We characterize the weighted Hardy's inequalities for monotone functions in
${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of
$B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In
fact, our main theorem is proved in the more general setting of partially
ordered measure spaces.Comment: 14 page

Arcozzi, Nicola

Barza, Sorina

Garcia-Domingo, Josep L.

Soria, Javier

Proceedings of the Royal Society of Edinburgh Section A Mathematics

English

arXiv

Nicola Arcozzi

Sorina Barza

J. L. Garcia-Domingo

Javier Soria

Crossref

Hardy's inequalities for monotone functions on partly ordered measure spaces

The Hardy inequalities for a class of partial orders are characterized

ARCOZZI, NICOLA

S. Barza

J. L. Garcia Domingo

J. Soria

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

We characterize the weighted Hardy inequalities for monotone functions in Rn
+. In
dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the
result was previously only known for the case p = 1. In fact, our main theorem is
proved in the more general setting of partly ordered measure spaces

Garcia Domingo, Josep Lluís

RIUVic

Proceedings of the Royal Society of Edinburgh, 136A, 909–919, 2006Hardy’s inequalities for monotone functions onpartly ordered measure spacesNicola ArcozziDepartment of Mathematics, University of Bologna,Piazza di Porta S. Donato 5, 540127 Bologna, Italy(arcozzi@dm.unibo.it)Sorina BarzaDepartment of Mathematics, Karlstad University,Universitetsgatan 2, 65188 Karlstad, Sweden(sorina.barza@kau.se)J. L. Garcia-DomingoDepartment of Economics, Mathematics and Computers,Universitat de Vic, 08500 Vic, Spain (jlgarcia@uvic.es)Javier SoriaDepartment of Applied Mathematics and Analysis,University of Barcelona, 08007 Barcelona, Spain (soria@ub.edu)(MS received 11 May 2005; accepted 5 October 2005)We characterize the weighted Hardy inequalities for monotone functions in Rn+. Indimension n = 1, this recovers the standard theory of Bp weights. For n > 1, theresult was previously only known for the case p = 1. In fact, our main theorem isproved in the more general setting of partly ordered measure spaces.1. IntroductionThe theory of weighted inequalities for the Hardy operator, acting on monotonefunctions in R+, was ﬁrst introduced in [2]. Extensions of these results to higherdimensions have been considered only in very speciﬁc cases: in particular, in thediagonal case, for p = 1 only (see [3]). The main diﬃculty in this context is thatthe level sets of the monotone functions are not totally ordered, contrary to theone-dimensional case, where one considers intervals of the form (0, a), a > 0. It isalso worth pointing out that, with no monotonicity restriction, the boundednessof the Hardy operator is known only in dimension n = 2 (see [11, 13] and for anextension in the case of product weights see [4]).In this work we characterize completely the weighted Hardy inequalities for allvalues of p > 0, namely, the boundedness of the operatorS : Lpdec(u) → Lp(u),909c© 2006 The Royal Society of Edinburgh910 N. Arcozzi and otherswhereSf(s, t) =1st∫ s0∫ t0f(x, y) dy dx,and Lpdec(u) is the cone of positive and decreasing functions, on each variable, inLp(u) = Lp(R2+, u(x) dx) (we consider, for simplicity, n = 2, although the resultholds in any dimension).The techniques we will use were introduced in [6] for the one-dimensional case,and also apply to a more general setting, which we now deﬁne.We will consider a family of σ-ﬁnite measure spaces (X,µx) (where µx is a mea-sure on X, for each x ∈ X), with a partial order ‘’ satisfying the following condi-tions.(i) If Xx := {u ∈ X : u  x}, then  restricted to Xx is a total order.(ii) If D is a decreasing set, with respect to the order  (i.e. χD is a decreasingfunction), then D is measurable.(iii) µx(Xx) = 1 (observe that Xx is a decreasing set).(iv) If u ∈ Xx, then dµx(y) = µx(Xu) dµu(y). In particular, µx(Xu)µu(Xx) = 1.The main examples are as follows.(a) X = R+ with the usual order, and µx(E) = x−1|E|. This is the case consid-ered in [2].(b) X is a tree with the usual order on geodesics, and µx(E) = Card(E)/|x|,where |x| = Card([o, x]) and [o, x] is the geodesic path joining the origin owith any vertex x of the tree. (For more information on this case, see [10] andthe references quoted therein.)(c) R2+ with the order given by (a1, b1)  (a2, b2) if and only if a1 = a2 and b1  b2(we could also choose to ﬁx the second coordinate). For x = (a, b) ∈ R2+,µx(E) = b−1∫E∩({a}×R+)dt.(d) In many cases, we can easily obtain the existence of the family of measures µxby taking a non-negative measure µ on X, and deﬁning µx = µ/µ(Xx).We now deﬁne the Hardy operator as follows:Sf(x) =∫Xxf(u) dµx(u).This deﬁnition is similar to the one considered in [3]. For the case of R+,Sf(x) =1x∫ x0f(t) dt.Hardy’s inequalities for monotone functions 911On a tree,Sf(x) =∑y∈[o,x]f(y)|x|and, for R2+,Sf(a, b) =1b∫ b0f(a, t) dt.One of the main techniques we will use is the following lemma. This is a kind ofintegration by parts.Lemma 1.1. Let (X,µ,) be a ﬁnite measure space with a total order , and α ∈ R.There then exists a constant, Cα, which depends only on α (and not on (X,µ,)),such that (∫Xdµ)α Cα∫X(∫Xudµ)α−1dµ(u).Proof. Since µ(X) < ∞, by dividing both sides by (µ(X))α it suﬃces to show that1  Cα∫Xϕα−1(u) dµ(u),where ϕ(u) = µ(Xu) and µ(X) = 1.If α  1, then, using the fact that 0  ϕ  1, we obtain ϕα−1(u)  1, and∫Xϕα−1(u) dµ(u)  µ(X) = 1.If 1 < α  2, then ϕα−1(u)  ϕ(u) and, hence, it suﬃces to prove it for α = 2.If 2 < α, using Jensen’s inequality,(∫Xϕ(u) dµ(u))α−1∫Xϕα−1(u) dµ(u),and as before, this reduces to the case α = 2.Finally, if α = 2,∫X(∫Xudµ(x))dµ(u) =∫X(∫{xu}dµ(u))dµ(x)=∫X(1 −∫{x>u}dµ(u))dµ(x)= 1 −∫X∫{ux}dµ(u) dµ(x) +∫X∫{u=x}dµ(u) dµ(x)(here we have used the fact that the order is total). Thus,∫X(∫Xudµ(x))dµ(u)  12.912 N. Arcozzi and others2. The weighted Hardy inequalityIn this section we will prove the main theorem. In order to include all the examples,we need to consider a second, weaker order ≺ satisfying the following conditions.(i) If x  y, then x ≺ y.(ii) If f is ≺-decreasing, then Sf is ≺-decreasing.We recall that we still keep  to deﬁne the operator S.Remark 2.1. We can (and will in some cases) take ≺ to be . In fact, we needonly to check that the second condition holds for : if f is -decreasing, and u  x,then Sf(x)  Sf(u) if and only if∫Xuf(y)[1 − µx(Xu)] dµu(y) −∫Xx\Xuf(y)µx(Xu) dµu(y)  0(here we have used the fact that dµx(y) = µx(Xu) dµu(y)), and this follows fromthe fact that inf f |Xu  sup f |Xx\Xu .If a function f , or a set D, is ≺-decreasing, it is also -decreasing.The main example we have in mind is  in R2+ as before, and ≺ the order givenby the rectangles (which is clearly a weaker order): (a1, b1) ≺ (a2, b2), if and onlyif a1  a2 and b1  b2 (i.e. the rectangle in R2+ determined by the origin and(a1, b1) is contained in that determined by the origin and (a2, b2)). To show thesecond condition, assume that f is a function decreasing on each variable. It is thenobvious thaty−1∫ y0f(x, t) dtis also decreasing on each variable.We denote by Lp≺(dν) the class of ≺-decreasing functions in Lp(dν). As a generalassumption, we will only consider cases for which measurability of the functionsinvolved always holds.Theorem 2.2. Let (X,µx,) and ≺ satisfy the conditions given above, for all x ∈X. Let dν be a measure on X, and p > 0. The Hardy operator then is bounded:S : Lp≺(dν) → Lp(dν),if and only if there exists a constant C > 0 such that, for all ≺-decreasing sets D,∫X\Dµpx(D ∩ Xx) dν(x)  Cν(D). (2.1)Proof. Consider f = χD, where D is ≺-decreasing. Then (Sf)p(x) = µpx(D ∩ Xx),and hence‖Sf‖pLp(dν) =∫Xµpx(D ∩ Xx) dν(x)=∫Dµpx(D ∩ Xx) dν(x) +∫X\Dµpx(D ∩ Xx) dν(x) Cν(D).Hardy’s inequalities for monotone functions 913Thus, ∫X\Dµpx(D ∩ Xx) dν(x)  Cν(D).Observe that, since D is also -decreasing, if x ∈ D, then Xx ⊂ D, and soµpx(D ∩ Xx) = µpx(Xx) = 1. Therefore,∫Dµpx(D ∩ Xx) dν(x) = ν(D).Conversely, if p  1 and f ∈ Lp≺(dν), using the lemma with (Xx, µ,) anddµ(u) = f(u) dµx(u), for a constant C that does not depend on either f or x, wehave(Sf)p(x) =(∫Xxf(u) dµx(u))p C∫Xx(∫Xuf(y) dµx(y))p−1f(u) dµx(u)= C∫Xx(∫Xuf(y) dµu(y))p−1f(u)µp−1x (Xu) dµx(u)= C∫ ∞0∫{g>t}∩Xxµp−1x (Xu) dµx(u) dt,where g(u) = (Sf)p−1(u)f(u). Hence,‖Sf‖pLp(dν)  C∫X∫ ∞0∫{g>t}∩Xxµp−1x (Xu) dµx(u) dtdν(x)≈∫X∫ g(x)0∫{g>t}∩Xxµp−1x (Xu) dµx(u) dtdν(x)+∫X∫ ∞g(x)∫{g>t}∩Xxµp−1x (Xu) dµx(u) dtdν(x)= I+ II .Since Xu ⊂ Xx, if u ∈ Xx, and p  1, thenI ∫X∫ g(x)0µp−1x (Xx)µx(Xx) dtdν(x) =∫Xg(x) dν(x).Since both Sf and f are ≺-decreasing, and p  1, g is ≺-decreasing and {g > t} isa ≺-decreasing set. Also, if u ∈ {g > t} ∩ Xx, then Xu ⊂ {g > t} ∩ Xx, and hence∫{g>t}∩Xxµp−1x (Xu) dµx(u)  µpx({g > t} ∩ Xx).Therefore, using the hypothesis,II ∫ ∞0∫X\{g>t}µpx({g > t} ∩ Xx) dν(x) dt C∫ ∞0∫{g>t}dν(x) dt = C∫Xg(x) dν(x).914 N. Arcozzi and othersTherefore, using Ho¨lder’s inequality,‖Sf‖pLp(dν)  C∫X(Sf)p−1(x)f(x) dν(x)  C‖Sf‖p−1Lp(dν)‖f‖Lp(dν).From this a priori estimate, one may obtain the general result by a standarddensity argument.If 0 < p < 1, and f ∈ Lp≺(dν), set Dt = {f > t}, andgx(t) =∫Dt∩Xxdµx(u).Then, using the embedding Lpdec(tp−1) ↪→ L1 (see [14]), we have (observe that gxis a decreasing function)(∫X(∫Xxf(u) dµx(u))pdν(x))1/p=(∫X(∫ ∞0gx(t) dt)pdν(x))1/p C(∫X∫ ∞0tp−1(gx(t))p dtdν(x))1/p.Since gx(t)  µx(Xx) = 1, we then have∫ ∞0tp−1∫Dt(gx(t))p dν(x) dt ∫ ∞0tp−1∫Dtdν(x) dt =1p‖f‖pLp(dν).On the other hand, using the hypothesis (observe that Dt ∩ Xx is a decreasingset),∫ ∞0tp−1∫X\Dt(gx(t))p dν(x) dt  C∫ ∞0tp−1∫Dtdν(x) dt =Cp‖f‖pLp(dν).Therefore, (∫X(∫Xxf(u) dµx(u))pdν(x))1/p C‖f‖Lp(dν).The following results follow easily by particularizing on each case condition (2.1)of theorem 2.2.Corollary 2.3.Case 1 (equality of orders ≺ and ). Let (X,µx,) satisfy the conditions givenabove. Let dν be a measure on X, and p > 0. Then, the Hardy operator is boundedasS : Lp(dν) → Lp(dν)if and only if there exists a constant C > 0 such that, for all -decreasing sets D,∫X\Dµpx(D ∩ Xx) dν(x)  Cν(D).Hardy’s inequalities for monotone functions 915Case 2 (R+). Condition (2.1) of theorem 2.2 is∫ ∞r(rx)pdν(x)  C∫ r0dν(x),for all r > 0, which is Bp (see [2]).Case 3 (tree T ). Condition (2.1) of theorem 2.2 is∑x∈T\D|x ∨ D|p|x|p ν(x)  C∑x∈Dν(x),where x ∨ D is the largest vertex in [o, x] ∩ D.Case 4 (R2+). Condition (2.1) of theorem 2.2 is∫R2+\D|Dx|ptpdν(x, t)  C∫Ddν(x, t),where Dx = {t > 0 : (x, t) ∈ D}, and D is any decreasing set (on each variable).Remark 2.4. As mentioned above, the case of R+ was ﬁrst considered in [2].Bp weights are well understood and enjoy a very rich structure (see also [7, 8, 14]for an account of Bp and normability properties of weighted Lorentz spaces).The discrete case N is a particular case of a tree, and can be found in [8]. Weightsfor a general tree were studied, without the monotonicity condition, in [1, 9]. It iseasy to prove that a weight satisfying case 3 of corollary 2.3 must necessarily be inBp(N) (uniformly) on each geodesic (see [8]), but the converse is not true in general.3. Weights in Bp(Rn+)We will now show how to apply our previous result to obtain the weighted inequal-ities for the multidimensional Hardy operator, acting on decreasing functions. Forsimplicity, we will only consider the case n = 2, the general case being an easyextension. We ﬁrst introduce the following notation:S1f(x, y) =1x∫ x0f(s, y) ds, S2f(x, y) =1y∫ y0f(x, t) dt,Sf(x, y) =1xy∫ x0∫ y0f(s, t) dtds = S1(S2f)(x, y) = S2(S1f)(x, y).⎫⎪⎪⎬⎪⎪⎭(3.1)We denote by Dx = {t > 0 : (x, t) ∈ D}, and Dyx = D ∩ ([0, x] × [0, y]). Lpdec(u) isthe usual cone of functions in Lp(u), which are decreasing on each variable. Wethen have the following theorem.Theorem 3.1. If 0 < p < ∞, the following conditions are equivalent:(a) S : Lpdec(u) → Lp(u);(b) there exists a constant C > 0 such that, for every decreasing set D,∫R2+\D|Dyx|p(xy)pu(x, y) dxdy  C∫Du(x, y) dxdy; (3.2)(c) S1, S2 : Lpdec(u) → Lp(u).916 N. Arcozzi and othersProof. The fact that (a) implies (b) follows as usual: taking f = χD, and using thefact thatSf(x, y) =|Dyx|xy,we then see that (3.2) is a consequence of the hypothesis.To show that (b) implies (c), we observe that if (x, y) ∈ D, then[0, x] × Dx ⊂ Dyxand, hence, x|Dx|  |Dyx|. Therefore,∫R2+\D|Dx|pypu(x, y) dxdy ∫Dc|Dyx|p(xy)pu(x, y) dxdy C∫Du(x, y) dxdy,and the result follows from case 4 of corollary 2.3. A similar result holds for S1.(c) implies (a): iterate and observe that Sjf is decreasing if f is decreasing.Remark 3.2. The iteration technique used to prove theorem 3.1 can be extendedvery easily to other settings. For example, we could consider in N2 the operatorS({an,m}n,m) = 1nmn∑j=1m∑k=1aj,k,acting on decreasing two-index sequences, and obtain the characterization of theboundedness of S on the weighted sequence spaces p({un,m}n,m), for generalweights {un,m}n,m, which improves some of the results in [4] proved only for productweights.Condition (3.2) in Rn+ takes the form∫Rn+\D|D ∩ ([0, x1] × · · · × [0, xn])|p(x1 · · ·xn)p u(x1, . . . , xn) dx1 · · ·dxn C∫Du(x1, . . . , xn) dx1 · · ·dxn,which will be denoted by u ∈ Bp(Rn+). Observe that, since |D ∩ ([0, x1] × · · · ×[0, xn])|  x1 · · ·xn, we obtain Bp(Rn+) ⊂ Bq(Rn+), if p < q.We will now prove that, as in the one-dimensional case (see [2] for the originalresult and [12] for a diﬀerent proof, related to the one we will use), Bp(Rn+) satisﬁesthe p − ε condition.Theorem 3.3. If u ∈ Bp(Rn+), 1  p < ∞, then there exists an ε > 0 such thatu ∈ Bp−ε(Rn+).Proof. We will consider the case n = 2 only (n  3 follows similarly). Using theo-rem 3.1, it suﬃces to show that Sj : Lp−εdec (u) → Lp−ε(u), j = 1, 2, and, by symmetry,Hardy’s inequalities for monotone functions 917we may consider only the case j = 2. Take any decreasing set D ⊂ R2+. There thenexists a decreasing function h : R+ → R+ such thatD = {(s, t) ∈ R2+; 0 < t < h(s)}.Therefore, case 4 of corollary 2.3 gives∫ ∞0∫ ∞h(s)hp(s)tpu(s, t) dtds  C∫ ∞0∫ h(s)0u(s, t) dtds.With f = χD, we haveS2f(s, t) =⎧⎪⎨⎪⎩1 if 0 < t  h(s),h(s)tif 0  h(s) < t.By iterating, we can prove that, for every m ∈ N,Sm2 f(s, t) = S2 ◦m· · · ◦S2f(s, t) =⎧⎪⎨⎪⎩1 if 0 < t  h(s),h(s)tm−1∑j=01j!logjth(s)if 0  h(s) < t.Hence, if h(s) < t, we ﬁnd that the inequality(Sm2 f(s, t))p (h(s)t)p 1(m − 1)! logm−1 th(s)follows easily and that∫ ∞0∫ ∞h(s)(h(s)t)p 1(m − 1)! logm−1 th(s)u(s, t) dtds  Cm∫ ∞0∫ h(s)0u(s, t) dtds.Thus, taking σ > max(C, 1/p), and summing over m gives∫ ∞0∫ ∞h(s)(h(s)t)p ∞∑m=11σm−1(m − 1)! logm−1 th(s)u(s, t) dtds=∫ ∞0∫ ∞h(s)(h(s)t)p−1/σu(s, t) dtds∞∑m=1(Cσ)m ∫ ∞0∫ h(s)0u(s, t) dtds= C ′∫ ∞0∫ h(s)0u(s, t) dtds,and the result follows with ε = 1/σ.It was proved in [3] that, for the case of the identity operator (i.e. when consid-ering embeddings), one cannot, in general, replace the condition on all decreasingsets just by taking rectangles of the form [0, a1] × · · · × [0, an], aj > 0. However, in918 N. Arcozzi and othersthe case of product weights, both conditions were equivalent (see [3, theorem 2.5]).We will now show that, in this context,u(x) =n∏j=1uj(xj) ∈ Bp(Rn+)factorizes very nicely as uj ∈ Bp, for all j ∈ {1, . . . , n}.Theorem 3.4. Letu(x) =n∏j=1uj(xj)be a product weight. The following conditions are then equivalent:(a) u ∈ Bp(Rn+);(b) for every aj > 0, j ∈ {1, . . . , n},∫Rn+\([0,a1]×···×[0,an])|([0, a1]×· · ·×[0, an]) ∩ ([0, x1]×· · ·×[0, xn])|p(x1 · · ·xn)p u(x) dx1 · · ·dxn C∫[0,a1]×···×[0,an]u(x) dx1 · · ·dxn;(c) uj ∈ Bp, j ∈ {1, . . . , n}.Proof. As before, and by simplicity, we will work the details only for n = 2.If u ∈ Bp(R2+), then by evaluating (3.2) for rectangles of the form [0, a1]× [0, a2]we get (b).Assuming now that (b) holds, we evaluate the condition and obtain∫R2+\([0,a1]×[0,a2])|([0, a1] × [0, a2]) ∩ ([0, x1] × [0, x2])|p(x1x2)pu1(x1)u2(x2) dx1 dx2=(∫ a10u1(x1) dx1)ap2∫ ∞a2u2(x2)xp2dx2 +(∫ a20u2(x2) dx2)ap1∫ ∞a1u1(x1)xp1dx1+ (a1a2)p(∫ ∞a1u1(x1)xp1dx1)(∫ ∞a2u2(x2)xp2dx2) C(∫ a10u1(x1) dx1)(∫ a20u2(x2) dx2),from which we easily deduce that, for j = 1, 2,apj∫ ∞ajuj(xj)xpjdxj  C∫ aj0uj(xj) dxj ,and, hence, uj ∈ Bp.Finally, by iterating the one-dimensional Hardy operator, and using the fact thatu is a product weight, we deduce thatS : Lpdec(u) → Lp(u),which is equivalent to (a).Hardy’s inequalities for monotone functions 919Remark 3.5. The equivalence between theorem 3.4(c) and the boundedness of theHardy operator S : Lpdec(u1u2) → Lp(u1u2), for the range p  1, was proved in [5],by using an indirect argument related to the characterization of the normabilityproperty of some multidimensional analogues of the weighted Lorentz spaces (inparticular this proof did not make use of the Bp(Rn+) condition). For the case p = 1,one can even prove a quantitative estimate of the constant in the B1 condition,namely, if we set‖u‖B1(R2+) = supD decreasing∫R2+SχD(s, t)u(s, t) dsdt∫Du(s, t) dsdt,then ‖u1(x1)u2(x2)‖B1(R2+) = ‖u1‖B1‖u2‖B1 .As we pointed out in remark 3.2, similar results to theorem 3.4 may be obtained,for product weights, in more general settings (for example, in N2 [4]).AcknowledgmentsN.A. was partly supported by the COFIN project ‘Harmonic Analysis’ of the ItalianMinistry of Education, University and Research. J.L.G.-D. was partly supported byGrant nos MTM2004-02299 and 2005SGR00556 and J.S. by Grant nos MTM2004-02299 and 2005SGR00556.References1 N. Arcozzi, R. Rochberg and E. Sawyer. Carleson measures for analytic Besov spaces. Rev.Mat. Iber. 18 (2002), 443–510.2 M. A. Arin˜o and B. Muckenhoupt. Maximal functions on classical Lorentz spaces andHardy’s inequality with weights for nonincreasing functions. Trans. Am. Math. Soc. 320(1990), 727–735.3 S. Barza, L. E. Persson and J. Soria. Sharp weighted multidimensional integral inequalitiesfor monotone functions. Math. Nachr. 210 (2000), 43–58.4 S. Barza, H. Heinig and L. E. Persson. Duality theorem over the cone of monotone functionsand sequences in higher dimensions. J. Inequal. Applicat. 7 (2002), 79–108.5 S. Barza, A. Kamin´ska, L. E. Persson and J. Soria. Mixed norm and multidimensionalLorentz spaces. Positivity. doi:10.1007/s11117-005-0004-3. (In the press.)6 M. J. Carro and J. Soria. Boundedness of some integral operators. Can. J. Math. 45 (1993),1155–1166.7 M. J. Carro, A. Garc´ıa del Amo and J. Soria. Weak-type weights and normable Lorentzspaces. Proc. Am. Math. Soc. 124 (1996), 849–857.8 M. J. Carro, J. A. Raposo and J. Soria. Recent developments in the theory of Lorentzspaces and weighted inequalities. Mem. Am. Math. Soc. (In the press.)9 W. D. Evans, D. J. Harris and L. Pick. Weighted Hardy and Poincare´ inequalities on trees.J. Lond. Math. Soc. 52 (1995), 121–136.10 J. L. Garcia-Domingo and J. Soria. Lorentz spaces for decreasing rearrangements of func-tions on trees. Math. Inequal. Applicat. 7 (2004), 471–490.11 A. Kufner and L. E. Persson. Weighted inequalities of Hardy type (World Scientiﬁc, 2003).12 C. J. Neugebauer. Weighted norm inequalities for averaging operators of monotone func-tions. Publ. Mat. 35 (1991), 429–447.13 E. Sawyer. Weighted inequalities for the two-dimensional Hardy operator. Studia Math. 82(1985), 1–16.14 E. Sawyer. Boundedness of classical operators on classical Lorentz spaces. 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Hardy’s inequalities for monotone functions on partly ordered measure spaces

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