Lp−Lq–boundedness of the mapf →w(x)Rb(x) a(x)k(x, y)f(y)v(y)dy is described by different types of criteria expressed in terms of given parameters 0&lt

Banach J. Math. Anal. 4 (2010), no. 1, 28–52

B

J

M

A

www.emis.de/journals/BJMA/

ON BOUNDEDNESS OF A CERTAIN CLASS OF HARDY–STEKLOV TYPE OPERATORS IN LEBESGUE

SPACES

V. D. STEPANOV^1∗ AND E. P. USHAKOVA²

Dedicated to Professor Lars-Erik Persson on the occasion of his 65th birthday Communicated by L. Maligranda

Abstract. Lp−Lq–boundedness of the mapf →w(x)Rb(x)

a(x)k(x, y)f(y)v(y)dy is described by different types of criteria expressed in terms of given parameters 0< p, q < ∞,strictly increasing boundaries a(x) andb(x), locally integrable weight functionsv, w and a positive continuous kernel k(x, y) satisfying some growth conditions.

1. Introduction and preliminaries Let 0 < p < ∞, kfk_p := R∞

0 |f(x)|^pdx1/p

and L_p denote the Lebesgue space of all measurable functions on R⁺ := [0,∞) such that kfk_p < ∞. Here and throughout the paper the mark := is applied for introducing new notations and quantities. For the same purposes we make use of the mark =:.

Date: Received: 1 December 2009; Accepted: 21 January 2010.

∗ Corresponding author.

2000Mathematics Subject Classification. Primary 26D10; Secondary 26D15, 26D07.

Key words and phrases. Integral operators, Lebesgue spaces, weights, boundedness.

The work was partially supported by the Russian Foundation for Basic Research (Projects 09-01-00093, 09-01-98516, 07-01-00054 and 09-01-98516-p vostok a) and by the Far-Eastern Branch of the Russian Academy of Sciences (Projects 09-I-OMH-02 and 09-II-CO-01-003).

28

Assume w, v be locally integrable non-negative weight functions. We study L_p−L_q boundedness of Hardy-Steklov type operator

Kf(x) :=w(x) Z b(x)

a(x)

k(x, y)f(y)v(y)dy (1.1)

with border functions a(x) andb(x) satisfying the conditions

(i) a(x) and b(x) are differentiable and strictly increasing on (0,∞);

(ii) a(0) =b(0) = 0, a(x)< b(x) for 0< x < ∞, a(∞) = b(∞) =∞, (1.2) and a continuous kernel k(x, y) > 0 on R := {(x, y) : x >0, a(x)< y < b(x)}

satisfying at least one of two generalized Oinarov’s conditions O_b and O_a. Definition 1.1. A kernel k(x, y) is satisfying Oinarov’s condition O_b if there exists a constant D₁ ≥1 independent on x, y, z such that for z ≤x and a(x)≤ y≤b(z) we have

D⁻¹₁ k(x, y)≤k(x, b(z)) +k(z, y)≤D₁k(x, y). (1.3) Definition 1.2. We say k(x, y) ∈ O_a if there exists an independent on x, y, z constant D2 ≥1 such that for x≤z, a(z)≤y≤b(x) it holds that

D⁻¹₂ k(x, y)≤k(x, a(z)) +k(z, y)≤D₂k(x, y). (1.4) Operators of the type (1.1) have been studied by many authors (see, for in- stance, [1] – [4], [5, 9, 11]). In the limiting cases a(x) = 0 or b(x) = ∞ the operator (1.1) is reduced to the Hardy type operators with only one variable boundary a(x) or b(x). This fact stands behind a block-diagonal method, which we use in this work for investigation of K.The method consists of decomposition (1.1) into a sum of operators with non-overlapping domains and regulated by the following key lemma.

Lemma 1.1. [10, Lemma 1] Let U = F

kU_k and V = F

kV_k be unions of non- overlapping measurable sets andT =P

kT_k,where T_k: L_p(U_k)→L_q(V_k).Then kTk_{Lp(U)→Lq(V)} = sup

k

kT_kk_{Lp(Uk)→Lq(Vk)},

if 0< p≤q <∞. For 0< q < p <∞ with r:=pq/(p−q) one has kTk_Lp(U)→Lq(V)≈ X

k

kT_kk^r_Lp(U

k)→Lq(V_k)

!1/r

. (1.5)

In (1.5) and throughout the paper we write A ≈ B instead of A B A or A = cB, where relations like A B mean A ≤ cB with some constant c depending only on parameters of summations and, possibly, on constants of equivalence in the inequalities of the type (1.3). We shall also make shortening w_k(x, t) :=k(x, t)w(x) and v_k(t, y) :=k(t, y)v(y).

Note that operators T_k in Lemma 1.1 in our case have the forms K_bf(x) =w(x)

Z b(x) b(c)

v_kb(x, y)f(y)dy, 0≤c≤x≤d≤ ∞, (1.6)

or

K_af(x) = w(x) Z a(d)

a(x)

v_ka(x, y)f(y)dy, 0≤c≤x≤d≤ ∞, (1.7) wherek_b(x, y) andk_a(x, y) inherit properties of the original kernel functionk(x, y).

In particular, if the kernelk(x, y) in (1.1) is from Oinarov’s classO_b thenk_b(x, y) is also equipped by (1.3) for 0≤c≤z ≤x≤d≤ ∞ and 0≤b(c)≤y≤b(z). It is known (see [6, 7, 8]) that for 1 < q < p <∞ the Lp −Lq–boundedness of Kb

in this case is guaranteed by finiteness of the constantsBb,0, Bb,1, where B_b,0 :=

Z b(d) b(c)

Z d b⁻¹(t)

w_k^q(x, t)dx

r q Z t

b(c)

v^p0(y)dy _q^r0

v^p0(t)dt

!¹_r

, (1.8)

B_b,1 :=



 Z d

c

Z d t

w^q(x)dx

r p"

Z b(t) b(c)

v_k^p0(t, y)dy

#_p^r0

w^q(t)dt





1 r

, (1.9)

p⁰ :=p/(p−1), q⁰ :=q/(q−1) andkK_bk_Lp(b(c),b(d))→Lq(c,d) ≈B_b,0+B_b,1.Moreover, there are some known estimates for functionals (1.8) – (1.9) (see [8] or [12] for details):

B_b,0 ≈Bb,0 :=



 Z b(d)

b(c)

"

Z b(d) t

Z d b⁻¹(y)

w_k^q(x, y)dx ^p

0

v^p0(y)dy

#_q^r0

× Z d

b⁻¹(t)

w^q_k(x, t)dx ^p0−_q^r0

v^p0(t)dt

!¹_r

, (1.10)

B_b,1 ≈Bb,1 :=



 Z d

c

"

Z t c

(Z b(x)

b(c)

v_k^p0(x, y)dy )q

w^q(x)dx

#^r_p

×

"

Z b(t) b(c)

v_k^p0(t, y)dy

#q−^r

p

w^q(t)dt





1 r

. (1.11)

Similar characterizations and estimates are true forK_a if k_a(x, y) is satisfying (1.4) as ever 0 ≤ c ≤ x ≤ z ≤ d ≤ ∞ and 0 ≤ a(z) ≤ y ≤ a(d). Namely, for 1 < q < p < ∞ the operator K_a is bounded from L_p to L_q if and only if B_a,0, B_a,1 <∞,where

Ba,0 :=



 Z a(d)

a(c)

"

Z a⁻¹(t) c

w^q_k(x, t)dx

#^r_q "

Z a(d) t

v^p0(y)dy

#_q^r0

v^p0(t)dt





1 r

, (1.12)

B_a,1 :=



 Z d

c

Z t c

w^q(x)dx ^r_p"

Z a(d) a(t)

v_k^p0(t, y)dy

#_p^r0

w^q(t)dt





1 r

(1.13)

and kK_ak_Lp(a(c),a(d))→L_q(c,d) ≈B_a,0+B_a,1. We have also

B_a,0 ≈Ba,0 :=





 Z a(d)

a(c)



 Z t

a(c)

(Z a⁻¹(y) c

w^q_k(x, y)dx )p⁰

v^p0(y)dy





r q0

×

"

Z a⁻¹(t) c

w^q_k(x, t)dx

#p⁰−^r

q0

v^p0(t)dt





1 r

, (1.14)

B_a,1 ≈Ba,1 :=



 Z d

c

"

Z d t

(Z a(d) a(x)

v_k^p0(x, y)dy )q

w^q(x)dx

#^r_p

×

"

Z a(d) a(t)

v^p_k⁰(t, y)dy

#q−^r_p

w⁽t)dt





1 r

. (1.15)

If k(x, y) = 1 in (1.1) then, obviously, A_b,0 = A_b,1 and B_b,0 = B_b,1. Besides, properties of (1.6) and (1.7) are described by functionals similar to B_b,0 = B_b,1 and by alternative boundedness constants even in more general case when 0 <

q < p <∞ and K¯_b =w(x)

Z b(x) a

f(y)v(y)dy, 0≤a≤b(x)<∞, x∈[c, d) (1.16) or

K¯_a =w(x) Z b

a(x)

f(y)v(y)dy, 0< a(x)≤b≤ ∞, x∈(c, d). (1.17) Lemma 1.2. [11, Lemma 1] Let 0< q < p < ∞, p >1 and the operator K¯_b be defined by (1.16). Then K¯_b is bounded from L_p(a, b(d)) to L_q(c, d) if and only if B¯b <∞ or B¯^b <∞, where kK¯bkLp(a,b(d))→Lq(c,d) ≈B¯b ≈B¯^b and

B¯_b :=



 Z d

c

Z d t

w^q(x)dx ^r_p "

Z b(t) a

v^p0(y)dy

#_p^r0

w^q(t)dt





1 r

, (1.18)

B¯b :=



 Z d

c

"

Z t c

(Z b(x) a

v^p0(y)dy )q

w^q(x)dx

#^r_p

×

"

Z b(t) a

v^p0(y)dy

#q−^r

p

w^q(t)dt





1 r

. (1.19)

Lemma 1.3. [11, Lemma 2] Let 0 < q < p < ∞, p >1 and the operator B¯_a be defined by (1.17). Then K¯_a :L_p(a(c), b))→L_q(c, d) iff B¯_a <∞ or B¯a<∞ with

kK¯_ak_Lp(a(c),b)→L_q(c,d) ≈B¯_a ≈B¯a, where B¯_a :=

Z d c

Z t c

w^q(x)dx

^r_p Z b a(t)

v^p0(y)dy _p^r0

w^q(t)dt

!¹_r

(1.20)

B¯a:=



 Z d

c

"

Z d t

Z b a(x)

v^p0(y)dy ^q

w^q(x)dx

#^r_p

× Z b

a(t)

v^p0(y)dy ^q−

r p

w^q(t)dt

!¹_r

. (1.21)

In this work we adduce several results describing the boundedness of (1.1) with kernelk(x, y) satisfying Oinarov’s type conditionsO_b and/orO_a.The results are obtained with using enumerated characteristics (1.8) – (1.15), (1.18) – (1.21) and under conception of the fairway function introduced in [9]. We use the notion of the fairway in its original form (Definition 2.1) for the boundedness criteria in case k(x, y) = 1 (Theorem 2.1) and modify it in a proper way for deriving new necessary and sufficient conditions for the boundedness of (1.1) when k(x, y) is from Oinarov’s type classes (Theorem 2.5 or Theorem 2.6).

Throughout the paper we assume products of the form 0· ∞ be equal to 0.

The notation kTk_Lp(a,b)→Lq(c,d) means the norm of an operator T from Lp(a, b) to Lq(c, d). Z denotes the set of all integers and χE stands for a characteristic function (indicator) of a subset E ⊂R⁺.

2. Main result We start from the case k(x, y) = 1.

Definition 2.1. [9, Deinition 1] Given boundary functions a(x) and b(x), satis- fying the conditions (1.2), a numberp∈ (1,∞) and a weight function v(x) such that 0< v(x)<∞ a.e. x∈R⁺ and v^p0(x) is locally integrable on R⁺, we define the fairway-function σ(x) such that a(x)< σ(x)< b(x) and

Z σ(x) a(x)

v^p0(y)dy= Z b(x)

σ(x)

v^p0(y)dy for all x∈R⁺.

Under given conditions (1.2) on boundary functions a(x) andb(x) it is possible to prove that the fairwayσ is differentiable and strictly increasing function.

Put

∆(t) := [a(t), b(t)], δ(t) := [b⁻¹(σ(t)), a⁻¹(σ(t))], θ(t) := (σ⁻¹(a(t)), σ⁻¹(b(t))).

The following statement contains two forms of criteria for the boundedness of the operator (1.1) withk(x, y) = 1.

Theorem 2.1. Let the operator K of the form (1.1) be given with the boundary functions a(x) and b(x) satisfying the conditions (1.2). K is bounded from L_p to

L_q for 1< p≤q <∞ if and only if A_M := sup

t>0

A_M(t) = sup

t>0

Z

δ(t)

w^q(x)dx ¹_q Z

∆(t)

v^p0(y)dy _p¹0

<∞ or if and only if

A_T := sup

t>0

Z

θ(t)

Z

∆(x)

v^p0(y)dy q

w^q(x)dx ¹_q Z

∆(t)

v^p0(y)dy −¹_p

<∞.

Moreover, AM ≈ AT ≈ kKkLp→Lq.

If 0< q < p < ∞, p >1 then K is bounded if and only if B_{M R} <∞ or if and only if B_{P S} <∞, where

B_{M R} :=

Z ∞ 0

Z

δ(t)

w^q(x)dx ^r_pZ

∆(t)

v^p0(y)dy _p^r0

w^q(t)dt

!¹_r ,

B_{P S} :=

Z ∞ 0

Z

θ(t)

Z

∆(x)

v^p0(y)dy q

w^q(x)dx ^r_pZ

∆(t)

v^p0(y)dy q−_p^r

w^q(t)dt

!¹_r

and BM R≈ BP S ≈ kKkLp→Lq.

Less general form of this statement can be found in [9] and [11]. The dual form of Theorem 2.1 reads

Theorem 2.2. Let the operatorK of the form (1.1)be given withk(x, y) = 1 and a(x), b(x) satisfying the conditions (1.2). Then K:L_p →L_q for 1< p≤q <∞ iff A^∗_M <∞ or iff A^∗_T <∞, where

A^∗_M := sup

t>0

Z b(ψ(t)) a(ψ(t))

v^p0(y)dy

!_p¹0

Z a⁻¹(t) b⁻¹(t)

w^q(x)dx

!¹_q ,

A^∗_T := sup

t>0





Z ψ⁻¹(a⁻¹(t)) ψ⁻¹(b⁻¹(t))

"

Z a⁻¹(y) b⁻¹(y)

w^q(x)dx

#p⁰

v^p0(y)dy





1 p0

Z a⁻¹(t) b⁻¹(t)

w^q(x)dx

!−¹

q0

,

Z ψ(y) b⁻¹(y)

w^q(x)dx=

Z a⁻¹(y) ψ(y)

w^q(x)dx for all y∈R⁺ (2.1) and A^∗_M ≈ A^∗_T ≈ kKk_Lp→Lq.

If 0 < q < p < ∞, p > 1 one has K : L_p → L_q iff B_{M R}^∗ < ∞ or B^∗_{P S} < ∞, where

B_{M R}^∗ :=



 Z ∞

0

"

Z b(ψ(t)) a(ψ(t))

v^p0(y)dy

#_q^r0 "

Z a⁻¹(t) b⁻¹(t)

w^q(x)dx

#^r_q

v^p0(t)dt





1 r

,

B^∗_{P S} :=





 Z ∞

0





Z ψ⁻¹(a⁻¹(t)) ψ⁻¹(b⁻¹(t))

(Z a⁻¹(y)

b⁻¹(y)

w^q(x)dx )p⁰

v^p0(y)dy





r q0

×

"

Z a⁻¹(t) b⁻¹(t)

w^q(x)dx

#p⁰−^r

q0

v^p0(t)dt





1 r

with fairway-function ψ(t) defined by (2.1) and B^∗_{M R} ≈ B_{P S}^∗ ≈ kKk_Lp→Lq. The next statement was obtained for Kwith k(x, y) from Oinarov’s classO_a. Theorem 2.3. [9, Theorem 2]Let the boundaries a(x), b(x)of the operator (1.1) be satisfying (1.2) and k(x, y)∈ O_a. If 1< p≤q <∞, then

kKkLp→Lq ≈ Aa,0+Aa,1, (2.2) where

A_a,0 := sup

s>0

sup

s≤t≤a⁻¹(b(s))

Z t s

w^q_k(x, a(t))dx

¹_q Z b(s) a(t)

v^p0(y)dy

!_p¹0

, (2.3)

A_a,1 := sup

s>0

sup

s≤t≤a⁻¹(b(s))

Z t s

w^q(x)dx

¹_q Z b(s) a(t)

v_k^p0(t, y)dy

!_p¹0

. (2.4)

If 1< q < p <∞, then kKk_Lp→L_q ≈ X

k

B^∗_k,1r

+ B^∗_k,2r

+ B_k,3^∗ r

+ B_k,4^∗ r

!¹_r

, (2.5) where

B_k,1^∗ :=





Z a(ξ_k+1) a(ξk)

"

Z a⁻¹(t) ξk

w_k^q(x, t)dx

#^r_q "

Z a(ξ_k+1) t

v^p0(y)dy

#_q^r0

v^p0(t)dt





1 r

,

B_k,2^∗ :=



 Z ξk+1

ξk

Z t ξk

w^q(x)dx _p^r "

Z a(ξk+1) a(t)

v_k^p0(t, y)dy

#_p^r0

w^q(t)dt





1 r

,

B_k,3^∗ :=

Z b(ξk+1) b(ξk)

Z ξk+1

b⁻¹(t)

w_k^q(x, a(ξ_k+1))dx

^rq Z t b(ξk)

v^p0(y)dy _q^r0

v^p0(t)dt

!¹_r ,

B_k,4^∗ :=



 Z ξk+1

ξk

Z ξk+1

t

w^q(x)dx ^rp"

Z b(t) b(ξk)

v_k^p0(ξ_k+1, y)dy

#_p^r0

w^q(t)dt





1 r

and ξ₀ = 1, ξ_k = (a⁻¹◦b)^k(ξ₀), k∈Z.

Similar result is true for Kwith k(x, y)∈ O_b.

Theorem 2.4. [9, Theorem 3] Let the operator K be defined by (1.1) with a(x), b(x) satisfying (1.2) and let k(x, y)∈ O_b. If 1< p≤q <∞, then

kKk_Lp→Lq ≈ A_b,0+A_b,1, (2.6) where

A_b,0 := sup

t>0

sup

b⁻¹(a(t))≤s≤t

Z t s

w_k^q(x, b(s))dx

¹_q Z b(s) a(t)

v^p0(y)dy

!_p¹0

, (2.7)

A_b,1 := sup

t>0

sup

b⁻¹(a(t))≤s≤t

Z t s

w^q(x)dx

¹_q Z b(s) a(t)

v^p_k⁰(s, y)dy

!_p¹0

. (2.8) If 1< q < p <∞, then

kKk_Lp→Lq ≈ X

k

B^r_k,1+B^r_k,2+B^r_k,3+B^r_k,4

!¹_r

, (2.9)

where

B_k,1 :=





Z a(ξ_k+1) a(ξ_k)

"

Z a⁻¹(t) ξ_k

w_k^q(x, b(ξ_k))dx

#^r_q "

Z a(ξ_k+1) t

v^p0(y)dy

#_q^r0

v^p0(t)dt





1 r

,

B_k,2 :=



 Z ξk+1

ξk

Z t ξk

w^q(x)dx ^r_p"

Z a(ξk+1) a(t)

v^p_k⁰(ξ_k, y)dy

#_p^r0

w^q(t)dt





1 r

,

B_k,3 :=

Z b(ξ_k+1) b(ξk)

Z ξ_k+1 b⁻¹(t)

w_k^q(x, t)dx

^r_q Z t b(ξk)

v^p0(y)dy _q^r0

v^p0(t)dt

!¹_r ,

B_k,4 :=



 Z ξk+1

ξ_k

Z ξk+1

t

w^q(x)dx

r p"

Z b(t) b(ξ_k)

v^p_k⁰(t, y)dy

#_p^r0

w^q(t)dt





1 r

and ξ₀ = 1, ξ_k = (a⁻¹◦b)^k(ξ₀), k∈Z.

Note that (2.6) was derived earlier in [3] and [4].

Double supremums in (2.3), (2.4), (2.7), (2.8) and a discrete form of (2.5), (2.9) gave a motivation for searching new necessary and sufficient boundedness conditions of K with more convenient forms.

Let functions φ(x) and ρ(y) on R⁺ ∪ {+∞}, where a(x) ≤ φ(x) ≤ b(x) and b⁻¹(y)≤ρ(y)≤a⁻¹(y), be fairway-functions satisfying the following

Definition 2.2. Given boundary functionsa(x) andb(x) satisfying the conditions (1.2), numbers p, q ∈ (1,∞), a continuous kernel 0 < k(x, y) <∞ a.e on R and weight functions 0 < v, w < ∞ a.e. on R⁺ such that for any fixed x > 0 the function v_k^p0(x, y) is locally integrable on R⁺ with respect to the variable y as well as for any y > 0 the function w^q_k(x, y) is locally integrable on R⁺ with

respect to x, we define two fairways – the functions φ(x) and ρ(y) such that a(x)< φ(x)< b(x), b⁻¹(y)< ρ(y)< a⁻¹(y) and

Z φ(x) a(x)

v_k^p0(x, y)dy= Z b(x)

φ(x)

v^p_k⁰(x, y)dy for all x >0, (2.10) Z ρ(y)

b⁻¹(y)

w^q_k(x, y)dx=

Z a⁻¹(y) ρ(y)

w_k^q(x, y)dx for all y >0. (2.11) By assumptions of the definitionφ(x) andρ(y) are continuous functions. Put

Θ(t) := Θ⁻(t)∪Θ⁺(t), Θ⁻(t) := [b⁻¹(t), ρ(t)), Θ⁺(t) := [ρ(t), a⁻¹(t)), ϑ(t) :=ϑ⁻(t)∪ϑ⁺(t), ϑ⁻(t) := [a(ρ(t)), t), ϑ⁺(t) := [t, b(ρ(t))), δ(t) := δ⁻(t)∪δ⁺(t), δ⁻(t) := [b⁻¹(φ(t)), t), δ⁺(t) := [t, a⁻¹(φ(t))),

∆(t) := ∆⁻(t)∪∆⁺(t), ∆⁻(t) := [a(t), φ(t)), ∆⁺(t) := [φ(t), b(t)) and denote

A^±_ρ := sup

t>0

A^±_ρ(t) = sup

t>0

Z

Θ(t)

w^q_k(x, t)dx ¹_q Z

ϑ^±(t)

v^p0(y)dy _p¹0

,

A^±_φ := sup

t>0

A^±_φ(t) = sup

t>0

Z

δ^±(t)

w^q(x)dx ¹_q Z

∆(t)

v_k^p0(t, y)dy _p¹0

,

B_ρ^±:=

Z ∞ 0

B_ρ^±(t)dt ¹_r

= Z ∞

0

Z

Θ(t)

w_k^q(x, t)dx ^r_qZ

ϑ^±(t)

v^p0(y)dy _q^r0

v^p0(t)dt

!¹_r ,

B_φ^±:=

Z ∞ 0

B_φ^±(t)dt ¹_r

= Z ∞

0

Z

δ^±(t)

w^q(x)dx ^r_pZ

∆(t)

v_k^p0(t, y)dy _p^r0

w^q(t)dt

!¹_r ,

Aρ := sup

t>0

Aρ(t) = sup

t>0

Z

Θ(t)

w^q_k(x, t)dx ¹_q Z

ϑ(t)

v^p0(y)dy _p¹0

,

A_φ:= sup

t>0

A_φ(t) = sup

t>0

Z

δ(t)

w^q(x)dx ¹_q Z

∆(t)

v_k^p0(t, y)dy _p¹0

,

B_ρ:=

Z ∞ 0

B_ρ(t)dt ¹_r

= Z ∞

0

Z

Θ(t)

w^q_k(x, t)dx ^r_qZ

ϑ(t)

v^p0(y)dy _q^r0

v^p0(t)dt

!¹_r ,

B_φ:=

Z ∞ 0

B_φ(t)dt ¹_r

= Z ∞

0

Z

δ(t)

w^q(x)dx ^r_pZ

∆(t)

v_k^p0(t, y)dy _p^r0

w^q(t)dt

!¹_r .

New boundedness conditions forKwith k(x, y)∈ O_b are proved in Section3and read

Theorem 2.5. Let the operator K be defined by (1.1) with the border functions a(x), b(x)satisfying (1.2) and a continuous positive kernel k(x, y)on R from the Oinarov’s type class O_b.Suppose that the functions ρ(y), φ(x) onR⁺ are strictly increasing fairways from Definition 2.2. If 1< p ≤q <∞, then

A⁻_ρ +A⁺_φ kKk_Lp→Lq A_ρ+A_φ. (2.12) If 1< q < p <∞, then

B⁻_ρ +B_φ⁺ kKk_Lp→Lq B_ρ+B_φ. (2.13) Analogously we obtain a similar result forK with k(x, y) satisfying the condi- tion (1.4).

Theorem 2.6. Let the operator K be defined by (1.1) with a(x), b(x) satisfying (1.2) and a continuous kernel k(x, y)> 0 on R from the class O_a. Suppose that ρ(y), φ(x) on R⁺ are strictly increasing fairways satisfying Definition 2.2. If 1< p≤q <∞, then

A⁺_ρ +A⁻_φ kKk_Lp→Lq A_ρ+A_φ. If 1< q < p <∞, then

B⁺_ρ +B_φ⁻ kKk_Lp→L_q B_ρ+B_φ.

In conclusion of the section we provide several cases when the results of Theo- rems 2.5 and 2.6 became of a criterion form.

Theorem 2.7. Let the operator K be defined by (1.1) with a(x), b(x) satisfying (1.2) and a continuous kernel k(x, y)>0 on R. Suppose that the functions ρ(x), φ(x) on R⁺ are strictly increasing fairways from Definition 2.2.

(a) If k(x, y)∈ Ob and Z

ϑ⁻(t)

v^p0(y)dy≈ Z

ϑ(t)

v^p0(y)dy, t >0, Z

δ⁺(t)

w^q(x)dx≈ Z

δ(t)

w^q(x)dx, t >0, then

kKk_Lp→L_q ≈

(A_ρ+A_φ, 1< p ≤q <∞,

B_ρ+B_φ, 1< q < p <∞, (2.14) (b) If k(x, y)∈ O_a and

Z

ϑ⁺(t)

v^p0(y)dy ≈ Z

ϑ(t)

v^p0(y)dy, t >0, Z

δ⁻(t)

w^q(x)dx≈ Z

δ(t)

w^q(x)dx, t >0, then the estimate (2.14) holds.

(c) If k(x, y)∈ O_a∩ O_b, then the equivalence (2.14) is true.

Proof of Theorem 2.7 easy follows from Theorems 2.5 and 2.6.

3. Proof of Theorem 2.5

We start fromthe lower estimate in (2.12). Let 1< p≤q <∞. We have from (2.6)

kKk_Lp→Lq ≈sup

t>0

sup

b⁻¹(a(t))≤s≤t

[A₀(s, t) +A₁(s, t)], (3.1) where

A₀(s, t) :=

Z t s

w^q_k(x, b(s))dx

¹_q Z b(s) a(t)

v^p0(y)dy

!_p¹0

,

A₁(s, t) :=

Z t s

w^q(x)dx

¹_q Z b(s) a(t)

v_k^p0(s, y)dy

!_p¹0

.

Using (2.11) we find that A⁻_ρ(t) = 2^1/q

Z

Θ⁻(t)

w^q_k(x, t)dx ¹_q Z

ϑ⁻(t)

v^p0(y)dy _p¹0

= 2^1/q Z s

b⁻¹(ρ⁻¹(s))

w_k^q(x, ρ⁻¹(s))dx

¹_q Z ρ⁻¹(s) a(s)

v^p0(y)dy

!_p¹0

= 2^1/qA0(b⁻¹(ρ⁻¹(s)), s)≤sup

t>0

sup

b⁻¹(a(t))≤s≤t

A0(s, t).

On the strength of (3.1) it implies A⁻_ρ kKkLp→Lq. Analogously, A⁺_φ(t)^(2.10)= 2^1/p0

Z

δ⁺(t)

w^q(x)dx ¹_q Z

∆⁺(t)

v_k^p0(t, y)dy _p¹0

= 2^1/p0A1(t, a⁻¹(φ(t)))≤sup

t>0

sup

b⁻¹(a(t))≤s≤t

A1(s, t) impliesA⁺_φ kKkLp→Lq,and the lower estimate in (2.12) is proved.

Forthe upper estimate in (2.12) we put τ₀ :=ρ(a(t)) and write sup

b⁻¹(a(t))≤s≤t

A0(s, t)≤ sup

b⁻¹(a(t))≤s≤ρ(a(t))<t

A0(s, t) + sup

ρ(a(t))≤s≤t

A0(s, t)

≤ sup

b⁻¹(ρ⁻¹(τ0))≤s≤τ₀

Z a⁻¹(ρ⁻¹(τ0)) s

w^q_k(x, b(s))dx

!¹_q

Z b(s) ρ⁻¹(τ0)

v^p0(y)dy

!_p¹0

+ sup

ρ(a(t))≤s≤t

Z a⁻¹(ρ⁻¹(s)) s

w^q_k(x, b(s))dx

!¹_q

Z b(s) a(s)

v^p0(y)dy

!_p¹0

=:H1(τ0) +H2(t).

Indeed, if b⁻¹(a(t)) ≤ s ≤ ρ(a(t)) < t, that is b⁻¹(ρ⁻¹(τ₀)) ≤ s ≤ τ₀, then (s, t) = (s, a⁻¹(ρ⁻¹(τ₀))) and (a(t), b(s)) = (ρ⁻¹(τ₀), b(s)).Ifρ(a(t))≤s≤t,then (s, t)⊂(s, a⁻¹(ρ⁻¹(s))) and (a(t), b(s))⊂(a(s), b(s)).

To estimate H₁(τ₀) we use (1.3) with y=ρ⁻¹(τ₀), z =s and obtain H₁(τ₀)

Z a⁻¹(ρ⁻¹(τ0)) b⁻¹(ρ⁻¹(τ0))

w^q_k(x, ρ⁻¹(τ₀))dx

!¹_q

Z b(τ0) ρ⁻¹(τ0)

v^p0(y)dy

!_p¹0

=

Z a⁻¹(z)) b⁻¹(z)

w_k^q(x, z)dx

!¹_q

Z b(ρ(z)) z

v^p0(y)dy

!_p¹0

=A⁺_ρ(z)≤ A⁺_ρ.

Since s ≤ x and a(x) ≤ ρ⁻¹(s) ≤ b(s) in H₂(t) we obtain by using (1.3) with z =s and y=ρ⁻¹(s) :

H₂(t) sup

ρ(a(t))≤s≤t

Z a⁻¹(ρ⁻¹(s)) s

w_k^q(x, ρ⁻¹(s))dx

!¹_q

Z b(s) a(s)

v^p0(y)dy

!_p¹0

= sup

a(t)≤z≤ρ⁻¹(t)

Z a⁻¹(z) ρ(z)

w^q_k(x, z)dx

!¹_q

Z b(ρ(z)) a(ρ(z))

v^p0(y)dy

!_p¹0

≤ A_ρ.

Thus,

sup

t>0

sup

b⁻¹(a(t))≤s≤t

A₀(s, t) A_ρ. (3.2)

Analogously, we put τ₁ :=φ⁻¹(a(t)) and write sup

b⁻¹(a(t))≤s≤t

A₁(s, t)≤ sup

b⁻¹(a(t))≤s≤φ⁻¹(a(t))<t

A₁(s, t) + sup

φ⁻¹(a(t))≤s≤t

A₁(s, t)

≤ sup

b⁻¹(φ(τ1))≤s≤τ1

Z a⁻¹(φ(τ1)) s

w^q(x)dx

!¹_q

Z b(s) φ(τ1)

v_k^p0(s, y)dy

!_p¹0

+ sup

φ⁻¹(a(t))≤s≤t

Z a⁻¹(φ(s)) s

w^q(x)dx

!¹_q

Z b(s) a(s)

v_k^p0(s, y)dy

!_p¹0

=:H₃(τ₁) +H₄(t).

Obviously, H4(t) ≤ sup_φ⁻¹_{(a(t))≤s≤t}A⁺_φ(s)≤ A⁺_φ. For H3(τ1) we apply (1.3) with z =s ≤τ₁ =x and a(τ₁)< φ(τ₁)≤y≤b(s) :

H₃(τ₁) sup

b⁻¹(φ(τ1))≤s≤τ₁

Z a⁻¹(φ(τ1)) s

w^q(x)dx

!¹_q

Z b(s) φ(τ1)

v_k^p0(τ₁, y)dy

!_p¹0

≤ Aφ(τ1)≤ Aφ. Thus,

sup

t>0

sup

b⁻¹(a(t))≤s≤t

A₁(s, t) A_φ. (3.3)

By combining (3.1), (3.2) and (3.3) we obtain the upper estimate in (2.12).

Now we consider the case 1< q < p <∞.Let us prove firstthe upper estimate in (2.13). To this end we take a point sequence {ξ_k}k∈Z ⊂(0,∞) such that

ξ₀ = 1, ξ_k = (a⁻¹◦b)^k(1), k ∈Z,

and put

ηk=a(ξk) = b(ξk−1), ∆k= [ξk, ξk+1), δk = [ηk, ηk+1), k∈Z. Breaking the semiaxis (0,∞) by points of the sequence {ξ_k}k∈Z we decompose the operator K into the sum

K=T +S (3.4)

of block-diagonal operators T and S such that T =X

k∈Z

T_k, S =X

k∈Z

S_k, (3.5)

where

T_kf(x) =w(x)

Z a(ξk+1) a(x)

v_k(x, y)f(y)dy, T_k:L_p(δ_k)→L_q(∆_k),

Skf(x) =w(x) Z b(x)

b(ξk)

vk(x, y)f(y)dy, Sk:Lp(δk+1)→Lq(∆k).

Kernels k(x, y) of the operators Tk and Sk satisfy the condition (1.3) for z ≤x, x∈[ξk, ξk+1] and

a(x)≤y≤b(ξ_k), b(ξ_k)≤y≤b(z), (3.6) respectively.

To estimate a norm of the operator S_k we take into account two key points s_ρ := b⁻¹(ρ⁻¹(ξ_k+1)), s_φ := φ⁻¹(b(ξ_k)) = φ⁻¹(a(ξ_k+1)) and consider three only possible variants:

(i) s_ρ< s_φ, (ii) s_ρ =s_φ, (iii) s_ρ> s_φ. In the case (i) we have

S_kf =

3

X

i=1

S_k,if +

3

X

i=1

H_k,if, (3.7)

where

S_k,1f =χ_[ξk,sρ]S_kf, L_p(b(ξ_k), b(s_ρ))→L_q(ξ_k, s_ρ), H_k,1f =χ_[sρ,sφ]S_k f χ_{[b(ξk),b(sρ)]}

, L_p(b(ξ_k), b(s_ρ))→L_q(s_ρ, s_φ), Sk,2f =χ[sρ,s_φ]Sk f χ[b(sρ),b(s_φ)]

, Lp(b(sρ), b(sφ))→Lq(sρ, sφ), H_k,2f =χ_[sφ,ξk+1]S_k f χ_{[b(ξk),b(sρ)]}

, L_p(b(ξ_k), b(s_ρ))→L_q(s_φ, ξ_k+1), Hk,3f =χ_[sφ,ξk+1]Sk f χ_{[b(sρ),b(sφ)]}

, Lp(b(sρ), b(sφ))→Lq(sφ, ξk+1), S_k,3f =χ_[sφ,ξk+1]S_k f χ_{[b(sφ),b(ξk+1)]}

, L_p(b(s_φ), b(ξ_k+1))→L_q(s_φ, ξ_k+1).