,∞ k Λu (w1) Λ (w1) 1 u1 k

1 Remark 3.1 : (1) We say that l ∈ BΨ [11, Definition 1.2] if l : R+ → R+, l(1) = 1, l ∈ C (the class of functions having continuous first derivatives), and tl0(t) tl0(t) 0 < ξl = inf ≤ sup = ηl < 1. t>0 l(t) t>0 l(t)

³ 0 ´p0 t1/p0 If l ∈ BΨ, 1 < p0 ≤ σ, 1 ≤ σ(1 − ηl), then w0(t) = l(t) satisfies (3.1) [5, Proposition 4.3]. The classes of weights satisfying (3.3) can be found in [24, Remark 3.2].

(2) If k · k p0 (resp. k · k p1 ) is a quasi-norm then (3.2) (resp. (3.4)) means that the space Λv0 (w0) Λv1 (w1) p0 p1 Λv0 (w0) (resp. Λv1 (w1)) satisfies the lower (resp. upper) σ-estimate for σ ≥ p0 (resp. p1 ≥ σ). p0 p1 There exist other characterizations of Λv0 (w0) (resp. Λv1 (w1)) satisfying the lower (resp. upper)

σ-estimate for σ ≥ p0 (resp. p1 ≥ σ) [18].

p1 p1,∞ Proposition 3.1 — Let 0 < p0 ≤ p1 < ∞ and Y = Λv1 (w1) or Y = Λv1 (w1). Assuming that G ∈ ∇0, the inequality

Gp0 kG(|T f|)kY ≤ CkG(|f|)k p0 , f ∈ Λv (w0), (3.5) Λv0 (w0) 0

Gp0 implies that there is a constant C > 0 such that for all a > 0, f ∈ Λv0 (w0), R G(| f(y)u2(y)v0(y)χ (y)dµ(y)|) ° ¡ ¢° X {φ(y)≤t} °G u χ ° ≤ C. 1 {ψ(x)≥t} Y (3.6) kG(|f|)k p0 Λv0 (w0)

0 p1 Conversely, if G ∈ ∆ and there exists p0 ≤ σ ≤ p1 when Y = Λu1 (w1), and σ ≥ p0 when p1,∞ Y = Λu1 (w1) such that (3.1) and (3.3) hold, then (3.6) implies (3.5).

PROOF : Let G ∈ ∇0 and (3.5) hold. Without loss of generality, let f ≥ 0. Since for each a > 0 and ψ(s) ≥ a, Z T f(s) ≥ u1(s) u2(y)f(y)v0(y)dµ(y), {φ(y)≤a} LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1113 by the modular inequality, ° Ã !° ° Z ° ° ° kG(f)k p0 ≥ CkG(T f)k ≥ C °G u χ u (y)f(y)v (y)dµ(y) ° Λv (w0) Y 1 ψ(s)≥a 2 0 0 ° {φ(y)≤a} ° ° Ã ! ° Y ° Z ° ° ° 0 ≥ C °G u2(y)f(y)dµ(y) G(u1χψ(s)≥a)° , by G ∈ ∇ ° {φ(y)≤a} ° Y ÃZ ! ° ° = CG u (y)f(y)dµ(y) °G(u χ )° 2 1 ψ(s)≥a Y {φ(y)≤a} which is (3.6).

Let Z I(s) = f(y)u2(y)v0(y)dµ(y). {φ(y)≤s}

Then I(s) is right continuous. Let Z m m+1 f(y)u2(y)v0(y)dµ(y) ∈ [2 , 2 ) {φ(y)6=0}

j for m ∈ Z and sj = sup{s : I(s) < 2 }, j ≤ m. Then

j j I(sj) ≥ 2 , and I(s) < 2 , s < sj.

Let Jm = {j ≤ m + 1 : sj+1 > sj} and β = limj→−∞ sj and sm+1 = ∞. Then

(0, ∞) = (∪j∈Jm [sj, sj+1)) ∪ (0, β)

if Jm is finite, and

(0, ∞) = (∪j∈Jm [sj, sj+1)) ∪ (0, β] if Jm is infinite.

Thus

{x ∈ X : ψ(x) 6= 0} = (∪j∈Jm Ej) ∪ F, where Ej = {sj ≤ ψ(x) < sj+1}, F = {0 < ψ(x) ≤ β}. Let F1 = {x : ψ(x) = 0}. If s ∈ Ej, 1114 QINXIU SUN, XIAO YU AND HONGLIANG LI

j j+1 then 2 ≤ I(s) < 2 ; if x ∈ F , then I(ψ(x)) = 0, that is, T f(x) = u1(x)I(ψ(x)) = 0. Now °  ° ° °σ ° X ° σ °  ° kG(|T f|)kY = °G (T f)χEj + (T f)χF1 ° ° j∈Jm ° ° Y° ° °σ ° X ° ° ° = ° G((T f)χEj ) + G((T f)χF1 )° °j∈Jm °  Y  X  σ σ  ≤ C kG((T f)χEj )kY + kG((T f)χF1 )kY , Lemma 3.1 (ii) and (iii) j∈Jm   X  j+1 σ σ  ≤ C kG(u12 )χEj kY + CkG((T f)χF1 )kY j∈Jm

≤ CI1 + CI2.

But X σ j+1 σ I1 ≤ C G (2 )kG(u1χEj )kY , j∈J Xm σ j−1 σ 0 ≤ C G (2 )kG(u1χEj )kY , by G ∈ ∆ , j∈J m à ! X Z σ σ ≤ C G f(y)u2(y)v0(y)dµ(y) kG(u1)χEj kY j∈J {sj−1≤φ(y)≤sj } m à ! X Z σ σ ≤ C G f(y)χ{sj−1≤φ(y)≤sj }u2(y)v0(y)dµ(y) kG(u1)χ{ψ(x)≥sj }kY j∈J {φ(y)≤sj } Xm σ p ≤ C kG(fχ{sj−1≤φ(y)≤sj })k 0 , by (3.6) Λv0 (w0)) j∈J ° m ° ° X °σ ≤ C° G(fχ )° , by Lemma 3.3 (i) {sj−1≤φ(y)≤sj } p0 Λv0 (w0)) j∈Jm σ ≤ CkG(f)k p0 , Λv0 (w0) and Z σ σ I2 ≤ kG(u1(x)χ{ψ(x)=0})k p0 G( f(y)u2(y)v0(y)) Λv0 (w0) Z{φ(y)=0} σ σ ≤ kG(u1(x)χ{ψ(x)≥0})kΛp0 (w )G( f(y)u2(y)v0(y)) v0 0 {φ(y)≤0} σ ≤ CkG(f)k p0 , by (3.6).2 Λv0 (w0)

We may express the inequality in Proposition 3.1 in different terms by using the following lemma which can be obtained analogously to [13, Theorem 3.1 and Corollary 3.2]. LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1115

Lemma 3.2 — Let b > 0. For a non-negative function g on (0, b) and 0 < p0 < ∞ we have R G( b f(t)g(t)dt) sup 0 ≈ A1/p0 , p f↓ kG(f)kL 0 (w0) where P R xj+1 Z n G0( ²j g(t)dt) xj o j Rxj k A = sup P xj+1 : {xj} is a covering sequence, w0 = 2 , k ∈ Z, ²j ↓ G0(²j) w0(t)dt j xj 0 (3.7) p with G0 = G 0 .

If additionally ³ X ´ X G0 aj ≤ C G0(aj) for all non-negative sequences {aj}j∈Z, (3.8) then −1 ² R r G0(G0 ( W (r) ) 0 g(t)dt) A ≈ sup 0 . (3.9) 0

Applying Lemmas 3.1 and 3.2, we get the following result.

p1 p1,∞ Lemma 3.3 — Let 0 < p0 ≤ p1 < ∞, Y = Λv1 (w1) or Y = Λv1 (w1). Let also B(a) be the right hand side of (3.7) and if additionally (3.8) holds, then B(a) is the right hand side of (3.9), with ∗ 0 g = (u2χ{φ(y)≤a})v0 and b = v0(X). If G ∈ ∇ , then (3.5) implies ° ¡ ¢° sup B(a)1/p0 °G u χ ° < ∞. 1 ψ(x)>a Y (3.10) a>0

0 p1 Conversely if G ∈ ∆ and there exists p0 ≤ σ ≤ p1 when Y = Λv1 (w1), and σ ≥ p0 when p1,∞ Y = Λv1 (w1), such that (3.1) and (3.3) hold, then (3.10) implies (3.5).

PROOF : By the property of rearrangement of function [2, Theorem 2.7], which implies R R ∞ ∗ ∗ G(| f(y)u2(y)v0(y)χ (y)dµ(y)|) G( f (t)(u2χ ) (t)dt) sup X {φ(y)≤a} = sup 0 v0 φ(y)≤a v0 p p p ∗ p 0 kG(f)k 0 0 kG(fv )kL 0 (w0) f∈Λv0 (w0) Λv0 (w0) f∈Λv0 (w0) 0 R ∞ ∗ G( f(t)(u2χ ) (t)dt) = sup 0 φ(y)≤a v0 , p f↓ kG(f)kL 0 (w0) and Lemmas 3.1-3.2, the Lemma holds. 2

In view of Lemma 3.3 and the fact that modular inequalities can deduce norm inequalities, we get the following sufficient condition of the boundedness of T between Orlicz-Lorentz spaces. 1116 QINXIU SUN, XIAO YU AND HONGLIANG LI

0 Corollary 3.1 — Let 0 < p0 ≤ p1 < ∞, G ∈ ∆ . Then

p1 (i) If (3.10) holds with Y = Λv1 (w1), and there exists p0 ≤ σ ≤ p1 such that (3.1), (3.3) hold, Gp0 Gp1 then T :Λv0 (w0) → Λv1 (w1) is bounded.

p1,∞ (ii) If (3.10) holds with Y = Λv1 (w1), and there exists σ ≥ p0 such that (3.1), (3.3) hold, then p Gp0 G 1 ,∞ T :Λv0 (w0) → Λv1 (w1) is bounded.

Letting G(t) = tα, α > 0, clearly G ∈ ∆0 ∩ ∇0. Taking without loss of generality α = 1 we obtain the next corollary as a consequence of Lemma 3.3.

Corollary 3.2 — Let 0 < p0 ≤ p1 < ∞, (X, µ) be a nonatomic resonance measure space, p1 p1,∞ p1 Y = Λv1 (w1) or Y = Λv1 (w1), and there exists p0 ≤ σ ≤ p1 when Y = Λv1 (w1), σ ≥ p0 p1,∞ when Y = Λv1 (w1), such that (3.1) and (3.3) hold. Then a necessary and sufficient condition for p0 T :Λv0 (w0) → Y is ° ° sup D(a)1/p0 °u χ ° < ∞, 1 {ψ(x)>a} Y a>0 where D(a) is the right hand side of (3.7) and if p0 ≤ 1, then D(a) can be the right hand side of (3.9),

∗ p0 with g = (u2χ{φ(y)≤a})v0 and G0(t) = t .

p0 Recall now the formulas of the associate spaces of Λv0 (w0), for 0 < p0 < ∞ [4, Definition 2.4.1]:  p0  0 1  Γv0 (wf0), if w0 6∈ L , p0 > 1, 0 p0 (Λp0 (w )) = Γ 0 (wf) ∩ L1, if w ∈ L1, p > 1, v0 0  v0 0 0 0  1,∞ Γv0 (dΦ), if 0 < p0 ≤ 1,

0 0 −p −1/p0 p0 0 where wf0(t) = t W0 (t)w0(t), Φ(t) = tW0 (t) (see [4, Theorem 2.4.7]). In the rest of this ¢0 p0 paper, the notation (Λv0 (w0) is meant as above.

The next theorem is another characterization of boundedness of the operator T between weighted Lorentz spaces. It has a simpler pattern than that in Corollary 3.2 and it generalizes [8, Theorem 2.3] and [9, Theorem 1.1.3] by weakenning its assumptions.

p1 Theorem 3.1 — (Charaterization of Boundedness) Let 0 < p0 ≤ p1 < ∞, Y = Λv1 (w1) or p1,∞ p1 p1,∞ Y = Λv1 (w1), and there exists p0 ≤ σ ≤ p1 when Y = Λv1 (w1), σ ≥ p0 when Y = Λv1 (w1), p0 such that (3.1) and (3.3) hold. Then T :Λv0 (w0) → Y is bounded if and only if there exists C > 0 such that for all a > 0,

I(a) := ku2χ{φ(y)≤a}k p0 0 · ku1χ{ψ(x)≥a}kY ≤ C. (3.11) (Λv0 (w0)) LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1117

PROOF : The theorem establishes by using Proposition 3.1 and the following equality R X f(y)φ(y)v0(y)χ{φ(y)≤a}(y)dµ(y) sup = kφ(y)χ k p0 0 . {φ(y)≤a} (Λv (w0)) p0 kfk p0 0 f∈Λv0 (w0) Λv0 (w0)

Remark 3.2 : (i) [24, Theorem 3.9] gave a characterization of the boundedness of

p0 H :Λv0 (w0) → Y when the underlying space is X = Rn, that is,

I(a) := ku2χB(0,a)k p0 0 · ku1χRn\B(0,a)kY ≤ C. (3.12) (Λv0 (w0))

Clearly, it is a special case in Theorem 3.1 when X = Rn, φ(y) = |y| and ψ(x) = |x|.

(ii) [8, 9] gave a characterization of the boundedness of H : X → Y when X,Y are Banach function spaces. But Since the weighted Lorentz spaces in Theorem 3.1 are not required to be Banach spaces, our results remain true for a wider class of spaces than before. Furthermore, it is obvious that the method of the proof of Theorem 3.1 is quite different from that of [8, 9].

Let us consider the dual operator of T Z ∗ T f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X. {φ(y)≥ψ(x)} Similarly to Proposition 3.1 and Theorem 3.1, we get the following results.

p1 p1,∞ Corollary 3.3 — Let 0 < p0 ≤ p1 < ∞ and Y = Λv1 (w1) or Y = Λv1 (w1). Assuming that G ∈ ∇0, the inequality

∗ Gp0 kG(|T f|)kY ≤ CkG(|f|)k p0 , f ∈ Λv (w0), (3.13) Λv0 (w0) 0

Gp0 implies that there is a constant C > 0 such that for all a > 0, f ∈ Λv0 (w0), R G(| f(y)u2(y)v0(y)χ (y)dµ(y)|) ° ¡ ¢° X {φ(y)≥t} °G v χ ° ≤ C. 1 {ψ(x)≤t} Y (3.14) kG(|f|)k p0 Λv0 (w0)

0 p1 Conversely, if G ∈ ∆ and there exists p0 ≤ σ ≤ p1 when Y = Λv1 (w1), and σ ≥ p0 when p1,∞ Y = Λv1 (w1) such that (3.1) and (3.3) hold, then (3.14) implies (3.13).

∗ p0 Corollary 3.4 — Let the assumptions be as in Theorem 3.1. Then T :Λv0 (w0) → Y is bounded if and only if there exists C > 0 such that for all a > 0,

I(a) := ku2χ{φ(y)≥a}k p0 0 · ku1χ{ψ(x)≤a}kY ≤ C. (3.15) (Λv0 (w0)) 1118 QINXIU SUN, XIAO YU AND HONGLIANG LI

Next we investigate the boundedness of the operator P defined by Z P f(x) = u1(x) f(y)u2(y)v0(y)dµ(y). {aψ(x)<φ(y)

p0 Theorem 3.2 gives a characterization of the boundedness of P :Λv0 (w0) → Y which extends [8, Theorem 1.1.5] since the weighted Lorentz spaces considered here are only quasi-Banach spaces. First introduce two operators Z

P1f(x) = χ{a<ψ(x)

P2f(x) = χ{a<ψ(x)

p1 p1,∞ Lemma 3.4 — Let 0 < p0 ≤ p1 < ∞, Y = Λv1 (w1) or Y = Λv1 (w1), and there exists p1 p1,∞ p0 ≤ σ ≤ p1 when Y = Λv1 (w1), σ ≥ p0 when Y = Λv1 (w1), such that (3.1) and (3.3) hold.

Let µ{φ(y) = t} = 0 for every t ∈ [λa, λb],

µ{ψ(x) = t} = 0 for every t ∈ [a, b].

Let 1 A = sup ku χ k p0 0 ku χ k . ab 2 {λt<φ(y)<λb} Λv (w0) 1 {a<ψ(x)

Then there exist a constant C > 0 such that

kP1fkY ≤ Ckfχ{λa<φ(y)<λb}k p0 (3.16) Λv0 (w0) if and only if 1 Aab < ∞.

PROOF : It is apparent that Z

P1f(x) = u1(x)χ{λa<λψ(x)<λb}(x) f(y)u2(y)v0(y)dµ(y) Z{λψ(x)<φ(y)<λb}

= u1(x)χ{λa<λψ(x)<λb}(x) χ{λψ(x)<φ(y)<λb}(x)f(y)u2(y)v0(y)dµ(y) Z {φ(y)>λψ(x)}

= u1(x)χ{λa<ψ1(x)<λb}(x) χ{λa<φ(y)<λb}(x)f(y)u2(y)v0(y)dµ(y), {φ(y)>ψ1(x)} LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1119

where ψ1(x) = λψ(x). Thus by Corollary 3.4 we get that (3.16) holds if and only if

sup ku χ χ k p0 0 ku χ χ k < C, 2 {φ(y)>s} {λa<φ(y)<λb} Λv (w0) 1 {λa<ψ1(x)<λb} {ψ1(x)

sup ku χ k p0 0 ku χ k < C, 2 {λt<φ(y)<λb} Λv (w0) 1 {a<ψ(x)

Similarly to Lemma 3.4, it follows that

Lemma 3.5 — Let φ, ψ satisfy the conditions of Lemma 3.4. Let

2 A = sup ku χ k p0 0 ku χ k . ab 2 {λa<φ(y)<λt} Λv (w0) 1 {t<ψ(x)

Then there exists a constant C > 0 such that

kP2fkY ≤ Ckfχ{λa<φ(y)<λb}k p0 Λv0 (w0) if and only if 2 Aab < ∞.

Let ° ° ° ° ° ° ° ° A1k = sup °u1χ b k ° °u2χ b k+1 ° p , {( a ) <ψ(x)

A1 = sup{A1k},A2 = sup{A2k},A = max{A1,A2}. k∈Z k∈Z

p1 p1,∞ Theorem 3.2 — Let 0 < p0 ≤ p1 < ∞, Y = Λv1 (w1) or Y = Λv1 (w1), and there exists p1 p1,∞ p0 ≤ σ ≤ p1 when Y = Λv1 (w1), σ ≥ p0 when Y = Λv1 (w1), such that (3.1) and (3.3) hold. Suppose that µ{x : φ(y) = t} = µ{x : ψ(x) = t} = 0

p0 for any t ∈ [0, ∞). Then the operator P :Λv0 (w0) → Y is bounded if and only if A < ∞. 1120 QINXIU SUN, XIAO YU AND HONGLIANG LI

b k b k+1 PROOF : Let Fk = {x :( a ) ≤ ψ(x) < ( a ) }. Then X = ∪k∈Z Fk. Thus ° ° ° °σ °X ° X kP fkσ = ° (P f)χ ° ≤ k(P f)χ kσ , by Lemma 3.1 (ii) and (iii) Y ° Fk ° Fk Y ° Ãk Y k ! ° ° Z °σ X ° ° = °u1(x) f(y)u2(y)v0(y)dµ(y) χ{( b )k≤ψ(x)<( b )k+1}° ° {aψ(x)<φ(y)

Estimate I1 as follows: ° " # ° ° Z °σ X ° ° I1 = °u1(x) f(y)u2(y)v0(y)dµ χ{( b )k≤ψ(x)<( b )k+1}° ° {aψ(x)<φ(y)

Similarly, by Lemma 3.5, σ σ I2 ≤ CA kfk p0 . Λv0 (w0) Thus σ σ σ kP fkY ≤ CA kfk p0 . Λv0 (w0) LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1121

4. COMPACTNESS OF HARDY OPERATORS

We next consider the compactness of T on weighted Lorentz spaces. Edmund, Kokilashvili and Meskhi [8, 9] gave a characterization for T to be compact from Xe to Ye where Xe and Ye are Banach function spaces when the spaces X,e Ye satisfy certain conditions. This part studies the compactness

p0 p0 of T :Λv0 (w0) → Y when Λv0 (w0) is a quasi-Banach space. Thus Theorem 4.1 improves (7)-(9) p0 n since the normability of Λv0 (w0) is not compulsory. On the other hand, if X = R , φ(y) = |y| and ψ(x) = |x| Theorem 4.1 reduces to [24, Theorem 4.1].

In this section, we let

µ{x ∈ X : φ(x) = t} = µ{x ∈ X : ψ(x) = t} = 0.

For 0 < a < b < ∞, let Z

Taf(x) = χ{ψ(x)

Tbf(x) = χ{ψ(x)>b}(x)u1(x) χ{φ(y)>b}f(y)u2(y)v0(y)dµ(y), {φ(y)<ψ(x)} Z

Tabf(x) = χ{a<ψ(x)≤b}(x)u1(x) χ{a<φ(y)

p0 We first give characterization of the compactness of the operator Tab :Λv0 (w0) → Y inspired by [9].

p1 Lemma 4.1 — Let W0 ∈ ∆2, 1 ≤ p1 < ∞ and w1 ∈ Bp1,∞ if Y = Λv1 (w1); and w1 ∈ Bp1 if p1,∞ p Y = Λv1 (w1). Let ku1χ{a<ψ(x)

lim ku1χ{α<ψ(x)

lim ku1χ{R<ψ(x)<α}kY = 0 or lim ku2χ{R<φ(y)<α}k(Λp0 (w ))0 = 0. (4.2) R→α− R→α− v0 0

p0 PROOF : Sufficiency. The operator Tab is bounded from Λv0 (w0) to Y since Z

kTabfkY ≤ ku1χ{a<ψ(x)

≤ ku1χ{α<ψ(x)

Next prove that Tab is a limit of finite rank operators and thus is compact. Let ² > 0. Then for every α ∈ [a, b] there exist c and d with c < α < d, such that

ku1χ{α<ψ(x)

ku1χ{c<ψ(x)<α}kY < ² or ku2χ{c<φ(y)<α}k p0 0 < ². (4.4) (Λv0 (w0))

Thus we obtain an open covering of the segment [a, b] by such intervals (c, d) from which there exists a finite subcovering {(ci, di)} having appropriate interior point αi. The points ci, αi, di form a partition of [a, b], and we obtain closed intervals [βj, βj+1], j = 0, 1, ..., N such that N ∪j=0[βj, βj+1] = [a, b] with (βi, βi+1) ∩ (βj, βj+1) = ∅ for i 6= j. Obviously in view of the property of Banach function spaces there holds that

ku1χ{β <ψ(x)<β }kY < ² or ku2χ{β <φ(y)<β }k p0 0 < ². j j+1 j j+1 (Λv0 (w0))

Let XN Z

Sf(x) = χ{βj <ψ(x)<βj+1}(x)u1(x) u2(y)f(y)v0(y)dµ(y) j=0 {a<φ(y)<βj }

Then

XN Z

Tabf(x) − Sf(x) = χ{βj <ψ(x)≤βj+1}(x)u1(x) u2(y)f(y)v0(y)dµ(y). j=0 {βj <φ(y)<ψ(x)}

p1 p1,∞ According to the assumptions, Λv1 (w1) (resp. Λv1 (w1)) is a Banach function space with a norm p1 00 p1 p1,∞ 00 p1,∞ p p ,∞ k · k 1 (resp. k · k 1 ) which implies Γv1 (w) = Γv1 (w) (resp. Γv1 (w) = Γv1 (w)) [2, Γv1 (w) Γv1 (w) Theorem 1.2.7] and then Z

kfk p1 ≤ kfk p1 = sup f(x)g(x)v1(x)dµ(x) (4.5) Λv1 (w) Γv1 (w) kgk p ≤1 X Γ 1 (w)0 v1 Z

(resp. kfk p1,∞ ≤ kfk p1,∞ = sup f(x)g(x)v1(x)dµ(x)). Λv1 (w) Γv1 (w) kgk p ,∞ ≤1 X Γ 1 (w)0 v1 But since

kgk p1 ≤ Ckgk p1 , Γv1 (w) Λv1 (w) LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1123 we get Z

kgk p1 0 = sup f(x)g(x)v1(x)dµ(x) Γv1 (w) kgk p ≤1 X Γ 1 (w) v1 Z ≥ sup f(x)g(x)v1(x)dµ(x) Ckgk p ≤1 X Λ 1 (w) v1 Z 1 = sup f(x)(Cg(x))v(x)dµ(x) C kCgk p ≤1 X Λ 1 (w) v1 Z 1 1 = sup f(x)g(x)v(x)dµ(x) = kgk p1 0 . (4.6) Λv1 (w) C kgk p ≤1 X C Λ 1 (w) v1

Thus by (4.5) and (4.6) it follows that Z Z

kfk p1 ≤ sup f(x)g(x)v (x)dµ(x) = C sup f(x)g(x)v (x)dµ(x), Λv (w) 1 1 1 1 kgk p ≤1 X kgk p1 ≤1 X C Λ 1 (w)0 Λ (w)0 v1 v1 (4.7) as does the evaluation of kfk p1,∞ . This implies that Λv1 (w)

kTab − Sk p0 = sup k(Tab − S)fkY Λv0 (w0)→Y kfk p ≤1 Λ 0 (w ) v0 0 Z

≤ C sup sup (Tab − S)f(x)g(x)v1(x)dµ(x) kfk p ≤1 kgk 0 ≤1 X Λ 0 (w ) Y v0 0 XN Z = C sup sup u1(x)g(x)v1(x) kfk p ≤1 kgk 0 ≤1 {β <ψ(x)≤β } Λ 0 (w ) Y j=0 j j+1 v0 0 ÃZ ! · u2(y)f(y)v0(y)dµ(y) dµ(x) βj <φ(y)≤ψ(x) XN Z ≤ C sup sup u1(x)g(x)v1(x)dµ(x) kfk p ≤1 kgk 0 ≤1 {β <ψ(x)≤β } Λ 0 (w ) Y j=0 j j+1 Z v0 0 · u2(y)f(y)v0(y)dµ(y) {βj <φ(y)≤βj+1} XN j = C sup sup Iψφ. kfk p ≤1 kgk 0 ≤1 Λ 0 (w ) Y j=0 v0 0

If

A1 = {j : ku1χ{βj <ψ(x)<βj+1}kY < ²} 1124 QINXIU SUN, XIAO YU AND HONGLIANG LI and

A2 = {j : ku2χ{β <φ(y)<β }k p0 0 < ²}, j j+1 (Λv0 (w0))

p0 then we obtain by Holder¨ inequality in Λv0 (w0) and Y that X j sup sup Iψφ ≤ ²Cc1 kfk p ≤1 kgk 0 ≤1 Λ 0 (w ) Y j∈A v0 0 1 and X j sup sup Iψφ ≤ ²Cc2. kfk p ≤1 kgk 0 ≤1 Λ 0 (w ) Y j∈A v0 0 2 Thus we have

kTab − Sk ≤ ²C(c1 + c2) which yields that Tab is a limit of finite rank operators.

Necessity : Use contradiction. Let there exist numbers α ∈ [a, b) and ²0 > 0 and a sequence {tn}, + tn → α such that

p ku1χ{a<ψ(x)

For γ ∈ (0, 1) there exist functions fn, gn with support in [α, tn] such that kfnk p0 ≤ 1, Λv0 (w0) kgnkY 0 ≤ 1, Z

p u2(y)fn(y)v0(y)dµ(y) ≥ γku2χ{α<φ(y)

Furthermore there exist numbers βn ∈ (α, tn) such that Z

2 p u2(y)fn(y)v0(y)dµ(y) ≥ γ ku2χ{a<φ(y)

Let Fn = fnχ{βn<φ(y)

tm < βk < tk < βn is fulfilled for them. Then we obtain that kTabFm − TabFnkY ≥ kχ (TabFm − TabFn)kY ° {βk<ψ(x)

= u2(y)fm(y)v0(y)dµ(y)ku1χ{βk<ψ(x) 0. C

Since any subsequence of {TabFk} does not converge in X2, Tab is noncompact. Thus (4.2) is proved. Analogously verify (4.2).

Theorem 4.1 — (Characterization of Compactness) Let W0 ∈ ∆2, 1 ≤ p1 < ∞ and w1 ∈ Bp1,∞ p1 p1,∞ p0 if Y = Λv1 (w1); and w1 ∈ Bp1 if Y = Λv1 (w1). Then the operator T :Λv0 (w0) → Y is compact if and only if (3.15) and the following conditions (a) and (b) hold:

(a)

lim sup ku χ k p0 0 · ku χ k = 0 (4.9) 2 {φ(y)

lim sup ku χ k p0 0 · ku χ k = 0, (4.10) 2 {a<φ(y)R} Y a→∞ a

(b) for every α ∈ [0, ∞) the following two alternatives are satisfied

lim ku1χ{α<ψ(x)

lim ku1χ{R<ψ(x)<α}kY = 0 or lim ku2χ{R<φ(y)<α}k(Λp0 (w ))0 = 0. (4.12) R→α− R→α− v0 0

PROOF : Sufficiency. We prove it through three steps.

Step 1 : Define the operator TD by replacing u1 and u2 by u1χD1 and u2χD2 in the operator T , where

D1 = {y : r2 < ψ(x) < r1},D2 = {y : r2 < φ(y) < r1}, 0 ≤ r2 < r1 ≤ ∞. 1126 QINXIU SUN, XIAO YU AND HONGLIANG LI

By Theorem 3.1,

BD ≤ kTDk ≤ KBD, (4.13) where

B = sup ku χ χ k p0 0 · ku χ χ k D 2 D2 {φ(y)a} Y a>0 v0 and K is a constant independent of v0, u1, φ, ψ, D1 and D2.

Step 2 : For 0 < a < b < ∞, the operator T can be decomposed as

T f = Taf + Tbf + Tabf + (QbTPabf + QaTPaf), (4.14) where Paf = χφ(y)bf, Pabf = χa<φ(y)

Step 3 : Each of the two operators in parentheses in (4.14) is one-dimensional and so is compact.

By using Lemma 4.1 we get that Tab is a if and only if (4.11) and (4.12) hold. Let ² > 0. By (4.13) we get

kT k ≤ K sup ku χ k p0 0 · ku χ k a 2 {φ(y)

kT k ≤ K sup ku χ k p0 0 · ku χ k , b 2 {b<φ(y)R} Y b

In the light of (4.9) and (4.10), there exist a, b, with 0 < a < b < ∞, such that kTak < ², kTbk < ². Hence H is compact, since it is a limit of compact operators when a → 0, and b → ∞.

Necessity : The proof of necessity is similar to that of Lemma 4.1. We omit the details. 2

Remark 4.1 : (1) A function f in a quasi-normed space X is said to have absolutely continuous

(AC) norm if limn→∞ kfχAn kX = 0, for every decreasing sequence of measurable sets (An)n with

χAn ↓ ∅. If every function in X has this property, we say that X has an (AC) norm (see [4, Definition p1 p0 0 p1 2.3.2]). If Λv1 (w1) and (Λv0 (w0)) are spaces with (AC) norms (the conditions which make Λv1 (w1) p0 0 and (Λv0 (w0)) have (AC) norms can be found in [24, Remark 4.2]), then it is obvious that (a) may be replaced by a stronger condition

lim I(a) = lim I(a) = 0, (4.15) a→0 a→∞

p1 when Y = Λv1 (w1). Furthermore, in this circumstance (b) is superfluous since it is automatically satisfied. LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1127

n p0 p1 In the case when the underlying space X = R , the weighted Lorentz space Λv0 (w0) or Λv1 (w1) becomes the Lorentz space Lr,s and the operator T has some special forms and then the characteri-

p0 p1 zation of the compactness of T from Λv0 (w0) to Λv1 (w1) achieves simpler forms. Let the operators

T1,T2 be defined by Z Z f(y) 1 T1f(x) = u1(x) α dy, T2f(x) = α u2(y)f(y)dy. {|y|≤ψ(x)} |y| |x| {|y|≤ψ(x)}

The next result characterizes compactness of type T1.

1 α 1 1 Corollary 3.1 — Let min(r, s) > 1, max(r, s) ≤ p1, r0 > n , w1 ∈ Bp1 , u1 ∈ L or w1 6∈ L , r,s and there exist max(r, s) ≤ σ ≤ p1 such that (3.3) hold. Then the operator T1 is compact from L p1 to Λv1 (w1) if and only if there exists C > 0 such that for every k ∈ Z,

n/r0−α Ak := ku1ψ χ2k<ψ(x)<2k+1 k p1 ≤ C (4.16) Λv1 (w1) and

lim Ak = lim Ak = 0. (4.17) k→−∞ k→∞

p1 PROOF : (i) Note that under the assumptions Λv1 (w1) is a Banach space with (AC) norm [4, r0,s0 s/r−1 Theorem 2.3.4] and L has (AC) norm. Furthermore, (3.1) holds with w0(t) = t and p0 = s and ° ° ° 1 ° n −α ° ° r0 ° α χB(0,a)° = a . | · | Lr0,s0

r,s p1 Thus by Theorem 4.1 and Remark 4.1, for T1 to be compact from L to Λv1 (w1), it is sufficient and necessary that there exists C > 0 such that for all a > 0

n 0 −α I(a) := a r · ku1χ{ψ(x)>a}k p1 ≤ C, (4.18) Λv1 (w1) and lim I(a) = lim I(a) = 0. (4.19) a→0 a→∞ So it suffices to prove that (4.16) and (4.17) are equivalent to (4.18) and (4.19). We prove it by four steps.

Let

sup I(a) = I, sup Ak = A. a>0 k 1128 QINXIU SUN, XIAO YU AND HONGLIANG LI

k We verify I ≈ A. Indeed, it is easy to see that Ak ≤ CI(2 ), and thus A ≤ CI. On the other hand, let 2k < a ≤ 2k+1, k ∈ Z. Then

n k( 0 −α) I(a) ≤ C2 r · ku1χ{ψ(x)>2k}k p1 Λv1 (w1) X∞ k( n −α) r0 p ≤ C2 · ku1χ{2j <ψ(x)<2j+1}k 1 , by w1 ∈ Bp1 Λv1 (w1) j=k ∞ X n 0 (k−j)( 0 −α) n/r −α ≤ C 2 r · ku1ψ χ{2k<ψ(x)<2k+1}k p1 Λv1 (w1) j=k ≤ CA,

k i.e., I ≤ CA. Secondly, Taking into account Ak ≤ CI(2 ), we get that (4.19) implies (4.17). Third, if limk→∞ Ak = 0, it is similar to prove that lima→∞ I(a) = 0 to the arguments of the proof above that I ≤ CA. Finally, letting limk→−∞ Ak = 0 and A < ∞, we prove limt→0 I(t) = 0. Indeed, limk→−∞ Ak = 0 yields that given ² > 0 there exist K1 ∈ Z, K2 ∈ N such that

X∞ n/r0−α −m (2 ) < ², and Ak < ² for k < K1. (4.20)

m=K2

K −K k k+1 Since for t < 2 1 2 there exists k ∈ Z, k < K1 − K2 with 2 ≤ t ≤ 2 , we obtain

k n/r0−α I(t) ≤ C(2 ) ku1χ{ψ(x)>2k}k p1 Λv1 (w1) ∞ 0 X 0 0 k n/r −α −j n/r −α n/r −α p ≤ C(2 ) (2 ) ku1ψ χ{2j <ψ(x)<2j+1}k 1 , by w1 ∈ Bp1 Λv1 (w1) j=k X∞ −m n/r0−α n/r0−α = C (2 ) ku1ψ χ{2k+m<ψ(x)<2k+m+1}k p1 Λv1 (w1) m=0  K2−1 ∞ K2−1 X X X 0 = C  +  ≤ C² (2−m)n/r −α + ²A, by (4.20)

m=0 m=K2 m=0 1 α ≤ C², by > , r0 n which ends the proof. 2

Remark 4.2 : Prokhorov [33, Theorem 3] proved that if 1 < p ≤ q < ∞, l is a non-negative R l(x) x p q function on R+, T f(x) = x 0 f(y)dy, x > 0, then T is compact from L to L if and only if

sup Dk < ∞, lim Dk = lim Dk = 0, k k→0 k→∞ LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES 1129

R k+1 where D = ( 2 l(x) dx)1/q. Obviously Corollary 4.1 generalizes this result. It also extends [24, k 2k xq/p Corollary 4.3] corresponding to the case ψ(x) = |x|.

Let us show the analogous characterization of compactness of operators T2, which we leave with- out proof.

α 1 1 1 Corollary 4.2 — Let 1 < p0 ≤ min(r, s), n > r , W0 ∈ ∆2, v0 ∈ L or w˜0 6∈ L , and there 0 −p0 p0 0 exists p0 ≤ σ ≤ min(r, s) such that (3.1) holds where wf0(t) = t W0 (t)w0(t). Then the operator p0 r,s T2 is compact from Λv0 (w0) to L if and only if there exists C > 0 such that for every k ∈ Z,

n/r−α Bk := ku2ψ χ{2k<ψ(x)<2k+1}k p0 0 ≤ C (Λv0 (w0)) and

lim Bk = lim Bk = 0. k→−∞ k→∞

5. SOME NOTESON SPACES OF HOMOGENEOUSAND NONHOMOGENEOUS TYPE

Definition 5.1 — A space of homogeneous type (SHT) (X; d; µ) is a topological space X endowed with a complete measure µ such that: (a) the space of compactly supported continuous functions is dense in L1(X; µ), and (b) there exists a non-negative real-valued function (quasimetric) d : X × X → R1 satisfying:

(i) d(x; x) = 0 for arbitrary x ∈ X;

(ii) d(x; y) > 0 for arbitrary x; y ∈ X, x 6= y;

(iii) there exists a positive constant a0 such that the inequality d(x; y) ≤ a0d(y; x) holds for all x; y ∈ X;

(iv) there exists a positive constant a1 such that the inequality

d(x; y) ≤ a1(d(x; z) + d(z; y)) holds for arbitrary x; y; z ∈ X;

(v) for every neighborhood V of any point x ∈ X there exists a number r > 0 such that the ball B(x; r) = {y ∈ X : d(x; y) < r} with center in x and radius r is contained in V ;

(vi) the balls B(x; r) are measurable for all x ∈ X, r > 0 and, moreover, 0 < µB(x; r) < ∞;

(vii) there exists a positive constant b such that the inequality (doubling condition) µB(x; 2r) ≤ bµB(x; r) is true for all x ∈ X and for all positive r. 1130 QINXIU SUN, XIAO YU AND HONGLIANG LI

Definition 5.2 — By a space of nonhomogeneous type we mean a measure space with a quasi- metric (X, d, µ) satisfying conditions (i)-(v) of Definition 5.1, i.e., the doubling condition may fail.

Let u1, u2 and w be positive µ−measurable functions on X and x0 ∈ X. Suppose that there exists a point x0 such that for all numbers t1, t2 with 0 < t1 < t2 < ∞ we have

µ(B(x0, t2)\B(x0, t1)) > 0.

Assume that Z R1f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X, {d(x0,y)≤ψ(x)} Z R2f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X {d(x0,y)≥ψ(x)} and Z R3f(x) = u1(x) f(y)u2(y)v0(y)dµ(y), 0 < a < b < ∞, x ∈ X {aψ(x)

0 < {x ∈ X : t1 < ψ(x) < t2} < ∞, 0 < t1, t2 < ∞.

Then for the operators R1-R3 by using the preceding results in Chapter 3 and Chapter 4 we can p0 p1 get the boundedness and compactness criteria from Λv0 (w0) to Λv1 (w1) defined on (X, d, µ). The same method is also applicable to the following operators: Z R4f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X, {d(x0,y)<ψ(x)} and Z R5f(x) = u1(x) f(y)u2(y)v0(y) dµ(y), x ∈ X. {d(x0,y)>ψ(x)}

n When (X, µ) = (R , | · |), x0 = 0 and ψ(x) = d(x0, x) = |x|, the operators R1 − R5 have been studied in [7-9], [24, 25, 28] and so on.

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