Discretization and anti-discretization of rearrangement-invariant norms

Publ. Mat. 47 (2003), 311–358 DISCRETIZATION AND ANTI-DISCRETIZATION OF REARRANGEMENT-INVARIANT NORMS Amiran Gogatishvili and Luboˇ s Pick Abstract We develop a new method of discretization and anti-discretization of weighted inequalities which we apply to norms in classical Lorentz spaces and to spaces endowed with the so-called Hilbert norm. Main applications of our results include new integral conditions characterizing embeddings Γp(v)→Γq(w) and Γp(v)→Λq(w) and an integral characterization of the associate space to Γp(v), where p, q ∈(0,∞), v,ware weights on [0,∞) and fΛp(v)=∞ 0 f∗(t)pv(t)dt1/p , fΓp(v)=∞ 0 f∗∗ (t)pv(t)dt1/p . 1. Introduction Let (R,µ)beatotally σ-finite measure space with a non-atomic measure µ, and let M(R,µ)bethe set of all extended complex-valued µ-measurable functions on R.Weshall throughout assume that µ(R)=∞. For f∈M(R,µ), let f∗(t)=µ({x∈R;|f(x)|>t}), t∈(0,∞), be the distribution function of f. The non-increasing rearrangement of f is defined by f∗(t)=inf {s>0; f∗(s)≤t},t∈[0,∞).(1.1) 2000 Mathematics Subject Classification. 26D10, 46E20. Key words. Discretizing sequence, Hilbert norm, classical Lorentz space, duality theorems. This research was partly supported by the grant no. 201/01/0333 of the Grant Agency of the Czech Republic and by the Leverhulme Trust Grant no. F/00407/E. The research of the second author was partly supported by the grant no. MSM 113200007 of the Czech Ministry of Education. 312 A. Gogatishvili, L. Pick We further set f∗∗(t)=1 tt 0 f∗(s)ds, t ∈[0,∞). When vis a non-negative measurable function on [0,∞), we say that v is a weight. Definition 1.1. Let p∈(0,∞) and let vbe aweight. Then the classical Lorentz space Λp(v)isdefined as Λp(v)=f∈M(R,µ); fΛp(v):= ∞ 0 f∗(t)pv(t)dt1/p <∞. Let us recall that · Λp(v)is not always a norm (consider, for example, the cases when p∈(0,1)), and, indeed, not even a quasinorm (cf. [8, Corollary 2.2]). The spaces Λp(v)were introduced by Lorentz in 1951 in [21]. For appropriate values of pand for appropriate weight vthis space is a rearrangement-invariant Banach function space (for definitions and detailed study of rearrangement-invariant Banach function spaces see e.g. [3]). Another type of a function space is obtained by replacing in the definition on · Λp(v)the f∗with f∗∗. This space, denoted by Γp(v), is defined as Γp(v)=f∈M(R,µ); fΓp(v):= ∞ 0 f∗∗(t)pv(t)dt1/p <∞. The spaces Γp(v) proved to be quite useful in many branches of functional analysis, for example in interpolation theory. They are particularly popular since 1990 when Sawyer [26] used them in order to characterize certain duality properties of spaces Λp(v), but they can be traced in earlier works of Calder´on, Hunt, O’Neil and others. Naturally, we have seen an extensive research of classical Lorentz spaces during the 1990’s. Above all, the authors concentrated on seeking manageable and easily-verifiable necessary and sufficient conditions for embedding theorems involving both Λ and Γ types of spaces. Such results have an intimate connection to other challenging problems. Let us name some of them: Is a given classical Lorentz space (either of type Λ or of type Γ) a Banach space? Is the Hardy-Littlewood maximal operator bounded on a given classical Lorentz spaces? What is the associate space of Γp(v)? Discretization and Anti-Discretization 313 All these questions could be answered if only we knew necessary and sufficient conditions on embeddings between Λ and Γ types of spaces. The research brought plenty of deep results, cf. e.g. [1], [26], [8], [9], [10], [16], [30], [31], [32], [28]. A summary of the results on embeddings of classical Lorentz spaces known by the end of 1990’s, as well as some more references, can be found in [7]. For some cases of the parameters, the characterization of the corresponding embedding is still not known. In some other cases necessary and sufficient conditions have been established, but formulated in a way which is not entirely satisfactory, as they might be quite difficult to verify. Typical examples of such conditions are those expressed in terms of the Halperin level function [7, Section 7] or in terms of discretizing sequences [16]. In [16], a new approach based on discretization techniques of [25] and [15]was applied to classical Lorentz spaces in order to obtain necessary and sufficient conditions on parameters p, q ∈(0,∞) and weights v,wsuch that the embedding Γp(v)→Γq(w)orthe embedding Γp(v)→Λ1(w) hold. The former embedding is useful in interpolation theory while a standard argument applied to the latter provides a characterization of the associate space to Γq(w). The results of [16] meant a considerable step ahead. However, to verify the conditions formulated through the discretizing sequences is almost impossible. After [16]was published many authors tried to obtain more manageable conditions (expressed, if possible, in an “integral form” —such conditions have been successfully used for example to characterize weighted Hardy inequalities). As far as we know, no such results have been found so far. In this paper we develop a new method that leads to such results. Our approach is based on discretization and (more importantly) antidiscretization methods combined with the blocking technique from [17]. Let us outline our approach and the structure of the paper. We start with general discretization and anti-discretization formulae for weighted integral norms. This is done in Section 2. Next, in Section 3, we present a discretization of the so-called Hilbert norm. Let us recall that, in a very general form, the Hilbert norm of a function fis the quantity ∞ 0 ϕ(x)∞ 0 |f(y)| u(x)+u(y)dyq ϕ(x)dx1 q when q∈(0,∞), and sup x∈(0,∞) ϕ(x)∞ 0 |f(y)| u(x)+u(y)dy, 314 A. Gogatishvili, L. Pick when q>0; here uand ϕare weights satisfying certain conditions (see Sections 2 and 3 below). Let us recall that, in a very general form, the Hilbert norm of a function fis the quantity ∞ 0 ϕ(x)∞ 0 |f(y)| u(x)+u(y)dyq ϕ(x)dx1 q when q∈(0,∞), and sup x∈(0,∞) ϕ(x)∞ 0 |f(y)| u(x)+u(y)dy, when q>0; here uand ϕare weights satisfying certain conditions (see Sections 2 and 3 below). We point out three of the major applications of our results. In Sections 4 and 5 we establish integral necessary and sufficient conditions for the embeddings Γp(v)→Λq(w) and Γp(v)→Γq(w), respectively. Finally, in Section 6, we give a precise integral characterization of the associate space of Γp(v). The results on spaces Γp(v) can be extended to the more general context of K-spaces (defined through the Peetre K-functional). Everywhere below, u,vand ware weights. We shall throughout denote U(t)=t 0u(s)ds,V(t)=t 0v(s)ds and W(t)=t 0w(s)ds for t∈(0,∞). By ABwe mean that A≤CB with some positive C independent of appropriate quantities. If ABand BA,wewrite A≈B.Wesay that two functions f,gare equivalent on (0,∞)ifthere exists a constant C>0 such that C−1f(t)≤g(t)≤Cf(t) for all t∈(0,∞). 2. General anti-discretization theorems We start with some basic definitions. We follow [15], [25] and [16]. Definition 2.1. Let {ak}beasequence of positive real numbers. We say that {ak}is strongly increasing or strongly decreasing and write {ak}or {ak}when inf k∈Z ak+1 ak >1or sup k∈Z ak+1 ak <1, respectively. Definition 2.2. Let ϕbeacontinuous strictly increasing function on [0,∞) such that ϕ(0) = 0 and limt→∞ ϕ(t)=∞. Then we say that ϕ is admissible. Discretization and Anti-Discretization 315 Let ϕbe an admissible function. We say that a function his ϕ-quasiconcave if his equivalent to a non-decreasing function on [0,∞) and h ϕis equivalent to a non-increasing function on (0,∞). We say that a ϕ-quasiconcave function his non-degenerate if lim t→0+ h(t)= lim t→∞ 1 h(t)= lim t→∞ h(t) ϕ(t)= lim t→0+ ϕ(t) h(t)=0.(2.1) The family of non-degenerate ϕ-quasiconcave functions will be denoted by Ωϕ. We say that his quasiconcave when h∈Ωϕwith ϕ(t)=t. Remarks 2.3.(i) It will be useful to note that h∈Ωϕ⇐⇒ ϕ h∈Ωϕ.(2.2) (ii) Some authors add the restriction h(t)=0if and only if t=0to the definition of a quasiconcave function. However, the only difference is that our definition recognizes the zero function as quasiconcave. (iii) Note that any non-degenerate ϕ-quasiconcave function is necessarily continuous on [0,∞). Definition 2.4. Assume that ϕis admissible and h∈Ωϕ.Wesay that {µk}k∈Zis a discretizing sequence for hwith respect to ϕif (i) µ0=1and ϕ(µk); (ii) h(µk)and h(µk) ϕ(µk); (iii) there is a decomposition Z=Z1∪Z2such that Z1∩Z2=∅and, for every t∈[µk,µ k+1], h(µk)≈h(t)ifk∈Z1,(2.3) h(µk) ϕ(µk)≈h(t) ϕ(t)if k∈Z2.(2.4) Remark 2.5.In [16], a special case ϕ(t)=tis treated. Examples 2.6. (i) Let ϕ(t)=tα,α>0. Then we recover the situation which has been in one way or another treated by several authors (cf. e.g. [5], [22]or[4]). (ii) When uis a positive function on [0,∞) such that ∞ 0u(s)ds = ∞and fis a locally integrable function, then Ku(f,t)∈ΩU, where Ku(f,t)=t 0f∗(s)u(s)ds and U(t)=t 0u(s)ds. The operator Kuis a particular case of the well-known Peetre K-functional. 316 A. Gogatishvili, L. Pick (iii) Recently, a considerable attention is being paid to the Hardy operators involving suprema such as Ru, defined at a locally integrable function fby (Ruf)(t)= sup s∈[t,∞) u(s)s 0 f∗(y)dy, t ∈(0,∞), where uis a given weight on (0,∞) (see for example [11], [13]or[19]). If moreover uis such that the function ϕ, defined by ϕ(t)=sup s∈[t,∞) u(s)−1 ,t∈(0,∞), is admissible, then, for any f,ϕRufis ϕ-quasiconcave. Indeed, this follows from the readily verified relation ϕ(t)(Ruf)(t)≈sup s∈(0,∞) ϕ(t) ϕ(s)+ϕ(t)s 0 f∗(y)dy, t ∈(0,∞). (iv) Let 0 <p 0<p 1<∞,0<q 0≤q1<∞and let m= 1 q0−1 q1 1 p0−1 p1 . Then the Calder´on operator (cf. [3, Chapter 3, Definition 5.1]) (Sf∗)(t)=t−1 q0tm 0 s1 p0−1f∗(s)ds +t−1 q1∞ tm s1 p1−1f∗(s)ds satisfies, for any fixed (appropriate) f,(Sf∗)(t)t1 q0∈Ωϕ, where ϕ(t)= t1 q0−1 q1. Lemma 2.7. Let ϕbe an admissible function on [0,∞), let h∈Ωϕand let a>1.Wedefine the sequence {µk}by µ0=1and µk+1 = inf t; min h(t) h(µk),h(µk)ϕ(t) ϕ(µk)h(t)=a,when k≥0;(2.5) µk−1= inf t; min h(µk) h(t),h(t)ϕ(µk) ϕ(t)h(µk)=a,when k≤0.(2.6) Then {µk}is a discretizing sequence for hwith respect to ϕ. Proof: We have to show that the properties (i), (ii), (iii) from Definition 2.4 are satisfied. Set Z1={k∈Z;ah(µk)=h(µk+1)},Z2=Z\Z1.(2.7) Discretization and Anti-Discretization 317 Then h(µk) ϕ(µk)=ah(µk+1) ϕ(µk+1)for k∈Z2.(2.8) Since h∈Ωϕ,itfollows from (2.5) and (2.6) that µk+1 >µ kfor every k∈Z. Hence, using also (2.5) and (2.6), we get for every k∈Z, h(µk) ϕ(µk)≥ah(µk+1) ϕ(µk+1)≥ah(µk) ϕ(µk+1) and therefore, for k∈Z, ϕ(µk+1) ϕ(µk)≥a>1. This shows that ϕ(µk). Next, by (2.5) and (2.6), we have, for k∈Z, h(µk+1) h(µk)≥a>1,and h(µk+1) ϕ(µk+1) ϕ(µk) h(µk)≤1 a<1, so h(µk)and h(µk) ϕ(µk). Finally, h∈Ωϕ, whence, by (2.7), for t∈[µk,µ k+1],      h(t)≤h(µk+1)=ah(µk)≤ah(t) when k∈Z1; h(t) ϕ(t)≤h(µk) ϕ(µk)=ah(µk+1) ϕ(µk+1)≤ah(t) ϕ(t)when k∈Z2, showing (iii). Lemma 2.8. Let ϕbe an admissible function. Then the following statements are equivalent: (i) h∈Ωϕ; (ii) there exists a non-increasing function ψsuch that h(t)≈t 0 ψ(s)dϕ(s),t∈(0,∞) (with the Lebesgue-Stieltjes integral on the right); (iii) there is a non-negative Borel measure ηon [0,∞)such that h(t)≈[0,t] ϕ(s)dη(s)+ϕ(t)[t,∞) dη(s),t∈(0,∞).(2.9) Proof: (i) ⇒(ii) Let h∈Ωϕ. Then his a non-degenerate ϕ-quasiconcave function, and so h(ϕ−1)isquasi-concave with lim t→0+ h(ϕ−1)(t)= lim t→∞ h(ϕ−1)(t) t=0. 318 A. Gogatishvili, L. Pick Thus, there exists a non-increasing function θon (0,∞) such that h(ϕ−1)(t)≈t 0 θ(s)ds, t ∈(0,∞). Hence, h(t)≈ϕ(t) 0 θ(s)ds =t 0 θ(ϕ(s)) dϕ(s),t∈(0,∞). Denoting ψ=θ◦ϕ,weget (ii). (ii) ⇒(iii) Since θis non-increasing, it follows that ψis non-increasing, too. Integrating by parts, we get h(t)≈ψ(t)ϕ(t)−t 0 ϕ(s)dψ(s),t∈(0,∞), and, to get (iii), it suffices to denote dη =d(−ψ). (iii) ⇒(i) Assume that hcan be represented as in (2.9). Then, denoting dν(s)=ϕ(s)dη(s), we have h(t)≈ϕ(t)[0,∞) dν(s) ϕ(s)+ϕ(t),t∈(0,∞).(2.10) Now, the monotonicity properties required in order to verify (i) are obvious. Definition 2.9. Let ϕbe an admissible function and let νbeanonnegative Borel measure on [0,∞). We say that the function hdefined as h(t)=ϕ(t)[0,∞) dν(s) ϕ(s)+ϕ(t),t∈(0,∞),(2.11) is the fundamental function of the measure νwith respect to ϕ.Wewill also say that νis a representation measure of hwith respect to ϕ. We say that νis non-degenerate if the following conditions are satisfied for every t∈(0,∞): [0,∞) dν(s) ϕ(s)+ϕ(t)<∞,t∈(0,∞),[0,1] dν(s) ϕ(s)=[1,∞) dν(s)=∞. Remark 2.10.(i) Let ϕbe an admissible function and let νbe a nonnegative non-degenerate Borel measure on [0,∞). Let hbe the fundamental function of νwith respect to ϕ. Then h(t)≈t 0[s,∞) dν(y) ϕ(y)dϕ(s),t∈(0,∞),(2.12) Discretization and Anti-Discretization 319 and also h(t)≈[0,t] dν(s)+ϕ(t)[t,∞) dν(s) ϕ(s),t∈(0,∞).(2.13) Moreover, h∈Ωϕ.Inparticular, by Lemma 2.7, there always exists a discretizing sequence for hwith respect to ϕ. Indeed, (2.12) follows immediately from (2.11) and the Fubini theorem, while (2.13) is an immediate consequence of (2.11) and the monotonicity of ϕ.From (2.12) it is clear that his non-decreasing and also that h ϕis non-increasing because it is an integral mean of a nonincreasing function. Finally, the non-degeneracy requirements in (2.1) follow from (2.12) and (2.13) combined with the Monotone Convergence Theorem. (ii) Conversely, if ϕis an admissible function and h∈Ωϕ, then there exists a representation measure ν. This follows from Remark 2.10 (i) and Lemma 2.8; indeed, we can put dν =ϕdη. Theorem 2.11. Let p, q, r ∈(0,∞).Letube an admissible function. Let νbe a non-negative non-degenerate Borel measure on [0,∞), and let hbe the fundamental function of νwith respect to uq.Letσ∈Ωup. Let {µk}beadiscretizing sequence for hwith respect to uq. Then [0,∞) h(t)r q−1 σ(t)r p dν(t)≈ k∈Z h(µk)r q σ(µk)r p . Proof: Let a>2, and let Z1,Z2be defined by (2.7) (with ϕ=uq). Then [0,∞) h(t)r q−1 σ(t)r p dν(t)= k∈Z[µk,µk+1) h(t)r q−1 σ(t)r p dν(t)=  k∈Z1 + k∈Z2 =I+II, say. For k∈Z1,h(µk+1)=ah(µk), and [µk,µk+1) dν(t)h(µk+1)h(µk). Thus, since σis equivalent to a non-decreasing function, we get from (2.3) I k∈Z1 h(µk)r q−1 σ(µk)r p[µk,µk+1) dν(t) k∈Z1 h(µk)r q−1 σ(µk)r p h(µk) k∈Z1 h(µk)r q σ(µk)r p . For k∈Z2,by(2.8) we have ah(µk+1) u(µk+1)q=h(µk) u(µk)q, 326 A. Gogatishvili, L. Pick Lemma 3.3. Let {ak}k∈Z,{σk}k∈Z, and {τk}k∈Zbe sequences of nonnegative numbers. Let p∈(0,∞). (i) If σk, then sup k∈Z  m≥k am  p σk≈sup m∈Z ap mσm. (ii) If τk, then sup k∈Z  m≤k am  p τk≈sup m∈Z ap mτm. Lemma 3.4. Let {ak}k∈Z,{σk}k∈Z, and {τk}k∈Zbe sequences of nonnegative numbers. (i) If σk, then sup k∈Zsup m≥k amσk≈sup m∈Z amσm. (ii) If τk, then sup k∈Zsup m≤k amτk≈sup m∈Z amτm. Next we shall prove an important lemma on an equivalence of two discretizing sequences. Lemma 3.5. Let p, q, r ∈(0,∞).Letube an admissible function. Let h∈Ωuqand g∈Ωup.Let{µk}beadiscretizing sequence of hwith respect to uqand let {λk}be a discretizing sequence of gwith respect to up. Then  k∈Z h(µk)r q g(µk)r p ≈ ∈Z h(λ)r q g(λ)r p and sup t∈(0,∞) h(t)1 q g(t)1 p ≈sup k∈Z h(µk)1 q g(µk)1 p ≈sup ∈Z h(λ)1 q g(λ)1 p . Discretization and Anti-Discretization 327 Proof: Since h(µk)and h(µk) uq(µk),wehave  k∈Z h(µk)r q g(µk)r p = ∈Z λ≤µk<λ+1 h(µk)r q g(µk)r p = ∈Z1 + ∈Z2 . For the first sum, we have by (2.3), (2.7) and (2.8)  ∈Z1 λ≤µk<λ+1 h(µk)r q g(µk)r p  ∈Z1 1 g(λ+1)r p µk<λ+1 h(µk)r q ∈Z h(λ+1)r q g(λ+1)r p . Similarly,  ∈Z2 λ≤µk<λ+1 h(µk)r q g(µk)r p  ∈Z2 u(λ)r g(λ)r p λ≤µk h(µk)r q u(µk)r ∈Z h(λ)r q g(λ)r p . On replacing the roles of λkand µk,weget the converse inequality. As for the second assertion, observe that sup t∈(0,∞) h(t)1 q g(t)1 p = sup k∈Z sup µk≤t<µk+1 h(t)1 q g(t)1 p . If now k∈Z1, then, by (2.3) for hand (2.7) for g, sup µk≤t<µk+1 h(t)1 q g(t)1 p h(µk)1 q g(µk)1 p . If k∈Z2, then sup µk≤t<µk+1 h(t)1 q g(t)1 p h(µk)1 q u(µk)sup µk≤t<µk+1 u(t) g(t)1 p h(µk)1 q g(µk)1 p . We thus conclude that sup t∈(0,∞) h(t)1 q g(t)1 p sup k∈Z h(µk)1 q g(µk)1 p and by the same way we get sup t∈(0,∞) h(t)1 q g(t)1 p sup ∈Z h(λ)1 q g(λ)1 p . Now we shall prove a discretization lemma for a Hilbert norm, a main result of this section. We work within the following fixed scheme: uis an admissible function, νis a positive measure on [0,∞), q∈(0,∞) and h(t)=[0,∞) u(t)q (u(s)+u(t))qdν(s),t∈(0,∞). 328 A. Gogatishvili, L. Pick Recall that h(t)≈[0,t] dν(s)+u(t)q[t,∞) u(s)−qdν(s),t∈(0,∞). We shall further assume that νis non-degenerate. In this setup, we have Lemma 3.6. Let q∈(0,∞), let ube an admissible function and let ν beanon-degenerate positive Borel measure. Let hbe the fundamental function of νwith respect to uqand let fbe ameasurable function on [0,∞).Let{xk}beadiscretizing sequence for hwith respect to uq. Then ∞ 0[0,∞) |f(y)| u(x)+u(y)dyq dν(x) ≈ k∈Z[0,∞) |f(y)| u(xk)+u(y)dyq h(xk) ≈ k∈Zu−1(xk)xk xk−1 |f(y)|dy +xk+1 xk |f(y)|u(y)−1dyq h(xk) ≈ k∈Z1xk+1 xk |f(y)|u(y)−1dyq h(xk) + k∈Z2xk+1 xk |f(y)|dyq u(xk)−qh(xk) ≈ k∈Zxk+1 xk |f(y)|u(y)−1h(y)1 qdyq . Proof: Clearly, the function g(x)=u(x)∞ 0 |f(y)| u(x)+u(y)dy belongs to Ωu. Using Corollary 2.13 with the measure u(x)−qdν,weobtain the first relation. Discretization and Anti-Discretization 329 Since ∞ 0 |f(y)| u(xk)+u(y)dy ≈u(xk)−1xk 0 |f(y)|dy +∞ xk |f(y)|u(y)−1dy, h(xk)and h(xk) u(xk)q,weget the second relation by Lemma 3.1. The converse inequality is trivial. To prove the third equivalence, it suffices to show that  k∈Zu−1(xk)xk xk−1 |f(y)|dy +xk+1 xk |f(y)|u(y)−1dyq h(xk)  k∈Z1xk+1 xk |f(y)|u(y)−1dyq h(xk) + k∈Z2xk+1 xk |f(y)|dyq u(xk)−qh(xk). When k−1∈Z1,wehave h(xk)=ah(xk−1). Thus,  k∈Z u−q(xk)xk xk−1 |f(y)|dyq h(xk) ≤a k−1∈Z1xk xk−1 |f(y)|u(y)−1dyq h(xk−1) + k−1∈Z2xk xk−1 |f(y)|dyq u(xk−1)−qh(xk−1) =a k∈Z1xk+1 xk |f(y)|u(y)−1dyq h(xk) + k∈Z2xk+1 xk |f(y)|dyq u(xk)−qh(xk). 330 A. Gogatishvili, L. Pick Using the monotonicity of u,weobtain the remaining estimate  k∈Zxk+1 xk |f(y)|u(y)−1dyq h(xk)  k∈Z1xk+1 xk |f(y)|u(y)−1dyq h(xk) + k∈Z2xk+1 xk |f(y)|dyq u(xk)−qh(xk). Finally, the last equivalence is a simple consequence of (2.3) and (2.4). Lemma 3.7. Let q∈(0,∞), let ube an admissible function and let ν beanon-degenerate positive Borel measure. Let hbe the fundamental function of νwith respect to uqand let fbe ameasurable function on [0,∞).Let{xk}beadiscretizing sequence for hwith respect to uq. Then [0,∞)sup y∈(0,∞) |f(y)| u(x)+u(y)q dν(x) ≈ k∈Zsup y∈(0,∞) |f(y)| u(xk)+u(y)q h(xk) ≈ k∈Zu−1(xk) sup xk−1≤y<xk |f(y)|+ sup xk≤y<xk+1 |f(y)|u(y)−1q h(xk) ≈ k∈Z1 sup xk≤y<xk+1 |f(y)|qu(y)−qh(xk) + k∈Z2 sup xk≤y<xk+1 |f(y)|qu(xk)−qh(xk) ≈ k∈Z sup xk≤y<xk+1 |f(y)|qu(y)−qh(y). Proof: This follows on using Corollary 2.13 and Lemma 3.2. Discretization and Anti-Discretization 331 Lemma 3.8. Let q∈(0,∞).Letube an admissible function and let ϕ∈Ωuq.Let{xk}be a discretizing sequence for ϕwith respect to uq. Let fbe ameasurable function on [0,∞). Then sup x∈(0,∞) ϕ(x)∞ 0 |f(y)| u(x)+u(y)dyq ≈sup k∈Z ϕ(xk)∞ 0 |f(y)| u(xk)+u(y)dyq ≈sup k∈Z ϕ(xk)u(xk)−qxk xk−1 |f(y)|dyq + sup k∈Z ϕ(xk)xk+1 xk |f(y)|u(y)−1dyq ≈sup k∈Z1 ϕ(xk)xk+1 xk |f(y)|u(y)−1dyq + sup k∈Z2 ϕ(xk)u(xk)−qxk+1 xk |f(y)|dyq ≈sup k∈Zxk+1 xk |f(y)|u(y)−1ϕ(y)1 qdyq . Proof: This follows from using Lemmas 3.3 and 3.5. Lemma 3.9. Let ube an admissible function and let ϕ∈Ωu.Let{xk} be a discretizing sequence for ϕwith respect to u. Then sup x∈(0,∞) ϕ(x) sup 0<y<∞ |f(y)| u(x)+u(y) ≈sup k∈Z ϕ(xk) sup 0<y<∞ |f(y)| u(xk)+u(y) ≈sup k∈Z ϕ(xk)u(xk)−1sup xk−1<y<xk |f(y)| + sup k∈Z ϕ(xk) sup xk<y<xk+1 |f(y)|u(y)−1 ≈sup k∈Z1 ϕ(xk) sup xk≤y<xk+1 |f(y)|u(y)−1 + sup k∈Z2 ϕ(xk)u(xk)−1sup xk<y<xk+1 |f(y)| ≈sup k∈Z sup xk≤y<xk+1 |f(y)|u(y)−1ϕ(y). Proof: This follows from using Lemmas 3.4 and 3.5. 332 A. Gogatishvili, L. Pick 4. Embeddings of classical Lorentz spaces, type Γ →Λ As our first main application we shall establish necessary and sufficient conditions such that the inequality ∞ 0 (f∗(t))qw(t)dt1 q ≤C∞ 0 (f∗∗ u(t))pv(t)dt1 p (4.1) holds for every f∈M(R,µ), where f∗∗ u(t)= 1 U(t)t 0f∗(s)u(s)ds,u,v,w are weights, U(t)=t 0u(s)ds, and p, q ∈(0,∞). In the proof of the necessity part of our main theorem we will need the following version of the classical Landau resonance theorem. Proposition 4.1. Let {wk}and {vk},k∈Z,betwo (double-infinite) sequences of positive real numbers. Let p, q ∈(0,∞)and assume that the inequality  k∈Z aq kvk1 q  k∈Z ap kwk1 p is satisfied for every sequence {ak}of non-negative real numbers. (i) If p≤q, then sup k∈Z w−q p kvk<∞. (ii) If p>q, then  k∈Z w−r p kv r q k1 r <∞,where r=pq p−q. This assertion is well known. For the sake of completeness, let us just point out that a simple direct proof of (i) is seen from setting, for a given k∈Z, aj=   w−1 p kwhen j=k; 0 when j=k; while (ii) follows on putting ak=    vk wk1 p−q when k∈[−N,N]; 0 otherwise. Discretization and Anti-Discretization 333 This shows N  k=−N w−r p kv r q k1 r ≤C with Cindependent of N, and we just have to let N→∞. Our main result reads as follows. Theorem 4.2. Let u,v,wbe locally integrable weights on [0,∞).Let p, q ∈(0,∞). When p>q,weset r=pq p−q. Assume that uis such that Upis admissible and the measure v(t)dt is non-degenerate with respect to Up. (i) If 0<p≤q<∞and 1≤q<∞, then (4.1) holds for some C>0 and all fif and only if A(1) = sup t∈(0,∞) W(t)1 q V(t)+U(t)p∞ tU(s)−pv(s)ds1 p <∞. (ii) If 1≤q<p<∞, then (4.1) holds for some C>0and all fif and only if A(2) =  ∞ 0 U(t)rsupy∈[t,∞)U(y)−rW(y)r q V(t)+U(t)p∞ tU(s)−pv(s)dsr p+2 ×V(t)∞ t U(s)−pv(s)ds d(Up(t))  1 r <∞. (iii) If 0<p≤q<1, then (4.1) holds for some C>0and all fif and only if A(3) = sup t∈(0,∞) W(t)1 q+U(t)∞ tW(s)q 1−qw(s)U(s)−q 1−qds1−q q V(t)+U(t)p∞ tU(s)−pv(s)ds1 p <∞. 334 A. Gogatishvili, L. Pick (iv) If 0<q<1and 0<q<p, then (4.1) holds for some C>0and all fif and only if A(4) <∞, where A(4)=   ∞ 0W(t)1 1−q+U(t)q 1−q∞ tW(s)q 1−qw(s)U(s)−q 1−qds r(1−q) q−1 V(t)+U(t)p∞ tU(s)−pv(s)dsr p ×W(t)q 1−qw(t)dt    1 r . Moreover, A(4) ≈A(5), where A(5)=   ∞ 0W(t)1 1−q+U(t)q 1−q∞ tW(s)q 1−qw(s)U(s)−q 1−qds r(1−q) q V(t)+U(t)p∞ tU(s)−pv(s)dsr p+2 ×V(t)∞ t U(s)−pv(s)ds d(Up(t))    1 r . Proof: We start with the upper bounds (sufficiency). First, a standard argument shows that it is enough to prove (4.1) for fsatisfying f∗(t)= ∞ th(s)ds, where his some positive measurable function on (0,∞). That is, we only have to prove (4.2) ∞ 0∞ t h(s)dsq w(t)dt1 q ≤C∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt1 p for every h≥0. Define ϕ(t)=U(t)p∞ 0 v(s) (U(s)+U(t))pds.(4.3) Then ϕ∈ΩUp, and therefore there exists a discretizing sequence for ϕ with respect to Up. Let {xk}be one such sequence. Then ϕ(xk)and ϕ(xk) U(xk)p.Furthermore, there is a decomposition Z=Z1∪Z2,Z1∩Z2=∅ such that for every k∈Z1and t∈∆k=[xk,x k+1], ϕ(t)≈ϕ(xk) and Discretization and Anti-Discretization 335 for every k∈Z2and t∈∆k,ϕ(t) U(t)p≈ϕ(xk) U(xk)p.ByLemma 3.6, applied to f=hU and h=ϕ,wehave ∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt1 p ≈ k∈Z∞ 0 U(s)h(s) U(xk)+U(s)dsp ϕ(xk)1 p ≈ k∈Zxk+1 xk h(s)ϕ(s)1 pdsp1 p . (4.4) For the left side of (4.2), we get ∞ 0∞ t h(s)dsq w(t)dt = k∈Zxk+1 xk∞ t h(s)dsq w(t)dt ≈ k∈Zxk+1 xkxk+1 t h(s)dsq w(t)dt + k∈Zxk+1 xk w(t)dt ∞ xk+1 h(s)dsq =I+II. (4.5) Now we shall distinguish several cases. Assume first that 1 ≤q<∞. We shall use the Hardy inequality (cf. [24]) xk+1 xkxk+1 t h(s)dsq w(t)dt xk+1 xk h(s)ϕ(s)1 pdsq sup t∈[xk,xk+1] ϕ(t)−q pt xk w(s)ds, where the constant does not depend on {xk}.Weobtain I k∈Zxk+1 xk h(s)ϕ(s)1 pdsq sup t∈[xk,xk+1] ϕ(t)−q pt xk w(y)dy.(4.6) 342 A. Gogatishvili, L. Pick Let 0 <q<1. For k∈Z, define hkso that supp hk∈[xk,x k+1], xk+1 xkhk(s)ϕ(s)1 pds =1,and xk+1 xkxk+1 t hk(s)dsq w(t)dt xk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt1−q . This is possible thanks to the saturation of the Hardy inequality and due to a simple modification of [28, Theorem 3.3]. Let {ak}be a sequence of non-negative real numbers. We define h(s)= k∈Z akhk(s). From (4.15) we get, using the definition of tkand h, (4.17)   k∈Z aq kxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt1−q  1 q  k∈Z ap k1 p . Let p≤q. Then, by Proposition 4.1, sup k∈Zxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt1−q <∞.(4.18) By Lemma 3.8 and Lemma 3.3 we get A(3) sup k∈Zxk+1 xk Wq 1−q(t)w(t)ϕ(t)−q p(1−q)dt1−q q sup k∈Z W(xk)xk+1 xk w(t)ϕ(t)−q p(1−q)dt1−q q + sup k∈Zxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt 1−q q . (4.19) Discretization and Anti-Discretization 343 By (4.18), the second summand on the right hand side of (4.19) is finite. It remains to estimate the first term, that is, to show that also sup k∈Z W(xk)xk+1 xk w(t)ϕ(t)−q p(1−q)dt1−q q <∞. This is obvious when xk+1 xkw(s)ds≤W(xk). Assume that xk+1 xkw(s)ds> W(xk), and let tk∈[xk,x k+1]besuch that tk xkw(s)ds =W(xk). Then, by (4.18) sup k∈Z W(xk)xk+1 xk w(t)ϕ(t)−q p(1−q)dt1−q q sup k∈Z W(xk)tk xk w(t)ϕ(t)−q p(1−q)dt 1−q q + sup k∈Z W(xk)xk+1 tk w(t)ϕ(t)−q p(1−q)dt1−q q sup k∈Z ϕ(xk)−1 pW(xk)1 q + sup k∈Zxk+1 tkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt 1−q q . Now, sup k∈Z ϕ(xk)−1 pW(xk)1 q≈sup k∈Z ϕ(xk)−1 pxk xk−1 w(s)ds1 q ≤sup k∈Z ϕ(xk)−1 p xk xk−1t xk−1 w(s)dsq 1−q w(t)dt  1−q q ≤sup k∈Z xk xk−1 ϕ(t)−q p(1−q)t xk−1 w(s)dsq 1−q w(t)dt  1−q q , 344 A. Gogatishvili, L. Pick and sup k∈Zxk+1 tkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt 1−q q ≤xk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt 1−q q <∞. Hence, A(3) <∞. Let now q<p, then we get, by (4.17) and Proposition 4.1,    k∈Zxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q   1 r <∞.(4.20) By Theorem 2.11 and Lemma 3.5, we get A(4) ≈    k∈Z W(xk)r q+U(xk)r∞ xkW(t)q 1−qw(t)U(t)−q 1−qdt(1−q)r q ϕ(xk)r p    1 r ≈  k∈Zxk+1 xk W(t)q 1−qw(t)ϕ(t)−q p(1−q)dt(1−q)r q  1 r ≈  k∈Z W(xk)rxk+1 xk w(t)ϕ(t)−q p(1−q)dt(1−q)r q  1 r +   k∈Zxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q   1 r . (4.21) By (4.20), the second term is finite. Discretization and Anti-Discretization 345 Next, note that W(xk)rxk+1 xk w(t)ϕ(t)−q p(1−q)dt(1−q)r q ϕ(xk)−r pW(xk)r q+ xk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q . Indeed, if xk+1 xkw(s)ds<W(xk), then W(xk)rxk+1 xk w(t)ϕ(t)−q p(1−q)dt(1−q)r q ≤ϕ(xk)−r pW(xk)rxk+1 xk w(t)dt(1−q)r q ≤ϕ(xk)−r pW(xk)r q. Now assume that xk+1 xkw(s)ds ≥W(xk). Let tk∈[xk,x k+1]besuch that tk xkw(s)ds =W(xk). Then we have W(xk)rxk+1 xk w(t)ϕ(t)−q p(1−q)dt(1−q)r q ≤W(xk)rtk xk w(t)dt (1−q)r q ϕ(xk)−r p +xk+1 tkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q ϕ(xk)−r pW(xk)r q +xk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q . 346 A. Gogatishvili, L. Pick We thus have   k∈Z W(xk)rxk+1 xk w(t)ϕ(t)−q p(1−q)dt(1−q)r q  1 r ≤ k∈Z ϕ(xk)−r pW(xk)r q1 r +   k∈Zxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q   1 r , (4.22) and, using Lemma 3.1 (i),  k∈Z W(xk)r qϕ(xk)−r p1 r ≈ k∈Zxk+1 xk w(s)dsr q ϕ(xk+1)−r p1 r    k∈Zxk+1 xkt xk w(s)ds q 1−q w(t)ϕ(t)−q p(1−q)dt (1−q)r q   1 r , (4.23) which is finite by (4.20). Finally, combining (4.21), (4.23) and (4.22), we obtain A(4) <∞. The proof is complete. 5. Embeddings of classical Lorentz spaces, type Γ →Γ In this section we characterize the inequality ∞ 0 f∗∗ u(t)qw(t)dt1 q ≤C∞ 0 f∗∗ u(t)pv(t)dt1 p ,(5.1) where p, q ∈(0,∞) and u,v,ware weights. Discretization and Anti-Discretization 347 Our main result reads as follows. Theorem 5.1. Let p, q ∈(0,∞)and let u,v,wbe weights. Assume that v(t)dt is a non-degenerate measure with respect to Up. (i) Let 0<p≤q<∞. Then (5.1) holds if and only if A(6) = sup t∈(0,∞)W(t)+U(t)q∞ tU(s)−qw(s)ds1 q V(t)+U(t)p∞ tU(s)−pv(s)ds1 p <∞. (ii) Let 0<q<p<∞. Then (5.1) holds if and only if A(7) = ∞ 0W(t)+U(t)q∞ tU(y)−qw(y)dyr q V(t)+U(t)p∞ tU(s)−pv(s)dsr p+2 ×V(t)∞ t U(s)−pv(s)ds d(Up)(t)1 r <∞, where, again, r=pq p−q. Moreover, A(7) ≈A(8), where A(8) = ∞ 0W(t)+U(t)q∞ tU(s)−qw(s)dsr q−1w(t) V(t)+U(t)p∞ tU(s)−pv(s)dsr p dt1 r <∞. Proof: Sufficiency: First, in order to prove (5.1) it is enough to show (5.2) ∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsq w(t)dt1 q ≤∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt1 p . Let ϕbe defined as in (4.3) and let {xk}be a discretizing sequence for ϕ. Recall that then ϕ(xk),ϕ(xk)U−p(xk), and there is a decomposition Z=Z1∪Z2such that Z1∩Z2=∅and for k∈Z1and t∈∆k:= [xk,x k+1] we have ϕ(t)≈ϕ(xk), while for k∈Z1and t∈∆kwe have ϕ(t)U(t)−p≈ 348 A. Gogatishvili, L. Pick ϕ(xk)U(xk)−p.By(4.4), ∞ 0∞ 0 U(s)h(s) U(s)+U(t)q w(t)dt  k∈Zxk+1 xk∞ t h(s)dsq w(t)dt +xk+1 xkt 0 h(s)U(s)dsq U(t)−qw(t)dt  k∈Z1xk+1 xk w(t)dt ∞ xk h(s)dsq + k∈Z2xk+1 xk w(t)dt ∞ xk+1 h(s)dsq + k∈Z2xk+1 xkxk+1 t h(s)dsq w(t)dt + k∈Z2xk+1 xk U(t)−qw(t)dt xk+1 0 U(s)h(s)dsq + k∈Z1xk+1 xk U(t)−qw(t)dt xk 0 U(s)h(s)dsq + k∈Z1xk+1 xkt xk U(s)h(s)dsq U(t)−qw(t)dt ≈ k∈Z1xk+1 xk w(t)dt ∞ xk h(s)dsq + k∈Z1xk+1 xk U(t)−qw(t)dt xk 0 U(s)h(s)dsq + k∈Z2xk+1 xk U(t)−qw(t)dt xk+1 0 U(s)h(s)dsq + k∈Z2xk+1 xk w(t)dt ∞ xk+1 h(s)dsq =I+II+III+IV, say. Discretization and Anti-Discretization 349 Let first p≤q. Then I=  k∈Z1xk+1 xk w(t)dt ∞ xk h(s)dsq ≤ k∈Z1∞ xk h(s)dsp ϕ(xk)q p sup k∈Z1 ϕ(xk)−q pxk+1 0 w(t)dt, II =  k∈Z1xk+1 xk U(t)−qw(t)dt xk 0 U(s)h(s)dsq ≤ k∈Z1xk 0 U(s)h(s)dsp U(xk)−pϕ(xk)q p ×sup k∈Z1 ϕ(xk)−q pU(xk)q∞ xk w(t)U(t)−qdt, III =  k∈Z2xk+1 xk U(t)−qw(t)dt xk+1 0 U(s)h(s)dsq ≤ k∈Z2xk+1 0 U(s)h(s)dsp U(xk+1)−pϕ(xk+1)q p ×sup k∈Z2 ϕ(xk+1)−q pU(xk+1)q∞ xk w(t)U(t)−qdt, and IV =  k∈Z2xk+1 xk w(t)dt ∞ xk+1 h(s)dsq ≤ k∈Z2∞ xk+1 h(s)dsp ϕ(xk+1) q p ×sup k∈Z2 ϕ(xk+1)−q pxk+1 0 w(t)dt. 350 A. Gogatishvili, L. Pick Using the fact that ϕ(xk)≈ϕ(xk+1) when k∈Z1 and ϕ(xk)U(xk)−p≈ϕ(xk+1)U(xk+1)−pwhen k∈Z2, we get ∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsq w(t)dt A(6)q∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dtq . Now assume that q<p. Using the H¨older inequality with parameters p qand p p−q, Lemma 3.1 (i), and (4.4), we obtain I≤ k∈Z1∞ xk h(s)dsp ϕ(xk)q p × k∈Z1 W(xk+1)r qϕ(xk)−r pq r ∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt q p × k∈Z1 W(xk+1)r qϕ(xk+1)−r pq r ∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt q p × k∈Z W(xk)r qϕ(xk)−r pq r , Discretization and Anti-Discretization 351 II ≤ k∈Z1xk 0 U(s)h(s)dsp U(xk)−pϕ(xk)q p × k∈Z1∞ xk w(t)U(t)−qdtr q ϕ(xk)−r pU(xk)rq r ≤∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt q p × k∈Z1∞ xk w(t)U(t)−qdtr q ϕ(xk)−r pU(xk)rq r , III ≤ k∈Z2xk+1 0 U(s)h(s)dsp U(xk+1)−pϕ(xk+1)q p × k∈Z2∞ xk w(t)U(t)−qdtr q ϕ(xk+1)−r pU(xk+1)rr q ∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt q p × k∈Z2∞ xk w(t)U(t)−qdtr q ϕ(xk)−r pU(xk)rq r and IV ≤ k∈Z2∞ xk+1 h(s)dsp ϕ(xk+1) q p × k∈Z2xk+1 0 w(t)dtr q ϕ(xk+1)−r pq r ≤∞ 0∞ 0 U(s)h(s) U(s)+U(t)dsp v(t)dt q p × k∈Zxk 0 w(t)dtr q ϕ(xk)−r pq r . 358 A. Gogatishvili, L. Pick [32] V. D. Stepanov,Integral operators on the cone of monotone functions and embeddings of Lorentz spaces, (Russian), Dokl. Akad. Nauk SSSR 317(6) (1991), 1308–1311; translation in Soviet Math. Dokl. 43(2) (1991), 620–623. Amiran Gogatishvili: Mathematical Institute Czech Academy of Sciences ˇ Zitn´a25 115 67 Praha 1 Czech Republic E-mail address:[email protected] Luboˇs Pick: Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovsk´a83 186 75 Praha 8 Czech Republic E-mail address:[email protected] Primera versi´o rebuda el 18 d’abril de 2002, darrera versi´o rebuda el 10 d’octubre de 2002.