Journal of Inequalities in Pure and
Applied Mathematics
NORMS OF CERTAIN OPERATORS ON WEIGHTED `p SPACES AND
LORENTZ SEQUENCE SPACES
volume 3, issue 1, article 6,
2002.
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
Received 07 May, 2001;
accepted 21 September, 2001.
Communicated by: B. Opic
Abstract
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School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
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Abstract
The problem addressed is the exact determination of the norms of the classical
Hilbert, Copson and averaging operators on weighted `p spaces and the corresponding Lorentz sequence spaces d(w, p), with the power weighting sequence
wn = n−α or the variant defined by w1 + · · · + wn = n1−α . Exact values are
found in each case except for the averaging operator with wn = n−α , for which
estimates deriving from various different methods are obtained and compared.
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
G.J.O. Jameson and
R. Lashkaripour
2000 Mathematics Subject Classification: 26D15, 15A60, 47B37.
Key words: Hardy’s inequality, Hilbert’s inequality, Norms of operators.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
The Hilbert Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
The Copson Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
The Averaging Operator: Results for General Weights . . . . . .
6
The Averaging Operator: Results for Specific Weights . . . . . .
References
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3
6
14
18
22
31
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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 sequence wn = 1/nα or the
closely related sequence (equally natural in the context of Lorentz spaces) given
by Wn = n1−α , where Wn = w1 +· · ·+wn . In the present paper, we address the
problem of finding the norms of these operators in the case p > 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 take wn = 1, and the results of [13] when
we take p = 1. The methods used for the case p = 1 no longer apply. The norms
of these operators on d(w, p) are determined by their action on decreasing, nonnegative 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 weight wn = nα , although such weights do not
generate a Lorentz sequence space. This case cannot always be treated together
with 1/nα , because of the reversal of some inequalities at α = 0.
Our two special choices of w are alternative analogues of the weighting function 1/xα in 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
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
G.J.O. Jameson and
R. Lashkaripour
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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
indices p, 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 consideration. Any particular theorem can be effective in one context and ineffective in
another.
For the Hilbert operator H, the “Schur" method can be adapted to show that
the value from the continuous case is reproduced: for either choice of w, we
have
π
kHkp,w = ∆p,w (H) =
.
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 wn = 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 general w), kCkp,w ≤ p supn≥1 (Wn /nwn ), and hence
that kCkp,w = ∆p,w (C) = p/(1 − α) for both our decreasing weights (reproducing the value in the continuous case).
For the averaging operator A, a similar method gives the value p/(p − 1 − α)
(reproducing the continuous case) for the increasing weight nα (where α < p −
1). For Wn = n1−α , classical methods can be adapted to show that ∆p,w (A) =
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
G.J.O. Jameson and
R. Lashkaripour
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p/(p − 1 + α), suggesting that this weight is the “right" analogue of 1/xα in this
context (though we do not know whether kAkp,w has the same value). However, for wn = 1/nα , 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 is pζ(p + α), and we show that this can be replaced
by the scale of estimates [rζ(r + α)]r/p for 1 ≤ r ≤ p. The case r = 1 occurs
as a point on another scale of estimates derived by the Schur method. A precise
solution would have to reproduce the known values p∗ when α = 0 and ζ(1+α)
when p = 1: it seems unlikely that it can be expressed by a single reasonably
simple formula in terms of p and α.
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
G.J.O. Jameson and
R. Lashkaripour
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2.
Preliminaries
Let w = (wn ) be a sequence of positive numbers. We write Wn = w1 +· · ·+wn
(and similarly for sequences denoted by (xn ), (yn ), etc.). Let p ≥ 1. By `p (w)
we mean the space of sequences x = (xn ) with
Sp =
∞
X
wn |xn |p
n=1
1/p
convergent, with norm kxkp,w = Sp . When wn = 1 for all n, we denote the
norm by k.kp .
P
Now suppose that (wn ) is decreasing, limn→∞ wn = 0 and ∞
n=1 wn is divergent. The Lorentz sequence space d(w, p) is then defined as follows. Given a
null sequence x = (xn ), let (x∗n ) be the decreasing rearrangement of |xn |. Then
d(w, p) is the space of null sequences x for which x∗ is in `p (w), with norm
kxkd(w,p) = kx∗ kp,w .
We denote by en the sequence having 1 in place n and 0 elsewhere.
P
Let A be the operator defined by Ax = y, where yi = ∞
j=1 ai,j xj . We write
kAkp for the norm of A as an operator on `p , and kAkp,w,v for its norm as an
operator from `p (w) to `p (v) (or just kAkp,w when v = w). This norm equates
to the norm of another operator on `p itself: by substitution, one has kAkp,w,v =
−1/p
1/p
kBkp , where B is the operator with matrix bi,j = vi ai,j wj .
We assume throughout that ai,j ≥ 0 for all i, j, which implies in each case
that the norm is determined by the action of A on non-negative sequences. Next,
we establish conditions, adequate for the operators considered below, ensuring that kAkd(w,p) is determined by decreasing, non-negative sequences (more
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
G.J.O. Jameson and
R. Lashkaripour
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general conditions are given in [13, Theorem 2]). Denote by δp (w) the set of
decreasing, non-negative sequences in `p (w), and define
∆p,w (A) = sup{kAxkp,w : x ∈ δp (w) : kxkp,w = 1}.
Lemma 2.1. Suppose that (wn ) is decreasing,
that ai,j ≥ 0 for all i,j, and A
Pm
maps δp (w) into `p (w). Write cm,j = i=1 ai,j . Suppose further that:
(i)
limj→∞ ai,j = 0 for each i;
and either
(ii)
ai,j decreases with j for each i,
or
(iii)
ai,j decreases with i for each j and cm,j decreases with j
for each m.
Then kA(x∗ )kd(w,p) ≥ kA(x)kd(w,p) for non-negative elements x of d(w, p).
Hence kAkd(w,p) = ∆p,w (A).
Proof. Let y = Ax and z = Ax∗ . As before, write Xj = x1 + · · · + xj , etc.
First, assume condition (ii). By Abel summation and (ii), we have
yi =
∞
X
ai,j xj =
j=1
∞
X
(ai,j − ai,j+1 )Xj ,
Ym =
i=1 j=1
ai,j xj =
∞
X
j=1
cm,j xj =
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j=1
and similarly for zi with Xj∗ replacing Xj . Since Xj ≤ Xj∗ for all j, we have
yi ≤ zi for all i, which implies that kykd(w,p) ≤ kzkd(w,p) .
Now assume (iii). Then yi and zi decrease with i, and
m X
∞
X
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
∞
X
j=1
(cm,j − cm,j+1 )Xj
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and similarly for Zm . Hence Ym ≤PZm for all P
m. By the majorization principle
m
p
p
(e.g. [3, 1.30]), this implies that i=1 yi ≤ m
i=1 zi for all m, and hence by
Abel summation that kykd(w,p) ≤ kzkd(w,p) .
The evaluations in [13] are based on the property, special for p = 1, that
kAk1,w is determined by the elements en , and ∆1,w (A) by the elements e1 +
· · · + en . These statements fail when p > 1 (with or without weights). For kAkp
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 if xn = e1 + · · · + en , then
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
kAxn kp
= (p∗ )1/p .
sup
n≥1 kxn kp
G.J.O. Jameson and
R. Lashkaripour
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.
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Lemma 2.2. Let p > 1 and p∗ = p/(p − 1). Let B be the operator with matrix
(bi,j ), where bi,j ≥ 0 for all i, j. Suppose that (si ), (tj ) are two sequences of
strictly positive numbers such that for some C1 , C2 :
1/p
si
∞
X
−1/p
bi,j tj
≤ C1
j=1
for all i,
1/p∗
tj
∞
X
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−1/p∗
bi,j si
≤ C2
for all j.
Page 8 of 38
i=1
1/p∗
Then kBkp ≤ C1
1/p
C2 .
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Proof. Let yi =
P∞
j=1 bi,j xj .
yi =
∞
X
By Hölder’s inequality,
1/p∗ −1/pp∗
(bi,j tj
1/p 1/pp∗
)(bi,j tj
xj )
j=1
∞
X
≤
!1/p∗
−1/p
bi,j tj
j=1
≤
∞
X
1/p∗
bi,j tj
!1/p
xpj
j=1
−1/p
C1 si
1/p∗
∞
X
1/p∗ p
xj
Norms of certain operators on
weighted `p spaces and Lorentz
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!1/p
bi,j tj
,
G.J.O. Jameson and
R. Lashkaripour
j=1
so
∞
X
i=1
yip
≤
p/p∗
C1
∞
X
1/p∗
xpj tj
j=1
∞
X
−1/p∗
bi,j si
≤
p/p∗
C1 C2
i=1
∞
X
xpj .
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j=1
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This result has usually been applied with sj = tj = j. As we will see, useful
consequences can be derived from other choices of sj , tj .
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, operators S, T are defined by
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(Sx)i = ai
∞
X
j=i
bj x j ,
(T x)i = ai
i
X
bj x j ,
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where ai , bj ≥ 0 for all i, j. Note that S is upper-triangular, T lower-triangular.
For p > 1, we define
αm =
m
X
i=1
api ,
α̃m =
∞
X
api ,
i=m
βm =
m
X
∗
bpj ,
β̃m =
j=1
∞
X
∗
bpj .
j=m
Proposition 2.3. (“head” version).
P
∗
p∗
(i) If m
≤ K1p αm for all m (in particular, if bj αj ≤ K1 ap−1
for
j
j=1 (bj αj )
all j),then kSkp ≤ pK1 .
Pm
p
p∗ −1
p
(ii) If
for
i=1 (ai βi ) ≤ K1 βm for all m (in particular, if aj βj ≤ K1 bj
all j), then kT kp ≤ p∗ K1 .
Proposition 2.4. (“tail” version).
P∞
p
p∗ −1
p
(i) If
for
i=m (ai β̃i ) ≤ K2 β̃m for all m, (in particular, if aj β̃j ≤ K2 bj
all j), then kSkp ≤ p∗ K2 .
P∞
∗
p∗
(ii) If
≤ K2p α̃m for all m, (in particular, if bj α̃j ≤ K2 ap−1
j
j=m (bj α̃j )
for all j) then kT kp ≤ pK2 .
1/p 1/p∗
(ii) If
1/p 1/p∗
α̃m βm
∗
≤ K3 for all m, then kSkp ≤ p1/p (p∗ )1/p K3 .
≤ K3 for all m, then kT kp ≤ p
G.J.O. Jameson and
R. Lashkaripour
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Proposition 2.5. (“mixed” version).
(i) If αm β̃m
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1/p
∗ 1/p∗
(p )
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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 Proposition 2.3. (i) Note first the following fact, easily proved
P by Abel
summation: if sj , : tj (1 ≤ j ≤ N ) are real numbers such that m
sj ≤
Pm
PN j=1
j=1 tj for 1 ≤ m ≤ N , and u1 ≥ . . . ≥ uN ≥ 0, then
j=1 sj uj ≤
PN
j=1 tj uj .
It is enough to consider
PN finite sums with i, j ≤ N , and to take xi ≥ 0. Let
y = Sx. Write Ri = j=i bj xj , so that yi = ai Ri . By Abel summation,
N
X
yip =
i=1
N
X
i=1
api Rip =
N
−1
X
i=1
p
Rip − Ri+1
≤ pbi xi Rip−1
i=1
yip ≤ p
i=1
αi bi xi Rip−1 ≤ p
N
X
i=1
!1/p
xpi
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p−1
p
, since RN = bN xN . Hence
≤ pbN xN RN
for such i. Also, RN
N
X
G.J.O. Jameson and
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p
p
αi (Rip − Ri+1
) + αN RN
.
By the mean-value theorem, y p − xp ≤ p(y − x)y p−1 for any x, y ≥ 0. Since
Ri − Ri+1 = bi xi for 1 ≤ i ≤ N − 1, we have
N
X
Norms of certain operators on
weighted `p spaces and Lorentz
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N
X
i=1
!1/p∗
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∗
(bi αi )p Rip
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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
p∗
(bi αi )
Rip
≤
∗
K1p
N
X
api Rip
=
∗
K1p
i=1
N
X
yip .
i=1
Together, these inequalities give kykp ≤ pK1 kxkp .
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 of w, defined respectively by wn = 1/nα (for α ≥ 0) and by Wn = n1−α (for 0 < α < 1). Note that
in the second case,
Z n
1−α
1−α
1−α
wn = n
− (n − 1)
=
dt,
tα
n−1
and hence
1−α
1−α
≤ wn ≤
.
α
n
(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) ≤ 1 and r(α) → γ (Euler’s constant) as α → 0. Also, for the evaluation
of various suprema that arise, we will need the following lemmas.
P
Lemma 2.6. [9, Proposition 3]. Let An = nj=1 j α . Then An /n1+α decreases
with n if α ≥ 0, and increases if α < 0. If α > −1, it tends to 1/(1 + α) as
n → ∞.
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
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R. Lashkaripour
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Lemma 2.7. [8,
4.10] (without proof), [13, Proposition 6]. Let α > 0
PRemark
∞
1+α
and let Cn =
. Then nα Cn decreases with n, and (n − 1)α Cn
k=n 1/k
increases.
Lemma 2.8. [7, Lemma 8], more simply in [9, Proposition 1]. The expression
n
X
1
kα
α
n(n + 1) k=1
increases with n when α ≥ 1 or α ≤ 0, and decreases with n if 0 ≤ α ≤ 1.
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
G.J.O. Jameson and
R. Lashkaripour
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3.
The Hilbert Operator
We consider the Hilbert operator H, with matrix ai,j = 1/(i + j). This satisfies
conditions (i) and (ii) of Lemma 2.1. Hilbert’s classical inequality states that
kHkp = π/ sin(π/p) for p > 1. For the case p = 1, with either of our choices
of w, it was shown in [13] that kHk1,w = ∆1,w (H) = π/ sin απ.
For the analogous operator in the continuous case, with w(x) = 1/xα , it is
quite easily shown by the Schur method that kHkp,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 for wn = 1/nα , but rather
more delicate for Wn = n1−α .
Let 0 < a < 1. As with most studies of the Hilbert operator, we use the
well-known integral
Z ∞
1
π
dt = a
.
a
t (t + c)
c sin aπ
0
Write
Z
n
gn (a) =
n−1
1
dt.
ta
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
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a
Note that gn (a) ≥ 1/n .
Close
Lemma 3.1. With this notation, we have for each j ≥ 1,
∞
X
i=1
∞
X gi (a)
1
π
≤
≤ a
.
a
i (i + j)
i+j
j sin aπ
i=1
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Proof. Clearly,
gi (a)
1
=
i+j
i+j
Z
i
i−1
1
dt ≤
ta
Z
i
i−1
1
dt.
ta (t + j)
The statement follows, by the integral quoted above.
Now let 0 < α < 1. Write vn = gn (α) and vn0 = (1 − α)vn . In our previous
terminology, v 0 is our “second choice of w". Clearly, an operator will have
the same norm on d(v 0 , p) and on d(v, p). Note that by Hölder’s inequality for
integrals, vnr ≤ gn (αr) for r > 1.
Theorem 3.2. Let H be the operator with matrix ai,j = 1/(i+j), and let p > 1.
Let wn = n−α , where 1 − p < α < 1. Then
kHkp,w = ∆p,w (H) =
If 0 < α < 1 and vn =
value stated.
Rn
n−1
π
.
sin[(1 − α)π/p]
t−α dt, then kHkp,v and ∆p,v (H) also have the
Proof. Write M = π/ sin[(1 − α)π/p]. Let wn = n−α , where 1 − p < α < 1.
Now kHkp,w = kBkp , where B has matrix bi,j = (j/i)α/p /(i + j). In Lemma
2.2, take si = ti = i, and let C1 , C2 be defined as before. Then
Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
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1/p −1/p
bi,j si tj
(1−α)/p
1
i
=
.
i+j j
By Lemma 3.1, it follows that C1 ≤ 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 all x. Note that kHkp,w,v = kBkp , where now
bi,j =
−1/α
Take si = vi
1
(j α vi )1/p .
i+j
and tj = j. Then
bi,j (si /tj )1/p =
By Lemma 3.1,
1
(j α vi )(α−1)/αp .
i+j
∞
X
j (α−1)/p
j=1
i+j
1/α
≤ i(α−1)/p M.
(1−α)/αp
Since i−1 ≤ vi , we have i(α−1)/p ≤ vi
Also,
1/p∗
bi,j (tj /si )
, and hence C1 ≤ M .
1
=
(j α vi )t ,
i+j
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where
t=
1
1
+ ∗.
p αp
Note that t > 1, so as remarked above, we have vit ≤ gi (αt). By Lemma 3.1
again,
∞
X
gi (αt)
M0
≤ αt ,
i+j
j
i=1
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where M 0 = π/ sin αtπ. Now
αt =
α
1
1−α
+ ∗ =1−
,
p p
p
so that M 0 = M , and hence C2 ≤ M . The statement follows.
To show that ∆p,w (H) ≥ M , take r = (1 − α)/p, so that α + rp = 1. Fix
N , and let
1/j r for j ≤ N,
xj =
0
for j > n.
P
PN 1
p
Then (xj ) is decreasing and ∞
j=1 wj xj =
j=1 j . Let y = Hx. By routine
methods (we omit the details), one finds that
N
X
i=1
wi yip
≥M
p
N
X
1
i=1
i
− g(r),
where g(r) is independent of N . Clearly, the required statement follows. Minor
modifications give the same conclusion for v.
Note. A variant H0 of the discrete Hilbert operator has matrix 1/(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 for kH0 k1,w : see [13].
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4.
The Copson Operator
The “Copson” operator C is defined by y = Cx, where yi =
given by the transpose of the Cesaro matrix:
1/j for i ≤ j
ai,j =
.
0
for i > j
P∞
j=i (xj /j).
It is
This matrix satisfies conditions (i) and (iii) of Lemma 2.1. Copson’s inequality
[11, Theorem 331] states that kCkp ≤ p (Copson’s original result [10] was in
fact the reverse inequality for the case 0 < p < 1).
R ∞The corresponding operator in theα continuous case is defined by (Cf )(x) =
[f (y)/y] dy. When w(x) = 1/x , it is quite easily shown, for example by
x
the Schur method, that ∆p,w (C) = kCkp,w = p/(1 − α).
For the discrete case, we will show that this value is correct for either decreasing choice of w. However, the Schur method does not lead to the right constant in the discrete version of Copson’s inequality, and instead we use Proposition 2.3. First we formulate a result for general weights. The 1-regularity
constant of a sequence (wn ) is defined to be r1 (w) = supn≥1 Wn /(nwn ).
Norms of certain operators on
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Theorem 4.1. Suppose that (wn ) is 1-regular and p ≥ 1. Then the Copson
operator C maps d(w, p) into itself , and kCkp,w ≤ pr1 (w).
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Proof. We have kCkp,w = kSkp , where S is as in Proposition 2.3, with ai =
1/p
1/p
wi and bj = 1/(jwj ). In the notation of Proposition 2.3, αm = Wm , so
Quit
bj αj =
Wj
1/p
jwj
=
Wj 1/p∗
1/p∗
p/p∗
wj ≤ r1 (w)wj = r1 (w)aj .
jwj
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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 kCkp,w,v ≤ pK1 , where K is such
1/p 1/p∗
that Vn ≤ K1 nwn vn
for all n.
An example in [15, Section 2.3] shows that kCkp,w is not necessarily equal
to pr1 (w). However, equality does hold for our special choices of decreasing w,
as we now show.
Theorem 4.2. Let C be the Copson operator. Suppose that p ≥ 1 and that w is
defined either by wn = 1/nα or by Wn = n1−α , where 0 ≤ α < 1. Then
∆p,w (C) = kCkp,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, consider wn = 1/nα . By comparison with the integrals of 1/tα
on [1, n] and [0, n], we have
1
n1−α
1−α
(n
− 1) ≤ Wn ≤
,
1−α
1−α
from which the required statement follows easily. Now consider the case Wn =
n1−α . Then nwn /Wn = nα wn . By the inequalities for wn mentioned in Section
2,
nα
1 − α ≤ nα wn ≤ (1 − 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. Choose ε > 0 and define r by α +
rp = 1 + ε. Let xn = 1/nr for all n; note that (xn ) is decreasing. Then
n−α xpn = 1/nα+rp = 1/n1+ε , so x is in `p (w) for either choice of (wn ) (note
that for the second choice, wn ≤ c/nα for a constant c). Also, again by integral
estimation,
∞
X
1
1
xn
yn =
≥ r =
1+r
k
rn
r
k=n
for all n, so
p
1
kxkp,w .
kykp,w ≥ kxkp,w =
r
1−α+ε
The statement follows.
Theorem 4.1 has a very simple consequence for increasing weights:
Proposition 4.3. Let (wn ) be any increasing weight sequence. Then kCkp,w ≤
p.
Proof. If (wn ) is increasing, then Wn ≤ nwn , with equality when n = 1. Hence
r1 (w) = 1.
Clearly, ∆p,w (C) ≥ 1, since C(e1 ) = e1 . For the case wn = nα , the method
of Theorem 4.2 shows that also ∆p,w (C) ≥ p/(1+α). A simple example shows
that ∆p,w (C) can be greater than both 1 and p/(1 + α).
Example 4.1. Let p = 2 and α = 1, so that p/(1 + α) = 1. Take x =
(4, 2, 0, 0, . . .). Then y = (5, 1, 0, 0, . . .), and kxk22,w = 24, while kyk22,w = 27.
Norms of certain operators on
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We leave further investigation of this case to another study; it is analogous
to the problem of the averaging operator with wn = 1/nα , 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
yn = n1 (x1 + · · · + xn ). It is given by the Cesaro matrix
1/i for j ≤ i
ai,j =
,
0
for j > 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 that kAkp = p∗ for p > 1. It is easy to deduce
that the same upper estimate applies for any decreasing w:
Proposition 5.1. If (wn ) is any decreasing, non-negative sequence, then
kAkp,w ≤ p∗ .
1/p
Proof. Let x be a non-negative element of `p (w), and let uj = wj xj . Then
kukp = kxkp,w , and since (wj ) is decreasing,
wn1/p Xn
≤
n
X
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1/p
wj xj
= Un
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(where, as usual, Xn means x1 + · · · + xn ). Hence
kAxkpp,w =
∞
X
n=1
wn p
X ≤
np n
∞
X
n=1
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U
np n
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≤ (p∗ )p
∞
X
upn
by Hardy’s inequality
n=1
= (p∗ )p kxkpp,w .
The next result records the estimates for the averaging operator derived from
Propositions 2.3, 2.4 and 2.5. Instead of the fully general statements, we give
simpler forms pertinent to our objectives. For 1 ≤ r ≤ p, define
Um (r) =
∞
X
wj
j=m
j
,
r
mr−1 Um (r)
.
wm
m≥1
V (r) = sup
Proposition 5.2. Let A be the averaging operator and let p > 1. Then:
Pm
1−p∗
1−p∗
(i) kAkp,w ≤ p∗ K1 , where
≤ K1 mwm
for all m;
j=1 wj
(ii) kAkp,w ≤ pV (p);
∗
(iii) if (wn ) is decreasing, then kAkp,w ≤ p1/p (p∗ )1/p V (p)1/p .
Proof. Recall that kAkp,w = kT kp , where T is as in Proposition 2.3, with ai =
P
1/p
−1/p
wi /i and bj = wj . With notation as before, we have α̃m = j≥m wj /j p =
P
−p∗ /p
Um (p) and βm = m
. Noting that p∗ /p = p∗ − 1, one checks easily
j=1 wj
that Propositions 2.3(ii) and 2.4(ii) (with the simpler, alternative hypotheses)
translate into (i) and (ii).
−p∗ /p
For (iii), note that if (wn ) is decreasing, then βm ≤ mwm . Hence
p−1
α̃m βm
≤ mp−1 Um (p)/wm , so the condition of Proposition 2.5(iii) is satisfied
with K3 = V (p)1/p .
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For decreasing (wn ), (i) will give an estimate no less than p∗ , hence no advance on Proposition 5.1 (one need only take m = 1 to see that K1 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 defining V (p) is attained
at m = 1. In such a case, nothing is lost by the inequality used in (iii) for βm .
Furthermore, nothing would be gained by using the more general [1, Theorem
4.1] instead of Proposition 2.5.
Another result relating to our V (p) is [12, Corollary of Theorem 3.4] (repeated, with an extra condition removed, in [2, Corollary 4.3]). The quantity
considered is C = supn≥1 np Un (p)/Wn . Again, in our case, C coincides with
V (p). It is shown in [12] that if C 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.
P
r
Theorem 5.3. Suppose that 1 ≤ r ≤ p and ∞
n=1 wn /n is convergent. Define
V (r) as above. Then kAkp,w ≤ [rV (r)]r/p .
Proof. Write Un (r) = Un and V (r) = V . Let zn =
Zn = z1 + · · · + zn . By Hölder’s inequality,
Xn ≤ n
1−r/p
p/r
xn
and (as usual)
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Znr/p ,
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hence Xnp ≤ np−r Znr . If y = Ax, then yn = Xn /n, so ynp ≤ n−r Znr . For any N ,
we have
N
X
wn
n=1
n
Zr =
r n
=
N
X
n=1
N
X
(Un − Un+1 )Znr
r
Un (Znr − Zn−1
) − UN +1 ZNr
n=1
N
X
≤r
Un zn Znr−1
by the mean-value theorem
Norms of certain operators on
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G.J.O. Jameson and
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n=1
N
X
wn
zn Znr−1 by the definition of V
r−1
n
n=1
!1/r N
!1/r∗
N
X
X wn
≤ rV
wn znr
Znr
by Hölder’s inequality.
r
n
n=1
n=1
≤ rV
Since znr = xpn , it follows that (for all N )
N
X
n=1
wn ynp ≤
N
X
wn
n=1
nr
Znr ≤ (rV )r
N
X
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wn xpn .
n=1
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The case r = 1 (for which the proof, of course, becomes simpler), gives
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kAkp,w ≤ V (1)1/p , in which
∞
1 X wj
V (1) = sup
.
j
m≥1 wm
j=m
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), not kAkp,w .
G.J.O. Jameson and
R. Lashkaripour
Theorem 5.4. Let p ≥ 1, and let
c = sup
n≥1
nwn+1
.
Wn
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Assume that c < p. Then
p
.
∆p,w (A) ≤
p−c
Proof. Let x = (xn ) be a decreasing, non-negative element of `p (w), and let
y = Ax. Fix a positive integer N . By Abel summation,
N
X
n=1
wn ynp
Norms of certain operators on
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=
N
X
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p
Wn−1 (yn−1
−
ynp )
+
p
WN yN
.
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n=2
p
By the mean-value theorem, yn−1
− ynp ≥ pynp−1 (yn−1 − yn ). Also, for n ≥ 2,
xn = nyn − (n − 1)yn−1 = yn − (n − 1)(yn−1 − yn ),
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so
yn−1 − yn =
1
(yn − xn ).
n−1
Hence
N
X
wn ynp
≥ p
n=1
= p
N
X
n=2
N
X
n=2
Wn−1 ynp−1 (yn−1 − yn )
Wn−1 p−1
y (yn − xn ).
n−1 n
Now since (xn ) is decreasing, yn ≥ xn for all n. Also, y1 = x1 . Since wn ≤
cWn−1 /(n − 1), we deduce that
N
X
N
N
pX
pX
wn ynp−1 (yn − xn ) =
wn ynp−1 (yn − xn ),
wn ynp ≥
c
c
n=2
n=1
n=1
so that
(p − c)
N
X
n=1
wn ynp
≤p
N
X
wn ynp−1 xn .
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n=1
A standard application of Hölder’s inequality to the right-hand side finishes the
proof.
We now consider lower estimates. First, an obvious one:
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Proposition 5.5. We have
∆p,w (A) ≥
∞
1 X wn
w1 n=1 np
!1/p
.
Proof. Take x = e1 . Then yn = 1/n for all n, so the statement follows. (In
particular, the stated series must converge in order for A(e1 ) to be in `p (w).)
Secondly, we formulate the lower estimate derived by the classical method
of considering xn = n−s for a suitable s.
P∞
Lemma 5.6. Suppose that an , bn are non-negative numbers such that
n=1 an
is divergent and limn→∞ bn = 0. Then
PN
an b n
→ 0 as N → ∞.
Pn=1
N
n=1 an
Proof. Elementary.
Theorem 5.7. Let A be the averaging operator
P and let p q> 1. Let (wn ) be any
positive sequence, and let q < p be such that ∞
n=1 wn /n is divergent. Then
∆w,p (A) ≥
p
.
p−q
Proof. Write p/(p − q) = r, so that 1/r = 1 − q/p. Fix N , and let
−q/p
n
for 1 ≤ n ≤ N
xn =
.
0
for n > N
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Then
∞
X
(5.1)
wn xpn
n=1
Also, for n ≤ N ,
Z
=
N
X
wn
n=1
nq
.
n
t−q/p dt = r(n1/r − 1),
Xn ≥
1
so that
r
Xn
≥ q/p (1 − n−1/r ).
n
n
p
Since (1 − t) ≥ 1 − pt for 0 < t < 1, we have
yn =
ynp
rp
≥ q (1 − pn−1/r ),
n
(5.2)
≥
wn
rp q
n
− pr
wn
p
nq+1/r
Contents
(2).
Take ε > 0. By (5.1) and (5.2), together with Lemma 5.6 (with an = wn /nq
and bn = n−1/r ), it is clear that for all large enough N ,
N
X
n=1
wn ynp ≥ (1 − ε)rp
∞
X
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and hence
wn ynp
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P∞
Corollary 5.8. If
n=1 wn /n is divergent (in particular, if (wn ) is increasing),
∗
then ∆p,w (A) ≥ p .
Remarks on the relation between kAkp,w and ∆p,w (A). If (wn ) is increasing,
then kAkp,w = ∆p,w (A), for if x∗ = (x∗n ) is the decreasing rearrangement of
x, then it is easily seen that kx∗ kp,w ≤ kxkp,w , while kAx∗ kp,w ≥ kAxkp,w .
The following example shows that this is not true for decreasing weights in
general (though it may possibly be true for 1/nα ), and indeed that the estimate
in Theorem 5.4 does not apply to kAkp,w .
Example 5.1.
P Let Ek = {i : k! ≤ i < (k + 1)!}, andPlet wi = 1/(k!k) for i ∈
Ek . Then i∈Ek wi = 1 and, by integral estimation,P i∈Ek (1/i) > log(k +1).
−p
Now fix k, and choose p close enough to 1 to have
> log k. Write
i∈Ek i
p
k k for k kp,w . Take n = k!. Then ken k = wn = 1/(k!k), while
Norms of certain operators on
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X wi
log k
kAen kp >
>
.
p
i
k!k
i∈E
Contents
k
kAkpp,w
2
3
Hence
> log k. We show that nwn+1 /Wn ≤
for all n, so that
∆p,w (A) ≤ 3, by Theorem 5.4. If k ≥ 3 and n ∈ Ek , then Wn ≥ k − 1, so
(k + 1)! 1
k+1
2
nwn+1
≤
=
≤ .
Wn
k!k k − 1
k(k − 1)
3
The required inequality is easily checked for n in E1 and E2 .
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6.
The Averaging Operator: Results for
Specific Weights
We now explore the extent to which the results of Section 5 solve the problem
for our chosen weighting sequences.
The analogous operator in the continuous
R
1 x
case is given by (Af )(x) = x 0 f . When w(x) = x−α , one can show by the
Schur method (or as in [17, Theorem 1.9.16]) that
kAkp,w = ∆p,w (A) = p/(p − 1 + α).
In the discrete case, when p = 1, it was shown in [13] that:
(i) if wn = 1/nα , then kAk1,w = ∆1,w (A) = ζ(1 + α),
(ii) if Wn = n1−α , then ∆1,w (A) = 1/α.
As suggested by (ii) and Hardy’s inequality, the continuous case is reproduced when Wn = n1−α :
Theorem 6.1. Let A be the averaging operator and let p ≥ 1. Let (wn ) be
defined by Wn = n1−α , where 0 ≤ α < 1. Then
p
∆p,w (A) =
.
p−1+α
Proof. Recall that
wn+1
Hence
1−α
≤ wn .
≤
nα
nwn+1
= nα wn+1 ≤ 1 − α
Wn
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for n ≥ 1, so Theorem 5.4 applies with c = 1 − α. Also, Theorem 5.7 applies
with q = 1 − α. The stated equality follows.
We do not know whether kAkp,w has the same value; however, ∆p,w (A) is of
more interest for this choice of w, since it is motivated by Lorentz spaces.
For the increasing weight wn = nα , our problem is solved by the method of
Proposition 2.3 (which, it will be recalled, was effective for the Copson operator
with decreasing weights).
α
Theorem 6.2. Let wn = n , where 0 ≤ α < p − 1. Then
p
kAkp,w = ∆p,w (A) =
.
p−1−α
Proof. By Theorem 5.7, we have ∆p,w (A) ≥ p/(p − 1 − α). We use Proposition
5.2(i) to prove the reverse inequality. The condition is
m
X
j=1
1
j α(p∗ −1)
≤ K1
m
mα(p∗ −1)
.
∗
Note that α(p − 1) = α/(p − 1) < 1. By Lemma 2.6, the condition is satisfied
with
p−1
1
K1 =
=
.
1 − α(p∗ − 1)
p−1−α
So kAkp,w ≤ p∗ K1 = p/(p − 1 − α).
Note. For α < 1, this result also follows from Theorem 5.4 and Lemma 2.8.
We now come to the hard case, wn = 1/nα . The trivial lower estimate 5.5
is enough to show that ∆p,w (A) can be greater than p/(p − 1 + α). Indeed, we
have at once from Proposition 5.5 and Theorem 5.7:
Norms of certain operators on
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G.J.O. Jameson and
R. Lashkaripour
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Proposition 6.3. Let p > 1 and wn = 1/nα , where α ≥ 0. Then ∆p,w (A) ≥
max(m1 , m2 ), where
m1 =
p
,
p−1+α
m2 = ζ(p + α)1/p .
Either lower estimate can be larger (even when α < 1), as the following
table shows:
p
α
m1
m2
1.1 0.9 1.1 1.572
2 0.1 1.818 1.249
No improvement on m2 is obtained by considering elements of the form en
or e1 + · · · + en . However, the next example shows that ∆p,w (A) can be greater
than both m1 and m2 .
Norms of certain operators on
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sequence spaces
G.J.O. Jameson and
R. Lashkaripour
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Example 6.1. Let p = 2, α = 1. Then m1 = 1 and m2 = ζ(3) ≈ 1.096. Let
x = (2, 1, 0, 0, . . .). Then y1 = 2 and yn = 3/n for n ≥ 2. Hence kxk22,w = 4 21 ,
while kyk22,w = 4 + 9[ζ(3) − 1] ≈ 5.818, so that kyk2,w /kxk2,w ≈ 1.137.
(One could formulate another general lower estimate using this choice of x,
but it is too unpleasant to be worth stating explicitly.)
We now record the upper estimates derived from the various results above.
Of course, by Proposition 5.1, we have kAkp,w ≤ p∗ .
Proposition 6.4. Let p > 1, and let wn = 1/nα , where α ≥ 0. Then
p
.
∆p,w (A) ≤ M1 =:
p − 2−α
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Proof. We have
n
Wn
(n + 1)α X 1
=
.
nwn+1
n
kα
k=1
By Lemma 2.8, this expression increases with n, so has its least value 2α when
n = 1. Hence Theorem 5.4 applies with c = 2−α .
Proposition 6.5. Let p > 1 and 1 ≤ r ≤ p, and let wn = 1/nα , where α ≥ 0.
Then
kAkp,w ≤ M2 (r) =: [rζ(r + α)]r/p .
Proof. Apply Theorem 5.3. In the notation used there, we have
∞
X
1
mr−1 Um (r)
r+α−1
=m
.
r+α
wm
j
j=m
Norms of certain operators on
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By Lemma 2.7, this expression decreases with m, so is greatest when m = 1,
with the value ζ(r + α).
Note that in particular, M2 (1) = ζ(1 + α)1/p and M2 (p) = pζ(p + α). By
∗
the remark after Proposition 5.2, min[M2 (p), p∗ ] ≤ p1/p (p∗ )1/p m2 ≤ 2m2 .
To optimize the estimate in Proposition 6.5, we have to choose r (in the
interval [1, p]) to minimize [rζ(r+α)]r . Computations show that when α ≥ 0.4,
the least value occurs when r = 1, while for α = 0.1, it occurs when r ≈ 1.35.
The Schur method provides another scale of estimates, as follows.
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Proposition 6.6. Let wn = 1/nα , where α > 0. Let α ≤ r < p + α.
Then kAkp,w ≤ M3 (r), where
1/p∗ 1/p
p
r
α
M3 (r) =
ζ 1+ ∗ +
.
p−r+α
p
p
Proof. In Lemma 2.2, take sj = tj = j r . We have bi,j = ai bj for j ≤ i, where
ai = i−α/p−1 and bj = j α/p . So C1 is the supremum (as i varies) of
i(r−α)/p−1
i
X
j=1
1
j (r−α)/p
Norms of certain operators on
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.
G.J.O. Jameson and
R. Lashkaripour
By Lemma 2.6 (or trivially when r = α),
C1 =
1
p
=
.
1 − (r − α)/p
p−r+α
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Also, C2 is the supremum (as j varies) of
j
α/p+r/p∗
∞
X
i=j
1
i1+α/p+r/p∗
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∗
By Lemma 2.7, this is greatest when j = 1, with value ζ(1 + r/p + α/p).
Note that M3 (α) = M2 (1) = ζ(1 + α)1/p . Hence M3 will always be at least
as good as M2 when α > 0.4.
The following table compares the estimates in some particular cases. We
denote by m the greater of the lower estimates m1 , m2 . Recall that M1 estimates
∆p,w (A). The values of r used for M2 (r) and M3 (r) are indicated.
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p
1.1
1.1
1.2
2
α
m
p∗
0.1 5.5 11
0.9 1.572 11
0.3 2.4
6
0.5 1.333 2
M1
6.587
1.950
3.095
1.547
M2 (r)
6.151 (1.1)
1.663 (1)
3.084 (1.1)
1.616 (1)
M3 (r)
6.031 (0.98)
1.663 (0.9)
2.886 (0.9)
1.593 (0.75)
Table 6.1: Numerical values of upper and lower estimates
In most, if not all cases, M3 is a better estimate than M2 , at the cost of being
more complicated. Both M2 and M3 reproduce the correct value ζ(1 + α) when
p = 1. On the other hand, both can be larger than p∗ in other cases. Some
product of the type M3 (1) would reproduce the values ζ(1 + α) when p = 1
and p∗ when α = 0; the exponent would need to be of the form f (α, p), where
f (0, p) = 1 for p > 1 and f (α, 1) = 0 for α > 0.
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G.J.O. Jameson and
R. Lashkaripour
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References
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Norms of certain operators on
weighted `p spaces and Lorentz
sequence spaces
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R. Lashkaripour
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[10] E.T. COPSON, Notes on series of positive terms, J. London Math. Soc., 2
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AND
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