Factorizations of Weighted Hardy Inequalities...Factorizations of Weighted Hardy Inequalities 917...

Bull Braz Math Soc, New Series (2018) 49:915–932https://doi.org/10.1007/s00574-018-0087-7

Factorizations of Weighted Hardy Inequalities

Sorina Barza1 · Anca N. Marcoci2 ·Liviu G. Marcoci2

Received: 14 September 2017 / Accepted: 19 April 2018 / Published online: 28 April 2018© The Author(s) 2018

Abstract We present factorizations of weighted Lebesgue, Cesàro and Copsonspaces, for weights satisfying the conditions which assure the boundedness of theHardy’s integral operator between weighted Lebesgue spaces. Our results enhance,among other, the best known forms of weighted Hardy inequalities.

Keywords Factorization of function spaces · Hardy averaging integral operator ·Cesàro spaces · Copson spaces

Mathematics Subject Classification Primary 26D20; Secondary 46B25

1 Introduction

Let p > 1. The classical Hardy inequalities

( ∞∑n=1

(1

n

n∑k=1

|ak |)p)1/p

≤ pp − 1

( ∞∑n=1

|an|p)1/p

(1)

B Sorina [email protected]

Anca N. [email protected]

Liviu G. [email protected]

1 Department of Mathematics and Computer Science, Karlstad University, 65188 Karlstad, Sweden

2 Department of Mathematics and Computer Science, Technical University of Civil EngineeringBucharest, 020396 Bucharest, Romania

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s00574-018-0087-7&domain=pdfhttp://orcid.org/0000-0002-1172-0113

916 S. Barza et al.

and (∫ ∞0

(1

x

∫ x0

| f (t)|dt)p

dx

)1/p≤ p

p − 1(∫ ∞

0| f (t)|p

)1/p(2)

(see e.g. Kufner et al. 2007) can be interpreted as inclusions between the Lebesguespace and Cesàro space of sequences (respectively functions). The Cesàro space ofsequences is defined to be the set of all real sequences a = (an)n≥1 that satisfy

‖a‖ces(p) =( ∞∑n=1

(1

n

n∑k=1

|ak |)p)1/p

< ∞

and the Cesàro space of functions is defined to be the set of all Lebesgue measurablereal functions on [0,∞) such that

‖ f ‖Ces(p) =(∫ ∞

0

(1

x

∫ x0

| f (t)|dt)p

dx

)1/p< ∞.

The same interpretation is valid if the Hardy operator is substituted by its dual.In his celebrated book, Bennett (1996) “enhanced” the classical Hardy inequality bysubstituting it with an equality, factorizing the Cesàro space of sequences, with thefinal aim to characterize its Köthe dual. He proved that a sequence x belongs to theCesàro space of sequences ces(p) if and only if it admits a factorization x = y · zwith y ∈ l p and z p′1 + · · · z p

′n = O(n), where p′ = pp−1 is the conjugate index of

p. This factorization gives also a better insight in the structure of Cesàro spaces. Theanswer to the question of the dual space of the Cesàro space was given for the firsttime by Jagers (1974). This problem was posed by the Dutch Academy of Sciences.His description of the dual space of the Cesàro space is complicated, given in termsof the least decreasing majorant, namely

{x :

∞∑n=1

(n(x̃n − x̃n+1)p′

)< ∞

}

where x̃ is the least decreasing majorant of |x |, with the property that

x̃m − x̃n1mp + · · · + 1(n−1)p

, n > m

increase with n if m is fixed. By means of factorizations, a new isometric characteri-zation, an alternative description of the dual Cesàro space of sequences, as being thespace

d(p) ={x :

∞∑n=1

supk≥n

|xk |p < ∞}

123

Factorizations of Weighted Hardy Inequalities 917

was given by Bennett (1996). It was not proved yet directly, that Jagers and Bennet’scharacterizations are equivalent.

In the case of functions, the same factorization results as well as the dual spaceof Cesàro space are only mentioned in Bennett (1996, Ch. 20), for the unweightedspaces. A factorization result for the unweighted Cesàro function spaces was provedin Astashkin and Maligranda (2009) where also an isomorphic description of thedual space of the Cesàro space of functions was given. An isometric description forthe general weighted case, in the spirit of Jagers was given in Kamińska and Kubiak(2012). Although the characterization is given for general weights, the conditionwhichdefines the dual space is difficult. The results of Bennett (1996) have a big impact inmany parts of analysis but it seems that the corresponding results for weighted spacesare less studied. For a recent survey of results on classical Cesàro spaces see Astashkinand Maligranda (2014). In the newly, very interesting papers, Leśnik and Maligranda(2015), Kolwicz et al. (2014) similar results and motivations, in an abstract, verygeneral setting were presented. However, in Carton-Lebrun and Heinig (2003) can befound the following factorization result which is a weighted integral analogue of aresult obtained by Bennett in Bennett (1996) for the discrete Hardy operator in theunweighted case.

Theorem 1.1 (Carton-Lebrun and Heinig 2003) Let 1 < p < ∞ and w, v twoweights such that w > 0, v > 0 a.e. and assume that h is a non-negative function on[0,∞). Then the function h belongs to Cesp(w) if and only if it admits a factorizationh = f · g, f ≥ 0, g > 0 on [0,∞), with f ∈ L p(v) and g such that

‖g‖w,v = supt>0

(∫ ∞t

w(s)

s pds

)1/p (∫ t0

v1−p′(s)gp′(s)ds)1/p′

< ∞.

Moreover

inf ‖ f ‖L p(v)‖g‖w,v ≤ ‖h‖Cesp(w) ≤ 2(p′)1/p′p1/p inf ‖ f ‖L p(v)‖g‖w,v,

where the infimum is taken over all possible factorizations.

Throughout this paper, we use standard notations and conventions. The letters u,v, w, . . ., are used for weight functions which are positive a.e. and locally integrableon (0,∞). The function f is real-valued and Lebesgue measurable on (0,∞). Alsofor a given weight v we write V (t) = ∫ t0 v(s)ds, 0 ≤ t < ∞. By the symbol χAwe denote the characteristic function of the measurable set A. The symbol C · Dstays for the set of products of measurable, real valued functions defined on (0,∞),{ f · g : f ∈ C and g ∈ D}.

Observe that the best known form of the Hardy inequality does not follow fromTheorem 1.1. The aim of this paper is to prove factorization results of the same typefor the weighted Lebesgue, Cesàro and Copson spaces of functions, which enhancein the same manner the weighted Hardy inequality. The weights satisfy the naturalconditions which assure the boundedness of the Hardy, respectively the dual Hardyoperators as well as some reversed conditions.

123

918 S. Barza et al.

We denote by P the Hardy operator and by Q its adjoint

P f (t) = 1t

∫ t0

f (x)dx; Q f (t) =∫ ∞t

f (x)

xdx, (t > 0). (3)

For p ≥ 1, it is known that P is bounded on the weighted Lebesgue space L p(w) ifand only if w ∈ Mp (see Muckenhoupt 1972), where Mp is the class of weights forwhich there exists a constant C > 0 such that, for all t > 0 it holds

Mp :(∫ ∞

t

v(x)

x pdx

)1/p (∫ t0

v1−p′(x)dx)1/p′

≤ C. (4)

The least constant satisfying the condition Mp will be denoted by [v]Mp . Similarly,we denote by mp the class of weights satisfying the reverse inequality and by [v]mpthe biggest constant for which the reverse inequality holds.

The Hardy operator P is bounded on L1(v) if and only if there exists C > 0, suchthat

M1 :∫ ∞t

v(x)

xdx ≤ Cv(t), for every t > 0. (5)

We denote by [v]M1 the least constant for which the above inequality is satisfied.Similarly, [v]m1 is the biggest constant for which the reverse inequality of (5) issatisfied.

The corresponding condition for the boundedness of the adjoint operator Q onL p(v) (see Muckenhoupt (1972)) is given by

M∗p :(∫ t

0v(x)dx

)1/p (∫ ∞t

v1−p′(x)x p′

dx

)1/p′≤ C, for every t > 0. (6)

The least constant satisfying the M∗p condition will be denoted [w]M∗p . Similarly, wedenote by m∗p the class of weights satisfying the reverse inequality and by [w]m∗p thebiggest constant for which the reverse inequality holds.

The dual Hardy operator Q, (defined by (3)) is bounded on L1(v) if and only ifthere exists C > 0, such that

M∗1 :1

t

∫ t0

v(x)dx ≤ Cv(t), for all t > 0. (7)

We denote by [v]M∗1 the least constant for which the above inequality is satisfied.Similarly, [v]m∗1 is the biggest constant for which the reverse inequality of (7) issatisfied.

In Sect. 2 we prove a factorization result for the weighted Lebesgue spaces L p(v).This result is a natural extension of Theorem 3.8 from Bennett (1996).

In Sect. 3 we present some factorization theorems for the weighted Cesàro spacesin terms of weighted Lebesgue spaces and the spaces Gp(v), for p > 1. We treatseparately the case p = 1 which appears to be new. Moreover, our study is motivated

123


by similar factorization results established by Bennett (1996), in the unweighted case,for spaces of sequences and in Astashkin andMaligranda (2009, 2014), Kolwicz et al.(2014) in the unweighted, integral case or in abstract setting. Our study concentrateson the special weighted case of Cesàro spaces containing the Lebesgue spaces. As aconsequence we recover the best known form of the Hardy inequality for weightedLebesgue spaces.

We also present the optimal result for the power weights. Section 4 is devoted tothe same problems but for Copson spaces.

2 The Spaces Dp(v) and G p(v)

For 0 < p < ∞, the function spaces Gp(v) and Dp(v) are defined by

Gp(v) ={f : sup

t>0

(1

V (t)

∫ t0

| f (x)|pv(x)dx)1/p

< ∞}

(8)

and

Dp(v) ={f :

(∫ ∞0

esssupt≥x | f (t)|pv(x)dx)1/p

< ∞}

. (9)

If p = ∞, we clearly have

G∞(v) = D∞(v) = L∞,

where L∞ is the Lebesgue space of essentially bounded functions. The spaces Dp(1)and Gp(1) were introduced for the first time in Bennett (1996, page 124), whereanalogue integral results to the discrete ones were only formulated. Their weightedversions for 1 ≤ p < ∞, appeared firstly in Astashkin andMaligranda (2009, Remark2).

Using standard arguments such as Minkowski inequality and Fatou’s lemma (seeRudin 1987, Theorem 3.11), it is easy to see that Gp(v) and Dp(v) endowed with thenorms

‖ f ‖Gp(v) = supt>0

(1

V (t)

∫ t0

| f (x)|pv(x)dx)1/p

, (10)

respectively

‖ f ‖Dp(v) =(∫ ∞

0esssupt≥x | f (t)|pv(x)dx

)1/p

are Banach spaces, for p ≥ 1.We denote by f̂ (x) = essupt≥x | f (t)| the least decreasing majorant of the absolute

value of the function f . Obviously, the function f ∈ Dp(v) if and only if f̂ ∈ L p(v)and that ‖ f ‖Dp(v) = ‖ f̂ ‖L p(v).

In what follows we need the following two lemmas.

123

920 S. Barza et al.

Lemma 2.1 (Hardy’s lemma) Let f, g be two nonnegative real-valued functions andh be a nonnegative decreasing function. If

∫ t0

f (x)dx ≤∫ t0g(x)dx, for any t > 0

then ∫ ∞0

f (x)h(x)dx ≤∫ ∞0

g(x)h(x)dx .

Proof See Bennett and Sharpley (1988, Proposition 3.6).

Lemma 2.2 Let h be a nonnegative measurable function on (0,∞), such that

limx→∞

∫ x0 h(t)v(t)dt∫ x

0 v(t)dt= 0.

Then there exists a nonnegative decreasing function h◦ on (0,∞), called the levelfunction of h with respect to the measure v(x)dx satisfying the following conditions:

(1)∫ x0 h(t)v(t)dt ≤

∫ x0 h

◦(t)v(t)dt;(2) up to a set of measure zero, the set {x : h(x) = h◦(x)} = ∪∞k=1 Ik , where Ik are

bounded disjoint intervals such that∫Ikh◦(t)v(t)dt =

∫Ikh(t)v(t)dt

and h◦ is constant on Ik , i.e. h◦(t) =∫Ikhv∫

Ikv

.

Proof For a proof see e.g. Barza et al. (2009) or Sinnamon (1994).

We are now ready to prove the main theorem of this section, which contains one

of the possible factorizations of weighted Lebesgue spaces. Our proof of factorizationof L p(v) is based on Hardy’s lemma and some properties of the so-called ”levelfunction”, and is different than that given in Astashkin and Maligranda (2009) forthe unweighted case. This factorization is a natural extension to the weighted integralcase of the discrete, unweighted version (Bennett 1996, Theorem 3.8) and of theunweighted integral case proved in Astashkin and Maligranda (2009, Proposition 2).Also wemention here that the statement of the next Theorem, without a proof, appearsin Astashkin and Maligranda (2009, Remark 2).

Theorem 2.1 If 0 < p ≤ ∞, then a function h ∈ L p(v) if and only if f admits afactorization h = f · g such that f ∈ Dp(v) and g ∈ Gp(v). Moreover,

‖h‖L p(v) = inf{‖ f ‖Dp(v)‖g‖Gp(v)},

where the infimum is taken over all possible factorizations h = f · g with f ∈ Dp(v)and g ∈ Gp(v).

123


Proof The case p = ∞ is trivial. Observe that the spaces L p(v), Dp(v), Gp(v) arehomogeneous, namely f ∈ L p(v), Dp(v) or Gp(v) if and only if f p ∈ L1(v), D1(v)or G1(v). Hence, by homogeneity (or p-convexification), it is sufficient to prove thetheorem for p = 1.

We first prove that D1(v) · G1(v) ⊆ L1(v). Suppose that h admits a factorizationh = f · g with f ∈ D1(v), g ∈ G1(v). Then

‖h‖L1(v) =∫ ∞0

| f (x)g(x)|v(x)dx ≤∫ ∞0

f̂ (x)|g(x)|v(x)dx .

From definition (10) we have the inequality

∫ t0

|g(x)|v(x)dx ≤ ‖g‖G1(v)∫ t0

v(x)dx, for any t > 0

which together with Lemma 2.1 give

∫ ∞0

f̂ (x)|g(x)|v(x)dx ≤ ‖g‖G1(v)∫ ∞0

f̂ (x)v(x)dx = ‖g‖G1(v)‖ f ‖D1(v).

Thus we have that D1(v) · Gp(v) ⊆ L1(v) and that

‖h‖L1(v) ≤ inf{‖g‖G1(v)‖ f ‖D1(v)},

where the infimum is taken over all possible factorizations h = f · g.Conversely let h be a nonnegative function such that h ∈ L1(v). We set f (x) =

h◦(x), x > 0, where h◦(x) is the level function of h with respect to the measurev(x)dx , as in Lemma 2.2. Since h◦(x) is a decreasing function by the definition of thespace D1(v) and by Lemma 2.2 we have that

‖ f ‖D1(v) = ‖h◦‖D1(v) = ‖h◦‖L1(v) = ‖h‖L1(v).

We define g(x) = h(x)h◦(x) on {x > 0 : h◦(x) = 0} = [0, a), for some a > 0 andg(x) = 0 if x > a. If t ∈ In , for some n, we have

1

V (t)

∫ t0g(x)v(x)dx = 1

V (t)

∫ t0

h(x)

h◦(x)v(x)dx

= 1V (t)

(∫E

h(x)

h◦(x)v(x)dx +

∫∪n−1k=1 Ik

h(x)

h◦(x)v(x)dx

+∫ tan

h(x)

h◦(x)v(x)dx

),

123

922 S. Barza et al.

where E = {x ∈ (0, t) : h(x) = h◦(x)} and Ik = (ak, bk) are the disjoint intervalsfrom Lemma 2.2. Hence, by Lemma 2.2 we get that∫

E

h(x)

h◦(x)v(x)dx =

∫E

v(x)dx,∫Ik

h(x)

h◦(x)v(x)dx =

∫Ik

v(x)dx

and ∫ tan

h(x)

h◦(x)v(x)dx ≤

∫In

v(x)dx .

Hence

‖g‖G1(v) ≤ 1.

Since h = f · g, with f ∈ D1(v) and g ∈ G1(v) we have that L1(v) ⊆ D1(v) ·G1(v)and

‖h‖L1(v) = ‖ f ‖D1(v) ≥ ‖ f ‖D1(v) · ‖g‖G1(v) ≥ inf{‖ f ‖D1(v) · ‖g‖G1(v)},

where the infimum is taken over all possible factorizations h = f · g. It is easy to seefrom this proof that the infimum is actually attained and this concludes the proof ofthe theorem.

3 Factorization of the Weighted Cesàro Spaces

In this section we present a factorization of the weighted Cesàro spaces Cesp(v). Wetreat separately the cases p > 1 and p = 1. The weighted Cesàro spaces of functions,Cesp(v) is defined to be the space of all Lebesgue measurable real functions on [0,∞)such that

‖ f ‖Cesp(v) =(∫ ∞

0

(1

x

∫ x0

| f (t)|dt)p

v(x)dx

)1/p< ∞.

These spaces are obviously Banach spaces, for p ≥ 1 and if the weight v satisfies (4)we have that L p(v) ⊆ Cesp(v). We denote by

!h!p,v = inf{‖ f ‖L p(v)‖g‖Gp′ (v1−p′ )} (11)

where the infimum is taken over all possible decompositions of h = f · g, withf ∈ L p(v) and g ∈ Gp′(v1−p′).The following Theorem is an extension to the weighted case of Astashkin and

Maligranda (2009, Proposition 1). The discrete, unweighted case was proved in Ben-nett (1996, Theorem 1.5).

123


Theorem 3.1 Let p > 1 and v belongs to the classes Mp and mp. The function hbelongs to Cesp(v) if and only if it admits a factorization h = f · g, with f ∈ L p(v)and g ∈ Gp′(v1−p′). Moreover

[v]mp !h!p,v ≤ ‖h‖Cesp(v) ≤ (p′)1/p′p1/p[v]Mp !h!p,v.

Proof Let f ∈ L p(v) and g ∈ Gp′(v1−p′). First we prove that the function h =f · g ∈ Cesp(v) and the right-hand side inequality. Let u be an arbitrary decreasingfunction. By Hölder’s inequality we get

∫ t0

|h(x)|dx =∫ t0

| f (x)g(x)|dx ≤(∫ t

0| f (x)|pv(x)u−p(x)dx

)1/p

·(∫ t

0|g(x)|p′v1−p′(x)u p′(x)dx

)1/p′. (12)

On the other hand, by Lemma 2.1 we obtain

(∫ t0

|g(x)|p′v1−p′(x)u p′(x)dx)1/p′

≤ ‖g‖Gp′ (v1−p′ )

·(∫ t

0v1−p′(x)u p′(x)dx

)1/p′. (13)

Hence, by (12) and (13), integrating from 0 to ∞ and by applying Fubbini’s theoremwe have

∫ ∞0

(1

t

∫ t0

|h(x)|dx)p

v(t)dt ≤ ‖g‖pG p′ (v1−p

′)

∫ ∞0

(∫ t0

| f (x)|pv(x)u−p(x)dx)

·(∫ t

0v1−p′ (x)u p′(x)dx

)p−1t−pv(t)dt

= ‖g‖pG p′ (v1−p

′)

∫ ∞0

| f (x)|pv(x)u−p(x)

·(∫ ∞

xt−pv(t)

(∫ t0

v1−p′(x)u p′(x)dx)p−1

dt

)dx .

Taking u(t) =(∫ t

0 v1−p′(s)ds

)−1/(pp′), since v ∈ Mp we get

‖h‖Cesp(v) ≤ p1/p(p′)1/p′ [v]Mp‖g‖Gp′ (v1−p′ )‖ f ‖L p(v).

Hence h ∈ Cesp(v) and

‖h‖Cesp(v) ≤ p1/p(p′)1/p′ [v]Mp inf{‖ f ‖L p(v) · ‖g‖Gp′ (v1−p′ )},

123

924 S. Barza et al.

where the infimum is taken over all possible factorizations of h. This completes thefirst part of the proof of the theorem.

For the reversed embedding, i.e. Cesp(v) ⊆ L p(v) · Gp′(v1−p′), let h ∈ Cesp(v).Since v > 0 a.e. for t > 0, we may assume, without loss of generality that v(t) > 0,for any t > 0. Set now

w(t) := 1v(t)

∫ ∞t

v(x)

x

(1

x

∫ x0

|h(s)|ds)p−1

dx,

for t > 0.Wedefine f (t) = |h(t)|1/pw1/p(t) sign h(t) and g(t) = |h(t)|1/p′w−1/p(t).It is easy to see that

‖ f ‖L p(v) = ‖h‖Cesp(v) < ∞. (14)By Hölder’s inequality we have

(∫ t0gp

′(x)v1−p′(x)dx

)p≤

(∫ t0

|h(x)|dx)p−1

·(∫ t

0|h(x)|w−p′(x)v−p′(x)dx

). (15)

Multiplying the inequality (15) by∫ ∞t x

−pv(x)dx and using thatw(t)v(t) is a decreas-ing function we get

(∫ ∞t

v(x)

x pdx

) (∫ t0gp


)p

≤(∫ ∞

t

v(x)

x p

(∫ x0

|h(s)|ds)p−1

dx

)

·(∫ t


)

= W (t)v(t)(∫ t


)

≤∫ t0

|h(x)|w1−p′(x)v1−p′(x)dx .

Since gp′(x) = |h(x)|w1−p′(x) we obtain

(1∫ t

0 v1−p′(x)dx

∫ t0gp


)1/p′≤

(∫ t0

v1−p′(x)dx))−1/p′

·(∫ ∞

t

v(x)

x pdx

)−1/p.

123


Hence

supt>0

(1∫ t

0 v1−p′(x)dx

∫ t0gp


)1/p′≤ 1[v]mp

which shows that g belongs to Gp′(v1−p′) and

‖h‖Cesp(v) = ‖ f ‖L p(v) ≥ [v]mp‖ f ‖L p(v)‖g‖Gp′ (v1−p′ ).

In this way, we get the left-hand side inequality.

If we take g(x) = 1, x > 0 the right-hand side inequality implies the best form of

the weighted Hardy inequality for 1 < p < ∞ namely

‖ f ‖Cesp(v) ≤ (p′)1/p′p1/p[v]Mp‖ f ‖L p(v)

(see e.g. Kufner et al. 2007).Observe also that the infimum is attained.

In particular, we denote by L pα the weighted Lebesgue space with the power weightv(t) = tα and in a similar way the spacesGp,α and Cesp,α . In analogywith the generalcase we also denote by

!h!p,α = inf ‖ f ‖L pα ‖g‖Gp′,α(1−p′) ,

where the infimum is taken over all possible decompositions of h = f ·g, with f ∈ L pαand g ∈ Gp′,α(1−p′).Corollary 3.2 Let p > 1 and −1 < α < p − 1. The function h belongs to Cesp,αif and only if it admits a factorization h = f · g, with f ∈ L pα and g ∈ Gp′,α(1−p′).Moreover

(1

p

)1/p ( 1p′

)1/p′ pp − α − 1 !h!p,α ≤ ‖h‖Cesp,α ≤

p

p − α − 1 !h!p,α.

Proof Take v(t) = tα in Theorem 3.1. The constant in the right hand-side inequalityis optimal since it is the best constant in Hardy’s inequality with a power weight (seee.g. Kufner et al. 2007, p. 23).

For the sake of completeness, as well as for the independent interest we presentseparately the case p = 1, although the proof of the main result in this case followsthe same ideas as for p > 1.

By L∞ we denote, as usual, the space of all measurable functions which satisfy thecondition

‖g‖∞ := esssupx>0|g(x)| < ∞.

123

926 S. Barza et al.

As before,

!h!1,v = inf ‖ f ‖L1(v)‖g‖∞

where the infimum is taken over all possible factorizations of h = f ·g, with f ∈ L1(v)and g ∈ L∞.

Theorem 3.3 Let v belong to M1 and m1. The function h belongs to Ces1(v) if andonly if it admits a factorization h = f · g, with f ∈ L1(v) and g ∈ L∞. Moreover

[v]m1 !h!1,v ≤ ‖h‖Ces1(v) ≤ [v]M1 !h!1,v.

Proof Let f ∈ L1(v) and g ∈ L∞. We prove that the function h = f g belongs toCes1(v). By Hölder’s inequality and since g ∈ L∞ we get

∫ ∞0

(1

t

∫ t0h(s)ds

)v(t)dt ≤ ‖g‖∞

∫ ∞0

(1

t

∫ t0

f (x)dx

)v(t)dt. (16)

By Fubini’s theorem and taking into account that v ∈ M1 we have that

‖h‖Ces1(v) ≤ [v]M1‖g‖∞‖ f ‖L1(v),

for any f , g as above. Hence h ∈ Ces1(v) and

‖h‖Ces1(v) ≤ [v]M1 inf ‖g‖∞‖ f ‖L1(v),

where infimum is taken over all possible factorizations of h. This completes the firstpart of the proof.

Conversely, let h ∈ Ces1(v) and w(t) = 1v(t)∫ ∞t

v(x)x dx . We may assume, without

loss of generality that v(t) > 0, for all t > 0.Let f (t) = |h(t)|w(t) sign h(t) and g(x) = 1

w(x) . It is easy to see that

‖ f ‖L1(v) = ‖h‖Ces1(v) < ∞.

Since v ∈ m1, g belongs to L∞ and

‖g‖∞ ≤ 1[v]m1.

Moreover, ‖h‖Ces1(v) = ‖ f ‖L1(v) ≥ [v]m1‖ f ‖L1(v)‖g‖∞ and we get the left-handside inequality of the theorem. The proof is complete.

123


4 Factorization of the Weighted Copson Spaces

In the same manner, in this section we present the factorizations of the weightedCopson space, namely the space

Copp(v) ={f :

∫ ∞0

(∫ ∞t

| f (x)|x

dx

)pv(t)dt < ∞

}.

Let

G∗p(v) =⎧⎨⎩ f : supt>0

(1∫ ∞

t v(x)x−pdx

∫ ∞t

f (x)pv(x)x−pdx)1/p

< ∞⎫⎬⎭ (17)

To prove the main result we need the following Lemma.

Lemma 4.1 Let f, g be two non-negative real-valued functions and h be a non-negative increasing function. If

∫ ∞t

f (x)dx ≤∫ ∞t

g(x)dx, t > 0

then

∫ ∞0

f (x)h(x)dx ≤∫ ∞0

g(x)h(x)dx .

Proof The proof follows by a change of variable and Lemma 2.1.

We denote by!!h!!p,v = inf ‖ f ‖L p(v)‖g‖G∗

p′ (v1−p′ ) (18)

where the infimum is taken over all possible factorizations of h = f ·g. The followingtheorem extends to the weighted case a result formulated without proof in Bennett(1996, Theorem 21.6). The discrete case is proved in Bennett (1996, Theorem 5.5).

Theorem 4.1 Let p > 1 and v belong to the classes M∗p and m∗p.The function h belongs to Copp(v) if and only if it admits a factorization h = f · g,

with f ∈ L p(v) and g ∈ G∗p′(v1−p′). Moreover

[v]m∗p !!h!!p,v ≤ ‖h‖Copp(v) ≤ p′1/p′p1/p[v]M∗p !!h!!p,v.

Proof Let f ∈ L p(v) and g ∈ G∗p′(v1−p′).

123

928 S. Barza et al.

We show first that the function h = f g ∈ Copp(v). Let u be an arbitrary positiveincreasing function. Hölder’s inequality gives

∫ ∞t

f (x)g(x)

xdx

≤(∫ ∞

tf p(x)v(x)u−p(x)dx

)1/p (∫ ∞t

g p′(x)

v1−p′(x)x p′

u p′(x)dx

)1/p′. (19)

By Hardy’s Lemma 4.1 we obtain

(∫ ∞t

g p′(x)

v1−p′(x)x p′

u p′(x)dx

)1/p′≤ ‖g‖p

G∗p′ (v

1−p′ )

·(∫ ∞

t

v1−p′(x)x p′

u p′(x)dx

)1/p′.

Hence, multiplying (19) by v(t), raising to p and integrating from 0 to ∞, we get

∫ ∞0

(∫ ∞t

h(x)

xdx

)pv(t)dt ≤ ‖g‖p

G p′ (v1−p′)

·∫ ∞0

v(t)

(∫ ∞t

f p(x)v(x)u−p(x)dx) (∫ ∞

t

v1−p′(x)x p′

u p′(x)dx

)p−1dt.

By Fubini’s theorem we have

∫ ∞0

(∫ ∞t

h(s)

sds

)pv(t)dt ≤ ‖g‖p

G p′ (v1−p′)

·∫ ∞0

f p(x)v(x)

⎛⎝∫ x

0v(t)

(∫ ∞t

v1−p′(s)s p′

u p′(s)ds

)p−1dt

⎞⎠ u−p(x)dx .

Taking u(t) =(∫ ∞

tv1−p′ (x)

x p′

)−1/pp′, in the above inequality and since

∫ ∞t

v1−p′(s)s p′

u p′(s)ds = p′

(∫ ∞t

v1−p′(s)s p′

(s)ds

)1/p′dx

123


we have that

∫ ∞0

f p(x)v(x)

⎛⎝∫ x

0v(t)

(∫ ∞t

v1−p′(s)s p′

u p′(s)ds

)p−1dt

⎞⎠ u−p(x)dx

= (p′)p−1∫ ∞0

f p(x)v(x)

⎛⎝∫ x

0v(t)

(∫ ∞t

v1−p′(s)s p′

ds

) p−1p′

dt

⎞⎠

·(∫ ∞

x

v1−p′(s)s p′

ds

)1/p′.

By the definition of M∗p we get

∫ ∞0

(∫ ∞t

h(s)

sds

)pv(t)dt ≤ (p′)p−1‖v‖p−1M∗p

·∫ ∞0

f p(x)v(x)

(∫ x0

v(t)

(∫ t0

v(s)ds

)−1/p′dt

) (∫ ∞x

v1−p′(s)s p′

ds

)1/p′dx

= (p′)p−1 p‖v‖p−1M∗p∫ ∞0

f p(x)v(x)

(∫ x0

v(t)

)1/p (∫ ∞x

v1−p′(s)s p′

ds

)1/p′dx

≤ (p′)p−1 p‖v‖pM∗p∫ ∞0

f p(x)v(x)dx,

since

d

dx

(∫ x0

v(t)dt

)1/p= 1

p

(∫ x0

v(t)dt

)1/p−1v(x).

Hence

‖h‖Copp(v) ≤ p1/p p′1/p′ [v]M∗p‖g‖G∗

p′ (v1−p′ )‖ f ‖L p(v).

for any f , g as above. Hence h ∈ Copp(v) and

‖h‖Copp(v) ≤ p1/p p′1/p′ [v]M∗p inf ‖g‖G∗

p′ (v1−p′ )‖ f ‖L p(v)

where the infimum is taken over all possible factorizations of h which gives the left-hand side inequality of the theorem.

For the reverse embedding, i.e. Copp(v) ⊂ L p(v)G∗p′(v1−p′), let h ∈ Copp(v)

and

w(t) := 1tv(t)

∫ t0

v(x)

(∫ ∞x

h(s)

sds

)p−1dx,

123

930 S. Barza et al.

if v = 0 and w(t) = 0 on of Lebesgue measure possibly v = 0. Define f (t) =|h|1/p(t)w1/p(t) sign h(t) and g(t) = |h|1/p′(t)w−1/p(t). An easy application ofFubini theorem gives

‖ f ‖L p(v) = ‖h‖Copp(v) < ∞.

By Hölder’s inequality and the definition of g we have

(∫ ∞x

g p′(t)

v1−p′(t)t p′

dt

)p≤

(∫ ∞x

|h(t)|t

dt

)p−1 ∫ ∞x

|h(t)|t

w−p′(t)v−p′(t)t p′

dt.

(20)Weestimatefirst the right-hand side termof the inequality (20)multiplied by

∫ x0 v(t)dt .

(∫ x0

v(t)dt

)(∫ ∞x

|h(t)|t

dt

)p−1 ∫ ∞x

|h(t)|t


dt

≤∫ x0

v(t)

(∫ ∞t

|h(s)|s

ds

)p−1dt

∫ ∞x

|h(t)|t


dt

= xw(x)v(x)(∫ ∞

xh(t)w−p′(t)v−p′(t)dt

)

≤∫ ∞x

h(t)w1−p′(t)v1−p′(t)t−p′dt,

since, by definition, xw(x)v(x) is an increasing function. By using that gp′(x) =

h(x)w1−p′(x) we get

⎛⎝ 1∫ ∞

xv1−p′ (t)

t p′ dx

∫ ∞x

g p′(t)

v1−p′(t)t p′

dt

⎞⎠

1/p′

≤ 1(∫ x0 v(t)dt

)1/p (∫ ∞x t

−p′v1−p′(t)dt)1/p′ .

Hence

supt>0

(1∫ ∞

t v1−p′(x)x−p′dx

∫ ∞t

g p′(x)v1−p′(x)x−p′dx

)1/p′≤ 1[v]m∗p

(21)

which means that g belongs to G∗p′(v1−p′). Moreover,

‖h‖Copp(v) = ‖ f ‖L p(v) ≥ [v]m∗p‖ f ‖L p(v)‖g‖G∗p′ (v

1−p′ ). In this way the left-hand side

inequality is proved.

The space Copp,α is the Copson weighted space with the weight t

α . We have thefollowing result for the case of a power weight.

123


Corollary 4.2 Let p > 1 and α > −1. The function h belongs to Copp,α if and onlyif it admits a factorization h = f · g, with f ∈ L pα and g ∈ Gp′,α(1−p′). Moreover

(p − 1)1/p′α + 1 !h!p,α ≤ ‖h‖p,α ≤

p

α + 1 !h!p,α.

The constants in both inequalities are optimal.

Proof Take v(t) = tα in Theorem 4.1. The constant in the right hand-side inequality isoptimal since it is the best constant in Hardy inequality (see e.g. Kufner et al. 2007) andthe optimality of the constant in the left-hand side follows if we take h(x) = χ(a−ε,a+ε)and let then ε → 0 and a → ∞.

We present now the case p = 1.

The dual Hardy operator Q, (defined by (3)) is bounded on L1(v) if and only ifthere exists C > 0, such that

M∗1 :1

t

∫ t0

v(x)dx ≤ Cv(t). (22)

We denote by [v]M∗1 the least constant for which the above inequality is satisfied.Similarly, [v]m∗1 is the biggest constant for which the reverse inequality of (22) issatisfied.

Theorem 4.3 Let v belong to M∗1 and m∗1. The function h belongs to Cop1(v) if andonly if it admits a factorization h = f · g, with f ∈ L1(v) and g ∈ L∞. Moreover

[v]m∗1 !h!1,v ≤ ‖h‖Cop1(v) ≤ [v]M∗1 !h!1,v.

and the constants are optimal.

Proof The proof is similar with that of Theorem 3.3.

Acknowledgements We are very thankful to the referee for valuable suggestions and comments, whichhave improved the final version of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

References

Astashkin, S.V., Maligranda, L.: Structure of Cesàro function spaces. Indag. Math. (N.S.) 20(3), 329–379(2009)

Astashkin, S.V., Maligranda, L.: Structure of Cesàro function spaces: a survey. Banach Center Publ. 102,13–40 (2014)

Barza, S., Kolyada, V., Soria, J.: Sharp constants related to the triangle inequality in Lorentz spaces. Trans.Am. Math. Soc. 361(10), 5555–5574 (2009)

123

http://creativecommons.org/licenses/by/4.0/

932 S. Barza et al.

Bennett, G.: Factorizing the classical inequalities. Mem. Am. Math. Soc. 120(576), 1–130 (1996)Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)Carton-Lebrun, C., Heinig, H.P.: Weight characterization of an averaging operator. J. Math. Anal. Appl.

283(1), 236–243 (2003)Jagers, A.A.: A note on Cesàro sequence spaces. Nieuw Arch. voor Wiskunde (3) 22, 113–124 (1974)Kamińska, A., Kubiak, D.: On the dual of Cesàro function space. Nonlinear Anal. 75(5), 2760–2773 (2012)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factoriza-

tion. J. Funct. Anal. 266(2), 616–659 (2014)Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. About its History and Some Related

Results. Vydavatelsky Servis Publishing House, Pilsen (2007)Leśnik, K., Maligranda, L.: Abstract Cesàro spaces. Duality J. Math. Anal. Appl. 424(2), 932–951 (2015)Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)Sinnamon, G.: Spaces defined by their level functions and their duals. Studia Math. 111, 19–52 (1994)

123

Factorizations of Weighted Hardy InequalitiesAbstract1 Introduction2 The Spaces Dp(v) and Gp(v)3 Factorization of the Weighted Cesàro Spaces4 Factorization of the Weighted Copson SpacesAcknowledgementsReferences

Date post:	22-Oct-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Factorizations of Weighted Hardy Inequalities...Factorizations of Weighted Hardy Inequalities 917...

Documents