WEIGHTED DISCRETE HARDY'S INEQUALITIES

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WEIGHTED DISCRETE HARDY’S INEQUALITIES

Pascal Lefevre

To cite this version:

Pascal Lefevre. WEIGHTED DISCRETE HARDY’S INEQUALITIES. 2020. �hal-02528265�

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WEIGHTED DISCRETE HARDY’S INEQUALITIES

PASCAL LEF `EVRE

Abstract. The purpose of this note is to give a short proof of a weighted version of the Hardy’s inequality in the sequence case. This includes the known case of classical polynomial weights, with optimal constant. The proof relies on the ideas of the short direct proof given recently in [6].

In the sequel, we work withp >1 andp⁰= p

p−1 denotes its conjugate exponent.

The notationN0stands for the set of non negative integral numbers: 0,1,2, . . . Fory∈R⁺, we write [y] = max{k∈N⁰|k≤y}its integral part.

As usual, given a sequence (wn)_n≥0of positive numbers,`^p(w) is the space of sequences of complex numbersa= a_n

n≥0 such that X

n≥0

|a_n|^pw_n <∞, equipped with the normkak_p =^+∞X

n=0

|a_n|^pw_n_p¹ . Whenw_n= 1 for everyn∈N0, we simply write `^p.

Given a sequencea= a_k

k≥0 of complex numbers, we associate the sequence A_n = 1

n+ 1

n

X

k=0

a_k

We recall the famousdiscrete Hardy’s inequality.

Letp >1. For every a∈`^p, the sequenceA= An

n≥0belongs to`^p and kAkp≤p⁰kakp i.e.

(HI) X^+∞

n=0

1 n+ 1

n

X

k=0

ak

p¹_p

≤p⁰^+∞X

n=0

|an|^p_p¹ .

This inequality is equivalent to the boundedness of the Ces`aro operator defined by Γ(a) = A_n

n∈N, withkΓk ≤p⁰, viewed as an operator on`^p. Actually the constantp⁰ is optimal andkΓk=p⁰. See [2]

for the original result, [5] for a very interesting historical survey on the subject, [4] for a very recent nice extension, and [6] for a short proof.

The aim of this note is to give a short proof of the boundedness of Γ as an operator on weighted

`^p(w) spaces, under an homogeneity type assumption, following the same ideas than [6]. In particular, this includes the case of classical weights n^αp

n≥0 (with exact norm), so that we recover easily some results of [1] and [3].

Main Theorem 1. Let w= (w_n)_n≥0 andw⁰ = (w⁰_n)_n≥0 be sequences of non-negative real numbers and we assume that(w_n)_n≥0 is non-decreasing.

We assume that there exists a measurable positive functionf on(0,1) such that

• we have a sub-homogeneity property: for every n∈ m/s,(m+ 1)/s

∩N, w_n−1⁰ ≤f(s)wm, where s∈(0,1) andm∈N0.

• K= Z 1

0

f(s) s

¹_p

ds <∞.

ThenΓ is bounded from`^p(w) to`^p(w⁰)withkΓk ≤K:

(WHI) ^+∞X

n=0

w⁰_n

1 n+ 1

n

X

k=0

xk

p¹_p

≤K^+∞X

n=0

|xn|^pwn

¹_p .

1Universit´e d’Artois, UR 2462, Laboratoire de Math´ematiques de Lens (LML), F-62300 Lens, France pascal.lefevre@univ-artois.fr

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2 PASCAL LEF `EVRE

In particular, withf(s) =s^−αp, we immediately get the classical weights version:

Theorem 2. Let α∈[0,1/p⁰)andwn= (n+ 1)^αp forn∈N0. ThenΓ is bounded on`^p(w) withkΓk=1

p⁰ −α−1

, equivalently, for every a∈`^p:

(CWHI) ^+∞X

n=0

1 (n+ 1)^1−α

n

X

k=0

ak

(k+ 1)^α

p¹_p

≤1

p⁰ −α−1X^+∞

n=0

|an|^p¹_p .

It is well known and easy to check that this bound is sharp (see [6] for the caseα= 0). For instance, just testa_n= (n+ 1)⁻⁽^p¹^+ε) whenε→0⁺.

Before giving the proof of the main theorem, let us mention a general remark about monotone rearrangements of sequences.

We recall that we can define the monotone rearrangement of a vanishing sequence of non negative numbers b_k

k≥0 as the following non-increasing sequence:

∀N ∈N⁰, b^∗_N = inf

|E|=N sup

n /∈E

bn .

Remark. The following consequence of the Abel transform principle is well known:

Let ck

k≥0 a non-increasing sequence of non negative numbers. Let uk

k≥0 and u⁰_k

k≥0 be two sequence such that for everyn≥0,

n

X

k=0

u⁰_k ≥

n

X

k=0

u_k.

Then

N

X

n=0

cnu⁰_n≥

N

X

n=0

cnun for every N≥0.

Indeed write U_n⁰ =

n

X

k=0

u⁰_k andUn =

n

X

k=0

uk and defines for convenience U₋₁⁰ =U−1= 0. A simple Abel transform gives, for everyN ≥0,

N

X

n=0

cnu⁰_n=

N

X

n=0

cn U_n⁰ −U_n−1⁰

=cN+1U_N⁰ +

N

X

n=0

(cn−cn+1)U_n⁰ ≥cN+1UN +

N

X

n=0

(cn−cn+1)Un

and another Abel tranform gives the result.

In particular, we have the two following simple facts.

Fact 1. Let ck

k≥0 a non-increasing sequence of non negative real numbers. Let uk

k≥0 be a vanishing sequence of non negative numbers and u^∗_k

k≥0 its monotone rearrangement.

Then (RI)

+∞

X

n=0

cnun≤

+∞

X

n=0

cnu^∗_n.

Indeed we just point out that, by definition, for everyn≥0, we have

n

X

k=0

u^∗_k ≥

n

X

k=0

uk.

Fact 2. Let (α_m)_m≥0 be a non-increasing summable sequence of non-negative real numbers and λ >0. Then X

m≥0

αm [(m+ 1)λ]−[mλ]

≤λX

m≥0

αm.

Indeed, we only have to point out that [N λ]≤N λ for everyN ≥0.

Proof of the main Theorem. Leta∈`^p(w). We assume first that (|ak|^pwk)k is non-increasing.

Let us fix an arbitraryN ∈N0. For every n∈N0, we write An=

n

X

k=0

Z ^k+1_n+1

k n+1

ak ds= Z 1

0

a_[(n+1)s]ds .

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WEIGHTED DISCRETE HARDY’S INEQUALITIES 3

Thanks to the triangular inequality for integrals, we have X^N

n=0

|An|^pw⁰_n¹_p

≤ Z 1

0

X^N

n=0

a_[(n+1)s]

pw⁰_n¹_p ds≤

Z 1

0

X

n≥1

a_[ns]

pw_n−1⁰ _p¹ ds .

For every s∈(0,1) and form∈N0, in order to gather terms, we introduce I_m(s) =

n≥1|[ns] =m =hm s,m+ 1

s

∩N.

Clearly, (Im)m∈N0 is a partition ofN, so we have

(1) X

n≥1

a_[ns]

pw⁰_n−1= X

m≥0

am

p X

n∈Im(s)

w⁰_n−1≤f(s)X

m≥0

am

pwm(cardIm(s)) by hypothesis.

We point out that card (0, A)∩N

= [A] whenA /∈N. Therefore, for everys∈[0,1]\Q, we have for everym≥0,

(2) cardIm(s) =hm+ 1

s

i−hm s

i·

From (1), (2) and Fact 2, we obtain X

n≥1

a_[ns]

pw_n⁰¹_p

≤f(s)^p¹s⁻¹^p a

_p almost everywhere.

Integrating with respect tos, we getX^N

n=0

|An|^pw⁰_n¹_p

≤K a

p.

Since N ∈ N0 is arbitrary, the result is proved in the particular case when (|ak|^pwk)k is non- increasing.

Now, in the general case, takea∈`^p(w). The sequence (uk)k≥0= |ak|w

1 p

k

k is vanishing so we can consider its monotone rearrangement u^∗_k

k≥0. We define alsoc_k=w⁻

1 p

k fork≥0.

We point out that for every N≥0, thanks to Fact 1, we have

N

X

n=0

an

≤

N

X

n=0

|an|=

N

X

n=0

cnun ≤

N

X

n=0

cnu^∗_n Applying the first step to the sequence c_ku^∗_k

k≥0, we get kΓ(a)k`^p(w⁰)≤K^+∞X

n=0

w_n|cnu^∗_n|^p¹_p but

^+∞X

n=0

wn|cnu^∗_n|^p¹_p

=X^+∞

n=0

|u^∗_n|^p¹_p

=X^+∞

n=0

|un|^p¹_p

=kak`^p(w).

Final remark and acknowledgment.This work is partially supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front).

References

[1] G. Bennett, Some elementary inequalities,Quart. J. Math. Oxford, 38 (1987), 401-425.

[2] H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities, 2nd ed.,Cambridge University Press, Cambridge, 1967.

[3] G.J.O. Jameson, and R. Lashkaripour, Norms of certain operators on weighted`^p spaces and Lorentz sequence spaces,Journal of Inequalities in Pure & Applied Mathematics, (2002), Vol 3, Issue 1.

[4] F. Fischer, M. Keller and F. Pogorzelski, An improved discretep-Hardy inequality. arXiv:1910.03004.

[5] A. Kufner, L. Maligranda and L.E. Persson, The Prehistory of the Hardy Inequality.American Math Monthly113 (2006) 715–732.

[6] P. Lef`evre, A short direct proof of the discrete Hardy inequality.Archiv der Mathematik,(2020), 114(2), 195–198.