Skip to main content

Weighted kernel operators in variable exponent amalgam spaces

Abstract

The paper is devoted to weighted inequalities for positive kernel operators in variable exponent amalgam spaces. In particular, a characterization of a weight v governing the boundedness/compactness of the weighted kernel operators K v and K v , defined on R + and , respectively, under the log-Hölder continuity condition on exponents of spaces is established. These operators involve, for example, weighted variable parameter fractional integrals. The results are new even for constant exponent amalgam spaces.

MSC:46E30, 47B34.

1 Introduction

In the paper, we derive necessary and sufficient conditions on a weight function v governing the boundedness/compactness of the positive kernel operators

in variable exponent amalgam spaces (VEAS) under the log-Hölder continuity condition on exponents of spaces. It should be emphasized that the results are new even for constant exponent amalgam spaces.

Historically, the boundedness problem for the two-weighted Hardy transform ( H v , w f)(x)=v(x) 0 x f(t)w(t)dt from L p ( ) to L q ( ) was studied in the papers [1, 2] in different terms on weights (see also [3] for related topics). In [1], the authors explored also the compactness problem for H v , w . The boundedness for fractional integral operators in (weighted) variable exponent Lebesgue spaces defined on Euclidean spaces was investigated by many authors (see, e.g., the papers [414], etc.). The compactness (resp. non-compactness) of fractional and singular integrals in weighted L p ( ) spaces was studied in [15]. We refer also to the monograph [16] for related topics.

The space L p ( ) is a special case of the Musielak-Orlicz space (see [17, 18]). The first systematic study of modular spaces is due to Nakano [19].

Variable exponent Lebesgue and Sobolev spaces arise, e.g., in the study of mathematical problems related to applications to mechanics of the continuum medium (see [16, 20] and references cited therein).

The manuscript consists of four sections. In Section 2, we recall some well-known facts about variable exponent Lebesgue spaces L p ( ) . In Section 3, we recall the definition, history and some essential properties of amalgam spaces with a constant exponent, and also known results about the boundedness of some integral operators in these spaces; boundedness criteria for the operators K v and K v in VEAS are also established. The compactness of positive kernel operators in VEAS is studied in Section 4.

Throughout the paper, constants (often different constants in the same series of inequalities) will mainly be denoted by c or C; by the symbol p (x), we denote the function p ( x ) p ( x ) 1 , 1<p(x)<; the relation ab means that there are positive constants c 1 and c 2 such that c 1 ab c 2 a.

2 Preliminaries

We begin this section by the definition and essential properties of variable exponent Lebesgue spaces.

Let E be a measurable set in with positive measure. We denote

p (E):= inf E p, p + (E):= sup E p

for a measurable function p on E. Suppose that 1< p (E) p + (E)<. Denote by ρ a weight function on E (i.e., ρ is an almost everywhere positive measurable function). We say that a measurable function f on E belongs to L ρ p ( ) (E) (or to L ρ p ( x ) (E)) if

S p ( ) , ρ (f)= E | f ( x ) | p ( x ) ρ(x)dx<.

It is a Banach space with respect to the norm (see, e.g., [2124])

f L ρ p ( ) ( E ) =inf { λ > 0 : S p ( ) , ρ ( f / λ ) 1 } .

If ρconst, then we use the symbol L p ( ) (E) (resp. S p ( ) ) instead of L ρ p ( ) (E) (resp. S p ( ) , ρ ). It is clear that f L ρ p ( ) ( E ) = f ( ) ρ 1 / p ( ) ( ) L p ( ) ( E ) .

In the sequel, we will denote by and Z the set of all integers and the set of non-positive integers, respectively.

To prove the main results, we need some known statements.

Proposition A ([2224])

Let E be a measurable subset of . Suppose that 1< p (E) p + (E)<. Then

  1. (i)
    f L p ( ) ( E ) p + ( E ) S p ( f χ E ) f L p ( ) ( E ) p ( E ) , f L p ( ) ( E ) 1 ; f L p ( ) ( E ) p ( E ) S p ( f χ E ) f L p ( ) ( E ) p + ( E ) , f L p ( ) ( E ) 1 ;
  2. (ii)

    Hölder’s inequality

    | E f ( x ) g ( x ) d x | ( 1 p ( E ) + 1 ( p + ( E ) ) ) f L p ( ) ( E ) g L p ( ) ( E )

holds, where f L p ( ) (E), g L p ( ) (E).

Proposition B ([2224])

Let 1r(x)p(x) and let E be a bounded subset of . Then the following inequality

f L r ( ) ( E ) ( | E | + 1 ) f L p ( ) ( E )

holds.

Definition 2.1 We say that p satisfies the weak Lipschitz (log-Hölder continuity) condition on ER (pWL(E)), if there is a positive constant A such that for all x and y in E with 0<|xy|<1/2 the inequality

| p ( x ) p ( y ) | A/ ( ln | x y | )

holds.

Lemma A ([25])

Let I be an interval in . Then pWL(I) if and only if there exists a positive constant c such that

| J | p ( J ) p + ( J ) c

for all intervals JI with |J|>0. Moreover, the constant c does not depend on I.

For the next statement we refer to [2] in the case of finite interval, and [26] for infinite interval.

Proposition C Let p and q be measurable functions on I:=(a,b) (<a<b+) satisfying the condition 1< p (I)p(x)q(x)< q + (I)<, xI. Let p,qWL(I). Suppose also that if b=, then p(x) p c const, q(x) q c const outside some large interval (a,d). Then there is a positive constant c depending only on p and q such that for all f L p ( ) (I), g L q ( ) (I) and all sequences of intervals S k :=[ x k 1 , x k + 1 ), where [ x k , x k + 1 ) are disjoint intervals satisfying the condition k [ x k , x k + 1 )=I, the inequality

k f χ S k L p ( ) ( I ) g χ S k L q ( ) ( I ) c C a , b f L p ( ) ( I ) g L q ( ) ( I )

holds. Moreover, the value of C a , b is defined as follows: C a , b = [ ( b a ) + 1 ] 2 if b< and C a , = [ ( d a ) + 1 ] 2 +1 if b=.

Let v and w be a.e. positive measurable function on [a,b), <a<b, and let

( H v , w ( a , b ) f ) (x)=v(x) a x f(t)w(t)dt,x[a,b).

Further, we denote

Let us recall the two-weight criterion for the Hardy operator in classical Lebesgue spaces:

Theorem A ([27, 28])

Let r and s be constants such that 1<rs<. Suppose that 0a<b. Let v and w be non-negative measurable functions on [a,b). Then the Hardy inequality

( a b v ( x ) ( a x f ( t ) d t ) s d x ) 1 / s c ( a b w ( t ) ( f ( t ) ) r d t ) 1 / r ,f0,

holds if and only if

A:= sup a t b ( t b v ( x ) d x ) 1 / s ( a t w 1 r ( x ) d x ) 1 / r <.

Moreover, if c is the best constant in the Hardy inequality, then there are positive constants c 1 and c 2 depending only on r and s such that c 1 Ac c 2 A.

For the Hardy inequalities, we also refer the books [29, 30].

The following statement was proved in [2] for finite interval and in [12] for the case of infinite interval, but we give the proof because of the upper and lower bound of the norm of H v , w .

Theorem B Let <a<b+ and let p and q be measurable functions on I:=(a,b) satisfying the conditions: 1< p (I)p(x)q(x) q + (I)<, p,qWL(I). We assume that p p c const, q q c const outside some large interval (a,d) if b=. Then H v , w I is bounded from L p ( ) (I) to L q ( ) (I) if and only if

A a , b sup a < t < b χ ( t , b ) ( ) v ( ) L q ( ) ( I ) χ ( a , t ) ( ) w ( ) L p ( ) ( I ) <.

Moreover, there are positive constants c 1 and c 2 independent of the interval I such that

c 1 A a , b H v , w ( a , b ) L p ( ) ( I ) L q ( ) ( I ) c 2 C a , b A a , b ,

where the constant C a , b is defined in Proposition C.

Proof Sufficiency. Let f0. Suppose that b< and that a b f(t)dt[ 2 m 0 , 2 m 0 + 1 ) for some integer m 0 . We construct a sequence { x k } so that

a x k fw= x k x k + 1 fw= 2 k .

It is easy to check that (a,b)= k [ x k , x k + 1 ). Let g be a function satisfying the condition g L q ( ) ( [ a , b ] ) 1. Applying Hölder’s inequality for variable exponent Lebesgue spaces and Proposition C we have that

a b ( H v , w f ) g k ( x k x k + 1 g v ) ( 0 x k + 1 f w ) = 4 k ( x k x k + 1 g v ) ( x k 1 x k f w ) 4 k χ ( x k , x k + 1 ) ( ) g ( ) L q ( ) ( I ) χ ( x k , x k + 1 ) ( ) v ( ) L q ( ) ( I ) × χ ( x k 1 , x k ) ( ) f ( ) L p ( ) ( I ) χ ( x k 1 , x k ) ( ) w ( ) L p ( ) ( I ) 4 A a , b k χ ( x k , x k + 1 ) ( ) g ( ) L q ( ) ( I ) χ ( x k 1 , x k ) ( ) f ( ) L p ( ) ( I ) 4 C a , b A a , b f ( ) L p ( ) ( I ) g ( ) L q ( ) ( I ) ,

where C a , b is the constant defined in Proposition C. Taking now the supremum with respect to g, we have sufficiency for b<.

Let now b=. Then

H v , w ( a , ) f L q ( ) ( ( a , + ) ) v ( x ) a x f w L q ( ) ( ( a , d ) ) + v ( x ) a x f w L q c ( [ d , + ) ) : = I 1 + I 2 .

By applying already used arguments, we have that I 1 4 C a , A a , + , where C a , = [ ( d a ) + 1 ] 2 . Further, due to Hölder’s inequality and Theorem A, we find that

I 2 v ( x ) a d f w L q c ( [ d , + ) ) + v ( x ) d x f w L q c ( [ d , + ) ) v ( ) χ [ d , + ) ( ) L q ( ) w ( ) χ [ a , d ) ( ) L p ( ) f L p ( ) + 4 A a , + f L p ( ) ( I ) 5 A a , + f L p ( ) ( I ) .

To get the lower bound for H v , w ( a , b ) is trivial by choosing the appropriate test function f(x)= χ ( a , t ) (x), a<t<b in the boundedness of H v , w I from L p ( ) (I) to L q ( ) (I). □

Corollary A Let p and q be defined on R + and satisfy the conditions of Theorem B. Then for all nZ,

v ( x ) 2 n x f ( t ) w ( t ) d t L q ( ) ( [ 2 n , 2 n + 1 ] ) D f L p ( ) ( [ 2 n , 2 n + 1 ] ) ,

where D=max{c ( 2 d + 1 ) 2 ,4} sup n Z A 2 n , 2 n + 1 , A 2 n , 2 n + 1 is defined in Theorem B and the constant c depends only on p and q.

Proof By the hypothesis, p and q are constant outside some large interval (0,d). Let d[ 2 m 0 1 , 2 m 0 ) for some integer m 0 . Then by Theorem B for n m 0 , we have

H v , w ( 2 n , 2 n + 1 ) L p ( ) ( [ 2 n , 2 n + 1 ) ) L q ( ) ( [ 2 n , 2 n + 1 ) ) c ( 2 n + 1 ) 2 A 2 n , 2 n + 1 c ( 2 m 0 + 1 ) 2 A 2 n , 2 n + 1 c ( 2 d + 1 ) 2 sup n Z A 2 n , 2 n + 1 ,

where the positive constant c depends only on p and q. If n> m 0 , then p and q are constants on the intervals [ 2 n , 2 n + 1 ). In this case taking the proof of Theorem B into account, we find that

sup n > m 0 H v , w ( 2 n , 2 n + 1 ) L p ( ) ( [ 2 n , 2 n + 1 ] ) L q ( ) ( [ 2 n , 2 n + 1 ] ) 4 sup n Z A 2 n , 2 n + 1 .

 □

Theorem C ([5])

Let p(x) and q(x) be measurable functions on an interval I R + . Suppose that 1< p (I) p + (I)< and 1< q (I) q + (I)<. If

k ( x , y ) L p ( y ) ( I ) L q ( x ) ( I ) <,

where k is a non-negative kernel, then the operator

Kf(x)= I k(x,y)f(y)dy

is compact from L p ( ) (I) to L q ( ) (I).

Lemma B (see, e.g. [31])

Let 1<q< q ¯ < and 1 s = 1 q 1 q ¯ . Suppose that { u n } and { v n } are sequences of positive real numbers. The following statements are equivalent:

  1. (i)

    There exists C>0 such that the inequality

    { n Z ( | a n | u n ) q } 1 / q C { n Z ( | a n | v n ) q ¯ } 1 / q ¯

holds for all sequences { a n } of real numbers.

  1. (ii)
    { n Z ( u n v n 1 ) s } 1 / s <

    .

Lemma C (see e.g. [32])

Let p, q be constants such that 1<p,q<. Suppose that v k 0, w k >0, kZ. Then there exists a constant c>0 such that

{ n Z ( k = n v n a k ) q } 1 / q c ( n Z ( w n a n ) p ) 1 / p

holds for all non-negative sequence { a k } l { v n p } p , if and only if

  1. (i)

    in case 1<pq<,

    A 1 := sup m Z ( n = m v n q ) 1 / q ( n = m w n p ) 1 / p <;
  2. (ii)

    in case 1<q<p<,

    A 2 := { m Z ( n = m v n q ) r / q ( n = m w n p ) r / q w m p } 1 / r <,

where 1/r=1/q1/p.

Definition 2.2 Let I=(0,a), 0<a. We say that a kernel k:{(x,y):0<y<x<a}(0,) belongs to V(I) (kV(I)) if there exists a constant c 1 such that for all x, y, t with 0<y<t<x<a the inequality

k(x,y) c 1 k(x,t)

holds.

Definition 2.3 Let r be a measurable function on I=(0,a), 0<a with values in (1,+). We say a kernel k belongs to V r ( ) (I) if there exists a positive constant c 2 such that for a.e. x(0,a), the inequality

χ ( x 2 , x ) ( ) k ( x , ) L r ( ) ( I ) c 2 x 1 r ( x ) k ( x , x 2 )

is fulfilled.

Example 2.1 (Lemma 3 of [26])

Let I:=(0,a), where 0<a. Let α be a measurable function on I satisfying the condition 0< α (I) α + (I)1. Suppose that r is a function on I with values in (1,+) satisfying the condition rWL(I). Suppose that r(x) r 0 const outside some interval (0,b) when a=+. Then k(x,t)= ( x t ) α ( x ) 1 V(I) V r ( ) (I) when r(x)< 1 1 α ( x ) .

The next examples of kernels can be checked easily:

Example 2.2 Let I:=(0,a), where 0<a. Suppose that α is a measurable function on I satisfying the condition 0< α (I) α + (I)1. Let r be a function on I with the values in (1,+) satisfying the condition r, r ¯ WL(I) where r ¯ (t)=r( t 1 / σ ). Suppose that r(x) r 0 const outside some interval (0,b) when a=+. Then k(x,y)= ( x σ y σ ) α ( x ) 1 V(I) V r ( ) (I) when r(x)< 1 1 α ( x ) and σ>0.

Example 2.3 Let I:=(0,a), 0<a. Let r be a function on I with the values in (1,+) satisfying the condition rWL(I) and let r be increasing on I. Suppose that r(x) r 0 const outside some interval (0,b) when a=+. Further, let 0< α (I)α(x)1 and α(x)+β(x)>2 1 r ( x ) . Then k(x,y)= ( x y ) α ( x ) 1 ln β ( x ) 1 x y V(I) V r ( ) (I).

For other examples of kernel k satisfying the condition kV(I) V r (I), where r is constant, we refer to [33] (see also [34], p.163).

3 Boundedness on VEAS

This section is devoted to the boundedness of weighted kernel operators in VEAS.

3.1 Amalgam spaces

Let I be or R + and α={ I n ;nZ} be a cover of I consisting of disjoint half-open intervals I n , each of the form [ a 1 , a 2 ), whose union is I. Let

f ( L u p ( ) ( I ) , l q ) α := ( n Z χ I n ( ) f ( ) L u p ( ) ( I ) q ) 1 / q ,

we define the general amalgams with variable exponent

( L u p ( ) ( I ) , l q ) α = { f : f ( L u p ( ) ( I ) , l q ) α < } .

If uconst, then ( L u p ( ) ( I ) , l q ) α is denoted by ( L p ( ) ( I ) , l q ) α .

Let p p c const and uconst. Then we have the usual irregular amalgam (see [35]); if I=R and I n =[n,n+1), then ( L p c ( I ) , l q ) α is the amalgam space introduced by Wiener (see [36, 37]) in connection with the development of the theory of generalized harmonic analysis.

We call ( L u p ( ) ( I ) , l q ) α irregular weighted amalgam spaces with variable exponent. If I n =[n,n+1), then ( L u p ( ) ( I ) , l q ) α will be denoted by ( L u p ( ) (I), l q ).

Let d={[ 2 n , 2 n + 1 );nZ} and I= R + . We denote weighted dyadic amalgam with variable exponent by ( L u p ( ) ( I ) , l q ) d . Some properties regarding general amalgams with variable exponent can be derived in the same way as for usual irregular amalgams ( L u p ( R ) , l q ) α , where p is constant. Irregular amalgams were introduced in [38] and studied in [35].

Theorem D Let p be a measurable function on I with 1< p (I) p + (I)< and q be constant with 1<q<. The irregular amalgams with variable exponent ( L p ( ) ( I ) , l q ) α is a Banach space whose dual space is ( L p ( ) ( I ) , l q ) α = ( L p ( ) ( I ) , l q ) α . Further, Hölder’s inequality holds in the following form:

| I f ( t ) g ( t ) d t | f ( L p ( ) ( I ) , l q ) α g ( L ( p ( ) ) ( I ) , l q ) α .

Proof Since L p ( ) is a Banach space and ( L p ( ) ) = L p ( ) (see [22]), from general arguments (see [35, 3941]) we have the desired result. □

The next statement for more general case, i.e., when amalgams are defined with respect to Banach spaces, can be found in [35].

Theorem E Let p be measurable function on I and 1 q 1 q 2 , then

( L p ( ) ( I ) , l q 1 ) α ( L p ( ) ( I ) , l q 2 ) α .

Other structural properties of amalgams are investigated, e.g., in [41] and [35].

The next statement is a generalization of Theorem 4 in [35] for variable exponent amalgams with weights.

Proposition D Let p, q be measurable functions on I such that 1 q (I)q(x)<p(x) p + (I) and 1r<. Then the space ( L w p ( ) ( I ) , l r ) α is continuously embedded in ( L v q ( ) ( I ) , l r ) α if

S:= sup n Z I n ( v ( x ) w ( x ) ) p ( x ) p ( x ) q ( x ) dx<.
(3.1)

Conversely, if 1< q (I) q + (I)< p (I) p + (I)<, then condition (3.1) is also necessary for the continuous embedding of ( L w p ( ) ( I ) , l r ) α into ( L v q ( ) ( I ) , l r ) α .

Proof It is known (see [42]) that the continuous embedding L w p ( ) (I) L v q ( ) (I) (q(x)<p(x)) holds if and only if

I ( v ( x ) w ( x ) ) p ( x ) p ( x ) q ( x ) dx<.

Moreover, the estimate

holds, where the positive constant c depends only on p and q; Id is the identity operator.

Let condition (3.1) hold. Then

Id L w p ( ) ( I n ) L v q ( ) ( I n ) Id L w p ( ) ( I ) L v q ( ) ( I ) <.

Hence, ( L p ( ) , l r ) α ( L q ( ) , l r ) α .

Conversely, let the continuous embedding ( L p ( ) , l r ) α ( L q ( ) , l r ) α hold and let 1< q (I) q + (I)< p (I) p + (I)<. By taking functions supported in I n we can derive the estimate

sup n Z Id L p ( ) ( I n ) L v q ( ) ( I n ) Id ( L p ( ) ( I ) , l r ) α ( L v q ( ) ( I ) , l r ) α .

By applying the left-hand side inequality of (3.1′) and Proposition A, we conclude that condition (3.1) is satisfied. □

3.2 General operators in VEAS

We begin this subsection by the following definition.

Definition 3.1 ([31])

Let T be an operator defined on a set of real measurable functions f on . Define a sequence of local operators

( T n f)(x):=T(f χ ( n 1 , n + 2 ) )(x),x(n1,n+2),nZ.

Let us assume that there is a discrete operator T d satisfying the following conditions:

  1. (i)

    There exists a positive constant c such that for all non-negative functions f, x(n,n+1) and arbitrary nZ the inequality

    T(f χ ( , n 1 ) +f χ ( n + 2 , ) )(x)c T d ( m 1 m f ) (n)

holds.

  1. (ii)

    There is c>0 such that for all sequences { a k } of non-negative real numbers and nZ, the inequality

    T d ( { a k } ) (n)cTf(y)

holds for all y(n,n+1) and all non-negative f, where m 1 m f=: a m , mZ. It is also assumed that T satisfies the conditions

Tf=T|f|,T(λf)=|λ|Tf,T(f+g)Tf+Tg,TfTgif fg.

We will say that an operator T satisfying all the above mentioned conditions is admissible on .

For example, Hardy operators, Hardy-Littlewood maximal operators, fractional integral operators, fractional maximal operators are admissible on (see [31]). Carton-Leburn, Heinig and Hoffmann [32] established two weighted criteria for the Hardy transform (Hf)(x)= x f(t)dt in amalgam spaces defined on (see also [43, 44] for related topics). In [32], the authors derived some sufficient conditions for the two-weight boundedness of the kernel operator (Kf)(x):= x k(x,y)f(y)dy where k is non-decreasing in the second variable and non-increasing in the first one. In the paper [45], the two-weight problem for generalized Hardy-type kernel operators including the fractional integrals of order greater than one (without singularity) was solved.

General type results for the admissible operators read as follows.

Theorem F ([31])

Let 1<p, p ¯ ,q, q ¯ <, and let w and v be weight functions on . Suppose that T is an admissible operator on . Then the inequality

v T f ( L p ( R ) , l q ) c w f ( L p ¯ ( R ) , l q ¯ )

holds for all measurable f if and only if

  1. (i)
    T d

    is bounded from l q ¯ ({ w n }) to l q ({ v n }), where w n := ( n 1 n w p ¯ ) q ¯ p ¯ , v n := ( n n + 1 v p ) q p .(ii)

  2. (a)
    sup n Z T n [ L w p ¯ p ¯ ( n 1 , n + 2 ) L v p p ( n 1 , n + 2 ) ] <

    for 1< q ¯ q<.

  3. (b)
    T n [ L w p ¯ p ¯ ( n 1 , n + 2 ) L v p p ( n 1 , n + 2 ) ] l s

    , where 1 s = 1 q 1 q ¯ for 1<q< q ¯ <.

Our aim is to establish weighted characterization of the boundedness of kernel operators involving fractional integrals of variable parameter of order less than one in variable exponent amalgam spaces. For the continuous part of amalgam spaces, we take variable exponent Lebesgue spaces defined on I.

It should be emphasized that the following fact holds: by the change of variable z log 2 x it is possible to get appropriate boundedness or compactness results from dyadic amalgams ( L p ( ) ( R + ) , l q ) d to amalgams defined on .

Analyzing the proof of Theorem 1 of [31], we can formulate the next statement and give the proof for completeness.

Proposition 3.1 Let p ¯ (), p() be measurable functions on satisfying 1< p (R) p + (R)<, 1< p ¯ (R) p ¯ + (R)<. Suppose that q and q ¯ are constants satisfying 1<q, q ¯ <. Assume that w and v are weight functions on and that T is an admissible operator on . Then the inequality

v T f ( L p ( ) ( R ) , l q ) c w f ( L p ¯ ( ) ( R ) , l q ¯ )
(3.2)

holds if

  1. (i)
    T d

    is bounded from l q ¯ ({ w ¯ n }) to l q ({ v ¯ n }) where w ¯ n := χ ( n 1 , n ) ( ) w 1 ( ) L p ¯ ( ) q ¯ , v ¯ n := χ ( n , n + 1 ) ( ) v ( ) L p ( ) q . (ii)

  2. (a)
    sup n Z T n [ L w p ¯ ( ) p ¯ ( ) ( n 1 , n + 2 ) L v p ( ) p ( ) ( n 1 , n + 2 ) ] <

    for 1< q ¯ q<.

  3. (b)
    T n [ L w p ¯ ( ) ( n 1 , n + 2 ) L v p ( ) ( n 1 , n + 2 ) ] l s

    with 1 s = 1 q 1 q ¯ for 1<q< q ¯ <.

Conversely, let (3.2) hold. Then

  1. (1)

    conditions (ii) are satisfied;

  2. (2)

    condition (i) is satisfied for wconst or for p and p ¯ being constants outside some large interval [ m 0 , m 0 ], m 0 Z.

Proof

Let (i) and (ii) hold. We have

v T f ( L p ( ) ( R ) , l q ) c { n Z T [ f ( χ ( , n 1 ) + χ ( n + 2 , ) ) ] v ( ) L p ( ) ( n , n + 1 ) q } 1 / q + c { n Z v T n f L p ( ) ( n , n + 1 ) } 1 / q = : S 1 + S 2 .

Let a m := m 1 m f. By the hypothesis and Hölder’s inequality for variable exponents p() and p (), we have that

S 1 c { n Z ( T d ( { a m } ) ( n ) ) q χ ( n , n + 1 ) v L p ( ) ( n , n + 1 ) q } 1 / q c { n Z a n q ¯ χ ( n 1 , n ) w 1 L p ¯ ( ) q ¯ } 1 / q ¯ c w f ( L p ¯ ( ) ( R ) , l q ) .

Let us estimate S 2 . Suppose that 1< q ¯ q<. Since the operators T n are uniformly bounded, we find that

S 2 c { n Z f w L p ¯ ( ) ( n 1 , n + 1 ) q } 1 / q c { n Z f w L p ¯ ( ) ( n 1 , n + 1 ) q ¯ } 1 / q ¯ c f w ( L p ¯ ( ) ( R ) , l q ¯ ) .

If 1<q< q ¯ <, then by using Hölder’s inequality we derive

S 2 c { n Z T n [ L w p ¯ p ¯ ( ) ( n 1 , n + 2 ) L v p p ( ) ( n 1 , n + 2 ) ] q χ ( n 1 , n + 2 ) f w L p ¯ ( ) q } 1 / q c [ { n Z T n q q ¯ q ¯ q } q ¯ q q { n Z χ ( n 1 , n + 2 ) f w L p ¯ ( ) q ¯ } q q ¯ ] 1 / q c f w ( L p ¯ ( ) ( R ) , l q ¯ ) .

Conversely, suppose that (3.2) holds. Let nZ and let f be a non-negative function supported in (n1,n+2). Then

f w ( L p ¯ ( ) ( R ) , l q ¯ ) 3 f w χ ( n 1 , n + 2 ) ( L p ¯ ( ) ( R ) ) .

On the other hand,

v T f ( L p ( ) , l q ) v χ ( n 1 , n + 2 ) T f L p ( ) v χ ( n 1 , n + 2 ) T n f L p ( ) = v T n f L p ( ) .

Now due to inequality (3.2), we conclude that (a) of (ii) holds. Let us now show that if 1<q< q ¯ <, then (b) of (ii) is satisfied.

Since T n [ L w p ¯ ( ) p ¯ ( ) L v p ( ) p ( ) ] = sup { f : w f L p ¯ ( ) = 1 } v T n f L p ( ) , we have that for each n, there exists a non-negative measurable function f n , with the support in (n1,n+2) and with w χ ( n 1 , n + 2 ) f n L p ¯ ( ) =1, such that T n [ L w p ¯ ( ) p ¯ ( ) L v p ( ) p ( ) ] < v T n f n L p ( ) + 1 2 | n | . Thus, it is sufficient to prove that v T n f n L p ( ) l s .

Let { a n } be a sequence of non-negative real numbers and f= n a n f n . For each nZ, f(x)> a n f n (x) and then Tf(x) a n T n f n (x) for all x(n1,n+2).

Consequently,

v T f ( L p ( ) ( R ) , l q ) { n Z c a n q χ ( n 1 , n + 2 ) v T n f L p ( ) q } 1 / q =c { n Z a n q v T n f n L p ( ) q } 1 / q .

Hence, inequality (3.2) yields that

{ n Z a n q v T n f n L p ( ) q } 1 / q c { n Z χ ( n 1 , n + 2 ) w f L p ¯ ( ) q ¯ } 1 / q ¯ c { n Z a n q ¯ χ ( n 1 , n + 2 ) w f n L p ¯ ( ) q ¯ } 1 / q ¯ = c { n Z a n q ¯ } .

Finally, by Lemma B, we see that (b) of (ii) holds.

Now let us prove that (i) holds when wconst. If { a m } is a sequence of non-negative real numbers and

f= m Z a m χ ( m 1 , m ) ,

then m 1 m f= a m , and χ ( n , n + 1 ) f L p ¯ ( ) q ¯ = a n q ¯ χ ( n , n + 1 ) L p ¯ ( ) q ¯ = a n q ¯ and by the properties of T, we have

v T f ( L p ( ) , l q ) = { n Z χ ( n , n + 1 ) v T f L p ( ) q } 1 / q { n Z χ ( n , n + 1 ) v T d ( m 1 m f ) L p ( ) q } 1 / q c { n Z T d ( a m ) q ( n ) χ ( n , n + 1 ) v L p ( ) q } 1 / q = T d { a m } l q { v ¯ n q } .

Applying the two-weight inequality, we find that

T d { a m } l q { v ¯ n q } c { n Z χ ( n , n + 1 ) f L p ¯ ( ) q ¯ } 1 / q ¯ = c { n Z a n q ¯ } 1 / q ¯ = a n l q ¯ .

Hence, (i) holds.

Suppose now that w is a general weight and there is a positive integer m 0 such that p, p ¯ are constants outside [ m 0 , m 0 ]. Taking

f(x)= m Z a m χ ( m 1 , m ) (x) ( m 1 m w p ¯ ( y ) ( y ) d y ) 1 w p ¯ ( x ) (x),

it is easy to see that m 1 m f= a m . Moreover, by virtue of Proposition A and the fact that

m 1 m w p ¯ ( y ) (y)dy m 0 m 0 w p ¯ ( y ) (y)dy<,[m1,m][ m 0 , m 0 ],

we have for m m 0 +1,

χ ( m 1 , m ) f w L p ¯ ( ) = a m ( m 1 m w p ¯ ( y ) ( y ) d y ) 1 χ ( m 1 , m ) w ( 1 p ¯ ( ) ) L p ¯ ( ) c a m ( m 1 m w p ¯ ( y ) ( y ) d y ) 1 / p ¯ + ( [ m 1 , m ) ) ,

where the positive constant c depends on m 0 . Since

v T f ( L p ( ) ( R ) , l q ) C v ¯ n ( T d { a m } ) ( n ) l q ,

using again Proposition A, we find that

v ¯ n ( T d { a m } ) ( n ) l q C [ m χ ( m 1 , m ) f w L p ¯ ( ) ( R ) q ¯ ] 1 / q ¯ c [ m a m q ¯ ( m 1 m w p ¯ ( y ) ( y ) d y ) q ¯ / p ¯ + ( [ m 1 , m ) ) ] 1 / q ¯ = a m w ¯ m l q ¯ .

 □

Definition 3.2 Let T be an operator defined on a set of real measurable functions f on  R + . We say that an operator T is admissible on R + if the conditions of Definition 3.1 are satisfied replacing n by 2 n , nZ.

The next statement can be obtained in the similar manner as Proposition 3.1 was proved; therefore, we omit the proof.

Proposition 3.2 Let p ¯ (), p() be measurable functions on R + satisfying 1< p ( R + ) p + ( R + )<, 1< p ¯ ( R + ) p ¯ + ( R + )<. Suppose that q and q ¯ are constants satisfying 1<q, q ¯ <. Suppose also that w and v are weight functions on R + and that T is an admissible operator on R + .

Then the inequality

v T f ( L p ( ) ( R + ) , l q ) d c w f ( L p ¯ ( ) ( R + ) , l q ¯ ) d
(3.3)

holds if

  1. (i)
    T d

    is bounded from l q ¯ ({ w ¯ n }) to l q ({ v ¯ n }) where w ¯ n := χ ( 2 n 1 , 2 n ) ( ) w 1 ( ) L p ¯ ( ) q ¯ , v ¯ n := χ ( 2 n , 2 n + 1 ) ( ) v ( ) L p ( ) q . (ii)

  2. (a)
    sup n Z T n [ L w p ¯ ( ) p ¯ ( ) ( 2 n 1 , 2 n + 2 ) L v p ( ) p ( ) ( 2 n 1 , 2 n + 2 ) ] <

    for 1< q ¯ q<.

  3. (b)
    T n [ L w p ¯ ( ) p ¯ ( ) ( 2 n 1 , 2 n + 2 ) L v p ( ) p ( ) ( 2 n 1 , 2 n + 2 ) ] l s

    with 1 s = 1 q 1 q ¯ for 1<q< q ¯ <.

Conversely, if (3.3) holds, then

  1. (1)

    conditions (ii) are satisfied;

  2. (2)

    condition (i) is also satisfied but for wconst or for p and p ¯ satisfying the condition pconst, p ¯ const outside some large interval [0, 2 m 0 ], m 0 Z.

Proposition 3.2 gives criteria for the boundedness of H v , w in dyadic amalgams on R + but by the next statement we prove the two-weight inequality under slightly different conditions.

Proposition 3.3 Let I:= R + and let 1< p ¯ (I) p ¯ ()p() p + (I)<. Let 1< q ¯ ,q<. Suppose that p, p ¯ WL( R + ) and that p p c const outside some large interval (0,b). Then the inequality

H v , w f ( L p ( ) ( I ) , l q ) d c f ( L p ¯ ( ) , l q ¯ ) d

with a positive constant independent of f holds if

  1. (i)

    in the case 1< q ¯ q<,

  2. (a)
    sup m Z { n = m χ [ 2 n , 2 n + 1 ) ( ) v ( ) L p ( ) q } 1 / q { n = m χ [ 2 n 1 , 2 n ) ( ) w ( ) L p ¯ ( ) q ¯ } 1 / q ¯ <,
  3. (b)
    sup n Z sup 0 < α < 1 χ [ 2 n + α , 2 n + 1 ) ( ) v ( ) L p ( ) w ( ) χ ( 2 n , 2 n + α ) ( ) L p ¯ ( ) <;
  4. (ii)

    in the case 1<q< q ¯ <,

  5. (a)
    { C n } l s

    , where

    C n = sup β ( 0 , 1 ) χ [ 2 n + β , 2 n + 1 ) v ( ) L p ( ) w ( ) χ [ 2 n , 2 n + β ) L p ¯ ( ) ,
  6. (b)

where 1 s = 1 q ¯ 1 q .

Proof Let 1< q ¯ q<. Suppose that f0. We represent:

( H v , w f ) ( x ) = v ( x ) 0 2 n f ( t ) w ( t ) d t + v ( x ) 2 n x f ( t ) w ( t ) d t = : ( H v , w ( 1 ) f ) ( x ) + ( H v , w ( 2 ) f ) ( x ) , x [ 2 n , 2 n + 1 ] .
(3.4)

We have

( H v , w f ) χ [ 2 n , 2 n + 1 ) ( ) L p ( ) v ( ) χ [ 2 n , 2 n + 1 ) ( ) L p ( ) ( 0 2 n f ( t ) w ( t ) d t ) + v ( x ) 2 n x f ( t ) w ( t ) d t L p ( ) ( [ 2 n , 2 n + 1 ) ) = : S 1 ( n ) + S 2 ( n ) .

Let a k := 2 k 1 2 k fw. Then by the discrete Hardy inequality (see Lemma C) and Hölder’s inequality with respect to the exponents p ¯ () and ( p ¯ ( ) ) we derive

( n Z ( S 1 ( n ) ) q ) 1 / q = [ n Z v ( ) χ [ 2 n , 2 n + 1 ) ( ) L p ( ) q ( k = n 2 k 1 2 k f ( t ) w ( t ) d t ) q ] 1 / q c [ n Z ( 2 n 1 2 n f ( t ) w ( t ) d t ) q ¯ w ( ) χ [ 2 n 1 , 2 n ) ( ) L p ¯ ( ) q ¯ ] 1 / q c [ n Z χ [ 2 n 1 , 2 n ) ( ) f ( ) L p ¯ ( ) q ¯ ] 1 / q ¯ = c f ( L p ¯ ( ) , l q ¯ ) d .

Further, by Corollary A and Theorem E, we have that

( n Z ( S 2 ( n ) ) q ) 1 / q = [ n Z v ( x ) 2 n x f ( t ) w ( t ) d t L p ( ) ( 2 n , 2 n + 1 ) q ] 1 / q c [ n Z f ( ) χ ( 2 n , 2 n + 1 ) ( ) L p ¯ ( ) ( 2 n , 2 n + 1 ) q ] 1 / q c [ n Z f ( ) χ ( 2 n , 2 n + 1 ) ( ) L p ¯ ( ) ( 2 n , 2 n + 1 ) q ¯ ] 1 / q ¯ = c f ( L p ¯ ( ) , l q ¯ ) d .

Let 1<q< q ¯ <. Using representation (3.4), we derive

( H v , w f ) ( L p ( ) ( R + ) , l q ) d [ n Z χ [ 2 n , 2 n + 1 ) H v , w ( 1 ) f L p ( ) q ] 1 / q + [ n Z χ [ 2 n , 2 n + 1 ) H v , w ( 2 ) f L p ( ) q ] 1 / q = : S 1 + S 2 .

We estimate S 1 and S 2 .

S 1 = [ n Z χ [ 2 n , 2 n + 1 ) ( ) v ( ) L p ( ) q ( 0 2 n f w ) q ] 1 / q = [ n Z χ [ 2 n , 2 n + 1 ) ( ) v ( ) L p ( ) q ( k = n 2 k 1 2 k f w ) q ] 1 / q .

By the two-weight inequality for the discrete Hardy transform (see Lemma C), we have

S 1 c [ n Z χ [ 2 n 1 , 2 n ) ( ) w ( ) L p ¯ ( ) q ¯ ( 2 n 1 2 n f w ) q ¯ ] 1 / q ¯ c [ n Z χ [ 2 n 1 , 2 n ) ( ) w ( ) L p ¯ ( ) q ¯ χ [ 2 n 1 , 2 n ) f L p ¯ ( ) q ¯ χ [ 2 n 1 , 2 n ) w L p ¯ ( ) q ¯ ] 1 / q ¯ c f ( L p ¯ ( ) ( R + ) , l q ¯ ) d .

Now we estimate S 2 . Using Corollary A for intervals ( 2 n , 2 n + 1 ] and Hölder’s inequality, we find that

S 2 c { n Z C n q χ [ 2 n , 2 n + 1 ) f L p ¯ ( ) q } 1 / q c { ( n Z χ [ 2 n , 2 n + 1 ) f L p ¯ ( ) q ¯ ) q / q ¯ ( n Z C n q q ¯ q ¯ q ) q ¯ q q } 1 / q c ( n Z C n s ) 1 / s f ( L p ¯ ( ) ( R + ) , l q ¯ ) d .

 □

3.3 Kernel operators on amalgams ( L p ( ) ( R + ) , l q ) d and ( L p ( ) (R), l q )

The conditions of general-type statements (see Propositions 3.1 and 3.2) are not easily verifiable for general kernel operators as well as for some concrete fractional integral operators such as the Riemann-Liouville fractional integral transform with variable parameter. That is why we investigate mapping properties of general kernel operators independently from general-type statements.

Let

( K v f)(x)=v(x) 0 x f(t)k(x,t)dt,x>0.

One of our aims is to characterize a class of weights v governing the boundedness of K v from ( L p ¯ ( ) , l q ¯ ) d to ( L p ( ) , l q ) d .

We will use the notation:

(3.5)
(3.6)

Theorem 3.1 Let I:= R + , 1< p ¯ (I) p ¯ ()p() p + (I)< and let p ¯ ,pWL(I). Suppose that q ¯ and q are constants such that 1< q ¯ q<. Let p(x) p c const and p ¯ (x) p ¯ c const outside some large interval (0, 2 m 0 ), m 0 Z. Let kV(I) V p ¯ ( ) (I). Then K v is bounded from ( L p ¯ ( ) ( I ) , l q ¯ ) d to ( L p ( ) ( I ) , l q ) d if and only if B<, where B=max{ B 1 , B 2 }.

Proof Sufficiency. Using the representation:

( K v f ) ( x ) = v ( x ) 0 x / 2 k ( x , t ) f ( t ) d t + v ( x ) x / 2 x k ( x , t ) f ( t ) d t = : ( K v ( 1 ) f ) ( x ) + ( K v ( 2 ) f ) ( x )

we have that

K v f ( L p ( ) , l q ) d K v ( 1 ) f ( L p ( ) , l q ) d + K v ( 2 ) f ( L p ( ) , l q ) d .

Further, taking Proposition 3.3 and the condition kV(I) into account, we find that

K v ( 1 ) f ( L p ( ) ( I ) , l q ) d c v ( x ) k ( x , x 2 ) 0 x f ( t ) d t ( L p ( ) , l q ) d c B f ( L p ¯ ( ) ( I ) , l q ) d .

Now observe that by the condition k V p ¯ ( ) (I), Proposition A and Lemma A we obtain

where

Let us now observe that by Proposition A and Lemma A, B ¯ 1 A ¯ c B 1 , where

A ¯ := sup k Z v ( ) k ( x , x / 2 ) χ ( 2 k , 2 k + 1 ] L p ( ) χ ( 2 k 1 , 2 k ] ( ) L p ¯ ( ) .
(3.7)

Necessity. Let p ¯ n be the sequence defined above. Considering the test function f n = χ ( 2 n , 2 n + 1 ] 2 n / p ¯ n in the boundedness of K v from ( L p ¯ ( ) ( I ) , l q ¯ ) d to ( L p ( ) ( I ) , l q ) d and taking the condition kV(I) into account we have that

I n := χ ( 2 n , 2 n + 1 ] ( x ) v ( x ) k ( x , x / 2 ) L p ( x ) c 2 n / ( p ¯ n ) .
(3.8)

It is easy to see that

  1. (i)
    n = m I n c ( 2 m / p ¯ ( 0 ) + 2 m 0 / p ¯ c )
    (3.9)

for m m 0 ;

  1. (ii)
    n = m I n c 2 m 0 / p ¯ c
    (3.10)

for m m 0 +1.

Denoting S m := [ n = m I n q ] 1 / q [ n = m 1 χ ( 2 n , 2 n + 1 ] L p ¯ ( ) q ¯ ] 1 / q ¯ and taking (3.9), Proposition A and Lemma A into account we have for m m 0 ,

S m [ n = m I n q ] 1 / q 2 m / p ¯ ( 0 ) [ 2 m / p ¯ ( 0 ) + 2 m 0 / p ¯ c ] 2 m / p ¯ ( 0 ) 1 + 2 m / p ¯ ( 0 ) 2 m 0 / p ¯ c 1 + 2 m 0 / p ¯ ( 0 ) 2 m 0 / p ¯ c < .

Similarly if m m 0 +1, then by (3.10),

S m [ n = m I n q ] 1 / q [ 2 m 0 / p ¯ ( 0 ) + 2 m / p ¯ c ] 2 m / p ¯ c [ 2 m 0 / p ¯ ( 0 ) + 2 m / p ¯ c ] 1 + 2 m 0 / p ¯ ( 0 ) 2 m 0 / p ¯ c < .

Hence, B 1 <.

Let now f be a function supported in ( 2 m , 2 m + 1 ]. Then due to the boundedness of K v from ( L p ¯ ( ) ( I ) , l q ¯ ) d to ( L p ( ) ( I ) , l q ) d and the condition kV(I) we have that

χ ( 2 m , 2 m + 1 ] v ( x ) k ( x , x / 2 ) ( 2 m x f ( y ) d y ) ( L p ( ) ( I ) , l q ) d c χ ( 2 m , 2 m + 1 ] f ( L p ¯ ( ) ( I ) , l q ¯ ) d ,

where the positive constant c does not depend on n. Using Theorem B with respect to the intervals [ 2 m , 2 m + 1 ) and the weight pair ( v ¯ ,w), where v ¯ (x)=v(x)k(x,x/2) χ ( 2 m , 2 m + 1 ] and w ¯ const, it follows that B 2 <. □

Remark 3.1 We have noticed in the proof of Theorem 3.1 that B 1 A ¯ , where A ¯ is defined in the same proof.

Now we formulate the boundedness criteria for the kernel operator

( K v f)=v(x) x k(x,t)f(t)dt,xR,

on amalgams defined on .

Let k(x,y) be a kernel on {(x,y):y<x} and v, p, p ¯ be defined on . For the next statement we define k ˜ , v ˜ , p 0 and p ¯ 0 as follows:

Theorem 3.2 Let 1< p ¯ (R) p ¯ (x)p(x) p + (R)< and let p ¯ 0 , p 0 WL( R + ). Let q ¯ and q are constants such that 1< q ¯ q<. Assume that p ¯ (x) p ¯ c const and p(x) p c const outside some large interval (,b). Let k ˜ V( R + ) V ( p ¯ 0 ( ) ) ( R + ). Then K v is bounded from ( L p ¯ ( ) (R), l q ¯ ) to ( L p ( ) (R), l q ) if and only if

Proof The proof follows from Theorem 3.1 by the change of variable z log 2 t. □

Let

( R α ( ) f)(x)=v(x) x 2 t f ( t ) ( x t ) 1 α ( x ) dt,

where 0<infαsupα<1 and x R + .

By virtue of Theorem 3.2 and Example 2.1 we can easily deduce the next statement.

Corollary 3.1 Let p, p ¯ , q and q ¯ be constants. Suppose that α is a measurable function on and that 1< p ¯ p<, 1< q ¯ q<, 1 p ¯ <α(x)<1. Then the operator R α ( ) is bounded from ( L p ¯ , l q ¯ ) to ( L p , l q ) if and only if

Moreover, there are positive constants c 1 and c 2 depending on p, p ¯ , q, q ¯ and α such that c 1 max{ D ˜ 1 , D ˜ 2 } R α ( ) c 2 max{ D ˜ 1 , D ˜ 2 }.

4 Compactness of kernel operators on VEAS

In this section, we derive compactness necessary and sufficient conditions for kernel operators on VEAS. Since for the amalgam norm we have the property f n ( L p ( ) ( I ) , l q ) α 0 when f n 0 a.e. ( f n ( L p ( ) ( I ) , l q ) α ), therefore, the following statement holds (see [46], Chap. XI).

Proposition 4.1 Let p, p ¯ be measurable functions on I such that 1< p ¯ ,p<. Let q, q ¯ be constants satisfying the condition 1<q, q ¯ <. Then the set of all functions of the form

k n (s,t) i = 1 n η i (s) λ i (t),s,tI,

is dense in the mixed norm space ( L p ( ) ( I ) , l q ) α [ ( L p ¯ ( ) ( I ) , l q ¯ ) α ], where λ i χ B i , χ B i ( L p ¯ ( ) ( I ) , l q ¯ ) α ( B i are measurable disjoint sets of I) and η i ( L p ( ) ( I ) , l q ) α L (I).

The next statement gives sufficient condition for the kernel operator to be compact on amalgams defined on R + .

Proposition 4.2 Let p(x) and q(x) be measurable functions on an interval I R + . Suppose that 1< p (I) p + (I)<, 1< p ¯ (I) p ¯ + (I)<. Let q, q ¯ be constants such that 1< q ¯ ,q<. If

M:= k ( x , y ) ( L ( p ¯ ( y ) ) ( I ) , l ( q ¯ ) ) α ( L p ( x ) ( I ) , l q ) α <,

where k is a non-negative kernel, then the operator

Kf(x)= I k(x,y)f(y)dy

is compact from ( L p ¯ ( ) ( I ) , l q ¯ ) α to ( L p ( ) ( I ) , l q ) α .

Proof

By Proposition 4.1 the set of functions

k m (s,t)= i = 1 m η i (s) λ i (t),s,tI,

is dense in ( L p ( ) ( I ) , l q ) α [ ( L p ¯ ( ) ( I ) , l q ¯ ) α ]. By Hölder’s inequality for amalgam spaces (see Theorem D), we have

| K f ( x ) | = | I k ( x , y ) f ( y ) d y | f ( L p ¯ ( ) ( I ) , l q ¯ ) α k ( x , y ) ( L ( p ¯ ( ) ) ( I ) , l ( q ¯ ) ) α .

Hence,

K f ( L p ( ) ( I ) , l q ) α k ( x , y ) ( L ( p ¯ ( y ) ) ( I ) , l ( q ¯ ) ) α ( L p ( x ) ( I ) , l q ) α f ( L p ¯ ( ) ( I ) , l q ¯ ) α M f ( L p ¯ ( ) ( I ) , l q ¯ ) α .

This means that KM.

Now we prove the compactness of K. For each nN, let

( K n ϕ)(x)= I k n (x,y)ϕ(y)dy.

Note that

( K n ϕ)(x)= I k n (x,y)ϕ(y)dy= i = 1 n η i (x) I λ i (y)ϕ(y)dy=: i = 1 n η i (x) b i ,

where

b i = I λ i (y)ϕ(y)dy.

This means that K n is a finite rank operator, i.e., it is compact. Further, let ϵ>0. Using the above-mentioned arguments, we have that there is N 0 N such that for n> N 0 ,

K K n k ( x , y ) k n ( x , y ) ( L ( p ¯ ( y ) ) ( I ) , l ( q ¯ ) ) α ( L p ( x ) ( I ) , l q ) α <ϵ.

Thus K can be represented as a limit of finite rank operators. Hence, K is compact. □

Theorem 4.1 Let 1< p ¯ ( R + ) p ¯ (x)p(x) p + ( R + )< and let p ¯ ,pWL( R + ). Let q ¯ and q be constants such that 1< q ¯ q<. Assume that kV( R + ) V ( p ¯ ( ) ) ( R + ). Suppose that p ¯ (x) p ¯ c const and p(x) p c const outside some large interval (0, 2 m 0 ). Then K v is compact from ( L p ¯ ( ) , l q ¯ ) d to ( L p ( ) , l q ) d if and only if

(i) B 1 < ; B 2 < , (ii) lim m B 1 ( m ) = lim m + B 1 ( m ) = 0 , (iii) lim n B 2 ( n ) = lim n + B 2 ( n ) = 0 ,

where, B 1 and B 2 are defined by (3.5) and (3.6), respectively, and

Proof Sufficiency. Let k 0 , n 0 be integers such that k 0 < m 0 < n 0 . Then we represent K v as follows:

( K v f ) ( x ) = χ [ 0 , 2 k 0 ] ( x ) K v ( f χ [ 0 , 2 k 0 ) ) ( x ) + χ ( 2 k 0 , 2 n 0 ) ( x ) K v ( f χ [ 0 , 2 n 0 ) ) ( x ) + χ [ 2 n 0 , ) ( x ) K v ( f χ [ 0 , 2 n 0 1 ) ) ( x ) + χ [ 2 n 0 , ) ( x ) K v ( f χ ( 2 n 0 1 , ) ) ( x ) = : ( K v ( 1 ) f ) ( x ) + ( K v ( 2 ) f ) ( x ) + ( K v ( 3 ) f ) ( x ) + ( K v ( 4 ) f ) ( x ) .

It is clear that

( K v ( 2 ) f ) (x)= R + k 2 (x,y)f(y)dy,

where k 2 (x,y)=v(x) χ ( 2 k 0 , 2 n 0 ) (x)k(x,y) if y<x and k 2 (x,y)=0 if yx. Then

Denoting I(x):= n = m χ ( 2 n , 2 n + 1 ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) , x[ 2 m , 2 m + 1 ), k 0 m n 0 1, we represent I(x) as

I ( x ) = n = m 2 χ ( 2 n , 2 n + 1 ) ( y ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) + χ ( 2 m 1 , 2 m ) ( y ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) + χ ( 2 m , x ) ( y ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) = : I 1 ( x ) + I 2 ( x ) + I 3 ( x ) .

Now we estimate I 1 (x), I 2 (x) and I 3 (x) separately

I 1 ( x ) c k ( q ¯ ) ( x , x 2 ) n = m 2 χ [ 2 n , 2 n + 1 ) ( y ) L ( p ¯ ) ( ) ( q ¯ ) c k ( q ¯ ) ( x , x 2 ) [ n = m 0 χ [ 2 n , 2 n + 1 ) ( ) L ( p ¯ ) ( ) ( q ¯ ) + n = m 0 + 1 m 2 χ [ 2 n , 2 n + 1 ) ( y ) L ( p ¯ ) ( y ) ( q ¯ ) ] c k q ¯ ( x , x 2 ) [ n = m 0 ( 2 n ) ( q ¯ ) / ( p ¯ ) ( 0 ) + m 0 + 1 n 0 ( 2 n ) ( q ¯ ) / ( p ¯ ) c ] c k ( q ¯ ) ( x , x 2 ) [ ( 2 m 0 ) ( q ¯ ) / ( p ¯ ) ( 0 ) + ( 2 n 0 ) ( q ¯ ) / ( p ¯ ) c ] .

Further,

I 2 ( x ) + I 3 ( x ) 2 χ ( 0 , x ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) c χ ( 0 , x / 2 ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) + c χ ( x / 2 , x ) k ( x , y ) L ( p ¯ ) ( y ) ( q ¯ ) k ( q ¯ ) ( x , x 2 ) [ χ ( 0 , 2 m ) ( y ) L p ¯ ( y ) ( q ¯ ) + x ( q ¯ ) / ( p ¯ ) ( x ) ] .

Considering separately the cases m m 0 and m> m 0 , by using Proposition A and Lemma A we find that

I 2 (x)+ I 3 (x)c k ( q ¯ ) ( x , x 2 ) [ ( 2 m ) ( q ¯ ) / ( p ¯ ) ( 0 ) + ( 2 m ) ( q ¯ ) / ( p ¯ ) c ] .

Consequently, since k 0 m< n 0 1, we have

I(x)c k ( q ¯ ) ( x , x 2 ) [ ( 2 n 0 ) ( q ¯ ) / ( p ¯ ) ( 0 ) + ( 2 n 0 ) ( q ¯ ) / ( p ¯ ) c ] =:c k ( q ¯ ) ( x , x 2 ) B n 0 .

Since B 1 < we find that

J(x) B n 0 1 / ( q ¯ ) [ m = k 0 n 0 1 χ [ 2 n , 2 n + 1 ) k ( x , x / 2 ) v ( x ) L p ( ) q ] 1 / q <.

So, by Proposition 4.2, we conclude that K v ( 2 ) is a compact operator. Further, write K v ( 3 ) as follows:

K v ( 3 ) f(x)= R + k 3 (x,y)f(y)dy,

where k 3 (x,y)=k(x,y) χ ( 0 , 2 n 0 1 ) (y) χ [ 2 n 0 , ) (x)v(x) if y<x and k 3 (x,y)=0 if yx. Then we have

Denoting F:= ( n = n 0 1 χ ( 2 n , 2 n + 1 ) ( y ) L p ¯ ( ) ( q ¯ ) ) 1 / ( q ¯ ) and considering both cases when m 0 n 0 2 and m 0 > n 0 2 separately, we derive as previously that

Fc [ ( 2 m 0 ) ( q ¯ ) / ( p ¯ ) ( 0 ) + ( 2 n 0 ) ( q ¯ ) / ( p ¯ ) c ] 1 / ( q ¯ ) =: B n 0 , m 0 ,

and since B 1 < we have

G B n 0 , m 0 [ m = n 0 χ [ 2 n , 2 n + 1 ) ( x ) k ( x , x / 2 ) v ( x ) L p ( x ) q ] 1 / q <.

Hence, by Proposition 4.2, K v ( 3 ) is compact.

Let us denote

I m := χ [ 2 m , 2 m + 1 ) ( x ) k ( x , x / 2 ) v ( x ) L p ( ) .
(4.1)

Following the proofs of Theorems 3.1, 3.2 and applying Proposition A and Lemma A, we have that

as k 0 0 because lim m B 1 (m)= lim m B 2 (m)=0. Further, applying Theorem 3.1, we find that

K v ( 4 ) ( L p ¯ ( ) ( I ) , l q ¯ ) ( L p ( ) ( I ) , l q ) max { sup m n 0 B 1 ( m ) , sup m n 0 B 2 ( m ) } 0

as n 0 +.

Hence,

K v f K v ( 2 ) f K v ( 3 ) f K v ( 1 ) f + K v ( 4 ) f 0

as B 1 (m)0, B i (m)0 , i=1,2. Hence K v is compact, since it is the limit of compact operators.

Necessity. First we show that lim m B 1 (m)=0. Let f n = χ ( 2 n 1 , 2 n + 1 ) 2 n / p ¯ n , where p ¯ n is defined in the proof of Theorem 3.1. Then f n 0 weakly in ( L p ¯ ( ) ( I ) , l q ¯ ) d as n. Indeed, let ϕ ( L ( p ¯ ( ) ) ( I ) , l ( q ¯ ) ) d . Then

| 0 f n ( y ) ϕ ( y ) d y | ( χ ( 2 n 1 , 2 n ] L p ¯ ( ) q ¯ + χ ( 2 n , 2 n + 1 ] L p ¯ ( ) q ¯ ) 1 / q ¯ 2 n / p ¯ c × ( ϕ χ ( 2 n 1 , 2 n ] L ( p ¯ ( ) ) q ¯ + ϕ χ ( 2 n 1 , 2 n ] L ( p ¯ ( ) ) q ¯ ) 1 / q ¯ 0

as n.

Observe now that

K v f n ( L p ¯ ( ) ( I ) , l q ¯ ) d χ ( 2 n , 2 n + 1 ) ( x ) v ( x ) k ( x , x / 2 ) L p ( ) 2 n / p ¯ n ,nZ.
(4.2)

Hence, lim n B 1 (n)0 because K v is compact and p ¯ n = p ¯ (0) if n< m 0 .

Further, (4.2) implies that

χ ( 2 n , 2 n + 1 ) ( x ) v ( x ) k ( x , x / 2 ) L p ( ) 2 n / ( p ¯ c ) 0

as n+.

To show that lim n + B 1 (n)0 we represent B 1 (n) as follows:

B 1 ( n ) = ( m = n I m q ) 1 / q ( m = n 1 χ ( 2 m , 2 m + 1 ] L ( p ¯ ( ) ) q ¯ ) 1 / q ¯ ( m = n I m q ) 1 / q ( m = m 0 1 2 m q ¯ / ( p ¯ ( 0 ) ) ) 1 / q ¯ + ( m = n I m q ) 1 / q ( m = m 0 n 1 2 m q ¯ / ( p ¯ c ) ) 1 / q ¯ = : J n ( 1 ) + J n ( 2 ) ,

where n m 0 and I m is defined by (4.1). Observe now that

J n ( 1 ) = ( m = n I m q ) 1 / q 2 m 0 / ( p ¯ ( 0 ) ) 0

as n+ because ( m = n I m q ) 1 / q 0 as n+. The latter convergence follows from the convergence of the series.

Further,

J n ( 2 ) c sup m n ( I m 2 m / ( p ¯ c ) ) 2 n / ( p ¯ c ) 2 n / ( p ¯ c ) c sup m n I m 2 m / ( p ¯ c ) 0

as n+ because I m 2 m / ( p ¯ c ) 0 as m+ (see (4.2)). Hence, lim m + B 1 (m)=0.

Further, it is easy to see that for 0<α<1 and f n ,

K v f n ( L p ( ) , l q ) d 2 n / p ¯ n χ ( 2 n , 2 n + 1 ) ( x ) v ( x ) k ( x , x / 2 ) x L p ( ) 2 n / ( p ¯ n ) χ ( 2 n , 2 n + 1 ) ( x ) v ( x ) k ( x , x / 2 ) L p ( ) c ( 2 n ( 2 α 1 ) ) 1 / ( p ¯ n ) χ ( 2 n + α , 2 n + 1 ) ( x ) v ( x ) k ( x , x / 2 ) L p ( ) .

Hence,

K v f n ( L p ( ) , l q ) d sup 0 < α < 1 ( 2 n ( 2 α 1 ) ) 1 / ( p ¯ n ) χ ( 2 n + α , 2 n + 1 ) ( x ) v ( x ) k ( x , x / 2 ) L p ( ) 0

as n+ or n.

The conditions B 1 < and B 2 < follow from the fact that every compact operator is bounded. □

Now we formulate the compactness criteria for the kernel operator K v defined on .

Theorem 4.2 Let 1< p ¯ (R) p ¯ (x)p(x) p + (R)< and let p ¯ 0 , p 0 WL( R + ). Let q ¯ and q be constants such that 1< q ¯ q<. Assume that p ¯ (x) p ¯ c const and p(x) p c const outside some large interval (, 2 m 0 ). Let k ˜ V( R + ) V ( p ¯ 0 ( ) ) ( R + ). Then K v is compact from ( L p ¯ ( ) , l q ¯ ) to ( L p ( ) , l q ) if and only if

(i) D 1 = sup m Z D 1 ( m ) < ; D 2 = sup n Z D 2 ( n ) < , (ii) lim m D 1 ( m ) = lim m D 1 ( m ) = 0 , (iii) lim n D 2 ( n ) = lim n D 2 ( n ) = 0 ,

where

k ˜

, v ˜ and p 0 and p ¯ 0 are defined in Section 3.

Proof The proof follows from Theorem 4.1 by the change of variable z log 2 t. □

References

  1. Edmunds DE, Kokilashvili V, Meskhi A:On the boundedness and compactness of the weighted Hardy operators in L p ( x ) spaces.Georgian Math. J. 2005, 12(1):27–44.

    MathSciNet  Google Scholar 

  2. Kopaliani TS: On some structural properties of Banach function spaces and boundedness of certain integral operators. Czechoslov. Math. J. 2004, 54(3):791–805. 10.1007/s10587-004-6427-3

    Article  MathSciNet  Google Scholar 

  3. Cruz-Uribe D, Mamedov FI: On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces. Rev. Mat. Complut. 2012, 25(2):335–367. 10.1007/s13163-011-0076-5

    Article  MathSciNet  Google Scholar 

  4. Samko S:Convolution type operators in L p ( x ) ( R n ). Integral Transforms Spec. Funct. 1998, 7(3–4):261–284. 10.1080/10652469808819204

    Article  MathSciNet  Google Scholar 

  5. Edmunds DE, Meskhi A:Potential-type operators in L p ( x ) spaces. Z. Anal. Anwend. 2002, 21: 681–690.

    Article  MathSciNet  Google Scholar 

  6. Diening L:Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p ( ) and W k , p ( ) . Math. Nachr. 2004, 268: 31–43. 10.1002/mana.200310157

    Article  MathSciNet  Google Scholar 

  7. Capone C, Cruz-Uribe D, Fiorenza A:The fractional maximal operator on variable L p spaces. Rev. Mat. Iberoam. 2007, 3(23):747–770.

    MathSciNet  Google Scholar 

  8. Cruz-Uribe D, Fiorenza A, Martell JM, Perez C:The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 2006, 31: 239–264.

    MathSciNet  Google Scholar 

  9. Kokilashvili V, Samko S:Maximal and fractional operators in weighted L p ( x ) spaces. Rev. Mat. Iberoam. 2004, 20(2):493–515.

    Article  MathSciNet  Google Scholar 

  10. Kokilashvili V, Samko S: On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Z. Anal. Anwend. 2003, 22(4):899–910.

    Article  MathSciNet  Google Scholar 

  11. Kokilashvili V, Meskhi A: Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent. Integral Transforms Spec. Funct. 2007, 18(9):609–628. 10.1080/10652460701445344

    Article  MathSciNet  Google Scholar 

  12. Kokilashvili V, Meskhi A:Two-weight inequalities for fractional maximal functions and singular integrals in L p ( ) spaces. J. Math. Sci. 2011, 173(6):656–673. 10.1007/s10958-011-0265-2

    Article  MathSciNet  Google Scholar 

  13. Kokilashvili V, Meskhi A, Sarwar M:One and two weight estimates for one-sided operators in L p ( ) spaces. Eurasian Math. J. 2010, 1(1):73–110.

    MathSciNet  Google Scholar 

  14. Kokilashvili V, Meskhi A, Sarwar M: Potential operators in variable exponent Lebesgue spaces: two-weight estimates. J. Inequal. Appl. 2010. doi:10.1155/2010/329571

    Google Scholar 

  15. Meskhi A: Measure of Non-Compactness for Integral Operators in Weighted Lebesgue Spaces. Nova Science Publishers, New York; 2009.

    Google Scholar 

  16. Diening L, Harjulehto P, Hästö P, Ružička M Lecture Notes in Mathematics 2017. In Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin; 2011.

    Chapter  Google Scholar 

  17. Musielak J Lecture Notes in Mathematics 1034. In Orlicz Spaces and Modular Spaces. Springer, Berlin; 1983.

    Google Scholar 

  18. Musielak J, Orlicz W: On modular spaces. Stud. Math. 1959, 18: 49–65.

    MathSciNet  Google Scholar 

  19. Nakano H: Topology of Linear Topological Spaces. Moruzen Co. Ltd, Tokyo; 1981.

    Google Scholar 

  20. Ružička M Lecture Notes in Mathematics 1748. In Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin; 2000.

    Google Scholar 

  21. Kokilashvili V: On a progress in the theory of integral operators in weighted Banach function spaces. In Function Spaces, Differential Operators and Nonlinear Analysis. Math. Inst. Acad. Sci. of Czech Republic, Prague; 2004. Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28-June 2

    Google Scholar 

  22. Kovácik O, Rákosník J:On spaces L p ( x ) and W k , p ( x ) . Czechoslov. Math. J. 1991, 41(4):592–618.

    Google Scholar 

  23. Samko S:Convolution type operators in L p ( x ) . Integral Transforms Spec. Funct. 1998, 7(1–2):123–144. 10.1080/10652469808819191

    Article  MathSciNet  Google Scholar 

  24. Sharapudinov II:The topology of the space L p ( t ) ([0,1]). Mat. Zametki 1979, 26(4):613–632. (Russian)

    MathSciNet  Google Scholar 

  25. Diening L:Maximal function on generalized Lebesgue spaces L p ( ) . Math. Inequal. Appl. 2004, 7(2):245–253.

    MathSciNet  Google Scholar 

  26. Ashraf U, Kokilashvili V, Meskhi A:Weight characterization of the trace inequality for the generalized Riemann-Liouville transform in L p ( x ) spaces. Math. Inequal. Appl. 2010, 13(1):63–81.

    MathSciNet  Google Scholar 

  27. Muckenhoupt B: Hardy’s inequality with weights. Stud. Math. 1972, 44: 31–38.

    MathSciNet  Google Scholar 

  28. Kokilashvili VM: On Hardy’s inequalities in weighted spaces. Soobsch. Akad. Nauk Gruz. SSR 1979, 96: 37–40. (Russian)

    MathSciNet  Google Scholar 

  29. Maz’ya VG: Sobolev Spaces. Springer, Berlin; 1985.

    Google Scholar 

  30. Kufner A, Persson LE: Weighted Inequalities of Hardy Type. World Scientific, River Edge; 2003.

    Book  Google Scholar 

  31. Cañestro MIA, Salvador PO: Boundedness of positive operators on weighted amalgams. J. Inequal. Appl. 2011. doi:10.1186/1029–242X-2011–13

    Google Scholar 

  32. Carton-Lebrun C, Heinig HP, Hofmann SC: Integral operators on weighted amalgams. Stud. Math. 1994, 109(2):133–175.

    MathSciNet  Google Scholar 

  33. Meskhi A: Criteria for the boundedness and compactness of integral transforms with positive kernels. Proc. Edinb. Math. Soc. 2001, 44(2):267–284. 10.1017/S0013091599000747

    Article  MathSciNet  Google Scholar 

  34. Edmunds DE, Kokilashvili V, Meskhi A: Bounded and Compact Integral Operators. Kluwer Academic, Dordrecht; 2002.

    Book  Google Scholar 

  35. Stewart J, Watson S: Irregular amalgams. Int. J. Math. Math. Sci. 1986, 9(2):331–340. 10.1155/S0161171286000418

    Article  MathSciNet  Google Scholar 

  36. Wiener N: On the representation of functions by trigonometrical integrals. Math. Z. 1926, 24: 575–616. 10.1007/BF01216799

    Article  MathSciNet  Google Scholar 

  37. Wiener N: Tauberian theorem. Ann. Math. 1932, 33: 1–100. 10.2307/1968102

    Article  MathSciNet  Google Scholar 

  38. Jakimovski A, Russell DC: Interpolation by functions with m th derivative in pre-assigned spaces. In Approximation Theory III. Edited by: Cheney EW. Academic Press, New York; 1980:531–536. Conference Proc. Texas, 1980

    Google Scholar 

  39. Day MM: Some more uniformly convex spaces. Bull. Am. Math. Soc. 1941, 47: 504–507. 10.1090/S0002-9904-1941-07499-9

    Article  Google Scholar 

  40. Köthe G I. In Topological Vector Spaces. Springer, New York; 1969.

    Google Scholar 

  41. Fournier JF, Stewart J:Amalgams of L p and l q . Bull. Am. Math. Soc. 1985, 13(1):1–21. 10.1090/S0273-0979-1985-15350-9

    Article  MathSciNet  Google Scholar 

  42. Edmunds DE, Fiorenza A, Meskhi A: On the measure of non-compactness for some classical operators. Acta Math. Sin. 2006, 22(6):1847–1862. 10.1007/s10114-005-0674-6

    Article  MathSciNet  Google Scholar 

  43. Salvador PO, Ramírez Torreblanca C: Hardy operators on weighted amalgams. Proc. R. Soc. Edinb. A 2010, 140(1):175–188. 10.1017/S0308210508001054

    Article  Google Scholar 

  44. Heinig HP, Kufner A: Weighted Friedrichs inequalities in amalgams. Czechoslov. Math. J. 1993, 43(118)(2):285–308.

    MathSciNet  Google Scholar 

  45. Cañestro MIA, Salvador PO: Boundedness of generalized Hardy operators on weighted amalgam spaces. Math. Inequal. Appl. 2010, 13(2):305–318.

    MathSciNet  Google Scholar 

  46. Kantorovich LP, Akilov GP: Functional Analysis. Pergamon, Oxford; 1982.

    Google Scholar 

Download references

Acknowledgements

The first and second authors were supported by the Shota Rustaveli National Science Foundation grant (Contract No. D/13-23). The part of this work is carried out at Abdus Salam School of Mathematical Sciences, GC University, Lahore. The second and third authors are thankful to the Higher Education Commission, Pakistan for the financial support. The authors are grateful to the anonymous referees for their remarks and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Muhammad Asad Zaighum.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Kokilashvili, V., Meskhi, A. & Zaighum, M.A. Weighted kernel operators in variable exponent amalgam spaces. J Inequal Appl 2013, 173 (2013). https://doi.org/10.1186/1029-242X-2013-173

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-173

Keywords