Weighted Hardy type inequalities for supremum operators on the cones of monotone functions

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

The complete characterization of the weighted ...... inequalities of supremum operators on the cones of monotone functions for all ...... is given.

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 DOI 10.1186/s13660-016-1168-z

Weighted Hardy type inequalities for supremum operators on the cones of monotone functions

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Web End = Lars-Erik Persson1,2*, Guldarya E Shambilova3 and Vladimir D Stepanov4

*Correspondence: [email protected]

1Department of Engineering Sciences and Mathematics, Lulea University of Technology, Lulea, 97187, Sweden

2UiT, The Arctic University of Norway, P.O. Box 385, Narvik, 8505, NorwayFull list of author information is available at the end of the article

1 Introduction

Let R+ := [, ). Denote M the set of all measurable functions on

R+, M+ M the subset

of all non-negative functions and M M+ (M M+) is the cone of all non-increasing

(non-decreasing) functions. Also denote by C M the set of all continuous functions on R+. If < p and v M+ we dene

Lpv := f M : f L

pv :=

f

(x)

pv(x)

dx

p< ,

Lv := f

M : f L

v := ess sup

x

v(x)

f

(x)

< .

Let w M+ and k(x, y) is a Borel function on [, ) satisfying Oinarovs condition: k(x, y) = if x < y, and there is a constant D independent of x z y such that

D

k(x, z) + k(z, y) k(x, y) D k(x,z) + k(z, y)

. (.)

The mapping properties between weighted Lp spaces of Hardy type operators involved are very well studied. See e.g. the books [, ] and [] and the references therein. We also mention the following examples of articles in this area: [] and []. Recently, it has been discovered that it is of great interest to study also some corresponding supremum operators instead of the usual such Hardy type (arithmetic mean) operators. The interest comes both from purely mathematical point of view but also from various applications where such kernels many times are the unit impulse answers to the problem at hand and

2016 Persson et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License

(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro

vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and

indicate if changes were made.

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 2 of 18

the best constants means the operator norms of the corresponding transfer of the energy of the signals measured in weighted Lp spaces.

We consider supremum operators of the form

(Tf )(x) = ess sup

yx

k(y, x)w(y)f (y), f M,

(Sf )(x) = ess sup

yx

k(y, x)w(y)f (y), f M,

(T f )(x) = ess sup

yx

k(x, y)w(y)f (y), f M,

(S f )(x) = ess sup

yx

k(x, y)w(y)f (y), f M.

Let < p, r and u, v M+. The paper is devoted to the necessary and sucient conditions for the inequalities

Tf L

ru CT f L

pv , f M, (.)

Sf L

ru CS f L

pv , f M, (.)

T f L

ru CT f L

pv , f M, (.)

pv , f M, (.)

where the constants CT and others are taken as the least possible.

This problem was rst studied for the inequality (.) in [], Theorem ., in a case when k(x, y) = , w C . This result was extended in [] for the case k(x, y) satisfying (.)

with a discrete form of a criterion for < r < p < . With dierent supremum operators some similar problems were studied in []. This area is currently developing intensively and nds many interesting applications.

Section is devoted to preliminaries. The border cases < r < p = , < p < r = and r = p = are solved in Section . In Section we characterize the case k(x, y) = , which is essentially used in Section with the main results of the paper.

We use signs := and =: for determining new quantities and Z for the set of all integers. For positive functionals F and G we write F [lessorsimilar] G, if F cG with some positive constant c, which depends only on irrelevant parameters. F G means F [lessorsimilar] G [lessorsimilar] F or F = cG. E denotes the characteristic function (indicator) of a set E. Uncertainties of the form ,

and are taken to be zero. stands for the end of a proof.

2 Preliminaries

We denote

V(t) :=

t

S f L

ru CS f L

v, V(t) :=

t v.

Let < p, r < . By [], Lemma ., and the monotone convergence theorem the inequality (.) is equivalent to

p

ess sup

yx

k(y, x)w(y)

y

h

r p

u(x) dx

p r

CpT

hV, h

M+. (.)

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 3 of 18

If

Tph(x) := ess sup

yx

k(y, x)w(y)

p

y

h,

then (.) is equivalent to

Tph L

r

p u

CpT h L

V , h

M+. (.)

Analogously, if

Sph(x) := ess sup

yx

k(y, x)w(y)

p

y h,

p

y h,

Tph(x) := ess sup

yx

k(x, y)w(y)

p

y

h,

Sph(x) := ess sup

yx

k(x, y)w(y)

then (.), (.), and (.) are equivalent to

Sph L

r

p u

CpS h L

V , h M+, (.)

Tph L

r

p u

CpT h L

V , h M+, (.)

and

Sph L

r

p u

CpS h L

V , h

M+, (.)

respectively.

For the border cases < p < r = , < r < p = , and r = p = we have the following four groups of inequalities:

ess sup

x

u(x)

pTph(x) CpT,p hV, h

M+, (.)

ess supyxk(y, x)w(y)f (y)

ru(x) dx

r

CT,r f L

v , f M, (.)

ess sup

x

ess supyxk(y, x)w(y)f (y) CT, f Lv , f M, (.)

for the operator T;

ess sup

x

u(x)

u(x)

pSph(x) CpS,p hV, h

M+, (.)

ess supyxk(y, x)w(y)f (y)

ru(x) dx

r

CS,r f L

v , f M, (.)

ess sup

x

u(x)

ess supyxk(y, x)w(y)f (y) CS, f Lv , f M, (.)

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 4 of 18

for the operator S;

ess sup

x

u(x)

pTph(x) CpT ,p hV, h

M+, (.)

ess supyxk(x, y)w(y)f (y)

ru(x) dx

r

CT ,r f L

v , f M, (.)

ess sup

x

ess supyxk(x, y)w(y)f (y) CT, f Lv , f M, (.)

for the operator T , and

ess sup

x

u(x)

u(x)

pSph(x) CpS,p hV, h

M+, (.)

ess supyxk(x, y)w(y)f (y)

ru(x) dx

r

CS,r f L

v , f M, (.)

ess sup

x

ess supyxk(x, y)w(y)f (y) CS, f Lv , f M, (.)

for the operator S . We characterize the inequalities (.)-(.) in the next section.

To deal with the inequalities (.)-(.) we study rst the case k(x, y) = and then a general case.

3 Border cases of summation parameters

For a measurable function v M+ we dene monotone envelopes (see [], Section ) as follows:

v(x) := ess sup

yx

v(y),

u(x)

v(x) := ess sup

yx

v(y).

Theorem . For the best possible constants of the inequalities (.)-(.) we have

CT,p sup

x

u(x) ess sup

yx

k(y, x)w(y)

V/p(y)

, (.)

CT,r =

ess sup

yx

k(y, x)w(y) v(y)

ru(x) dx

r, (.)

CT, = ess sup

x

u(x)

ess sup

yx

k(y, x)w(y) v(y)

. (.)

Proof Observe that if k(x, y) satises (.), then [k(x, y)]p satises (.) too with a constant Dp . If x t, then

Tph(t) = ess sup

yt

k(y, t)w(y)

p

y

h

p

Dp ess sup

yt

k(y, x)w(y)

y

h

p

y

h = DpTph(x).

Dp ess sup

yx

k(y, x)w(y)

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 5 of 18

Hence,

Tph(x) sup

tx

Tph(t) := (x) M.

It implies (see [], Proposition .)

ess sup

x

u(x)

pTph(x) ess sup x

u(x)

p(x)

p

= ess sup

x

u(x)

p suptx(t) = supt (t)

u(t)

sup

x

u(x)

pTph(x),

and (.) is equivalent to

sup

x

u(x)

p Hxh L(k(,x)w())p [lessorsimilar] CpT,p hV, h

M+, (.)

where

Hxh(y) := [x,)(y)

y

h.

Thus,

CpT,p sup

x

u(x)

p Hx LV L (k(,x)w())p .

Since by a well-known theorem ([], Theorem .)

Hx L

V L

(k(,x)w())p =

ess sup

yx

(k(y, x)w(y))p V(y)

,

we obtain (.).

Now, (.) is equivalent to the inequality

r

CT,r f L

v , f M. (.)

The lower bound of (.) follows from (.) with f =

v and the upper bound from the

estimate f (y)

f L

v v(y) .

ess supyxk(y, x)w(y)f (y)

ru(x) dx

The proof of (.) is the same.

Analogously, we can prove the following.

Theorem . For the best possible constants of the inequalities (.)-(.) we have

CS,p sup

x

u(x) ess sup

yx

k(y, x)w(y)

V/p(y) , (.)

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 6 of 18

CS,r =

ess sup

yx

k(y, x)w(y) v(y)

ru(x) dx

r, (.)

CS, = ess sup

x

u(x)

ess sup

yx

k(y, x)w(y) v(y)

. (.)

CT ,p sup

x

u(x) ess sup yx

k(x, y)w(y)V/p(y) , (.)

CT ,r =

ess sup yx

k(x, y)w(y) v(y)

ru(x) dx

r, (.)

CT , = ess sup

x

u(x)

ess sup yx

k(x, y)w(y) v(y)

, (.)

CS ,p sup

x

u(x) ess sup yx

k(y, x)w(y)

V/p(y)

, (.)

CS ,r =

ess sup yx

k(y, x)w(y) v(y)

ru(x) dx

r, (.)

CS , = ess sup

x

u(x)

ess sup

yx

k(y, x)w(y) v(y)

. (.)

4 The case k(x, y) = 1

Let u, v, w M+ be weights. We suppose for simplicity that <

t

u < , for all t > ,

u = and dene the functions : [; ) [; ), : [; ) [; ), by

(x) := inf

y > :

y

u

x

u ,

(x) := inf

y > :

y

u

x

u .

Let := (). For c < d < and h M+ we put

Hch(x) := [c,)(x)

x

h,

Hc,dh(x) := [c,d)(x)

x (c) h,

Hch(x) := [c,)(x)

x h,

Hc,dh(x) := [c,d)(x)

(d)

x h.

We need the following partial cases of [], Theorems . and . (see also [, ]).

Theorem . Let < r < . Then:(a) For validity of the inequality

y

ess sup

yx

w(y)

h ru(x) dx

r

C h L

v , h M+, (.)

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 7 of 18

it is necessary and sucient that the inequality

u(x) w

(x)

r

h r dx

r

x

A h L

v , h M+,

holds and the constant

A :=

supt>(

t

u)r Ht L

v L

w , r ,

(

u(x)(

x

u) rr H[

(x),(x)]

r r

Lv L

w dx)r

r , < r < ,

is nite. Moreover, C A + A.(b) For validity of the inequality

ess sup

yx

w(y)

y h ru(x) dx

r

C h L

v , h M+, (.)

it is necessary and sucient that the inequality

u(x)

ess sup

xy

(x) w(y) r

(x) h r dx

r

B h L

v , h M+,

holds and the constant

B :=

supt>(

u)r Ht L

v L

w , r ,

(

t

u(x)(

x

u) rr H[(x),(x)]

r r

Lv L

w dx)r

r , < r < ,

is nite. Moreover, C B + B.

Using Theorem . we characterize (.) and (.) with k(x, y) = .

Theorem . Let < p, r < and k(x, y) = . Then, for the best possible constants of the inequalities (.) and (.) the following equivalences hold:

CT A + A, CS B + B, (.)

where

A = sup

t>

V(t) p

t u w r

r

, r p,

V(x)

x u w r

r pr

u(x)

pr pr

, < r < p,

A =

w(x)

r dx

A = sup

t>

t

u

r sup

yt

w(y) [V(y)]

p , r p,

r pr

ess sup

(x)y(x)

A =

u(x)

x

u

[w(y)]p

V(y)

r pr

dx

prpr, < r < p,

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 8 of 18

B = sup

t>

V

(t)

p

t

u(x)

ess sup

xy

(x) w(y) r dx

r

, r p,

z

B =

V

(z)

u(x)

ess sup

xy

(x) w(y) r dx

r pr

u(z)

ess sup

zy

(z) w(y) r dz

pr pr

, < r < p,

B = sup

t>

t

u

r ess sup

yt

w(y)

[V(y)]

p , r p,

r pr

ess sup

(x)y(x)

u(x)

x

u

[w(y)]p V(y)

r pr

dx

B =

prpr, < r < p.

Proof Since (.) (.) and (.) (.), the proof follows by applying Theorem . with r replaced by rp , w = wp, v = V in (.) and v = V in (.). Thus, CT A + A , where A is the best constant in the inequality

u(x) w(x) r

x

h

r p

dx

p r

A

p h LV , h

M+, (.)

and

A

p =

supt>(

t

u)

p r

Ht L

V L

wp , r p,

(

u(x)(

x

u)

r pr

H[

(x),(x)]

r pr

LV L

wp dx)

prr , < r < p.

If k(x, y) is a measurable kernel on

R+

R+ and

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

then by well-known results ([], Chapter XI, Section ., Theorem , see also [], Theorem .)

K L

L

Kf (x) :=

Lq , q . (.)

If k(x, y) = w(x)[,x](y)u(y) and < q < , then ([], Theorem .)

K L

L

q = ess sup

s

k(,

s)

q q

x wq

q q

w(x)

q dx

q q

. (.)

Applying (.) and (.) to (.) we nd that A A . Again, applying (.), when

k(x, y) = wp(x)[t,)(x)

[,x](y) V(y)

q

ess sup yxu(y)

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 9 of 18

we obtain

Ht L

V L

wp = ess sup

s

k(,

s)

L = ess sup

s

V(s)

ess sup

{xt}{xs}

wp(x)

= sup

s

V(s) w max(t,

s)

p

= max

sup

st

[w(t)]p V(s)

, sup

st

[w(s)]p V(s)

= sup

st

[w(s)]p V(s)

.

Similarly, using the monotonicity of V, we nd

H[

(x),(x)]

r pr

LV L

wp =

ess sup

s

(x)

V(s)

ess sup

{(x)y(x)}{ys}

wp(y)

= sup

(x)s(x)

V(s)

sup

(x)y(x)

ess sup

sy(x)

wp(y)

= ess sup

(x)y(x)

wp(y) V(y)

and the estimate A A follows.

For the second part we observe that CS B + B , where B is the least constant in the inequality

u(x)

ess sup

xy

r p

dx

p r

(x) w(y) r

(x) h

B

p h LV , h M+, (.)

and

B

p =

supt>(

t

u)

p r

Ht L

V L

wp , r p,

(

u(x)(

x

u)

r pr

H[(x),(x)]

r pr

LV L

wp dx)

prr , < r < p.

By a change of variables we see that (.) is equivalent to

u(x)

ess sup

xy

r p

dx

p r

(x) w(y) r

x h

p h LV , h M+, (.)

where V (y) := V((t)). By the same argument as above it follows that B B and

B B.

Analogously, we obtain the sharp estimates for the best constants in (.) and (.). Suppose for simplicity that <

B

t u < for all t > ,

u = and dene the functions

: [; ) [; ), : [; ) [; ), by

(x) := sup

y > :

y u

x u ,

(x) := sup

y > :

y u

x u .

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 10 of 18

Let := (). For c < d < and h M+ we put

Hdh(x) := (,d](x)

x h,

(d)

x h, H ch(x) := (,d](x)

x

h,

Hc,dh(x) := (c,d](x)

H c,dh(x) := (c,d](x)

x (c) h.

We need the following partial cases of [], Theorems . and ..

Theorem . Let < r < . Then:(a) For validity of the inequality

ess sup yx

w(y)

y h ru(x) dx

r

C h L

v , h M+,

it is necessary and sucient that the inequality

u(x) w

(x)

r

x h r dx

r

D h L

v , h M+,

holds and the constant

D :=

supt>(

t u)r Ht L

v L

w , r ,

(

u(x)(

x u) rr H[

(x),(x)]

r r

Lv L

w dx)r

r , < r < ,

is nite. Moreover, C D + D.(b) For validity of the inequality

y

ess sup yx

w(y)

h ru(x) dx

r

C h L

v , h M+,

it is necessary and sucient that the inequality

u(x)

ess sup

(x)yx

(x)

w(y)

r

h r dx

r

E h L

v , h M+,

holds and the constant

E :=

supt>(

t u)r H t L

v L

w , r ,

(

u(x)(

x u) rr H [(x),(x)]

r r

Lv L

w dx)r

r , < r < ,

is nite. Moreover, C E + E.

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 11 of 18

Using Theorem . we characterize (.) and (.) with k(x, y) = .

Theorem . Let < p, r < and k(x, y) = . Then for the best possible constants of the inequalities (.) and (.) the following equivalences hold:

CT

D + D, CS E + E,

where

D = sup

t>

V(t)

p

u w r

r

, r p,

t

x

pr pr

, < r < p,

D =

V(x)

u w r

r pr

u(x)

w(x)

r dx

D = sup

t>

t u

r sup

<y<t

w(y) [V(y)]

p , r p,

r pr

ess sup

(x)y(x)

u(x)

x u

[w(y)]p V(y)

r pr

dx

D =

prpr, < r < p,

E = sup

t>

V (t)

p

t u(x)

ess sup

(x)yx

w(y)

r dx

r, r p,

E =

V

(z)

z u(x)

ess sup

(x)yx

w(y)

r dx

r pr

u(z)

ess sup

(z)yz

w(y)

rdz

prpr, < r < p,

E = sup

t>

t u

r ess sup

yt

w(y)

[V(y)]

p , r p,

r pr

ess sup

(x)y(x)

E =

u(x)

x u

[w(y)]p

V(y)

r pr

dx

prpr, < r < p.

5 Main results

To deal with the kernel transformation we need the following extension of Theorem . following from [], Theorems . and ..

Theorem . Let < r < , u, v, w M+ and k(x, y) satises Oinarovs condition (.). Then:(a) For validity of the inequality

y

ess sup

yx

k(y, x)w(y)

h ru(x) dx

r

C h L

v , h M+, (.)

it is necessary and sucient that the inequalities

u(x)

ess sup

yx

x

k(y, x)w(y)

r

h r dx

r

A h L

v , h M+,

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 12 of 18

and

u(x) k

(x), x

r

ess sup

y

(x) w(y)

y

h r dx

r

A h L

v , h M+,

hold and the constant

A :=

supt>(

t

u)r Ht L

v L

w()k(,t) , r ,

(

u(x)(

x

u) rr H[

(x),(x)]

r r

Lv L

w()k(,(x))

dx)rr , r < ,

is nite. Moreover, C A + A + A.(b) For validity of the inequality

ess sup

yx

k(y, x)w(y)

y h ru(x) dx

r

C h L

v , h M+, (.)

it is necessary and sucient that the inequalities

u(x)

ess sup

xy

r

(x) k(y, x)w(y) r

(x) h r dx

B h L

v , h M+,

and

u(x) k

r

(x), x

r

ess sup

y

(x) w(y)

y h r dx

B h L

v , h M+,

hold and the constant

B :=

supt>(

u)r Ht L

v L

w()k(,t) , r ,

(

t

u(x)(

x

u) rr H[(x),(x)]

r r

Lv L

w()k(,(x))

dx)rr , r < ,

is nite. Moreover, C B + B + B.

Using Theorem . we obtain the characterization of (.) and (.) for < p, r < . Denote

Wk(x) := ess sup

yx

k(y, x)w(y), Wk(x) := ess sup

xy

(x) k(y, x)w(y),

w (w)(x) := ess sup

xy

(x) w(y),

k (x) := k

(x), x

, gk (y) := g

k(y)

,

y

(x) := inf

y > :

u[k ]r

x

u[k ]r ,

(x) := inf y > :

y

u[k ]r

x

u[k ]r .

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 13 of 18

Theorem . Let < p, r < . Then, for the best possible constants of the inequalities (.) and (.) the following equivalences hold:

CT

A + A, + A, + A, CS

B + B, + B, + B, (.)

where

A = sup

t>

V(t) p

t u[Wk]r

r

, r p,

V(x)

pr pr

, < r < p,

A =

x u[Wk]r

r pr

u(x)

Wk(x)

r dx

A, = sup

t>

(V) (t) p

t u k

w

r

r, r p,

A, =

(V) (x)

x u k

w

r

r pr

u(x) k

(x)w(x)

r dx

pr pr

, < r < p,

A, = sup

t>

u[k ]r

r ess sup

yt

t

w (y)

[(V) (y)]

p , r p,

x

A, =

u(x) k

(x)

r

u[k ]r

r pr

ess sup

[w (y)]p (V) (y)

r pr

dx

prpr, < r < p,

(x)y(x)

A = sup

t>

t

u

r ess sup

yt

w(y)k(y, t)

[V(y)]

p , r p,

r pr

A =

u(x)

x

u

ess sup

(x)y

(x)

[w(y)k(y, (x))]p

V(y)

r pr

dx

prpr, < r < p,

p

B = sup

t>

V (t)

t

u[

Wk]r

r, r p,

pr pr

, < r < p,

B =

V (z)

z

u[

Wk]r

r pr

u(z)

Wk(z)

r dz

B, = sup

t>

V

(t)

p

t

u k

w(w )

r

r, r p,

z

r pr

B, =

V

(t)

(z)

u[k ]r w

(w )

r

u(z)

k (z)w(w )(z)

r dz

pr pr

, < r < p,

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 14 of 18

B, = sup

t>

u[k ]r

r ess sup

yt

t

w (y)

[V (y)]

p , r p,

x

B, =

u(x) k

(x)

r

u[k ]r

r pr

ess sup

[w (y)]p V (y)

r pr

dx

prpr, < r < p,

(x)y(x)

B = sup

t>

t

u

r ess sup

yt

w(y)k(y, t)

[V(y)]

p , r p,

r pr

B =

u(x)

x

u

ess sup

(x)y

(x)

[w(y)k(y, (x))]p V(y)

r pr

dx

prpr, < r < p.

Proof We start with the inequality (.). Since (.) (.), then applying Theorem . we see that

CT

A + A + A ,

where A and A are the best constants in the inequalities

u(x)

ess sup

yx

k(y, x)w(y)

r

x

h

r p

dx

p r

A

p h LV , h M+,

u(x) k

(.)

(x), x

r

ess sup

y

(x)

w(y)

y

h

r p

dx

p r

A

p

p h LV , h M+,

and

t

u)

p r

Ht L

A

p :=

supt>(

V L

[w()k(,t)]p , r p,

(

u(x)(

x

u)

r pr

H[

(x),(x)]

r pr

LV L

[w()k(,(x))]p

dx)

prr , < r < p.

Applying (.) and (.) we see that A

A and A

A. By a change of variable we nd

that (.) is equivalent to the inequality

u(x) k

(x)

r

ess sup

yx

w (y)

p

y

h

r p

dx

p r

A

p h L[V] , h

M+, (.)

which is governed by Theorem .. Arguing analogously to the proof of Theorem . we see that

A

A , + A ,,

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 15 of 18

where A , is the best constant of the inequality

u(x) k

(x)

r

w(x)

r

x

h

A

,

r p

dx

p r

p h L[V] , h M+,

and

t

A ,

p := sup t>

u[k ]r

p r

Ht L

[V] L

wp , r p,

A ,

p := u(x) k (x)

r

u[k ]r

r pr

H[

x

(x),(x)]

r pr

L[V] L

wp

dx

prr,

for < r < p. Again applying (.) and (.) we see that A ,

A, and A ,

A,.

The proof for the inequality (.) is similar.

Analogously, we obtain the sharp estimates for the best constants in (.) and (.). To this end we need the following extension of Theorem . from [], Theorems . and ..

Theorem . Let < r < , u, v, w M+ and k(x, y) satisfy Oinarovs condition (.). Then:(a) For validity of the inequality

ess sup yx

k(x, y)w(y)

y h ru(x) dx

r

C h L

v , h M+,

it is necessary and sucient that the inequalities

u(x)

ess sup yx

k(x, y)w(y)

r

x h r dx

r

A h L

v , h M+,

and

u(x)

k

x, (x)

r

ess sup

y

(x) w(y)

y h r dx

r

A h L

v , h M+,

hold and the constant

A :=

supt>(

t u)r Ht L

v L

w()k(t,) , r ,

(

u(x)(

x u) rr H[

(x),(x)]

r r

Lv L

w()k((x),)

dx)rr , r < ,

is nite. Moreover, C A + A + A.(b) For validity of the inequality

y

ess sup

yx

k(y, x)w(y)

h ru(x) dx

r

C h L

v , h M+,

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 16 of 18

it is necessary and sucient that the inequalities

u(x)

ess sup

(x)yx

(x)

k(x, y)w(y)

r

h r dx

r

B h L

v , h M+,

and

u(x) k

x, (x)

r

ess sup

y

(x) w(y)

y

h r dx

r

B h L

v , h M+,

hold and the constant

B :=

supt>(

t u)r H t L

v L

w()k(t,) , r ,

(

u(x)(

x u) rr H [(x),(x)]

r r

Lv L

w()k((x),)

dx)rr , r < ,

is nite. Moreover, C B + B + B.

Using Theorem . we obtain the characterization of (.) and (.) for < p, r < . Denote

Wk(x) := ess sup

yx

k(x, y)w(y), W k(x) := ess sup

(x)yx

k(x, y)w(y),

(w)(x) := ess sup

(x)yx

w(y),

k (x) := k

x, (x)

, gk (y) := g

k(y)

,

(x) := sup

y > :

y u[k ]r

x u[k ]r ,

x u[k ]r .

Theorem . Let < p, r < . Then for the best possible constants of the inequalities (.) and (.) the following equivalences hold:

CT

(x) := sup y > :

y u[k ]r

D + D, + D, + D, CS

E + E, + E, + E, (.)

where

D = sup

t>

V(t)

p

u W

k

t

r

r, r p,

r pr

u(x)

V(x)

x

pr pr

, < r < p,

D =

u W

k

r

Wk(x)

r dx

p

t

u k

D, = sup

t>

V (t)

w

r

r, r p,

D, =

V (x)

x k w

r

r pr

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 17 of 18

u(x) k

pr pr

, < r < p,

(x)w(x)

r dx

D, = sup

t>

t u[k ]r

r ess sup yt

w (y)

[V (y)]

p , r p,

r pr

D, =

u(x) k

(x)

r

x u[k ]r

ess sup

(x)y(x)

r pr

dx

[w (y)]p

V (y)

prpr, < r < p,

D = sup

t>

t u

r ess sup yt

w(y)k(t, y)

[V(y)]

p , r p,

r pr

D =

u(x)

x u

ess sup

(x)y

(x)

[w(y)k((x), y)]p V(y)

r pr

dx

prpr, < r < p,

E = sup

t>

(V) (t) p

t u

W k

r

r, r p,

r pr

u(z)

pr pr

, < r < p,

E =

(V) (z)

z u

W k

r

W k(z)

r dz

(V) (t)

p

E, = sup

t>

t u[k ]r

(w )

r

r, r p,

E, =

(V)

(t)

(z)

z u[k ]r

(w )

r

r pr

u(z) k

(z)

r

(w )

r dz

pr pr

, < r < p,

E, = sup

t>

t u[k ]r

r ess sup

yt

w (y)

[(V) (y)]

p , r p,

r pr

E, =

u(x) k

(x)

r

x u[k ]r

ess sup

(x)y(x)

[w (y)]p

(V) (y)

r pr

dx

prpr, < r < p,

E = sup

t>

t u

r ess sup

yt

w(y)k(t, y)

[V(y)]

p , r p,

r pr

E =

u(x)

x u

ess sup

(x)y

(x)

[w(y)k((x), y)]p

V(y)

r pr

dx

prpr, < r < p.

Persson et al. Journal of Inequalities and Applications (2016) 2016:237 Page 18 of 18

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All the authors contributed equally and signicantly in writing this paper. All the authors read and approved the nal manuscript.

Author details

1Department of Engineering Sciences and Mathematics, Lulea University of Technology, Lulea, 97187, Sweden. 2UiT, The Arctic University of Norway, P.O. Box 385, Narvik, 8505, Norway. 3Department of Mathematics, Financial University under the Government of the Russian Federation, Leningradsky Prospekt 49, Moscow, 125993, Russia. 4Department of Nonlinear Analysis and Optimization, Peoples Friendship University of Russia, Miklukho-Maklay 6, Moscow, 117198, Russia.

Acknowledgements

The research work of GE Shambilova and VD Stepanov was carried out at the Peoples Friendship University of Russia and nancially supported by the Russian Science Foundation (Project no. 16-41-02004).

Received: 24 March 2016 Accepted: 7 September 2016

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