Compactness of the Hardy operator and its limiting case

Volume 32, No. 1, pp. 21-35, January 2006

COMPACTNESS OF THE HARDY OPERATOR AND ITS LIMITING CASE

BY

AMIRAN GOGATISHVILI, ALOIS KUFNER, LARS-ERIK PERSSON AND ANNA WEDESTIG

Abstract. Let 1 < p ≤ q < ∞. Inspired by some recent results concerning Hardy type inequalities where the equivalence of four scales of integral conditions is proved, we use related ideas to prove some new compactness results for the Hardy operator, and we give the corresponding scales for the P´olya-Knopp inequality.

1. Introduction

We consider the general one-dimensional Hardy inequality

Z _b

0

Z _x

0

f (t)dt^qu(x)dx^1/q ≤ C^ Z _b

0

f^p(x)v(x)dx^1/p (1.1) with a fixed b, 0 < b ≤ ∞, for measurable functions f ≥ 0, weights u and v and for the parameters p, q satisfying

1 < p ≤ q < ∞.

In [2] the equivalence of four scales of integral conditions that characterize the inequality (1.1) and give conditions with the usual Muckenhoupt condition as a special case was proved. The proof was carried out by first proving an equivalence theorem of independent interest that also will be applied in this paper. We therefore recall the equivalence theorem (see Theorem 2.1 in [2]):

Received July 2, 2004; revised February 8, 2005.

AMS Subject Classification. Primary 26D10, 26D15; secondary 47B07, 47B38.

Key words. Hardy operator, compactness, Hardy’s inequality, Polya-Knopp’s inequality, Lorentz space.

This paper is in final form and no version of it will be submitted for publication elsewhere.

21

Theorem 1.1. For −∞ ≤ a < b ≤ ∞, α, β and s positive numbers and f , g measurable functions positive a.e. in (a, b), let us denote

F (x) :=

Z _b

x

f (t)dt, G(x) :=

Z _x

a

g(t)dt (1.2)

and

B₁(x; α, β) := F^α(x)G^β(x);

B₂(x; α, β, s) :=^R_x^bf (t)G^β−sα (t)dt^αG^s(x);

B₃(x; α, β, s) :=

 R_x

a g(t)F^α−sβ (t)dt

β

F^s(x);

B₄(x; α, β, s) :=^R_a^xf (t)G^β+sα (t)dt^αG^−s(x);

B₅(x; α, β, s) :=

 Rb

xg(t)F^α+sβ (t)dt

β

F^−s(x).

(1.3)

The numbers B₁ := supa<x<bB₁(x; α, β) and B_i(s) = supa<x<bB_i(x; α, β, s) (i = 2, 3, 4, 5) are mutually equivalent. The constants in the equivalence relations can depend on α, β and s.

The main result of this paper can be found in Section 2 where we use related ideas to prove some new scales of compactness results for the Hardy operator which complement the classical criterion (see (2.7)). In Section 3, we derive some scales of weight characterizations for the P´olya-Knopp inequality, analogous to that of Theorem 1.1.

2. Compactness Results

In the paper [4] some criteria of the continuity of the Hardy operator H : (Hf )(x) =

Z _x

0

f (t)dt, (2.1)

as a mapping from L^p(v) = L^p(v; 0, b) into L^q(u) = L^q(u; 0, b) have been derived with 0 < b ≤ ∞ and

L^r(w) :=ⁿf ≥ 0; kf kr,w:=^ Z b

0 f^r(t)w(t)dt^1/r < ∞^o.

Here r > 1, 1 < p ≤ q < ∞, and u, v, w are given weight functions (i.e. measur- able and positive a.e. in (0, b)).

Let us denote U (x) =

Z b

x u(t)dt, V (x) = Z x

0 v^1−p0(t)dt, p⁰ = p

p − 1. (2.2) We assume throughout the paper, that

U (x) < ∞ and V (x) < ∞ (2.3) for a.e. x ∈ (0, b). On the other hand it is allowed (and this is in fact the more interesting case) that either

U (0) = ∞ (i.e. u /∈ L¹(0, b)) (2.4) or

V (b) = ∞ (i.e. v^1−p0 ∈ L/ ¹(0, b)). (2.5) The necessary and sufficient condition for the continuity of H is the finiteness of the suprema (on (0, b)) of any of the following functions

A₁(x) := U^1/q(x)V^1/p0(x),

A₁(x; s) :=^R_x^bu(t)V^q(p01−s)(t)dt^1/qV^s(x), s > 0, A₂(x; s) :=^R₀^xu(t)V^q(p01+s)(t)dt^1/qV^−s(x), s > 0, A3(x; s) :=^R₀^xv^1−p0(t)U^p0(1q−s)(t)dt^1/p

0

U^s(x) s > 0, A₄(x; s) :=^R_x^bv^1−p0(t)U^p0(1q+s)(t)dt^1/p

0

U^−s(x) s > 0.

(2.6)

Notice that

(i) A₁(x) = A₁(x;_p¹0) = A₃(x;¹_q);

(ii) A₃(x; s) and A₄(x; s) are the ”dual” expressions to A₁(x; s) and A₂(x; s), respectively;

(iii) A₁(x) is the usual ”Muckenhoupt function” A_M(x) and A₂(x;¹_p) is the

”Persson-Stepanov function” A_{P S}(x) (and for p = q the ”Tomaselli func- tion” A_T(x)).

According to [7, Theorem 7.11], the operator H is compact if and only if

x→0+lim A₁(x) = lim

x→b−A₁(x) = 0 (2.7)

and in [4, Theorems 3.1 and 3.2], it was shown that for 0 < s < ¹_p, H is compact if and only if

x→0+lim A_i(x, s) = lim

x→b−A_i(x, s) = 0, i = 1, 3. (2.8) But since the continuity of H takes place for all s > 0, we can repeat almost literally the proof of Theorems 3.1 and 3.2 in [4] and hence extend the criterion (2.8) to a bigger scale of s, namely to s ∈ (0, ∞) :

Theorem 2.1. The Hardy operator is compact if and only if the function A₁(x; s) and/or the function A₃(x; s) has zero limits at x = 0 and x = b for any s > 0.

An analogue of Theorem 2.1 using the function A2(x; s) and/or A4(x; s) does not hold. This follows from the example given in [4] where − for some particular weights u, v for which H is compact − it was shown that A₂(x;¹_p) tends to zero for x → 0+, but

x→b−lim A₂(x;1 p) 6= 0.

This observation is confirmed also by Example 2.1 below, but first let us mention one obvious result.

Corollary 2.2. Suppose that the weight functions u and v^1−p0 are integrable over the whole interval (0, b) (i.e. u, v^1−p0 ∈ L¹(0, b)). Then the Hardy operator H is compact.

Proof. Since U (0) < ∞ and V (b) < ∞, and since U (b) = 0 and V (0) = 0, we have that A₁(0) = A₁(b) = 0 and the proof follows by using criterion (2.7).

Example 2.1. (i) Suppose that the Hardy operator is compact and v^1−p0 ∈ L¹(0, b), i.e. V (b) < ∞ (and, of course, V (b) > 0). Then

x→b−lim A₂(x; s) =^ Z b

0 u(t)V^q(

1 p0+s)

(t)dt^1/qV^−s(b) and hence

x→b−lim A₂(x; s) 6= 0.

(ii) Suppose that the Hardy operator is compact and u ∈ L¹(0, b), i.e. U (0) <

∞. Then

x→0+lim A₄(x; s) =^ Z _b

0

v^1−p0(t)U^p0(1q+s)(t)dt^1/p

0

U^−s(0) 6= 0.

On the other hand, we can extend Theorem 2.1 to A₂(x; s) supposing that V (b) = ∞, and to A₄(x, s) supposing that U (0) = ∞. But for this purpose, we must introduce some notation and modify the criteria of compactness.

Let us denote

U (y, α) :=

Z α

y u(t)dt, V (β, y) :=

Z y

β v^1−p0(t)dt. (2.9) Hence it is U (x) = U (x, b), V (x) = V (0, x).

Further denote

A₁(y; α, β) := U^1/q(y, α)V^1/p0(β, y), (2.10) so that

A1(x) = A1(x; b, 0).

Compactness according to R. K. Juberg. Let us denote J_α := sup0<x<αA₁(x; α, 0) = sup0<x<αU^1/q(x, α)V^1/p0(0, x),

J^β := supβ<x<bA₁(x; b, β) = supβ<x<bU^1/q(x, b)V^1/p0(β, x). (2.11) Supposing that the Hardy operator H is bounded (i.e. continuous), R. K. Juberg [1] has shown, that H is compact if and only if

α→0+lim J_α+ lim

β→b−J^β = 0. (2.12)

First, let us show that according to (2.3), the assumption about the boundedness of H can be avoided. Indeed, according to (2.12), there exists an α₀> 0 such that J_α0 < ∞, and a β₀ < b such that J^β0 < ∞. Due to (2.3), A₁(x) is continuous on (0, b) and consequently, bounded on any closed interval [α0, β0] ⊂ (0, b) : A1(x) ≤ C₀ for x ∈ [α₀, β₀]. Moreover, for x ∈ (0, α₀]

A₁(x) = U^1/q(x, b)V^1/p0(0, x)

= (U (x, α₀) + U (α₀, b))^1/qV^1/p0(0, x)

≤ U^1/q(x, α₀)V^1/p0(0, x) + U^1/q(α₀, b)V^1/p0(0, x)

≤ J_α0 + U^1/q(α₀, b)V^1/p0(0, α₀) = C₁< ∞,

and similarly we can show that for x ∈ [β₀, b), A₁(x) ≤ C₂ < ∞. Thus H is bounded.

Hence, we have shown that H is compact if and only if (2.12) holds, and using now the equivalence relations described in Theorem 1.1, we are able to rewrite the compactness criteria in terms of the functions A₁(x; s ) and A₃(x, s).

Namely, if we denote

A₁(x; s; α, β) =

Z _α

x

u(t)V^q(p01−s)(β, t)dt

1/q

V^s(β, x), (2.13) so that

A₁(x; s) = A₁(x; s; b, 0),

we obtain from Theorem 1.1 that J_α and J^β are equivalent to J_1,α(s) := sup

0<x<αA₁(x; s; α, 0) and J₁^β(s) := sup

β<x<b

A₁(x; s; b, β) respectively: It suffices to use the equivalence relations mentioned not for the interval (0, b), but for the interval (0, α) and (β, b), respectively.

In view of the criterion (2.12), we immediately have:

Theorem 2.3. The Hardy operator is compact if and only if

α→0+lim J_1,α(s) + lim

β→b−J₁^β(s) = 0 for any s > 0.

A similar Theorem can be formulated using the function A₃(x; s).

If we want to use the functions A₂(x; s) and A₄(x; s) the situation is a little different, since in the expression for A₂(x; s) only the left endpoint (zero) ap- pears explicitly, while in A4(x; s), we have explicitly only the right endpoint b.

Nevertheless, if we proceed analogously as in the foregoing case and denote A₂(x; s; α, β) =

Z _x

α

u(t)V^q(

1 p0+s)

(β, t)dt

1/q

V^−s(β, x) (2.14) so that

A₂(x; s) = A₂(x; s; 0, 0),

we obtain again according to Theorem 1.1 that J_α and J^β are equivalent to J_2,α(s) := sup

0<x<α

A₂(x; s; 0, 0) = sup

0<x<α

A₂(x; s) (2.15)

and

J₂^β(s) := sup

β<x<b

A₂(x; s; β, β),

respectively, where now the parameter α does appear in J2,α(s) only under the suprema sign. The corresponding theorem now states:

Theorem 2.4. The Hardy operator is compact if and only if

α→0+lim J_2,α(s) + lim

β→b−J₂^β(s) = 0 (2.16)

for every s > 0.

Again, a similar theorem could be formulated in terms of A₄(x, s), where now

J_4,α(s) := sup

0<x<α

Z _α

x

v^1−p0(t)U^{p0 1q+s}

(t, α)dt

1/p⁰

U (x, α)^−s, and

J₄^β(s) := sup

β<x<b

Z _b

x

v^1−p0(t)U^{p0 1q+s}

(t, b)dt

!1/p⁰

U^−s(x, b) = sup

β<x<b

A₄(x; s).

But on the other hand, we have also the following assertion:

Theorem 2.5. Suppose that v^1−p0 ∈ L¹(0, b), i.e. V (b) < ∞. Then H is compact if and only if

x→0+lim A₂(x; s) = 0 (2.17)

for every s > 0.

Proof. According to (2.15), J_2,α(s) = sup0<x<αA₂(x; s). If H is compact, then due to (2.16), J_2,α(s) → 0 for α → 0, and, consequently, we have (2.17).

Conversely, if (2.17) holds, then J_2,α(s) → 0 for α → 0. Since for β fixed, V (β, t) is increasing in t, we have V (β, t) ≤ V (β, x) for t ≤ x, and hence

J₂^β(s) = sup

β<x<b

Z _x

β

u(t)V^q(p01+s)(β, t)dt^1/qV^−s(β, x)

≤ sup

β<x<b

Z _b

β

u(t)dt^1/qV^p01+s(β, x)V^−s(β, x)

=^ Z _b

β

u(t)dt^1/qV^1/p0(β, b),

and since V (β, b) < V (b) < ∞, J₂^β(s) → 0 for β → b. Consequently H is compact due to (2.16).

Hence, if v^1−p0 is integrable over the whole interval (0, b), we need for the compactness only one of the limits of A₂(x; s), and similarly it holds:

Theorem 2.6. Suppose that u ∈ L¹(0, b), i.e. U (0) < ∞. Then H is compact if and only if

x→b−lim A₄(x; s) = 0 for every s > 0.

But if v^1−p0 (or u) is not integrable over the whole interval (0, b), i.e., if V (b) = ∞ (or U (0) = ∞), then we need for the compactness of H that both limits of A₂(x; s) (or A₄(x; s) are zero: for x → 0+ and for x → b−. The following theorem is then an analogue of Theorem 2.1 for A₂(x; s), and a similar assertion holds also for A₄(x; s).

Theorem 2.7. Suppose that v^1−p0 ∈ L/ ¹(0, b), i.e. V (b) = ∞. Then the Hardy operator H is compact if and only if the function A₂(x; s) has zero limits at x = 0 and x = b for any s > 0.

Proof. (i) Sufficiency. Due to Theorem 1.1, it is Jα .J2,α= sup

0<x<αA2(x; s).

In [2] it was shown that B₁(x; α, β) .supx<t<bB₂(t; α, β, s) (see estimates (2.4) and (2.5) in [2]). Putting here α = ¹_p, β = _p¹0, we have

A₁(x). sup

x<t<b

A₂(t; s)

and this estimate implies J^β = sup

β<x<b

A₁(x, b, β). sup

β<x<b

A₁(x). sup

β<x<b

A₂(x; s).

If A₂(x; s) → 0 for x → 0₊ and for x → b−, then J_α → 0 for α → 0₊, J^β → 0 for β → b−, and hence H is compact.

(ii) Necessity. Suppose that H is compact. Since A₂(x; s) ≤ J_2,α(s) for 0 < x < α, we have that limx→0+A2(x, s) = 0 due to Theorem 2.4. Hence, it remains to prove that

x→b−lim A₂(x, s) = 0. (2.18)

Using the notation W (x) = V^q(p01+s)(x), we have for β < x < b

A₂(x; s) = Z β

0 u(t)W (t)dt + Z x

β u(t)W (t)dt

!1/q

V^−s(x)

≤



 Z β

0

u(t)W (t)dt

!1/q

+

Z _x

β

u(t)W (t)dt

1/q

V^−s(x)

=: I₁+ I₂. Since W (t) = V^q(

1 p0+s)

(t) = V^q(

1 p0+s)

(0, t) = [V (0, β) + V (β, t)]^q(

1 p0+s)

, we have I₂ =

Z _x

β u(t) [V (0, β) + V (β, t)]^q(

1 p0+s)

dt

1/q

V^−s(0, x) .

Z _x

β u(t) [V (0, β)]^q(

1 p0+s)

dt

1/q

V^−s(0, x) +

Z _x

β

u(t) [V (β, t)]^q(p01+s)dt

1/q

V^−s(0, x)

=

Z _x

β

u(t)dt

1/q

V (0, β)^(p01+s)V^−s(0, x) +

Z _x

β

u(t)V (β, t)^q(p01+s)dt

1/q

V^−s(0, x)

=: I₂₁+ I₂₂.

Since V (0, x) = V (0, β) + V (β, x), it is V (0, x) ≥ V (0, β) as well as V (0, x) ≥ V (β, x). Consequently, it is V^−s(0, x) ≤ V^−s(0, β) and V^−s(0, x) ≤ V^−s(β, x). If we use these estimates in I₂₁ and I₂₂, we obtain

I₂₁≤

Z _x

β

u(t)dt

1/q

V (0, β)^(p01+s)V^−s(0, β)

= U^1/q(β, x)V^1/p0(0, β)

= A₁(β; x, 0) ≤ A₁(β; b, 0) = A₁(β),

I₂₂≤

Z _x

β u(t)V (β, t)^q(

1 p0+s)

dt

1/q

V^−s(β, x)

= A2(x; s; β, β) ≤ J₂^β(s).

Since H is compact, we have A₁(β) → 0 and J₂^β(s) → 0 for β → b−, and hence for a given small ε > 0 we can fix an β ∈ (0, b) such that

I₂ ≤ C₀^A₁(β) + J₂^β(s)^< ε 2. Then we fix an x₀ ∈ (β, b) such that

I₁= Z _β

0

u(t)V^q(p01+s)(t)dt

!1/q

V^−s(x₀) < ε 2

(notice that we assumed V (b) = ∞), and finally A₂(x; s) < ε for x ∈ (x₀, b), i.e.

we have (2.18).

Obviously, in a similar way we can prove:

Theorem 2.8. Suppose that u /∈ L¹(0, b), i.e. U (0) = ∞. Then the Hardy operator H is compact if and only if the function A₄(x; s) has zero limits at x = 0 and x = b for any s > 0.

3. Scales of Weight Characterization for the P´olya-Knopp Inequality In this section we will derive the corresponding scales of conditions for the weighted P´olya-Knopp inequality

Z ∞ 0

 exp

1 x

Z x

0 ln f (t)dt

q

u(x)dx

1/q

≤ C

Z ∞

0 f^p(x)v(x)dx

1/p

, (3.1) both for general measurable functions and for decreasing functions f ≥ 0. We consider the following known results. In [6] it was proved by B. Opic and P.

Gurka:

Theorem 3.1. Let 0 < p ≤ q < ∞. Then the inequality (3.1) holds for all f > 0 if and only if

D_OG(r) = sup

x>0

Z ∞ x

w(t)t^−qrpdt

1/q

x^r−1p < ∞, (3.2)

where r > 1, and

w(t) =

 exp

1 t

Z _t

0

ln 1 v(y)dy

q/p

u(t). (3.3)

In [8] L. E. Persson and V. D. Stepanov proved:

Theorem 3.2. Let 0 < p ≤ q < ∞. Then the inequality (3.1) holds for all f > 0 if and only if

D_{P S} = sup

x>0

Z _x

0

w(t)dt

1/q

x^−1/p< ∞ (3.4)

where w(t) is defined by (3.3).

Our main result in this section reads:

Theorem 3.3. Let 0 < p ≤ q < ∞, 0 < s < ∞, and define D1(s) = sup_x>0^R_x^∞w(t)t^−q(s+1/p)dt^1/qx^s,

D₂(s) = sup_x>0^R₀^xw(t)t^q(s−p1)dt^1/qx^−s, D₃(s) = sup_x>0

 Rx

0

R∞

t w(y)y^−qdy^p

0(¹_q−s)

dt



1 p0

(^R_x^∞w(t)t^−qdt)^s,

D₄(s) = sup_x>0

 R∞

x

R∞

t w(y)y^−qdy^p

0(¹_q+s)

dt

_p^s

(^R_x^∞w(t)t^−qdt)^−s,

(3.5)

with w(t) defined by (3.3). Then the P´olya-Knopp inequality (3.1) holds for all measurable functions f > 0 if and only if any of the quantities D_i(s), i = 1, 2, 3, 4 is finite. Moreover, for the best constant C in (3.1) we have C ≈ D_i(s), i = 1, 2, 3, 4.

Remark 3.1. The conditions (3.2) and (3.4) can be described in the follow- ing way

D_OG= D₁^r−1_p ^ with r > 1, D_{P S} = D₂^1_p^.

Hence Theorem 3.3 generalizes the corresponding results in [6] and [8].

Proof of Theorem 3.3. In Theorem 1.1 we put a = 0, b = ∞, f (x) = w(x)x^−q with w(x) defined by (3.3), g(x) = 1, and choose α = ¹_q, β = _p¹₀. Then the assertion follows from the fact that

D₁(s) = sup_x>0B₂(x,¹_q,_p¹0, s), D₂(s) = sup_x>0B₄(x,¹_q,_p¹₀, s), D₃(s) = sup_x>0B₃(x,¹_q,_p¹₀, s), D₄(s) = sup_x>0B₅(x,¹_q,_p¹₀, s),

are all equivalent to D_OG = sup_x>0B₁(x,¹_q,^r−1_p ) if we take r = sp + 1 in (3.2) (see Theorem 1.1) and the finiteness of (3.2) is necessary and sufficient for the inequality (3.1) to hold according to Theorem 3.1. Moreover, since for the least constant C in (3.1) we have C ≈ D_OG it is clear that C ≈ D_i(s), i = 1, 2, 3, 4 and the proof is complete.

In [2] also the Hardy inequality for decreasing functions f denoted f ↓ was characterized with scales of pairs of equivalent conditions by using Theorem 1.1 and the weight characterization given by E. Sawyer in [9] and by V. D. Stepanov in [10]. Our aim is now to characterize the weighted P´olya-Knopp inequality (3.1) for decreasing functions f , as far as we know this has not yet been done.

In [2] it was proved:

Theorem 3.4. Let 1 < p ≤ q < ∞, then the inequality

Z b 0

1 x

Z x

0 f (t)dt^qu(x)dx^1/q ≤ C^ Z b

0 f^p(x)v(x)dx^1/p (3.6) holds for all f ↓≥ 0 if and only if

A₀(p, q, w, v, s) = sup

x>0

Z ^∞

x w(t)t^−q Z t

0 y^p0V^−p0(y)v(y)dy^q(1−sp

0)/p⁰

dt^1/q

×^ Z _x

0 t^p0V^−p0(t)v(t)dt^s< ∞, (3.7) and

A₁(p, q, w, v, s) = sup

x>0

Z ∞ x

w(t)V^−qp(p0s+1)(t)dt

1/q

V^s(p0−1)(x) < ∞ (3.8)

where

V (t) = Z t

0 v(x)dx. (3.9)

Moreover, if C is the best possible constant in (3.6) then C ≈ A₀+ A₁.

We have now the following result:

Theorem 3.5. Let 0 < p ≤ q < ∞ and v(x) be non-increasing. Then the inequality (3.1) holds for all f ↓ ≥ 0 if and only if

D := D(p, q, w, s)= sup

x>0

Z ∞ x

w(t)t^−q(s+1p)dt^1/qx^s< ∞, s > 0. (3.10) Moreover, if C is the best possible constant in (3.1), then C ≈ D.

Remark 3.2. The assumption that v(x) is non-increasing comes here natural as it was pointed out by Lorentz in [5] that kf k_Λp(v) is a norm if and only if v(x) is non-increasing.

Proof. First we use the fact that as v(x) is non-increasing and if w(x) be defined as in (3.3). then (3.1) is equivalent to

Z ∞ 0

exp^1 x

Z _x

0

ln f (t)dt^qw(x)dx^

1

q ≤ C^

Z ∞ 0

f^p(x)dx^

1

p (3.11)

and also

Z ^∞

0

exp^1 x

Z x

0 ln f (t)dt^q(s+

1 p)

w(x)dx^

1

q ≤ C^

Z ^∞

0 f^ps+1(x)dx^

1

p. (3.12) Note that if s > 0 and 0 < p ≤ q < ∞ we have 1 < ps + 1 ≤ ^q(ps+1)_p < ∞.

Furthermore, by applying Jensen’s inequality, we can obtain inequality (3.12) from the following inequality for all non-increasing functions f (x):

Z ^∞

0

1 x

x

Z

0

f (t)dt^q(s+

1 p)

w(x)dx^

1

q ≤ C^

Z ^∞

0 f^ps+1(x)dx^

1

p (3.13)

Now we take s = _p¹₀ and v(x) = 1 in (3.7) and s = _p0(p¹₀₋₁₎ and v(x) = 1 in (3.8) and find that

A₀^p, q, w, 1, 1 p⁰

= A₁^p, q, w, 1, 1 p⁰(p⁰− 1)

= D^p, q, w, 1 p⁰

.

Moreover, if we replace p by sp + 1 and q by ^q(sp+1)_p , take v(x) = 1 in inequality (3.6) and apply Theorem 3.4, we find that the inequality (3.13) (and thus the upper bound for (3.12)) holds if and only if



D(ps + 1,q(ps + 1)

p , w, ps ps + 1)



ps+1 p

= D(p, q, w, s) < ∞.

For the lower estimate we apply the test function

f (x) = t^−(s+1p)χ_[0,t](x) + (xe)^−(s+1p)χ_[t,∞](x) to (3.11) and the proof follows.

Now, by using again the equivalence theorem (Theorem 1.1) it is possible to generalize Theorem 3.5 as follows:

Theorem 3.6. Let 0 < p ≤ q < ∞, 0 < s < ∞, and let D_i(s), i = 1, 2, 3, 4 be defined by (3.5). Then the inequality (3.1) holds for all f ↓ > 0 if and only if any of the quantities D_i(s), i = 1, 2, 3, 4 is finite. Moreover, for the best constant C in (3.1) we have C ≈ D_i(s), i = 1, 2, 3, 4.

Proof. In Theorem 1.1 we put a = 0, b = ∞, g(x) = 1 and f (x) = w(x)x^−q with w(x) defined by (3.3), and choose α = ¹_q, β = _p¹₀ as in the Proof of Theorem 3.3. Then the assertion follows from the fact that the functions in (3.5) are all equivalent to D = sup_x>0B₁(x,¹_q,_p¹0) defined in (3.10) (see Theorem 1.1), and the finiteness of (3.10) is necessary and sufficient for the inequality (3.1) to hold for all decreasing functions f according to Theorem 3.5. Moreover, since for the least constant C in (3.1) we have C ≈ D it is clear that C ≈ D_i(s), i = 1, 2, 3, 4 and the proof is complete.

Acknowledgments

The research of the second author was partially supported by the grants A1019305 of the Academy of Science of the Czech Republic and 201/03/0671 of the Grant Agency of the Czech Republic.

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[10] V. D. Stepanov, The weighted Hardy’s inequality for nonincreasing functions, Trans. Amer.

Math. Soc., 338:1(1993), 173-186.

Matematical Institute Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, CZECH REPUBLIC.

E-mail: gogatish@math.cas.cz E-mail: kufner@math.cas.cz

Department of Mathematics, Lule˚a University of Technology, SE-971 87 Lule˚a, SWEDEN.

E-mail: larserik@sm.luth.se E-mail: annaw@sm.luth.se