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volume 3, issue 1, article 6, 2002.

Received 07 May, 2001;

accepted 21 September, 2001.

Communicated by:B. Opic

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Journal of Inequalities in Pure and Applied Mathematics

NORMS OF CERTAIN OPERATORS ON WEIGHTED`_p SPACES AND LORENTZ SEQUENCE SPACES

G.J.O. JAMESON AND R. LASHKARIPOUR

Department of Mathematics and Statistics, Lancaster University,

Lancaster LA1 4YF, Great Britain.

EMail:g.jameson@lancaster.ac.uk

URL:http://www.maths.lancs.ac.uk/dept/people/gjameson.html Faculty of Science,

University of Sistan and Baluchistan, Zahedan, Iran.

EMail:lashkari@hamoon.usb.ac.ir

2000c School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

039-01

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Norms of certain operators on weighted`pspaces and Lorentz

sequence spaces

G.J.O. Jamesonand R. Lashkaripour

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J. Ineq. Pure and Appl. Math. 3(1) Art. 6, 2002

http://jipam.vu.edu.au

Abstract

The problem addressed is the exact determination of the norms of the classical Hilbert, Copson and averaging operators on weighted`_pspaces and the corresponding Lorentz sequence spacesd(w, p), with the power weighting sequence w_n =n^−αor the variant defined byw₁+· · ·+w_n = n^1−α. Exact values are found in each case except for the averaging operator withw_n=n^−α, for which estimates deriving from various different methods are obtained and compared.

2000 Mathematics Subject Classification:26D15, 15A60, 47B37.

Key words: Hardy’s inequality, Hilbert’s inequality, Norms of operators.

1. Introduction

In [13], the first author determined the norms and so-called “lower bounds"

of the Hilbert, Copson and averaging operators on `1(w) and on the Lorentz sequence space d(w,1), with the power weighting sequencew_n = 1/n^α or the closely related sequence (equally natural in the context of Lorentz spaces) given byW_n=n^1−α, whereW_n=w₁+· · ·+w_n. In the present paper, we address the problem of finding the norms of these operators in the casep > 1. The problem of lower bounds was considered in a companion paper [14].

The classical inequalities of Hilbert, Copson and Hardy describe the norms of these operators on `_p (where p > 1). Solutions to our problem need to reproduce these inequalities when we takewn= 1, and the results of [13] when we takep= 1. The methods used for the casep= 1no longer apply. The norms of these operators ond(w, p)are determined by their action on decreasing, non- negative sequences in `p(w): we denote this quantity by∆p,w. In most cases, it turns out to coincide with the norm on`_p(w)itself. In the context of`_p(w), we also consider the increasing weightw_n=n^α, although such weights do not generate a Lorentz sequence space. This case cannot always be treated together with1/n^α, because of the reversal of some inequalities atα= 0.

Our two special choices ofware alternative analogues of the weighting func- tion1/x^αin the continuous case. The solutions of the continuous analogues of our problems are well known and quite simple to establish. Best-constant estimations are notoriously harder for the discrete case, essentially because discrete sums may be greater or less than their approximating integrals.

There is an extensive literature on boundedness of various classes of operators on`_p spaces, with or without weights. Less attention has been given to the

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exact evaluation of norms. Our study aims to do this for the most “natural" operators and weights: as we shall see, the problem is already quite hard enough for these specific cases without attempting anything more general. Indeed, we fail to reach an exact solution in one important case. Problems involving two indicesp, q, or two weights, lead rapidly to intractable supremum evaluations.

Though we do formulate some estimates applying to general weights, our main objective is not to present new results of a general nature. Rather, given the wealth of known results and methods, the task is to identify the ones that lead to a solution, or at least a sharp estimate, for the problems under considera- tion. Any particular theorem can be effective in one context and ineffective in another.

For the Hilbert operatorH, the “Schur" method can be adapted to show that the value from the continuous case is reproduced: for either choice of w, we have

kHk_p,w = ∆_p,w(H) = π

sin[(1−α)π/p].

The Copson operator C and the averaging operator A are triangular instead of symmetric, and other methods are needed to deliver the right constant even when w_n = 1. A better starting point is Bennett’s systematic set of theorems on “factorable" triangular matrices [4, 5, 6]. For C, one such theorem can be applied to show that (for generalw),kCk_p,w ≤psup_n≥1(W_n/nw_n), and hence that kCk_p,w = ∆_p,w(C) = p/(1−α) for both our decreasing weights (reproducing the value in the continuous case).

For the averaging operatorA, a similar method gives the valuep/(p−1−α) (reproducing the continuous case) for the increasing weightn^α(whereα < p− 1). ForWn = n^1−α, classical methods can be adapted to show that∆p,w(A) =

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p/(p−1 +α), suggesting that this weight is the “right" analogue of1/x^αin this context (though we do not know whether kAkp,w has the same value). How- ever, for w_n = 1/n^α, the problem is much more difficult. A simple example shows that the above value is not correct. We can only identify and compare the estimates deriving from the various theorems and methods available; different estimates are sharper in different cases. The best estimate provided by the factorable-matrix theorems ispζ(p+α), and we show that this can be replaced by the scale of estimates [rζ(r+α)]^r/p for1≤r≤p. The caser = 1occurs as a point on another scale of estimates derived by the Schur method. A precise solution would have to reproduce the known valuesp^∗whenα= 0andζ(1 +α) when p = 1: it seems unlikely that it can be expressed by a single reasonably simple formula in terms ofpandα.

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2. Preliminaries

Letw= (w_n)be a sequence of positive numbers. We writeW_n =w₁+· · ·+w_n (and similarly for sequences denoted by(x_n),(y_n), etc.). Letp ≥1. By`_p(w) we mean the space of sequencesx= (x_n)with

S_p =

∞

X

n=1

w_n|x_n|^p

convergent, with norm kxk_p,w = Sp^1/p. Whenw_n = 1for alln, we denote the norm byk.k_p.

Now suppose that(w_n)is decreasing,limn→∞w_n = 0 andP∞

n=1w_n is di- vergent. The Lorentz sequence spaced(w, p)is then defined as follows. Given a null sequencex= (x_n), let(x^∗_n)be the decreasing rearrangement of|x_n|. Then d(w, p) is the space of null sequences x for which x^∗ is in `_p(w), with norm kxk_d(w,p) =kx^∗kp,w.

We denote bye_nthe sequence having 1 in placenand 0 elsewhere.

LetAbe the operator defined byAx=y, wherey_i =P∞

j=1a_i,jx_j. We write kAk_p for the norm of Aas an operator on `_p, and kAk_p,w,v for its norm as an operator from`_p(w)to`_p(v)(or justkAk_p,w whenv =w). This norm equates to the norm of another operator on`_pitself: by substitution, one has kAk_p,w,v = kBk_p, whereB is the operator with matrixb_i,j =v_i^1/pa_i,jw^−1/p_j .

We assume throughout thatai,j ≥ 0for all i, j, which implies in each case that the norm is determined by the action ofAon non-negative sequences. Next, we establish conditions, adequate for the operators considered below, ensur- ing that kAkd(w,p) is determined by decreasing, non-negative sequences (more

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general conditions are given in [13, Theorem 2]). Denote by δ_p(w)the set of decreasing, non-negative sequences in`p(w), and define

∆_p,w(A) = sup{kAxk_p,w :x∈δ_p(w) :kxk_p,w = 1}.

Lemma 2.1. Suppose that (w_n)is decreasing, that a_i,j ≥ 0for all i,j, andA mapsδp(w)into`p(w). Writecm,j =Pm

i=1ai,j. Suppose further that:

(i) limj→∞a_i,j = 0for each i;

and either (ii) a_i,jdecreases with j for each i,

or (iii) a_i,jdecreases with i for each j andc_m,j decreases with j for eachm.

Then kA(x^∗)k_d(w,p) ≥ kA(x)k_d(w,p) for non-negative elements x of d(w, p).

HencekAk_d(w,p)= ∆_p,w(A).

Proof. Let y = Ax andz = Ax^∗. As before, writeX_j = x₁ +· · ·+x_j, etc.

First, assume condition (ii). By Abel summation and (ii), we have y_i =

∞

X

j=1

a_i,jx_j =

∞

X

j=1

(a_i,j −a_i,j+1)X_j,

and similarly for z_i withX_j^∗ replacingX_j. SinceX_j ≤ X_j^∗ for allj, we have y_i ≤z_i for alli, which implies thatkyk_d(w,p) ≤ kzk_d(w,p).

Now assume (iii). Theny_iandz_idecrease withi, and Y_m =

m

X

i=1

∞

X

j=1

a_i,jx_j =

∞

X

j=1

c_m,jx_j =

∞

X

j=1

(c_m,j−c_m,j+1)X_j

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and similarly forZ_m. HenceY_m ≤Z_mfor allm. By the majorization principle (e.g. [3, 1.30]), this implies thatPm

i=1y_i^p ≤ Pm

i=1z_i^p for all m, and hence by Abel summation thatkyk_d(w,p) ≤ kzk_d(w,p).

The evaluations in [13] are based on the property, special for p = 1, that kAk_1,w is determined by the elementse_n, and ∆_1,w(A)by the elements e₁ +

· · ·+en. These statements fail whenp >1(with or without weights). ForkAkp

this is very well known. For ∆_p, let A be the averaging operator on`_p. The lower-bound estimation in Hardy’s inequality shows that ∆_p(A) = p^∗, while integral estimation shows that ifxn =e1 +· · ·+en, then

sup

n≥1

kAxnkp

kx_nk_p = (p^∗)^1/p.

The “Schur" method. By this (taking a slight historical liberty) we mean the following technique. It can be used to give a straightforward solution of the continuous analogues of all the problems considered here (cf. [11, Sections 9.2 and 9.3]). We state a slightly generalized form of the method for the discrete case.

Lemma 2.2. Letp >1andp^∗ =p/(p−1). LetB be the operator with matrix (bi,j), where bi,j ≥ 0for all i, j. Suppose that(si), (tj)are two sequences of strictly positive numbers such that for someC₁, C₂:

s^1/p_i

∞

X

j=1

b_i,jt^−1/p_j ≤C₁ for all i, t^1/p_j ^∗

∞

X

i=1

b_i,js^−1/p_i ^∗ ≤C₂ for all j.

Then kBkp ≤C₁^1/p^∗C₂^1/p.

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Proof. Let y_i =P∞

j=1b_i,jx_j. By Hölder’s inequality, y_i =

∞

X

j=1

(b^1/p_i,j ^∗t^−1/pp_j ^∗)(b^1/p_i,j t^1/pp_j ^∗x_j)

≤

∞

X

j=1

b_i,jt^−1/p_j

!1/p^∗ ∞

X

j=1

b_i,jt^1/p_j ^∗x^p_j

!1/p

≤

C₁s^−1/p_i 1/p^∗ ∞

X

j=1

b_i,jt^1/p_j ^∗x^p_j

!1/p

,

so ∞

X

i=1

y_i^p ≤C₁^p/p^∗

∞

X

j=1

x^p_jt^1/p_j ^∗

∞

X

i=1

b_i,js^−1/p_i ^∗ ≤C₁^p/p^∗C₂

∞

X

j=1

x^p_j.

This result has usually been applied withs_j =t_j =j. As we will see, useful consequences can be derived from other choices ofs_j, t_j.

Theorems on “factorable" triangular matrices. These theorems appeared in [4, 5,6], formulated for the more general case of operators from`p to`q. They can be summarized as follows. In each case, operatorsS,T are defined by

(Sx)i =ai

∞

X

j=i

bjxj, (T x)i =ai i

X

j=1

bjxj,

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wherea_i, b_j ≥0for alli, j. Note thatSis upper-triangular,T lower-triangular.

Forp >1, we define α_m =

m

X

i=1

a^p_i, α˜_m =

∞

X

i=m

a^p_i, β_m =

m

X

j=1

b^p_j^∗, β˜_m =

∞

X

j=m

b^p_j^∗.

Proposition 2.3. (“head” version).

(i) IfPm

j=1(bjαj)^p^∗ ≤K₁^p^∗αmfor allm (in particular, if bjαj ≤K1a^p−1_j for allj),thenkSk_p ≤pK₁.

(ii) If Pm

i=1(aiβi)^p ≤ K₁^pβm for all m (in particular, if ajβj ≤ K1b^p_j^∗⁻¹ for allj), thenkTk_p ≤p^∗K₁.

Proposition 2.4. (“tail” version).

(i) If P∞

i=m(a_iβ˜_i)^p ≤K₂^pβ˜_m for allm, (in particular, ifa_jβ˜_j ≤ K₂b^p_j^∗⁻¹ for allj), then kSk_p ≤p^∗K₂.

(ii) If P∞

j=m(b_jα˜_j)^p^∗ ≤ K₂^p^∗α˜_m for allm, (in particular, if b_jα˜_j ≤ K₂a^p−1_j for allj) then kTk_p ≤pK₂.

Proposition 2.5. (“mixed” version).

(i) Ifα^1/pm β˜m^1/p^∗ ≤K₃ for allm, then kSk_p ≤p^1/p(p^∗)^1/p^∗K₃. (ii) Ifα˜^1/pm βm^1/p^∗ ≤K3 for allm, then kTkp ≤p^1/p(p^∗)^1/p^∗K3.

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In each case, (ii) is equivalent to (i), by duality. Propositions 2.3 and 2.4 are [4, Theorem 2] and (with suitable substitutions) [5, Theorem 2⁰] (note: the right-hand exponent in Bennett’s formula (18) should be (r−1)/(r−s)). For the reader’s convenience, we include here a direct proof of Proposition 2.3(i), simplified from the more general theorem in [5]; the proof of 2.4(ii) is almost the same.

Proof of Proposition2.3. (i) Note first the following fact, easily proved by Abel summation: if sj,: tj (1 ≤ j ≤ N) are real numbers such that Pm

j=1sj ≤ Pm

j=1t_j for 1 ≤ m ≤ N, and u₁ ≥ . . . ≥ u_N ≥ 0, then PN

j=1s_ju_j ≤ PN

j=1t_ju_j.

It is enough to consider finite sums withi, j ≤ N, and to takex_i ≥ 0. Let y =Sx. WriteR_i =PN

j=ib_jx_j, so thaty_i =a_iR_i. By Abel summation,

N

X

i=1

y^p_i =

N

X

i=1

a^p_iR^p_i =

N−1

X

i=1

αi(R^p_i −R^p_i+1) +αNR^p_N.

By the mean-value theorem, y^p −x^p ≤ p(y−x)y^p−1 for anyx, y ≥ 0. Since R_i−R_i+1 =b_ix_ifor1≤i≤N −1, we have

R^p_i −R^p_i+1 ≤pbixiR^p−1_i

for suchi. Also,R_N^p ≤pb_Nx_NR^p−1_N , sinceR_N =b_Nx_N. Hence

N

X

i=1

y_i^p ≤p

N

X

i=1

α_ib_ix_iR^p−1_i ≤p

N

X

i=1

x^p_i

!^1/p _N X

i=1

(b_iα_i)^p^∗R^p_i

!^1/p^∗

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(note that p^∗(p−1) = p). By the “fact” noted above and our hypothesis (or directly from the alternative hypothesis),

N

X

i=1

(b_iα_i)^p^∗R^p_i ≤K₁^p^∗

N

X

i=1

a^p_iR^p_i =K₁^p^∗

N

X

i=1

y_i^p.

Together, these inequalities give kyk_p ≤pK₁kxk_p.

Proposition 2.5 is [6, Theorem 9]; with standard substitutions, it is also a special case of [1, Theorems 4.1 and 4.2]. The corresponding result for the continuous case was given in [16].

We shall be particularly concerned with two choices ofw, defined respec- tively byw_n = 1/n^α (forα≥0) and byW_n=n^1−α(for0< α <1). Note that in the second case,

w_n =n^1−α−(n−1)^1−α = Z n

n−1

1−α t^α dt, and hence

1−α

n^α ≤w_n≤ 1−α (n−1)^α.

Several of our estimations will be expressed in terms of the zeta function. It will be helpful to recall that ζ(1 +α) = 1/α+r(α)for α > 0, where ¹₂ ≤ r(α) ≤ 1andr(α) → γ (Euler’s constant) asα → 0. Also, for the evaluation of various suprema that arise, we will need the following lemmas.

Lemma 2.6. [9, Proposition 3]. LetA_n =Pn

j=1j^α. ThenA_n/n^1+αdecreases with n ifα ≥ 0, and increases ifα < 0. Ifα > −1, it tends to 1/(1 +α)as n → ∞.

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Lemma 2.7. [8, Remark 4.10] (without proof), [13, Proposition 6]. Letα > 0 and let Cn = P∞

k=n1/k^1+α. Then n^αCn decreases with n, and (n −1)^αCn

increases.

Lemma 2.8. [7, Lemma 8], more simply in [9, Proposition 1]. The expression

1 n(n+ 1)^α

n

X

k=1

k^α

increases withnwhenα≥1orα≤0, and decreases withnif0≤α≤1.

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3. The Hilbert Operator

We consider the Hilbert operatorH, with matrixa_i,j = 1/(i+j). This satisfies conditions (i) and (ii) of Lemma 2.1. Hilbert’s classical inequality states that kHk_p = π/sin(π/p)forp > 1. For the casep= 1, with either of our choices ofw, it was shown in [13] thatkHk_1,w= ∆_1,w(H) =π/sinαπ.

For the analogous operator in the continuous case, withw(x) = 1/x^α, it is quite easily shown by the Schur method that kHk_p,w = π/sin[(1− α)π/p].

We show that the method adapts to the discrete case, giving this value again for either of our choices of w. This is straightforward forw_n = 1/n^α, but rather more delicate forW_n=n^1−α.

Let 0 < a < 1. As with most studies of the Hilbert operator, we use the well-known integral

Z ∞

0

1

t^a(t+c) dt= π c^asinaπ. Write

g_n(a) = Z n

n−1

1 t^a dt.

Note thatg_n(a)≥1/n^a.

Lemma 3.1. With this notation, we have for eachj ≥1,

∞

X

i=1

1 i^a(i+j) ≤

∞

X

i=1

g_i(a)

i+j ≤ π j^asinaπ.

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Proof. Clearly, g_i(a) i+j = 1

i+j Z i

i−1

1 t^a dt ≤

Z i

i−1

1

t^a(t+j) dt.

The statement follows, by the integral quoted above.

Now let0< α <1. Writev_n =g_n(α)andv⁰_n= (1−α)v_n. In our previous terminology, v⁰ is our “second choice of w". Clearly, an operator will have the same norm ond(v⁰, p)and ond(v, p). Note that by Hölder’s inequality for integrals,v_n^r ≤g_n(αr)forr >1.

Theorem 3.2. LetHbe the operator with matrixa_i,j = 1/(i+j), and letp >1.

Letw_n =n^−α, where 1−p < α < 1. Then

kHkp,w = ∆p,w(H) = π

sin[(1−α)π/p]. If 0 < α < 1and v_n = Rn

n−1t^−αdt, then kHk_p,v and∆_p,v(H)also have the value stated.

Proof. WriteM = π/sin[(1−α)π/p]. Letwn = n^−α, where1−p < α < 1.

Now kHk_p,w =kBk_p, whereB has matrix b_i,j = (j/i)^α/p/(i+j). In Lemma 2.2, takes_i =t_i =i, and letC₁, C₂be defined as before. Then

b_i,js^1/p_i t^−1/p_j = 1 i+j

i j

(1−α)/p

. By Lemma3.1, it follows thatC1 ≤M. Similarly,C2 ≤M.

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Now let 0 < α < 1, and let v, w be as stated. We show in fact that kHkp,w,v ≤ M. It then follows that kHkp,v ≤ M, since kxkp,w ≤ kxkp,v

for allx. Note that kHk_p,w,v =kBk_p, where now b_i,j = 1

i+j(j^αv_i)^1/p.

Takes_i =v^−1/α_i andt_j =j. Then b_i,j(s_i/t_j)^1/p = 1

i+j(j^αv_i)^(α−1)/αp. By Lemma3.1,

∞

X

j=1

j^(α−1)/p

i+j ≤i^(α−1)/pM.

Sincei⁻¹ ≤v_i^1/α, we have i^(α−1)/p ≤v_i^(1−α)/αp, and hence C₁ ≤M. Also,

b_i,j(t_j/s_i)^1/p^∗ = 1

i+j(j^αv_i)^t, where

t= 1 p+ 1

αp^∗.

Note that t > 1, so as remarked above, we havev^t_i ≤ g_i(αt). By Lemma 3.1 again,

∞

X

i=1

g_i(αt) i+j ≤ M⁰

j^αt,

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where M⁰ =π/sinαtπ. Now αt= α

p + 1

p^∗ = 1−1−α p ,

so thatM⁰ =M, and henceC₂ ≤M. The statement follows.

To show that ∆_p,w(H)≥ M, taker = (1−α)/p, so thatα+rp = 1. Fix N, and let

x_j =

1/j^r forj ≤N, 0 forj > n.

Then (x_j)is decreasing and P∞

j=1w_jx^p_j = PN j=1

1

j. Let y = Hx. By routine methods (we omit the details), one finds that

N

X

i=1

w_iy^p_i ≥M^p

N

X

i=1

1

i −g(r),

whereg(r)is independent ofN. Clearly, the required statement follows. Minor modifications give the same conclusion forv.

Note. A variantH₀ of the discrete Hilbert operator has matrix1/(i+j−1).

This is decidedly more difficult! Already in the case p = 1, the norms do not coincide for our two weights, and it seems unlikely that there is a simple formula forkH₀k_1,w: see [13].

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4. The Copson Operator

The “Copson” operatorC is defined byy=Cx, where y_i =P∞

j=i(x_j/j). It is given by the transpose of the Cesaro matrix:

a_i,j =

1/j fori≤j 0 fori > j .

This matrix satisfies conditions (i) and (iii) of Lemma2.1. Copson’s inequality [11, Theorem 331] states thatkCk_p ≤ p (Copson’s original result [10] was in fact the reverse inequality for the case0< p <1).

The corresponding operator in the continuous case is defined by(Cf)(x) = R∞

x [f(y)/y] dy. Whenw(x) = 1/x^α, it is quite easily shown, for example by the Schur method, that∆_p,w(C) =kCk_p,w =p/(1−α).

For the discrete case, we will show that this value is correct for either decreasing choice ofw. However, the Schur method does not lead to the right constant in the discrete version of Copson’s inequality, and instead we use Propo- sition 2.3. First we formulate a result for general weights. The 1-regularity constant of a sequence(wn)is defined to be r1(w) = sup_n≥1Wn/(nwn).

Theorem 4.1. Suppose that (w_n) is 1-regular and p ≥ 1. Then the Copson operatorC mapsd(w, p)into itself , and kCk_p,w ≤pr₁(w).

Proof. We have kCk_p,w = kSk_p, where S is as in Proposition 2.3, with a_i = w^1/p_i andbj = 1/(jw^1/p_j ). In the notation of Proposition2.3,αm =Wm, so

b_jα_j = W_j

jw_j^1/p = W_j

jw_jw^1/p_j ^∗ ≤r₁(w)w^1/p_j ^∗ =r₁(w)a^p/p_j ^∗.

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Since p/p^∗ = p− 1, the simpler hypothesis of Proposition 2.3(i) holds with K1 =r1(w).

Note. The same reasoning shows that kCk_p,w,v ≤ pK₁, where K is such that V_n≤K₁nw^1/pn v^1/pn ^∗ for alln.

An example in [15, Section 2.3] shows that kCk_p,w is not necessarily equal topr₁(w). However, equality does hold for our special choices of decreasingw, as we now show.

Theorem 4.2. LetCbe the Copson operator. Suppose thatp≥1and thatwis defined either byw_n= 1/n^αor byW_n=n^1−α, where0≤α <1. Then

∆_p,w(C) = kCk_p,w = p 1−α.

Proof. We show first that in either case, r1(w) = 1/(1−α), so that kCkp,w ≤ p/(1−α). First, considerw_n= 1/n^α. By comparison with the integrals of1/t^α on[1, n]and[0, n], we have

1

1−α(n^1−α−1)≤W_n≤ n^1−α 1−α,

from which the required statement follows easily. Now consider the caseWn = n^1−α. Thennw_n/W_n=n^αw_n. By the inequalities forw_nmentioned in Section 2,

1−α≤n^αw_n≤(1−α) n^α (n−1)^α, from which it is clear that again r₁(w) = 1/(1−α).

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We now show that∆_p,w(C) ≥ 1−α. Chooseε > 0and define r by α+ rp = 1 + ε. Let xn = 1/n^r for all n; note that (xn) is decreasing. Then n^−αx^p_n = 1/n^α+rp = 1/n^1+ε, so x is in `_p(w) for either choice of(w_n) (note that for the second choice,w_n ≤c/n^α for a constantc). Also, again by integral estimation,

y_n =

∞

X

k=n

1

k^1+r ≥ 1

rn^r = x_n r for alln, so

kyk_p,w ≥ 1

rkxk_p,w = p

1−α+εkxk_p,w. The statement follows.

Theorem4.1has a very simple consequence for increasing weights:

Proposition 4.3. Let(w_n)be any increasing weight sequence. Then kCk_p,w ≤ p.

Proof. If(w_n)is increasing, thenW_n ≤nw_n, with equality whenn= 1. Hence r₁(w) = 1.

Clearly,∆p,w(C)≥1, sinceC(e1) =e1. For the casewn =n^α, the method of Theorem4.2shows that also∆_p,w(C)≥p/(1 +α). A simple example shows that∆_p,w(C)can be greater than both 1 andp/(1 +α).

Example 4.1. Let p = 2 and α = 1, so that p/(1 + α) = 1. Take x = (4,2,0,0, . . .). Theny= (5,1,0,0, . . .), andkxk²_2,w = 24, whilekyk²_2,w= 27.

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We leave further investigation of this case to another study; it is analogous to the problem of the averaging operator withwn = 1/n^α, considered in more detail below.

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5. The Averaging Operator: Results for General Weights

Henceforth, A will mean the averaging operator, defined by y = Ax, where y_n= _n¹(x₁+· · ·+x_n). It is given by the Cesaro matrix

a_i,j =

1/i forj ≤i 0 forj > i ,

which satisfies conditions (i) and (ii) of Lemma 2.1. In this section, we give some results for general weighting sequences. We consider upper estimates first. Hardy’s inequality states thatkAk_p = p^∗ forp > 1. It is easy to deduce that the same upper estimate applies for any decreasingw:

Proposition 5.1. If (w_n) is any decreasing, non-negative sequence, then kAk_p,w ≤p^∗.

Proof. Let x be a non-negative element of`_p(w), and letu_j = w^1/p_j x_j. Then kuk_p =kxk_p,w, and since(w_j)is decreasing,

w_n^1/pX_n ≤

n

X

j=1

w_j^1/px_j =U_n

(where, as usual,X_nmeansx₁+· · ·+x_n). Hence kAxk^p_p,w =

∞

X

n=1

w_n n^pX_n^p ≤

∞

X

n=1

1 n^pU_n^p

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≤ (p^∗)^p

∞

X

n=1

u^p_n by Hardy’s inequality

= (p^∗)^pkxk^p_p,w.

The next result records the estimates for the averaging operator derived from Propositions 2.3, 2.4 and2.5. Instead of the fully general statements, we give simpler forms pertinent to our objectives. For1≤r≤p, define

U_m(r) =

∞

X

j=m

w_j

j^r, V(r) = sup

m≥1

m^r−1U_m(r) w_m .

Proposition 5.2. LetAbe the averaging operator and letp >1. Then:

(i) kAk_p,w ≤p^∗K₁, where Pm

j=1w^1−p_j ^∗ ≤K₁mw^1−p_m ^∗ for allm;

(ii) kAk_p,w ≤pV(p);

(iii) if(w_n)is decreasing, thenkAk_p,w ≤p^1/p(p^∗)^1/p^∗V(p)^1/p.

Proof. Recall thatkAkp,w =kTkp, whereT is as in Proposition2.3, withai = w^1/p_i /iandb_j =w_j^−1/p. With notation as before, we have α˜_m =P

j≥mw_j/j^p = U_m(p) andβ_m =Pm

j=1w^−p_j ^∗^/p. Noting thatp^∗/p=p^∗ −1, one checks easily that Propositions 2.3(ii) and 2.4(ii) (with the simpler, alternative hypotheses) translate into (i) and (ii).

For (iii), note that if (w_n) is decreasing, then β_m ≤ mwm^−p^∗^/p. Hence

˜

α_mβ_m^p−1 ≤ m^p−1U_m(p)/w_m, so the condition of Proposition2.5(iii) is satisfied withK3 =V(p)^1/p.

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For decreasing(w_n), (i) will give an estimate no less thanp^∗, hence no ad- vance on Proposition 5.1(one need only take m = 1 to see thatK1 is at least 1). Also, (iii) adds nothing to (ii), since

[pV(p)]^1/p(p^∗)^1/p^∗ ≥min[pV(p), p^∗].

In the specific case we consider below, the supremum definingV(p)is attained atm = 1. In such a case, nothing is lost by the inequality used in (iii) forβ_m. Furthermore, nothing would be gained by using the more general [1, Theorem 4.1] instead of Proposition2.5.

Another result relating to our V(p) is [12, Corollary of Theorem 3.4] (re- peated, with an extra condition removed, in [2, Corollary 4.3]). The quantity considered is C = sup_n≥1n^pU_n(p)/W_n. Again, in our case,C coincides with V(p). It is shown in [12] that ifC is finite, then so is∆p,w(A), without giving an explicit relationship.

The next theorem improves on the estimate in (ii) by exhibiting it as one point in a scale of estimates. We are not aware that it is a case of any known result.

Theorem 5.3. Suppose that1≤r ≤pandP∞

n=1w_n/n^ris convergent. Define V(r)as above. ThenkAk_p,w ≤[rV(r)]^r/p.

Proof. Write U_n(r) = U_n and V(r) = V. Let z_n = x^p/rn and (as usual) Z_n =z₁+· · ·+z_n. By Hölder’s inequality,

X_n≤n^1−r/pZ_n^r/p,

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henceX_n^p ≤n^p−rZ_n^r. Ify=Ax, theny_n =X_n/n, soy_n^p ≤n^−rZ_n^r. For anyN, we have

N

X

n=1

w_n n^rZ_n^r =

N

X

n=1

(Un−Un+1)Z_n^r

=

N

X

n=1

U_n(Z_n^r−Z_n−1^r )−U_N+1Z_N^r

≤r

N

X

n=1

U_nz_nZ_n^r−1 by the mean-value theorem

≤rV

N

X

n=1

w_n

n^r−1z_nZ_n^r−1 by the definition ofV

≤rV

N

X

n=1

w_nz_n^r

!^1/r _N X

n=1

w_n n^rZ_n^r

!^1/r^∗

by Hölder’s inequality.

Since z_n^r =x^p_n, it follows that (for allN)

N

X

n=1

w_ny_n^p ≤

N

X

n=1

w_n

n^rZ_n^r ≤(rV)^r

N

X

n=1

w_nx^p_n.

The case r = 1 (for which the proof, of course, becomes simpler), gives

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kAk_p,w ≤V(1)^1/p, in which

V(1) = sup

m≥1

1 w_m

∞

X

j=m

wj

j .

Among the above results (including the Schur method) only Proposition 5.2(i) delivers the right constant p^∗ in Hardy’s original inequality. Another method that does so is the classical one of [11, Theorem 326]. The next result shows what is obtained by adapting this method to the weighted case. Note that it applies to∆_p,w(A), notkAk_p,w.

Theorem 5.4. Letp≥1, and let

c= sup

n≥1

nwn+1

W_n . Assume thatc < p. Then

∆_p,w(A)≤ p p−c.

Proof. Let x = (x_n) be a decreasing, non-negative element of `_p(w), and let y =Ax. Fix a positive integerN. By Abel summation,

N

X

n=1

w_ny_n^p =

N

X

n=2

Wn−1(y^p_n−1−y_n^p) +W_Ny_N^p.

By the mean-value theorem, y^p_n−1−y^p_n≥py^p−1_n (yn−1−yn). Also, forn≥2, xn=nyn−(n−1)yn−1 =yn−(n−1)(yn−1−yn),