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Local saturation of a positive linear convolution operator

Abstract

Let { H n (t)} be a sequence of non-negative, even, and continuous functions on . In this paper, we consider a convolution operator J n (f;x)= 0 f(t) H n (tx)dt, f L p ( R + ), and then investigate the local saturation of J n (f;x).

MSC:44A35.

1 Introduction and theorems

Through this paper we let { H n (t)}, n=1,2, , be a sequence of non-negative, even, and continuous functions on R:=(,), and there exist M,N>0, and T>0 such that { H n (t)} satisfied uniformly

sup | t | T H n (t)M,nN,
(1.1)
H n (t)dt=1,n=1,2,,
(1.2)
t 2 H n (t)dt= μ n 0as n
(1.3)

and there exist two positive constants α, β with 2βα>3 such that uniformly for n,

| t | α H n (t)dtC μ n β .
(1.4)

As an example for H n (t) we give the following:

Let

H n (t)= n π e ( n t ) 2 .

Then it satisfies (1.2),

μ n = 1 n 2 0as nand t 4 H n (t)dt= 3 4 1 n 4 .

Let us denote R + :=[0,). In what follows we assume that H n (t) satisfies the conditions (1.1)-(1.4). Using H n , we define the convolution operators for f L p ( R + ),

J n (f;x):= 0 f(t) H n (tx)dt,n=1,2,.
(1.5)

Swetits and Wood [1] studied the operators on a finite interval [0,r];

K n (f;x):= 0 r f(t) H n (tx)dt,f L p ( [ 0 , r ] ) ,n=1,2,,

where H n is defined on [r,r] as H n , and then they gave a local saturation theorem. Furthermore, there is a rich bibliography concerning the convergence of positive linear operators on [0,) (e.g. see [24] and the references cited therein).

In this paper we extend [1] to the infinite interval R + . Then we use a similar methods as [1]. For 1<p< and c0, we define L p 2 ([c,)) as the space of those functions f such that f L p ( R + ) and f is a locally absolutely continuous function on [c,), with f L p ([c,)) and f L p ([c,)). Let C c 2 ( R + ) be the space of continuous, compactly supported and continuously second differentiable functions on R + . Furthermore, the total variation of a real-valued function f defined on an interval [a,b]R is the quantity

V a b (f)= sup P P i = 0 n P 1 |f( x i + 1 )f( x i )|.

Here the supremum is taken over the set P of all partitions P={ x 0 , x 1 ,, x n P } of the interval considered. If F:[b,)R and x[b,), we define

T F (x)=sup { 1 n | F ( x j ) F ( x j 1 ) | : b = x 0 < x 1 < < x n = x , n N } .

If T F ():= lim x T F (x) is finite, we say that F is of bounded variation on [b,), and T F () is called the total variation of F on [b,). We define BV[b,) to be the set of all functions on [b,) whose total variation on [b,) is finite.

Then we first give the pointwise convergence theorem.

Theorem 1.1 (cf. [5])

Let f C c 2 ( R + ), and let x[a,), a>0. Then we have a pointwise convergence as follows:

lim n 1 μ n ( J n ( f ; x ) f ( x ) ) = 1 2 f (x).
(1.6)

Equation (1.6) holds uniformly on [a,).

Then the following is a direct convergence theorem.

Theorem 1.2 Let 0<b<a<.

  1. (i)

    If 1<p<, then we have for f L p 2 ([b,))

    J n ( f ; x ) f ( x ) L p ( [ a , ) ) =O( μ n )as n.
    (1.7)
  2. (ii)

    If p=1, then we have for f BV([b,)) with f L 1 ([0,))

    J n ( f ; x ) f ( x ) L 1 ( [ a , ) ) =O( μ n )as n.
    (1.8)
  3. (iii)

    Let 1p< and f be linear on [b,). Then we have

    J n ( f ; x ) f ( x ) L p ( [ a , ) ) =o( μ n )as n.
    (1.9)

Finally, we give an inverse theorem as follows.

Theorem 1.3 Let 0<b<a<. Let f L p ( R + ) and f L p ( R + ).

  1. (i)

    For 1<p<, the condition (1.7) implies f L p 2 ([a,)).

  2. (ii)

    For p=1, the condition (1.8) implies f BV([a,)).

  3. (iii)

    For 1p<, the condition (1.9) implies that f is linear on [a,).

This paper is organized as follows. In Section 2, we will give some fundamental lemmas in order to prove the main results and we will prove the results in Section 3.

2 Fundamental lemmas

Throughout this paper C, C 1 , C 2 , denote positive constants independent of n, x, t or function f(x). The same symbol does not necessarily denote the same constant in different occurrences.

To prove the theorems we need some lemmas.

Lemma 2.1 Let δ>0 and =0,1,2. Then for 2βα>3 defined in (1.4)

| u | δ | u | H n (u)duC μ n β δ α .
(2.1)

Proof Let =0,1,2. Then we have from (1.4)

| u | δ | u | H n ( u ) d u = δ | u | δ ( | u | δ ) H n ( u ) d u δ | u | δ ( | u | δ ) α H n ( u ) d u δ α | u | δ | u | α H n ( u ) d u = δ α | u | α H n ( u ) d u C μ n β δ α .

 □

Lemma 2.2 Let e 0 (x):=1, e 1 (x):=x and let δ be a positive constant. Then for n=1,2, , we have

| J n ( ( x ) ; x ) |C μ n β δ α 1 ,0<δx
(2.2)

and

| J n ( ( x ) 2 ; x ) | μ n ,x0.
(2.3)

Moreover, we have for 0<δx

| J n ( e 0 ;x) e 0 |C μ n β δ α and| J n ( e 1 ;x) e 1 |C μ n β δ α 1 .
(2.4)

Here, α and β are defined in (1.4).

Proof For 0<δx, we have by (2.1)

| J n ( ( x ) ; x ) | = | 0 ( t x ) H n ( t x ) d t | = | x y H n ( y ) d y | = | x y H n ( y ) d y | H n ( t )  is even y δ y H n ( y ) d y C μ n β δ α 1 .

For (2.3), we have from (1.3)

J n ( ( x ) 2 ; x ) = 0 ( t x ) 2 H n (tx)dt= x y 2 H n (y)dy μ n .

Since we know for 0<δx

J n ( e 0 ;x)= 0 H n (tx)dt= x H n (y)dy=1 x H n (y)dy=1 x H n (y)dy,

and from (2.1)

| x H n (y)dy|C μ n β δ α ,
(2.5)

we have for 0<δx,

| J n ( e 0 ;x) e 0 |C μ n β δ α .

Next, we give an estimate for e 1 . Since

J n ( e 1 ; x ) = 0 t H n ( t x ) d t = x ( x + y ) H n ( y ) d y = x ( 1 x H n ( y ) d y ) + x y H n ( y ) d y = x x x H n ( y ) d y + x y H n ( y ) d y

and by (2.1), for 0<δx,

|x x H n (y)dy|+| x y H n (y)dy|2 x y H n (y)dyC μ n β δ α 1 ,

we have for 0<δx,

| J n ( e 1 ;x) e 1 |C μ n β δ α 1 .

 □

Let b>0 and then we define

χ ( [ 0 , b ] ; t ) :={ 1 , t [ 0 , b ] , 0 , t [ 0 , b ] .

Lemma 2.3 Let 0<b<a, δ:=ab and let 1p<. If f L p ( R + ), then

J n ( χ ( [ 0 , b ] ) f ) L p [ a , ) C μ n β δ α f L p ( R + ) .

Proof Let p=1. Then we have

J n ( χ ( [ 0 , b ] ) f ) L 1 ( [ a , ) ) = a | 0 χ ( [ 0 , b ] ; t ) f ( t ) H n ( t x ) d t | d x a 0 b | f ( t ) | H n ( t x ) d t d x 0 b | f ( t ) | a H n ( t x ) d x d t ( sup t [ 0 , b ] a H n ( t x ) d x ) f L 1 ( R + ) .

For t[0,b] and xa, we see |tx|ab=:δ>0. From (1.4), we have the following:

sup t [ 0 , b ] a H n ( t x ) d x sup t [ 0 , b ] a ( | t x | δ ) α H n ( t x ) d x 1 δ α | t | α H n ( t ) d t C μ n β δ α .
(2.6)

Hence we have

J n ( χ ( [ 0 , b ] ) f ) L 1 [ a , ) C μ n β δ α f L 1 ( R + ) .

Let 1<p<, 0<b<ax and p+q=pq. By Hölder’s inequality,

| J n ( χ ( [ 0 , b ] ) f , x ) | = | 0 χ ( [ 0 , b ] ; t ) f ( t ) H n ( t x ) d t | C ( 0 χ ( [ 0 , b ] ; t ) H n ( t x ) d t ) 1 / q × ( 0 χ ( [ 0 , b ] ; t ) | f ( t ) | p H n ( t x ) d t ) 1 / p .

Here, from (2.1) we have

0 χ ( [ 0 , b ] ; t ) H n ( t x ) d t = 0 b H n ( t x ) d t = x b x H n ( y ) d y δ H n ( y ) d y C μ n β δ α .
(2.7)

Hence,

J n ( χ ( [ 0 , b ] ) f ) L p [ a , ) C ( μ n β δ α ) 1 / q ( a 0 χ ( [ 0 , b ] ; t ) | f ( t ) | p H n ( t x ) d t d x ) 1 / p .
(2.8)

Also, we see by (2.6)

a 0 χ ( [ 0 , b ] ; t ) | f ( t ) | p H n ( t x ) d t d x 0 a χ ( [ 0 , b ] ; t ) | f ( t ) | p H n ( t x ) d x d t sup t [ 0 , b ] a H n ( t x ) d x 0 | f ( t ) | p d t C f L p ( R + ) p μ n β δ α .

Thus, by (2.8) we conclude. □

3 Proof of theorems

Proof of Theorem 1.1 For x[a,) and t0, we set

f(t)=f(x)+ f (x)(tx)+ f ( η ) 2 ( t x ) 2 ,xηt.

Then we see

J n ( f ; x ) f ( x ) 1 2 f ( x ) μ n = f ( x ) ( J n ( e 0 ; x ) e 0 ) + f ( x ) 0 ( t x ) H n ( t x ) d t + 1 2 [ 0 f ( η ) ( t x ) 2 H n ( t x ) d t f ( x ) μ n ] , x η t : = J 1 + J 2 + J 3 .

For J 1 , we have from (2.4)

| J 1 |=|f(x)|O ( μ n β a α ) ,

and for J 2 , we have from (2.2)

| J 2 |=| f (x) J n ( ( x ) ; x ) |=| f (x)|O ( μ n β a α 1 ) .

Now, we will estimate J 3 . We have

0 f ( η ) ( t x ) 2 H n ( t x ) d t f ( x ) μ n = 0 ( f ( η ) f ( x ) ) ( t x ) 2 H n ( t x ) d t f ( x ) 0 ( t x ) 2 H n ( t x ) d t .

For the second term, we have by (2.1)

| f (x) 0 ( t x ) 2 H n (tx)dt|| f (x)| a u 2 H n (u)duC| f (x)| μ n β a α 2 .

For a given ε>0, there exists a positive constant δ 1 >0 (depending only on ε) such that for |xη|< δ 1

| f (x) f (η)|<ε.
(3.1)

For the first term, we have using (1.3)

| | x t | δ 1 ( f ( η ) f ( x ) ) ( t x ) 2 H n ( t x ) d t | ε | u | δ 1 u 2 H n ( u ) d u ε μ n ,

and by (2.1)

| | x t | δ 1 ( f ( η ) f ( x ) ) ( t x ) 2 H n ( t x ) d t | 2 f L ( R ) | u | δ 1 u 2 H n ( u ) d u f L ( R ) O ( μ n β δ 1 α 2 ) .

Then we have for some positive constants C 1 , C 2 , and C 3

| J n ( f ; x ) f ( x ) 1 2 f ( x ) μ n | C 1 | f ( x ) | μ n β a α + C 2 | f ( x ) | μ n β a α 1 + ε μ n + C 3 f L ( R ) μ n β δ 1 α 2 .

Therefore, we have for an arbitrary ε>0

lim n 1 μ n | J n (f;x)f(x) 1 2 f (x) μ n |ε.

Thus, (1.6) is proved. Moreover, noting (3.1), we see that (1.6) holds uniformly on [a,). □

Proof of Theorem 1.2 Let δ:=ab>0. (i) We start under the condition 1p< for the convenience of considering (ii). On the way of the proof we switch over to the assumption with 1<p<. Let 1<p< and let χ:=χ([0,b]) and let χ 1 (t)=1χ(t). Since

f(t)=(χf)(t)+( χ 1 f)(t),f(x)=( χ 1 f)(x),x[a,),

we have

J n ( f ) f L p ( [ a , ) ) J n ( χ f ) L p ( [ a , ) ) + J n ( χ 1 f ) χ 1 f L p ( [ a , ) ) =: K 1 + K 2 .
(3.2)

By Lemma 2.3,

K 1 = J n ( χ f ) L p ( [ a , ) ) C f L p ( R + ) μ n β δ α .
(3.3)

We estimate K 2 . Let f L p 2 ([a,)). Then

J : = J n ( ( χ 1 f ) ( t ) ( χ 1 f ) ( x ) , x ) L p ( [ a , ) ) = ( a | 0 ( χ 1 ( t ) f ( t ) f ( x ) ) H n ( t x ) d t | p d x ) 1 / p ( a | ( 0 b + b ) ( χ 1 ( t ) f ( t ) f ( x ) ) H n ( t x ) d t | p d x ) 1 / p ( a | f ( x ) | p | 0 b H n ( t x ) d t | p d x ) 1 / p + ( a | b ( f ( t ) f ( x ) ) H n ( t x ) d t | p d x ) 1 / p = : I 1 + I 2 .
(3.4)

Here, for the first term, using

0 b H n (tx)dtC μ n β δ α ,

which is shown in (2.7), we have

I 1 = ( a | f ( x ) | p | 0 b H n ( t x ) d t | p d x ) 1 / p C f L p ( [ a , ) ) μ n β δ α .
(3.5)

From this we suppose 1<p<. By

f(t)f(x)= f (x)(tx)+ x t (tu) f (u)du,

we have

I 2 = ( a | b ( f ( t ) f ( x ) ) H n ( t x ) d t | p d x ) 1 / p ( a | b f ( x ) ( t x ) H n ( t x ) d t | p d x ) 1 / p + ( a | b x t ( t u ) f ( u ) d u H n ( t x ) d t | p d x ) 1 / p = : I 2 , 1 + I 2 , 2 .

Here we note that y H n (y) is an odd function. Then we have by (2.1)

| b ( t x ) H n ( t x ) d t | = | b x y H n ( y ) d y | = x b y H n ( y ) d y y δ y H n ( y ) d y C μ n β δ α 1

and the first term I 2 , 1 is estimated as

I 2 , 1 = ( a ( | f ( x ) | | b ( t x ) H n ( t x ) d t | ) p d x ) 1 / p C μ n β δ α 1 ( a | f ( x ) | p d x ) 1 / p C f L p ( [ a , ) ) μ n β δ α 1 .
(3.6)

Now we set

θ ( f , x ) := sup b t , t x 1 t x x t | f (u)|du,ax,

and denote the Hardy-Littlewood majorant of f at x. Since f L p ([b,)) and 1<p<, we have

θ ( f , x ) L p ( [ a , ) ) A p f L p ( [ b , ) ) ,
(3.7)

where A p >0 depend only on p [[6], Theorem 1, p.201]. Since

| x t (tu) f (u)du|± x t |tu|| f (u)|du ( t x ) 2 1 t x x t | f (u)|du,

where ± x t du0, we have by (3.7) and (1.3)

I 2 , 2 ( a | b ( ( t x ) 2 1 t x x t | f ( u ) | d u ) H n ( t x ) d t | p d x ) 1 / p ( a | θ ( f ; x ) | p | b ( t x ) 2 H n ( t x ) d t | p d x ) 1 / p u 2 H n ( u ) d u ( a | θ ( f ; x ) | p d x ) 1 / p A p μ n f L p ( [ b , ) ) .
(3.8)

Hence, from (3.6) and (3.8), we have for some positive constant C>0

I 2 I 2 , 1 + I 2 , 2 C ( f L p ( [ a , ) ) + f L p ( [ b , ) ) ) μ n .
(3.9)

So, from (3.5) and (3.9),

J I 1 + I 2 C ( f L p ( R + ) + f L p ( [ a , ) ) + f L p ( [ b , ) ) ) μ n .
(3.10)

Now, we see by (3.10) and (2.4)

K 2 = J n ( χ 1 f ) ( χ 1 f ) L p ( [ a , ) ) J n ( ( χ 1 f ) ( t ) ( χ 1 f ) ( x ) , x ) L p ( [ a , ) ) + ( χ 1 f ) ( x ) ( J n ( e 0 , x ) e 0 ) L p ( [ a , ) ) = J + ( χ 1 f ) ( x ) ( J n ( e 0 , x ) e 0 ) L p ( [ a , ) ) C ( f L p ( R + ) + f L p ( [ a , ) ) + f L p ( [ b , ) ) ) μ n + J n ( e 0 , x ) e 0 L ( [ a , ) ) χ 1 f L p ( [ a , ) ) C 1 ( f L p ( R + ) + f L p ( [ a , ) ) + f L p ( [ b , ) ) ) μ n = O ( μ n ) .

Consequently, with (3.2) and (3.3) we conclude (i).

(ii) Let p=1 and f BV[b,). Then we have for x,t[b,),

f(t)f(x)= f (x)(tx)+ x t (tu)d f (u).
(3.11)

On the proof of (i) we recall the part which we assumed as p=1. From (3.3) and (3.5) we may only estimate

I 2 := a | b ( f ( t ) f ( x ) ) H n (tx)dtdx|
(3.12)

(see (3.4)). By (3.11) we see

I 2 | a b f ( x ) ( t x ) H n ( t x ) d t d x | + | a b x t ( t u ) d f ( u ) H n ( t x ) d t d x | = : I 2 , 1 + I 2 , 2 .
(3.13)

Now, we see that

I 2 , 1 = | a b f ( x ) ( t x ) H n ( t x ) d t d x | = | a f ( x ) x b u H n ( u ) d u d x | = | a b u H n ( u ) a u + b f ( x ) d x d u | sup a x < | f ( x ) | | δ ( u δ ) u H n ( u ) d u | sup [ b , ) | f ( x ) | O ( μ n )

by (1.3). We estimate I 2 , 2 . We fix an arbitrary η>0. Then we have by means of the substitution u=y+x with a new variable y

I 2 , 2 = | a b x t ( t u ) d f ( u ) H n ( t x ) d t d x | = | a b 0 t x ( t x y ) d f ( x + y ) H n ( t x ) d t d x | 2 a b 0 | t x | | t x | | d f ( x + y ) | H n ( t x ) d t d x | t x y | 2 | t x | 2 a j = 0 j η | t x | ( j + 1 ) η 0 ( j + 1 ) η | d f ( x + y ) | | t x | H n ( t x ) d t d x 2 j = 0 j η | u | ( j + 1 ) η | u | H n ( u ) d u a 0 ( j + 1 ) η | d f ( x + y ) | d x .

Then

a 0 ( j + 1 ) η | d f ( x + y ) | d x = a x x + ( j + 1 ) η | d f ( v ) | d x = a v a + ( j + 1 ) η a x v d x | d f ( v ) | + a + ( j + 1 ) η v < v ( j + 1 ) η x v d x | d f ( v ) | : = A 1 + A 2 .

Since we can see

A 1 = a v a + ( j + 1 ) η a x v dx|d f (v)|= a v a + ( j + 1 ) η (va)|d f (v)|(j+1)ηBV ( f ; R + )

and

A 2 = a + ( j + 1 ) η v < v ( j + 1 ) η x v dx|d f (v)|(j+1)ηBV ( f ; R + ) ,

we have

a 0 ( j + 1 ) η |d f (x+y)|dx2(j+1)ηBV ( f ; R + ) .

Now, we estimate j η | u | ( j + 1 ) η |u| H n (u)du for a non-negative integer j. Let j=0. Then from (1.2) and (1.3), we have

0 | u | η |u| H n (u)du ( 0 | u | η H n ( u ) d u ) 1 / 2 ( 0 | u | η u 2 H n ( u ) d u ) 1 / 2 =O ( μ n 1 / 2 ) .

Let j1. Then by (2.1) we have

j η | u | ( j + 1 ) η |u| H n (u)duC μ n β ( j η ) α 1 .

Therefore, we have

I 2 , 2 0 | u | η | u | H n ( u ) d u a 0 η | d f ( x + y ) | d x + j = 1 j η | u | ( j + 1 ) η | u | H n ( u ) d u a 0 ( j + 1 ) η | d f ( x + y ) | d x O ( μ n 1 / 2 ) η BV ( f ; R + ) + j = 1 1 ( j η ) α 1 O ( μ n β ) ( j + 1 ) η BV ( f ; R + ) = O ( μ n 1 / 2 ) η BV ( f ; R + ) + O ( μ n β ) 1 η α 2 BV ( f ; R + ) ( α > 3 ) .

If we let η= μ n 1 / 2 , then we have I 2 , 2 =O( μ n )BV( f ; R + ), because β(α2)/21. Consequently, (ii) is proved.

(iii) It follows from (2.4). Consequently, for a linear function f on [b,)

J n ( f ; x ) f ( x ) L p ( [ a , ) ) =O ( μ n β ) =o( μ n ).

 □

Proof of Theorem 1.3 (i), (ii) hold as follows: Let f L p ( R + ). First, we choose ψ C 2 ( R + ) with ψ L q ([a,)) (for any 1q) such that

ψ(a)= ψ (a)= ψ (a)=0, lim r ψ ( i ) (r)=0,i=0,1,2

and

ψ(x)=0for x[a,).

We use the bilinear functional;

A n (f,ψ)= 1 μ n 0 ( J n ( f , x ) f ( x ) ) ψ(x)dx.

We will show that for fixed ψ, A n (,ψ) is uniformly bounded on L p ( R + ). We see

0 J n (f,x)ψ(x)dx= 0 ψ(x) 0 f(t) H n (tx)dtdx.

For x,t[0,) we can write

ψ(x)=ψ(t)+ ψ (t)(xt)+ x t (xu) ψ (u)du.

Hence,

0 0 f ( t ) ψ ( x ) H n ( t x ) d x d t = 0 f ( t ) ψ ( t ) 0 H n ( t x ) d x d t + 0 f ( t ) ψ ( t ) 0 ( x t ) H n ( t x ) d x d t + 0 f ( t ) 0 x t ( x u ) ψ ( u ) d u H n ( t x ) d x d t = I 1 + I 2 + I 3 .

From (1.2)

I 1 = 0 f(t)ψ(t) t H n (u)dudt= 0 f(t)ψ(t) ( 1 t H n ( u ) d u ) dt.

Since ψ(t)=0 for t[a,) (so, we may take ta), by (2.1) we have

| 0 f ( t ) ψ ( t ) t H n ( u ) d u d t | | 0 f ( t ) ψ ( t ) t ( u t ) α H n ( u ) d u d t | ( u / t ) α 1 O ( μ n β ) a | f ( t ) ψ ( t ) | 1 t α d t by (1.4) O ( μ n β ) sup a t < | ψ ( t ) | a | f ( t ) | 1 t α d t O ( μ n β ) sup a t < | ψ ( t ) | f L p ( [ a , ) ) t α L q ( [ a , ) ) ,

where 1/p+1/q=1. Thus, we have

I 1 = 0 f(t)ψ(t)dt+O ( μ n β ) f L p ( [ a , ) ) .

We estimate I 2 . We may at, because ψ(t)=0 for t[a,). Noting (1.4) and α<3,

| I 2 | = a | f ( t ) ψ ( t ) | t u H n ( u ) d u d t a | f ( t ) ψ ( t ) | t u α t α 1 H n ( u ) d u d t = O ( μ n β ) a | f ( t ) ψ ( t ) | 1 t α 1 d t = O ( μ n β ) ψ L ( [ a , ) ) f L p ( [ a , ) ) t 1 α L q ( [ a , ) ) = O ( μ n β ) f L p ( [ a , ) ) .

Finally, we estimate I 3 . Let 1<p< and 1/p+1/q=1. As the estimation for I 2 , 2 , we have by (1.3)

| I 3 | = | 0 f ( t ) 0 x t ( x u ) ψ ( u ) d u H n ( t x ) d x d t | = | 0 f ( t ) 0 1 t x x t ψ ( u ) d u ( t x ) 2 H n ( t x ) d x d t | a | f ( t ) | θ ( ψ , t ) 0 ( t x ) 2 H n ( t x ) d x d t = O ( μ n ) a | f ( t ) | θ ( ψ , t ) d t O ( μ n ) f L p ( [ a , ) ) ψ L q ( [ a , ) ) = O ( μ n ) f L p ( [ a , ) ) .

We estimate I 3 for p=1. There exists η between x and t such that

| I 3 | | 0 f ( t ) 0 ψ ( η ) ( t x ) 2 H n ( t x ) d x d t | sup a t < | ψ ( t ) | a | f ( t ) | 0 ( t x ) 2 H n ( t x ) d x d t O ( μ n ) sup a t < | ψ ( t ) | f L 1 ( [ a , ) ) = O ( μ n ) f L 1 ( [ a , ) ) .

Here we used (1.3). Consequently, it follows that | A n (f,ψ)| is uniformly bounded on L p ( R + ). Next, from Theorem 1.1 we see that for f C c 2 ( R + ),

lim n A n (f,ψ)= 1 2 a f (x)ψ(x)dx= 1 2 a f(x) ψ (x)dx.
(3.14)

Since { A n (,ψ)} is uniformly bounded on L p ( R + ), and C c 2 ( R + ) is dense in L p ( R + ), (3.14) yields

lim n A n (f,ψ)= 1 2 a f(x) ψ (x)dx
(3.15)

for any f L p ( R + ). Now, for any fixed f L p ( R + ), we consider the sequence of linear functional { A n (f,)}. Since J n ( f ) f L p ( [ a , ) ) =O( μ n ), n, there exist h L p ([a,)) (p>1) and hBV[a,) (p=1) and a subsequence { A n j (f,)} such that

lim j A n j (f,ψ)={ 0 h ( x ) ψ ( x ) d x , p > 1 , 0 ψ ( x ) d h ( x ) , p = 1 .
(3.16)

From (3.15) and (3.16) we obtain

1 2 0 f(x) ψ (x)dx={ 0 h ( x ) ψ ( x ) d x , p > 1 , 0 ψ ( x ) d h ( x ) , p = 1 .
(3.17)

A particular solution to (3.17) is

1 2 f(x)={ η x η ξ h ( μ ) d μ d ξ , p > 1 , η x η ξ d h ( μ ) d ξ , p = 1 .

The homogeneous problem

a f(x) ψ (x)dx=0

has the general solution f(x)= C 1 x+ C 2 for ax<, since we can take ψ C 2 ([a,)) arbitrarily as ψ L q ([a,)), ψ(a)= ψ (a)=0, lim r ψ ( i ) (r)=0, i=0,1. Hence, if 1<p<, f L p 2 ([a,)), and if p=1 then f BV([a,)). Hence, (i) and (ii) hold. We will show (iii). Now, if

J n ( f ) f L p ( [ a , ) ) =o( μ n ),n,

then

| A n ( f , ψ ) | 1 μ n a | J n ( f , x ) f ( x ) | | ψ ( x ) | d x ( sup a x < | ψ ( x ) | ) C p μ n J n ( f ) f L p ( [ a , ) ) ,

where C p >0 is independent of n. Hence

lim n A n (f,ψ)=0.
(3.18)

Considering (3.15), (3.16), (3.17), and (3.18), we obtain

a f(x) ψ (x)dx=0,

and so

a f (x)ψ(x)dx=0,

consequently, we have f (x)=0, that is, f is linear on [a,). □

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Acknowledgements

The authors thank the referees for many valuable comments and corrections.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript and participated in the sequence alignment. All authors read and approved the final manuscript.

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Jung, H.S., Sakai, R. Local saturation of a positive linear convolution operator. J Inequal Appl 2014, 329 (2014). https://doi.org/10.1186/1029-242X-2014-329

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Keywords

  • convolution operator
  • local saturation