The case 0&lt

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OF MONOTONE FUNCTIONS

AMIRAN GOGATISHVILI AND VLADIMIR D. STEPANOV

Abstract. For a quasilinear operator on the semiaxis a reduction theorem is proved on the cones of monotone functions inL^p−L^q setting for 0< q <∞, 1 ≤p <∞. The case 0< p <1 is also studied for operators with additional properties. In particular, we obtain critera for three-weight inequalities for the Hardy-type operators with Oinarov’ kernel on monotone functions in the case 0< q < p≤1.

1. Introduction

Let R+ := [0,∞). Denote M⁺ the set of all non-negative measurable func- tions on R+ and M^↓ ⊂ M⁺ (M^↑ ⊂ M⁺) the subset of all non-increasing (non- decreasing) functions. For the last two decades the weighted norm L^p − L^q inequalities have extensively been studied. In particular, much attention was paid to the inequalities restricted to the cones of monotone functions, see for in- stance [1], [21], [25], [26], [6], [12], [22], survey [5], the monographs [15], [16] and references given there. At the initial stage the main tool was the Sawyer duality principle [21] (see also [23], [24]), which allowed to reduce an L^p−L^q inequality for monotone functions with 1 ≤ q ≤ ∞,0 < p < ∞ to a more menageable inequality for arbitrary non-negative functions. The case p ≤ q,0 < p ≤ 1 was alternatively characterized in [25], [26], [6], [3]. Later on some direct reduction theorems were found [9], [10] [4] involving the supremum operators which work for the case 0< q < p≤1.

LetT: M⁺ →M⁺ be a positive quasilinear operator such that (i) T(λf) =λT f for all λ≥0 and f ∈M⁺,

2010Mathematics Subject Classification. Primary 26D10; Secondary 26D15, 26D07.

Key words and phrases. Quasilinear operator, integral inequality, Lebesgue space, weight, Hardy operator, monotone functions.

This work was done as part of the research program Approximation Theory and Fourier Analysis at the Centre de Recerca Matematica (CRM), Bellaterra in the Fall semester of 2011.

The authors are grateful to CRM for its support and hospitality. The work of the first author was also partially supported by the grant 201/08/0383 of the Grant Agency of the Czech Republic, by the Institutional Research Plan no. AV0Z10190503 of AS CR. The work of the second author was also partially supported by the Russian Fund for Basic Research (Projects 09-01-00093, 09-01-00586 and 09-II-CO-01-003).

1

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(ii) T(f+g)≤c(T f+T g) for allf, g ∈M⁺with a constantc >0 independent onf and g,

(iii) T f(x)≤cT g(x) for almost everyx∈R⁺, iff(x)≤g(x) for almost every x∈R⁺ with a constant c >0 independent onf and g.

Let v and w be weights, that is non-negative locally integrable functions on R+. The first our result is a reduction of the inequality

Z ∞ 0

(T f(t))^qw(t)dt ¹_q

≤C Z ∞

0

(f(t))^pv(t)dt ¹_p

, f ∈M^↓ (1.1)

to a similar one on M⁺ in the case 0< q <∞,1≤p < ∞(see Teorems 2.1-2.4).

When 0< p≤q < ∞,0< p < 1 we supplement these results in Section 3 by an extension of [3] and [26].

It is well known that the case 0 < q < p ≤ 1 is the most difficult for a characterization of inequalities like (1.1) (see [2], [4], [5], [9], [7] [11], [13], [22]).

We study this case in Section 4 including the three-weight inequality of the form Z ∞

0

Z ∞ x

f(t)u(t)dt q

w(x)dx ¹_q

≤C Z ∞

0

(f(t))^pv(t)dt ¹_p (1.2)

for allf ∈M^↓ and give three alternative reductions and a criterion (see Theorem 4.1) and section 5 contains a characterization of (1.2) for 0 < p, q ≤ ∞ (see Theorems 5.1 and 5.3)

Also we study the inequality

Z ∞ 0

Z x 0

k(x, t)f(t)u(t)dt q

w(x)dx ¹_q

≤C Z ∞

0

(f(t))^pv(t)dt _p¹

, f ∈M^↓, (1.3)

where k(x, t)≥0 is Oinarov’s kernel and give a full description for 0 < p, q <∞ (see Theorems 4.5 and 5.7).

We use signs := and =: for determining new quantities and Z for the set of all integers. For positive functionals F and G we write F G, if F ≤ cG with some positive constant c, which depends only on irrelevant parameters. F ≈ G meansF GF orF =cG. χ_E denotes the characteristic function (indicator) of a set E. Uncertainties of the form 0· ∞,^∞_∞ and ⁰₀ are taken to be zero. We use notations C or C with lower indices for the constants (possibly different in different occasions) in the inequalities like (1.1). stands for the end of proof.

2. Quasilinear operators Put V(t) :=Rt

0v and denote 1 the function onR+ identically equal to 1.

Theorem 2.1. Let 0< q ≤ ∞,1 < p < ∞ and let T: M⁺ →M⁺ be a positive quasilinear operator, satisfying (i)-(iii). Then the inequality (1.1) holds iff the

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following two inequalities are valid:

Z ∞ 0

T

Z ∞ x

h q

w ¹_q

≤C Z ∞

0

h^pV^pv^1−p _p¹

, h∈M⁺ (2.1)

and

Z ∞ 0

(T1)^qw ¹_q

≤C Z ∞

0

v ¹_p

. (2.2)

Proof. Let 0 < q < ∞. Necessity. Let h ∈ M⁺ be integrable on [x,∞) for all x >0. Then f(x) =R∞

x h∈M^↓ and by (1.1) and Hardy’s inequality we have Z ∞

0

T

Z ∞ x

h q

w ¹_q

≤C Z ∞

0

Z ∞ x

h p

v(x)dx ¹_p

C Z ∞

0

h^pV^pv^{1−p 1}_p

. (2.2) follows from (1.1) with f =1.

Sufficiency. Suppose that V(∞) = ∞and f ∈M^↓. Then f(x) = f(x)V(x)

V(x) = Z ∞

x

v V²

f(x)V(x)

≤ Z ∞

x

v V²

Z x 0

f v≤ Z ∞

x

Z t 0

f v

v(t)dt V²(t). Applying (iii) and (2.1) with

h(t) = Z t

0

f v

v(t) V²(t) and applying Hardy’s inequality, we find

Z ∞ 0

(T f)^qw ¹_q

≤C Z ∞

0

Z t 0

f v ^p

v(t)dt V^p(t)

_p¹

C Z ∞

0

f^pv ¹_p

.

IfV(∞)<∞, then by H¨older’s inequality f(x) =

1

V(x) − 1 V(∞)

f(x)V(x) + V(x) V(∞)f(x)

≤ Z ∞

x

v V²

Z x 0

f v+ V^1/p0(x)V^1/p(x) V(∞) f(x)

≤ Z ∞

x

Z t 0

f v

v(t)dt V²(t)

+

R∞

0 f^pv1/p

V^1/p(∞) =:

Z ∞ x

h+λ1.

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Applying (i), (ii), (2.1), (2.2) and Hardy’s inequality, we obtain Z ∞

0

(T f)^qw ¹_q

Z ∞

0

Z ∞ x

h q

w(x)dx ¹_q

+λ Z ∞

0

(T1)^qw(x)dx ¹_q

C

Z ∞ 0

Z t 0

f v p

v(t)dt V^p(t)

¹_p +

Z ∞ 0

f^pv _p¹!

C Z ∞

0

f^pv ¹_p

.

The case q =∞is treated similarly.

To study the case p = 1 we suppose that an operator T: M⁺ →M⁺ satisfies the following axiom:

(iv) If {f_n} ⊂ M^↓ and f_n(x) ↑ f(x) ∈ M^↓ for almost every x ∈ R+, then T f_n(x)↑T f(x) for almost every x∈R+.

We also need the following simple case of ([23], Lemma 1.2).

Lemma 2.2. Let f ∈M^↓. Then there exist the sequence of non-negative finitely supported integrable functions {h_n} ⊂M⁺ such, that the functions

fn(x) := R∞

x hn(s)ds are increasing with respect to n for any x > 0 and f(x) =

n→∞lim R∞

x h_n(y)dy for almost all x >0.

Theorem 2.3. Let 0< q <∞, p= 1 and let T: M⁺ →M⁺ be a positive quasi- linear operator, satisfying(i)-(iv). Then the inequality (1.1) holds iff the inequal- ity (2.1) is valid.

Proof. The necessity is obvious. For sufficiency we suppose that f ∈M^↓ and by Lemma 2.2 there exists {h_n} ⊂L¹(R+) such that

fn(x) :=

Z ∞ x

hn(y)dy ↑f(x).

Then by (i)-(iv) and Fatou’s lemma Z ∞

0

(T f)^qw ¹_q

Z ∞

0

n→∞lim T f_n q

w ¹_q

≤ lim

n→∞

Z ∞ 0

(T f_n)^qw ¹_q

= lim

n→∞

Z ∞ 0

T

Z ∞ x

hn

q

w ¹_q

≤C lim

n→∞

Z ∞ 0

h_nV

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=C lim

n→∞

Z ∞ 0

f_nv

=C Z ∞

0

f v.

Analogously we reduce the inequality for non-decresing functions of the form Z ∞

0

(T f(t))^qw(t)dt ¹_q

≤C Z ∞

0

(f(t))^pv(t)dt ¹_p

, f ∈M^↑ (2.3)

provided the axiom (iv) is replaced by

(iv’) If {f_n} ⊂ M^↑ and f_n(x) ↑ f(x) ∈ M^↑ for almost every x ∈ R+, then T f_n(x)↑T f(x) for almost everyx∈R+.

PutV∗(t) := R∞ t v.

Theorem 2.4. Let 0 < q < ∞,1 < p <∞ and let T: M⁺ → M⁺ be a positive quasilinear operator, satisfying (i)-(iii). Then the inequality (2.3) holds iff (2.2) and the inequalitiy

Z ∞ 0

T

Z x 0

h q

w ¹_q

≤C Z ∞

0

h^pV_∗^pv^{1−p 1}_p

, h∈M⁺ (2.4)

are valid.

Theorem 2.5. Let 0 < q < ∞, p = 1 and let T: M⁺ → M⁺ be a positive quasilinear operator, satisfying (i)-(iii) and (iv’). Then the inequality (2.3) holds iff the inequalities (2.4) and (2.2) are valid.

3. The case 0< p≤q <∞ Letf ∈M^↓. Then there exist {x_n} ⊂R+ such that

f(x)≈X

n

2⁻ⁿχ_[0,xn](x)

= X

n:xn≥x

2⁻ⁿχ_[0,xn](x)

= Z

[x,∞)

X

n

2⁻ⁿδ_xn(s)

! ds

=:

Z

[x,∞)

h(s)ds, (3.1)

whereδ_t(s) is the Dirac delta-function at a pointt. Observe that [f(x)]^r ≈ X

n

2⁻ⁿχ_[0,xn](x)

!r

≈X

n

2^−nrχ_[0,xn](x), r >0.

(3.2)

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Theorem 3.1. Let 0 < p ≤ q < ∞ and let T: M⁺ → M⁺ be a positive quasi- linear operator, satisfying (i)-(iii), such that

T X

n

f_n

!

X

n

[T f_n]^p

!¹_p (3.3)

for any f_n ≥ 0. Then the inequality (1.1) is equivalent to the validity one of the following condtions:

Z ∞ 0

Z ∞ 0

T χ_[0,s](x)p

h(s)ds ^q_p

w(x)dx

!^p_q

≤C₂^p Z ∞

0

hV, h ∈M⁺, (3.4)

Z ∞ 0

sup

s>0

T χ_[0,s](x)f(s) q

w(x)dx ¹_q

≤C₃ Z ∞

0

f^pv ¹_p

, f ∈M^↓, (3.5)

Z ∞ 0

sup

s>0

T χ_[0,s](x)pZ ∞ s

h ^q_p

w(x)dx

!^p_q

≤C₄^p Z ∞

0

hV, h ∈M⁺, (3.6)

or

D := sup

t>0

Z ∞ 0

T χ_[0,t](x)q

w(x)dx ¹_q

V^−1p(t)<∞.

(3.7) Moreover,

C ≈C₂ =D≈C₃ =C₄. (3.8)

Proof. (3.5)⇔(3.6) follows by Lemma 2.2 with equalityC₃ =C₄. (3.5) =⇒ (3.7) follows by applying (3.5) to a test function f_t(s) :=χ_[0,t](s), t >0. Similarly, we obtain (1.1) =⇒ (3.7). From the properties (i)-(iii) we find, that for all s >0

T f(x)≥T(χ_[0,s]f)(x)≥T χ_[0,s](x)f(s) and (1.1) =⇒ (3.5) follows. Let

k(x, s) :=

T χ_[0,s](x)p

and

Kh(x) :=

Z ∞ 0

k(x, s)h(s)ds.

Then (3.4) is equivalent to the boundedness K: L¹_V →L

q

wp and C₂^p =kKk

L¹_V→L

q p w

=D^p.

Let us show that (3.4) =⇒ (1.1). It follows from (3.2) and (3.3), that

T f^p1

(x)≈



T X

n

2⁻ⁿχ[0,xn](x)

!¹_p

(x)

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≈T X

n

2^−npχ_[0,xn](x)

! (x)

X

n

2⁻ⁿ

T χ[0,xn](x)p

!¹_p . Observe that (1.1) is equivalent to

Z ∞ 0

T f^1pq

w ^p_q

≤C^p Z ∞

0

f v, f ∈M^↓.

Now, using (3.2), we find Z ∞

0

T f^1pq

w ^p_q



 Z ∞

0

X

n

2⁻ⁿ

T χ_[0,xn](x)p

!^q_p

w(x)dx





p q

= Z ∞

0

Z ∞ 0

T χ_[0,s](x)p

h(s)ds ^q_p

w(x)dx

!^p_q

≤C₂^p Z ∞

0

hV =C₂^pX

n

2⁻ⁿV(x_n)

≈C₂^pX

n

2⁻ⁿ Z

[xn,xn+1)

v ≈C₂^p Z ∞

0

f v.

Consequently, C C₂ and (3.8) follows.

Similarly, we characterize the case of non-decreasing functions.

Theorem 3.2. Let 0< p ≤q <∞and let T: M⁺ →M⁺ be a positive quasilin- ear operator, satisfying (i)-(iii) and (3.3). Then the inequality (2.3) is equivalent to one of the following conditions:

(3.9)

Z ∞ 0

T χ[s,∞)(x)p

h(s)ds^q_p

w(x)dx ^p_q

≤C₂^p Z ∞

0

hV∗, h∈M⁺,

(3.10)

Z ∞ 0

sup

s>0

T χ[s,∞)(x)f(s) q

w(x)dx ¹_q

≤C3

Z ∞ 0

f^pv _p¹

, f ∈M^↓,

(3.11)

Z ∞ 0

sup

s>0

T χ[s,∞)(x)pZ s 0

h ^q_p

w(x)dx

!^p_q

≤C₄^p Z ∞

0

hV, h∈M⁺,

(3.12) D∗ := sup

t>0

Z ∞ 0

T χ_[t,∞)(x)q

w(x)dx ¹_q

V⁻

1

∗ p(t)<∞.

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Moreover,

(3.13) C ≈C2 =D∗ ≈C3 =C4. Now we study the converse inequality

(3.14)

Z ∞ 0

f^qw ¹_q

≤C Z ∞

0

(T f)^pv ¹_p

, f ∈M^↓ Put W(t) :=Rt

0w, W∗(t) :=R∞ t w.

Theorem 3.3. Let 0< p≤q <∞ and let T: M⁺→M⁺ be a positive quasilin- ear operator, satisfying (i)-(iii), such that

(3.15) X

n

[T fn]^q

!¹_q

T X

n

fn

!

for any f_n ≥ 0. Then the inequality (3.14) is equivalent to the validity of the inequality

(3.16) Z ∞

0

hW ≤C₂^q Z ∞

0

Z ∞ 0

T χ_[0,s](x)p

h(s)ds ^p_q

v(x)dx

!^q_p

, h∈M⁺,

or

(3.17) D:= sup

t>0

W^1q(t) Z ∞

0

T χ_[0,t](x)p

v(x)dx ⁻_p¹

<∞.

Moreover,

(3.18) C≈C₂ =D.

Proof. The implication (3.14) =⇒ (3.17) is clear. Let us show (3.17) =⇒ (3.16).

By Minkowskii’s inequality we have Z ∞

0

hW ≤D^q Z ∞

0

Z ∞ 0

T χ_[0,t](x)p

v(x)dx ^q_p

h(t)dt

≤D^q Z ∞

0

v(x) Z ∞

0

T χ_[0,t](x)q

h(t)dt ^p_q

dx

!^q_p .

Now, (3.14) is equivalent to (3.19)

Z ∞ 0

f w ≤C Z ∞

0

T f^1qp

v ^q_p

, f ∈M^↓

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Using (3.16), (3.1), (3.2) and (3.15), we write Z ∞

0

f w= Z ∞

0

hW

≤C₂^q Z ∞

0

Z ∞ 0

T χ_[0,s](x)q

h(s)ds ^p_q

v(x)dx

!^q_p

=C₂^q



 Z ∞

0

X

n

2⁻ⁿ

T χ[0,xn](x)q

!^p_q

v(x)dx





q p

=C₂^q



 Z ∞

0

X

n

h T

2^−nqχ_[0,xn]

(x)iq!^p_q

v(x)dx





q p

≤C₂^q Z ∞

0

"

T X

n

2^−nqχ_[0,xn]

! (x)

#p

v(x)dx

!^q_p

≈C₂^q



 Z ∞

0



T X

n

2⁻ⁿχ_[0,xn]

!¹_q (x)





p

v(x)dx





q p

≈C₂^q Z ∞

0

T f^1qp

v ^q_p

and (3.18) follows.

Similarly we characterize the inequality (3.20)

Z ∞ 0

f^qw ¹_q

≤C Z ∞

0

(T f)^pv ¹_p

, f ∈M^↑

Theorem 3.4. Let 0 < p ≤ q < ∞ and let T: M⁺ → M⁺ be a positive quasi- linear operator, satisfying (i)-(iii) and (3.15). Then the inequality (3.20) is valid iff

(3.21)

Z ∞ 0

hW_∗ ≤C₂^q Z ∞

0

T χ_[s,∞)(x)p

h(s)ds^p_q

w(x)dx _p^q

, h∈M⁺, or

(3.22) D∗ := sup

t>0

W

1 q

∗ (t) Z ∞

0

T χ[t,∞)(x)p

v(x)dx −¹

p

<∞.

Moreover,

(3.23) C ≈C₂ =D∗.

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Remark 3.5. Let T be an integral operator

(3.24) T f(x) :=

Z ∞ 0

k(x, y)f(y)dy

with a non-negative kernel. Then the condition (3.3) is valid for all p ∈ (0,1]

and by Theorems 3.1 and 3.2 we obtain ([26], Theorem 4.1), ([18], Theorem 2.1 (a)) and ([3], Theorem 1). Analogously, the condition (3.15) holds for all q ≥ 1 and by Theorems 3.3 and 3.4 we obtain an extension of ([26], Theorem 4.2) ([18], Theorem 2.1 (b)) for a larger interval.

4. The case 0< q < p ≤1 Letu, v and wbe weights. Denote V(t) := Rt

0 v, W(t) :=Rt

0 w, U(y, x) := Ry x u.

For simplicity we suppose that 0 < V(t) < ∞,0< W(t) < ∞ for all t > 0 and V(∞) = ∞, W(∞) = ∞.

Theorem 4.1. Let0< q < p≤1,1/r:= 1/q−1/p. The following are equivalent:

(4.1)

Z ∞ 0

Z ∞ x

f u q

w(x)dx ¹_q

≤C₁ Z ∞

0

f^pv ¹_p

, f ∈M^↓,

(4.2)

Z ∞ 0

Z ∞ x

U^p(y, x)h(y)dy ^q_p

w(x)dx

!^p_q

≤C₂^p Z ∞

0

hV, h∈M⁺,

(4.3)

Z ∞ 0

sup

y≥x

U^p(y, x) Z ∞

x

h ^q_p

w(x)dx

!^p_q

≤C₃^p Z ∞

0

hV, h∈M⁺,

(4.4)

Z ∞ 0

sup

y≥x

U(y, x)f(y) q

w(x)dx ¹_q

≤C₄ Z ∞

0

f^pv ¹_p

, f ∈M^↓,

(4.5) B^r := sup

{x_k}

X

k∈Z

Z xk+1

x_k

U^q(x_k+1, y)w(y)dy ^r_q

V^−rp(x_k+1)<∞.

Moreover,

(4.6) C₁ ≈C₂ ≈C₃ =C₄ ≈B.

Proof. Observe first, that (4.4)⇐⇒(4.5) follows from ([27], Theorem 4.4).

(4.4)=⇒(4.3) is obvious and (4.3)=⇒(4.4) follows by applying Lemma 2.2 and Fatou’s lemma. For any f ∈M^↓

Z ∞ x

f u≥sup

y≥x

Z y x

f u≥sup

y≥x

U(y, x)f(y).

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Hence, (4.1) =⇒ (4.4). The inequality (4.1) is equivalent to (4.7)

Z ∞ 0

Z ∞ x

f^1pu q

w ^p_q

≤C₁^p Z ∞

0

f v ¹_p

, f ∈M^↓, Letf(x) = R∞

x h. Then by Minkowskii’s inequality Z ∞

x

f^p1u= Z ∞

x

Z ∞ z

h _p¹

u(z)dz ≤ Z ∞

x

U^p(y, x)h(y)dy ¹_p

and (4.2) =⇒ (4.1) follows by Lemma 2.2.

Thus, the only (4.3) =⇒ (4.2) remains to prove. To this end let us denote the left hand sides of (4.2) and (4.3) as A^pq and B^pq, respectively. Suppose, that (4.3) is true and denote {x_n} such a sequence, that W(x_n) = 2ⁿ, n ∈ Z. Put

∆_n := [x_n, x_n+1). Then A:=X

n

Z

∆n

Z ∞ x

U^p(y, x)h(y)dy ^q_p

w(x)dx

≈X

n

2ⁿ Z ∞

xn

U^p(y, x)h(y)dy ^q_p

=X

n

2ⁿ Z ∞

xn

U^p(y, x_n)h(y)dy ^q_p

=X

n

2ⁿ

∞

X

i=n

Z

∆i

U^p(y, x_n)h(y)dy

!^q_p

≈X

n

2ⁿ

∞

X

i=n

Z

∆i

U^p(y, x_i)h(y)dy

!^q_p

+X

n

2ⁿ

∞

X

i=n+1

U^p(x_i, x_n) Z

∆i

h(y)dy

!^q_p

=:A₁+A₂. Applying well known equivalence

(4.8) X

n

2ⁿ

∞

X

i=n

a_i

!s

≈X

n

2ⁿa^s_n

valid for any sequence {a_n} of non-negative numbers and s >0, we obtain

(4.9) A₁ ≈X

n

2ⁿ Z

∆n

U^p(y, x_n)h(y)dy ^q_p

CRMPreprintSeriesnumber1067

By Jensen’s inequality and (4.8)

A₂ :=X

n

2ⁿ

∞

X

i=n+1 i−1

X

j=n

U(x_j+1, x_j)

!p

Z

∆i

h(y)dy

!^q_p

≤X

n

2ⁿ

∞

X

i=n+1 i−1

X

j=n

U^p(x_j+1, x_j) Z

∆i

h(y)dy

!^q_p

=X

n

2ⁿ

∞

X

j=n

U^p(x_j+1, x_j) Z ∞

xj+1

h(y)dy

!^q_p

≈X

n

2ⁿ

U^p(x_n+1, x_n) Z ∞

xn+1

h(y)dy ^q_p

. (4.10)

Similarly, for the constant B we have

B :=

Z ∞ 0

sup

y≥x

U^p(y, x) Z ∞

y

h ^q_p

w(x)dx

=X

n

Z

∆n

sup

y≥x

U^p(y, x) Z ∞

y

h ^q_p

w(x)dx

≈X

n

2ⁿ

sup

y≥xn

U^p(y, x_n) Z ∞

y

h ^q_p

=X

n

2ⁿ

sup

i≥n

sup

y∈∆i

U^p(y, xn) Z ∞

y

h _p^q

≈X

n

2ⁿ

sup

i≥n

sup

y∈∆_i

U^p(y, x_i) Z ∞

y

h ^q_p

+X

n

2ⁿ

sup

i≥n+1

U^p(x_i, x_n) Z ∞

xi

h ^q_p

≈X

n

2ⁿ

sup

y∈∆n

U^p(y, x_n) Z ∞

y

h ^q_p

+X

n

2ⁿ

sup

i≥n+1

U^p(x_i, x_n) Z ∞

xi

h ^q_p

=:B₁+B₂.

CRMPreprintSeriesnumber1067

Here we applied P

n2ⁿ sup_i≥na_is

≈ P

n2ⁿa^s_n, valid for any sequence a_n ≥ 0 and s >0. Suppose, that (4.3) holds. Then

(4.11) B_i C₃^q

Z ∞ 0

hV ^q_p

, i= 1,2.

By (4.10)

(4.12) A₂ B₂ C₃^q

Z ∞ 0

hV ^q_p

.

By H¨older’s inequality A1 ≈X

n

2ⁿ Z

∆n

U^p(y, xn)V⁻¹(y)h(y)V(y)dy ^q_p

≤X

n

2ⁿ

sup

y∈∆_n

U^p(y, x_n)V⁻¹(y) ^q_pZ

∆n

hV _p^q

≤ X

n

2^nrq

sup

y∈∆n

U^p(y, xn)V⁻¹(y) ^r_p!^q_r

X

n

Z

∆n

hV

!^q_p

=:D^q Z ∞

0

hV ^q_p

It follows from (4.11), that

(4.13) X

n

2ⁿ

sup

y∈∆n

U^p(y, x_n) Z xn+1

y

h ^q_p

C₃^q Z ∞

0

hV ^q_p

.

LetH_n: L¹_V[∆_n]→L^∞[∆_n] be operator of the form H_nh(y) :=U^p(y, x_n)

Z xn+1

y

h.

Then

d_n :=kH_nk_L¹

V[∆n]→L^∞[∆n] = sup

y∈∆_n

U^p(y, x_n)V⁻¹(y).

Leth_n ∈L¹_V[∆_n] be such that sup

y∈∆n

U^p(y, x_n)V⁻¹(y)≥ d_n 2

Z

∆n

h_nV.

Then by (4.13)

C₃^qsup

h≥0

P

n2ⁿ

sup_y∈∆nU^p(y, x_n)Rxn+1

y h_p^q R∞

0 hV_p^q

CRMPreprintSeriesnumber1067

≥ sup

h=P

nanhn

P

n2ⁿa

q p

n

sup_y∈∆nU^p(y, x_n)Rxn+1

y h^q_p P

na_nR

∆nhV^q_p

sup

h=P

nanhn

P

n2ⁿd

q p

n

an

R

∆nhV ^q_p

P

nan

R

∆nhV

^q_p =D^q. Hence, DC₃ and

A₁ D^q Z ∞

0

hV _p^q

C₃^q Z ∞

0

hV ^q_p

.

This and (4.12) imply (4.3) =⇒ (4.2).

Symmetric version of the previous theorem is the following.

Theorem 4.2. Let 0< q < p≤1. The following are equivalent:

Z ∞ 0

Z x 0

f u q

w(x)dx ¹_q

≤C¯₁ Z ∞

0

f^pv ¹_p

, f ∈M^↑, (4.14)

Z ∞ 0

Z x 0

U^p(x, y)h(y)dy ^q_p

w(x)dx

!^p_q

≤C¯₂^p Z ∞

0

hV∗, h∈M⁺,

Z ∞ 0

sup

0<y≤x

U^p(x, y) Z x

0

h ^q_p

w(x)dx

!^p_q

≤C¯₃^p Z ∞

0

hV_∗, h∈M⁺,

Z ∞ 0

sup

0<y≤x

U(x, y)f(y) q

w(x)dx ¹_q

≤C¯₄ Z ∞

0

f^pv _p¹

, f ∈M^↑,

B¯^r:= sup

{x_k}

X

k∈Z

Z xk+1

x_k

U^q(y, x_k, y)w(y)dy ^r_q

V⁻

r p

∗ (x_k)<∞.

Moreover,

C¯₁ ≈C¯₂ ≈C¯₃ = ¯C₄ ≈B¯. Remark 4.3. The result of Theorem 4.2 supplements [12].

Definition 4.4. A measurable function k(x, y)≥ 0 on {(x, y) : x ≥y ≥ 0}, we name Oinarov kernel, k(x, y) ∈ O, if there exist a constant D ≥1, independent of x, y and z such, that

D⁻¹(k(x, z) +k(z, y))≤k(x, y)≤D(k(x, z) +k(z, y)) for all x≥z ≥y≥0.

CRMPreprintSeriesnumber1067

Letk(x, z)≥0 be a measurable kernel. Put K(x, y) =

Z y 0

k(x, z)u(z)dz.

Theorem 4.5. Let 0< q < p≤1,1/r:= 1/q−1/p. Let k(x, y) be a continuous Oinarov kernel. The following inequalities are equivalent:

Z ∞ 0

Z x 0

k(x, y)f(y)u(y)dy q

w(x)dx ¹_q

≤C₁ Z ∞

0

f^pv ¹_p

, f ∈M^↓, (4.15)

Z ∞ 0

Z x 0

K^p(x, y)h(y)dy+K^p(x, x) Z ∞

x

h(y)dy ^q_p

w(x)dx

!^p_q (4.16)

≤C₂^p Z ∞

0

hV, h ∈M⁺,

Z ∞ 0

sup

0<y≤x

K^p(x, y) Z ∞

y

h ^q_p

w(x)dx

!^p_q

≤C₃^p Z ∞

0

hV, h ∈M⁺, (4.17)

Z ∞ 0

sup

0<y≤x

K(x, y)f(y) q

w(x)dx ¹_q

≤C₄ Z ∞

0

f^pv ¹_p

, f ∈M^↓, (4.18)

B^r := sup

{xk}

X

k∈Z

Z xk+1

xk

K^q(y, x_k)w(y)dy ^r_q

V^−rp(x_k)<∞.

(4.19)

Moreover,

(4.20) C1 ≈C2 ≈C3 =C4 ≈B.

Proof. We will prove the following implications (4.16) =⇒(4.15) =⇒ (4.18)⇐⇒

(4.17) =⇒ (4.19) =⇒ (4.16).

The inequality (4.15) is equivalent to (4.21)

Z ∞ 0

Z x 0

k(x, y)f^1p(y)u(y) q

w ^p_q

≤C₁^p Z ∞

0

f v ¹_p

, f ∈M^↓, Letf(x) = R∞

x h. Then by Minkowskii’s inequality Z x

0

k(x, z)f^p1(z)u(z)dz = Z x

0

Z ∞ z

h ¹_p

k(x, z)u(z)dz

CRMPreprintSeriesnumber1067

≤ Z ∞

0

Z x 0

χ(z,∞)(y)k(x, z)u(z)dz p

h(y)dy ¹_p

≈ Z x

0

K^p(x, y)h(y)dy+K^p(x, x) Z ∞

x

h(y)dy ¹_p

and (4.16) =⇒ (4.15) follows by Lemma 2.2.

For any f ∈M^↓ Z x

0

k(x, z)f(z)u(z)dz ≥ sup

0<y≤x

Z y 0

k(x, z)u(z)dzf(y)≥ sup

0<y≤x

K(x, y)f(y).

Hence, (4.15) =⇒ (4.18). (4.18) =⇒ (4.17) is obvious and (4.17) =⇒ (4.18) follows by applying Lemma 2.2 and Fatou’s lemma.

Suppose, that (4.17) is true and let {x_n} ⊂ (0,∞) be an increasing sequence.

For any k ∈ Z, let ε_k ∈ (x_k, x_k+1) be such that V(ε_k) ≤ 2V(x_k) and for any sequence {a_k} ⊂ (0,∞) of positive numbers we define the function h(x) :=

P

k∈Z a_k

xk−εkχ_(xk,εk)(x). If we put the function in the inequality (4.17), we get

(4.22) X

k∈Z

a

q p

k

Z xk+1

xk

K^q(x, x_k)w(x)dx

!^p_q

≤2C₃^qX

k∈Z

a_kV(x_k) and by the Landau theorem it implies BC₃.

Thus, the only (4.19) =⇒ (4.16) it remains to prove. Using the definition of Oinarov’s kernel, we see that

K(x, y)≈k(x, y) Z y

0

u(z)dz+ Z y

0

k(y, z)u(z)dz =k(x, y)U(y) +K(y, y) and it implies, that (4.16) is equivalent to the following three inequalities:

Z ∞ 0

Z x 0

k(x, y)^pU(y)^ph(y)dy ^q_p

w(x)dx

!^p_q

≤C₂^p Z ∞

0

hV, h∈M⁺, (4.23)

Z ∞ 0

Z x 0

K(y, y)^ph(y)dy ^q_p

w(x)dx

!^p_q

≤C₂^p Z ∞

0

hV, h∈M⁺, (4.24)

Z ∞ 0

Z ∞ x

h(y)dy _p^q

K(x, x)^qw(x)dx

!^p_q

≤C₂^p Z ∞

0

hV, h∈M⁺, (4.25)

By [17, Theorem 5] (4.23) holds if and only if B₁^r := sup

{x_k}

X

k∈Z

Z xk+1

xk

k(y, x_k)^qw(y)dy ^r_q

sup

y∈(xk−1,xk)

U(y)^rV^−rp(y)<∞ (4.26)

B₂^r := sup

{xk}

X

k∈Z

Z xk+1

xk

w(y)dy ^r_q

sup

y∈(xk−1,xk)

k^r(xk, y)U(y)^rV^−rp(y)<∞ (4.27)

CRMPreprintSeriesnumber1067

as well as (4.24) holds if and only if B^r₃ := sup

{x_k}

X

k∈Z

Z xk+1

xk

w(y)dy ^r_q

sup

y∈(x_k−1,xk)

K(y, y)^rV^−rp(y)<∞ and by the dual form of [17, Theorem 5] (4.25) holds if and only if

B₄^r := sup

{x_k}

X

k∈Z

Z xk

xk−1

K(y, y)^qw(y)dy

!^r_q

V^−rp(x_k)<∞ Lety_k∈(xk−1, x_k) be such that

sup

y∈(xk−1,x_k)

U(y)^rV^−rp(y) = U(y_k)^rV^−rp(y_k).

Then

X

k∈Z

Z xk+1

xk

k(y, x_k)^qw(y)dy ^r_q

sup

y∈(x_k−1,xk)

U(y)^rV^−pr(y) X

k∈Z

Z yk+2

yk

K(y, y_k)^qw(y)dy ^r_q

V^−pr(y_k) X

k∈Z

Z y2k+2

y_2k

K(y, y_2k)^qw(y)dy ^r_q

V^−rp(y_2k)

+X

k∈Z

Z y_2k+3 y2k+1

K(y, y_2k+1)^qw(y)dy

!^r_q

V^−rp(y_2k+1)B^r. Therefore,

B1 B. Letyk∈(xk−1, xk) be such that

sup

y∈(x_k−1,xk)

k^r(x_k, y)U(y)^rV^−rp(y) =k^r(x_k, y_k)U(y_k)^rV^−pr(y_k).

Then

X

k∈Z

Z xk+1

xk

w(y)dy ^r_q

sup

y∈(xk−1,x_k)

k^r(x_k, y)U(y)^rV^−pr(y) X

k∈Z

Z yk+2

yk

K(y, y_k)^qw(y)dy ^r_q

V^−pr(y_k) X

k∈Z

Z y2k+2

y2k

K(y, y2k)^qw(y)dy ^r_q

V^−rp(y2k)

CRMPreprintSeriesnumber1067

+X

k∈Z

Z y2k+3

y_2k+1

K(y, y_2k+1)^qw(y)dy

!^r_q

V^−pr(y_2k+1)B^r. Hence,

B₂ B. Let yk ∈(xk−1, xk) be such that

sup

y∈(xk−1,xk)

K^r(y, y)U(y)^rV^−rp(y) = K(yk, yk)^rV^−rp(yk) Then

X

k∈Z

Z xk+1

xk

w(y)dy ^r_q

sup

y∈(xk−1,x_k)

K(y, y)^rV^−rp(y) X

k∈Z

Z yk+2

yk

K(y, y_k)^qw(y)dy ^r_q

V^−rp(y_k) X

k∈Z

Z y2k+2

y2k

K(y, y_2k)^qw(y)dy ^r_q

V^−rp(y_2k)

+X

k∈Z

Z y2k+3

y2k+1

K(y, y_2k+1)^qw(y)dy

!^r_q

V^−pr(y_2k+1)B^r. Consequently,

B₃ B

Since K(y, y) K(y, xk−1), y ∈ (xk−1, x_k), we have that B₄ B. Combining the above upper bounds we conclude that C₂ B and finish the proof.

Analogously, we obtain the dual version of the previous theorem.

Theorem 4.6. Let 0 < q < p≤1,1/r := 1/q−1/p and k(x, y) is a continuous Oinarov kernel and K∗(y, x) = R∞

y k(z, x)u(z)dz. Then the following inequalities are equivalent:

Z ∞ 0

Z ∞ x

k(y, x)f(y)u(y)dy q

w(x)dx ¹_q

≤C₁ Z ∞

0

f^pv ¹_p

, f ∈M^↑,

Z ∞ 0

Z ∞ x

K_∗^p(y, x)h(y)dy+K_∗^p(x, x) Z x

0

h(y)dy _p^q

w(x)dx

!^p_q

≤C₂^p Z ∞

0

hV∗, h∈M⁺,

Z ∞ 0

sup

y≥x

K_∗^p(y, x) Z y

0

h ^q_p

w(x)dx

!^p_q

≤C₃^p Z ∞

0

hV∗, h∈M⁺,