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volume 4, issue 4, article 81, 2003.

Received 23 June, 2003;

accepted 22 September, 2003.

Communicated by:T. Mills

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Journal of Inequalities in Pure and Applied Mathematics

WEIGHTED WEAK TYPE INEQUALITIES FOR THE HARDY OPERATOR WHENp= 1

TIELING CHEN

Mathematical Sciences Department University of South Carolina Aiken 471 University Parkway

Aiken, SC 29801, USA.

EMail:tielingc@usca.edu

c

2000Victoria University ISSN (electronic): 1443-5756 085-03

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Weighted Weak Type Inequalities For The Hardy

Operator Whenp= 1 Tieling Chen

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J. Ineq. Pure and Appl. Math. 4(4) Art. 81, 2003

http://jipam.vu.edu.au

Abstract

The paper studies the weighted weak type inequalities for the Hardy operator as an operator from weighted L^p to weighted weakL^qin the casep = 1. It considers two different versions of the Hardy operator and characterizes their weighted weak type inequalities whenp= 1. It proves that for the classical Hardy operator, the weak type inequality is generally weaker whenq < p= 1.

The best constant in the inequality is also estimated.

2000 Mathematics Subject Classification:26D15.

Key words: Hardy operator, Weak type inequality.

1. Introduction

The classical Hardy operator I is the integral operator If(x) = Rx

c f(t)dt, where the lower limit cin the integral is generally taken to be 0 or −∞, de- pending on the underlying space considered. In [4], Hardy first studied this operator from L^p to the weighted L^p_x−p when p > 1. The boundedness of this operator from L^p_u to L^q_v for general weights u, v and different pairs of indices p and q was considered in [12], [2], [11] and [16]. The boundedness of I is expressed by the strong type inequality

Z

If(x)^qv(x)dx ¹_q

≤C Z

f(y)^pu(y)dy ¹_p

, f ≥0,

which is also called the weighted norm inequality whenp, q ≥1. Whenp <1, the integral on the right hand side is no longer a norm, and the inequality is of little interest. Like other integral operators, the weighted strong type inequality forIalways implies the weighted weak type inequality

Z

{x:If(x)>λ}

v(x)dx ¹_q

≤ C λ

Z

f(y)^pu(y)dy ¹_p

, f ≥0, λ >0.

It is known that when1 ≤ p ≤ q < ∞, both the weighted strong type and weak type inequalities for the classical Hardy operator impose the same condition on the weightsuandv. That is, for givenuandv, either both inequalities hold or both fail. We say that the weighted strong type and weak type inequalities are equivalent. However, whenq < pand1< p <∞, the equivalence does not hold in general. Characteristics of weighted weak type inequalities for the

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Hardy operator and modified Hardy operators were studied in [1], [3], [5], [7], [9], [10], [13], and [14]. This paper looks at the Hardy Operator and considers the weighted weak type inequalities in the special casep= 1.

The case p = 1 is subtle, because in this case we need to consider two different operators. If p 6= 1, considering inequalities for I fromL^p_u to L^q_v is readily reduced to considering them for the operator

I_wf(x) = Z x

c

f(t)w(t)dt fromL^p_wtoL^q_v, wherew=u^1−p⁰ with ¹_p + _p¹0 = 1.

However, whenp = 1, the inequalities forI do not reduce to those for the operator I_w, so we need to deal with them separately. In Section 2, a more general operator than I_w is considered. Instead of consideringI_w, we consider the operatorI_µ,

(1.1) I_µf(x) =

Z x

c

f(t)dµ(t), whereµis theσ-finite measure of the underlying space.

In Theorem 2.2, we show that the weighted weak type and strong type inequalities forIµ are still equivalent. In Theorem2.4, the weak type inequality for I, when p = 1 and 0 < q < ∞, is considered. We will see that when 0< q <1 =p, the weighted weak type inequality is weaker in general.

Throughout the paper,λ is an arbitrary positive number, acting in the weak type inequalities. The conventions of0· ∞ = 0,0/0 = 0, and∞/∞ = 0are used.

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2. The Case p = 1 for the Hardy Operator

First let us consider the operatorI_µ defined in (1.1), withc = −∞for convenience. The strong type inequalities forIµwhenp= 1was studied in [15], and we state the result in the following proposition.

Proposition 2.1. Suppose 0 < q < ∞, and µ, ν are σ-finite measures on R. The strong type inequality

(2.1)

Z ∞

−∞

Iµf(x)^qdν ¹_q

≤C Z ∞

−∞

f(y)dµ, f ≥0,

holds if and only if

(2.2)

Z

E

dν <∞, whereE =

x∈R: Z x

−∞

dµ >0

.

In the next theorem, we show that condition (2.2) is also necessary and sufficient for the weak type inequality, in other words, the strong type and weak type inequalities forI_µare equivalent whenp= 1.

Theorem 2.2. Suppose0< q <∞, andµ,ν areσ-finite measures onR. Then the weak type inequality

(2.3) (ν{x:I_µf(x)> λ})¹^q ≤ C λ||f||_L¹

dµ, f ≥0, and the strong type inequality (2.1) are equivalent.

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Proof. Because the strong type inequality of an operator always implies the weak type inequality, we only need to prove (2.2) is also necessary for the weak type inequality (2.3).

Since Rx

−∞dµis a non-decreasing function, the set E is an interval of the formE = (z,∞)orE = [z,∞). SupposeE 6=∅, otherwise the proof is trivial.

If z 6= −∞, then we firstly suppose that z is an atom for µ. Set f(t) = (1/µ{z})χ_{z}(t). Sincez ∈ (−∞, x] for every x ∈ E we have I_µf(x) = 1.

Thus

Z

E

dν ¹_q

≤

x:I_µf(x)> 1 2

¹_q

≤2C||f||_L¹

dµ = 2C < ∞.

Secondly, suppose z is not an atom for µ. If we let > 0, and f(t) = [1/µ(z, z+)]χ_(z,z+)(t), then for every x ∈ [z+,∞), we have I_µf(x) = 1 and hence

Z ∞

z+

dν ¹_q

≤

x:I_µf(x)> 1 2

¹_q

≤2C||f||_L¹

dµ = 2C <∞.

As→0⁺, we have R

Edν¹_q

<∞, and (2.2) holds.

IfE = (−∞,∞), then we do the same discussion as above on the interval [z,∞)and then letz → −∞, and this completes the proof of Theorem2.2.

Now let us consider the weighted weak type inequality for the classical Hardy operator I (withc = 0 for convenience). We make use of some of the

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techniques in [17]. Notice that in Theorem 2.2, the conclusion forI_µ is inde- pendent of the relation between the indices q and p = 1. The operator I is a little bit more subtle. It does matter whetherq <1orq ≥1.

Definition 2.1. For a non-negative functionu, defineuby u(x) = essinf

0<t<x u(t).

It is easy to see thatuis non-increasing andu≤ualmost everywhere.

Lemma 2.3. Suppose that0< q <∞and thatk(x, t)is a non-negative kernel which is non-increasing in t for each x. Suppose u and v are non-negative functions. The best constant in the weighted weak type inequality

v

x:

Z ∞

0

k(x, t)f(t)dt > λ ¹_q

≤ C λ

Z ∞

0

f u forf ≥0,

is unchanged whenuis replaced byu.

Proof. LetC be the best constant in the above inequality and letC be the best constant in the above inequality with u replaced by u. Since u ≤ u almost everywhere,C ≤C. To prove the reverse inequality it is enough to show that (2.4)

v

x:

Z ∞

x

k(x, t)f(t)dt > λ ¹_q

≤ C λ

Z ∞

x

f u

for all non-negative f ∈ L¹(x,∞), wherex = inf{x ≥ 0 : u(x) < ∞}. The proof of Theorem 3.2 in [17] shows that for every non-negativef ∈ L¹(x,∞)

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and any >0, there exists anfsuch that Z ∞

x

fu≤ Z ∞

x

f u+ 2 Z ∞

x

f,

and Z ∞

x

k(x, t)f(t)dt ≤lim inf→0⁺

Z ∞

x

k(x, t)f(t)dt.

If lim inf→0⁺

R∞

x k(x, t)f(t)dt > λ, thenR∞

x k(x, t)f(t)dt > λ for all suffi- ciently small >0. Thus, for allx≥xand allλ >0,

χ{^{x:lim inf}→0+

R∞

x k(x,t)f(t)dt>λ}(x)≤lim inf_→0⁺χ{^x:^R_x^∞^k(x,t)f(t)dt>λ}(x).

We use these estimates to obtain v

x:

Z ∞

x

k(x, t)f(t)dt > λ

≤v

x: lim inf→0⁺

Z ∞

x

k(x, t)f(t)dt > λ

= Z ∞

0

χ{^{x:lim inf}→0+

R∞

x k(x,t)f(t)dt>λ}(x)v(x)dx

≤ Z ∞

0

lim inf_→0⁺χ{^x:^Rx^∞k(x,t)f(t)dt>λ}(x)v(x)dx

≤lim inf_→0⁺ Z ∞

0

χ{^x:^Rx^∞k(x,t)f(t)dt>λ}(x)v(x)dx

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= lim inf→0⁺v

x: Z ∞

x

k(x, t)f(t)dt > λ

≤lim inf_→0⁺C^qλ^−q Z ∞

x

fu q

≤C^qλ^−qlim inf_→0⁺ Z ∞

x

f u+ 2 Z ∞

x

f q

=C^qλ^−q Z ∞

x

f u q

,

which gives (2.4) and completes the proof.

Theorem 2.4. Suppose 0 < q < ∞, and u, v are non-negative functions on R. Then the weak type inequality for the classical Hardy operator If(x) = Rx

0 f(t)dt,

(2.5) (v{x:If(x)> λ})¹^q ≤ C λ

Z ∞

0

f(t)u(t)dt,

holds forf ≥0if and only if

(2.6) sup

y>0

v(y,∞)¹^q(u(y))⁻¹ =A <∞.

Moreover,C =Ais the best constant in (2.5).

Proof. Since If(x) = R∞

0 χ_(0,x)(t)f(t)dt, the kernel χ_(0,x)(t) satisfies the hy- potheses of Lemma2.3. By Lemma2.3, we only need to show thatAis the best

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constant in

(2.7) (v{x:If(x)> λ})¹^q ≤ C λ

Z ∞

0

f u.

We first consider the caseu=R∞

x bfor somebsatisfying (2.8)

Z ∞

x

b <∞ for allx >0, and Z ∞

0

b=∞.

Then the right hand side of (2.7) becomes C

λ Z ∞

0

f(t)u(t)dt = C λ

Z ∞

0

f(t) Z ∞

t

b(x)dx

dt

= C λ

Z ∞

0

Z x

0

f

b(x)dx.

Since any non-negative, non-decreasing functionF is the limit of an increasing sequence of functions of the formRx

0 f withf ≥ 0, it is sufficient to show that C =Ais also the best constant in the following inequality

(2.9) v{x:F(x)> λ}¹^q ≤ C λ

Z ∞

0

F b, forF ≥0, andF non-decreasing.

Suppose that A < ∞ and F is non-decreasing, then {x : F(x) > λ} is an interval of the form(y,∞)or[y,∞). Since the left end pointydoes not change the integral, we have

v{x:F(x)> λ}¹^q =v(y,∞)¹^q

≤Au(y) = A Z ∞

y

b ≤A Z ∞

y

F(x) λ b = A

λ Z ∞

y

F b,

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which gives (2.9) with the constantA.

Now suppose (2.9) holds. Fixy > 0. For a given > 0, letλ = 1−, and F(x) =χ(y,∞)(x), then

v(y,∞)¹^q =v{x:F(x)> λ}¹^q ≤ C λ

Z ∞

0

F b = C 1−

Z ∞

y

b = C

1−u(y).

Letting→0⁺, we get

v(y,∞)¹^qu(y)⁻¹ ≤C.

In the caseu(y) = 0, we use the convention0· ∞= 0. Then we obtainA≤C, and also get thatAis the best constant in (2.9).

Next we consider the case of generalu. We can assume thatu(x) < ∞for all x, since ifu = ∞on some interval(0, x)then we translateu to the left to get a smaller uand reduce the problem to one in which this does not happen.

Then for each n >0, the functionuχ_(0,n)is finite, non-increasing and tends to 0at∞. We can approximate it from above by functions of the formR∞

x bwithb satisfying (2.8). Let {u_m}be a non-increasing sequence of such functions that converges to uχ_(0,n) pointwise almost everywhere. Letv_n = vχ_(0,n), then the first part of the proof gives

v_n

x: Z x

0

f(t)dt > λ ¹_q

≤ 1 λsup

y>0

v_n(y,∞)¹^qu_m(y)⁻¹ Z ∞

0

f(t)u_m(t)dt, f ≥0,

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which implies v_n

x:

Z x

0

gu⁻¹_m > λ ¹_q

≤ 1 λsup

y>0

v_n(y,∞)¹^qu_m(y)⁻¹ Z ∞

0

g, g ≥0.

The Monotone Convergence Theorem, and the fact u_m(y)⁻¹ < u(y)⁻¹ when y ∈(0, n)give

v_n

x: Z x

0

gu⁻¹ > λ ¹_q

≤ 1 λsup

y>0

v_n(y,∞)¹^qu(y)⁻¹ Z ∞

0

g, g ≥0.

Letf =gu⁻¹to get v_n

x:

Z x

0

f > λ ¹_q

≤ 1 λsup

y>0

v_n(y,∞)¹^qu(y)⁻¹ Z ∞

0

f u

≤ A λ

Z ∞

0

f u, f ≥0.

Letn→ ∞, we get (2.7) with the constantC =A.

Conversely, suppose (2.7) holds for some constantC. Sincevn ≤v, then v_n{x:I(f χ_(0,n))(x)> λ}¹_q

≤ C λ

Z ∞

0

f χ_(0,n)u.

Note thatuχ_(0,n)≤u_m, then we have

(v_n{x:If(x)> λ})¹^q ≤ C λ

Z ∞

0

f u_m.

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The first part of the proof gives sup

y>0

v_n(y,∞)¹^qu_m(y)⁻¹ ≤C.

Then for everyy >0,

v_n(y,∞)¹^qu_m(y)⁻¹ ≤C, which gives

vn(y,∞)¹^qu(y)⁻¹ ≤C, whenm→ ∞. Thus

sup

y>0

v(y,∞)¹^qu(y)⁻¹ ≤C,

which isA≤C. SinceAitself is a constant such that (2.7) holds,Ais the best constant in (2.7). Theorem2.4is proved.

Remark 2.1. Theorem2.4characterizes the weighted weak type inequality for the classical Hardy operator in the casep= 1. The theorem imposes no restric- tion onq, except thatqis a positive number. In fact, differentqreveals different information on the equivalence of the weak and strong type inequalities. Re- call that when0 < q < p = 1, the weight characterization of the strong type inequality forI is (see [17])

Z ∞

0

u(x)^q/(q−1) Z ∞

x

v _1−q^q

v(x)dx <∞.

This condition is stronger than the condition (2.6) in general. For example, if we set u(x) = x^(α+1)/q and v(x) = x^α for some α < −1, then the condition

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(2.6) is satisfied but the above condition for the strong type inequality does not hold.

For the case 1 = p ≤ q < ∞, it is known that the weak and strong type inequalities for the operator I are equivalent. This conclusion can also be confirmed by 2.4. Recall that when 1 = p ≤ q < ∞, the necessary and sufficient condition of the strong type inequality forI is (see [2])

sup

r>0

Z ∞

r

v ¹_q

||u⁻¹χ_(0,r)||_L^∞ <∞.

It is easy to see that ||u⁻¹χ_(0,r)||_L^∞ coincides with u(r)⁻¹ and hence we get (2.6).

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[9] F.J. MARTIN-REYES, P. ORTEGA SALVADOR AND M.D. SARRION GAVILAN, Boundedness of operators of Hardy type in Λ^p,q spaces and weighted mixed inequalities for singular integral operators, Proc. Roy. Soc.

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