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

RESEARCH

Open Access

Weighted Hardy type inequalities for supremum operators on the cones of monotone functions Lars-Erik Persson1,2* , Guldarya E Shambilova3 and Vladimir D Stepanov4 *

Correspondence: [email protected] 1 Department of Engineering Sciences and Mathematics, Lulea University of Technology, Lulea, 97187, Sweden 2 UiT, The Arctic University of Norway, P.O. Box 385, Narvik, 8505, Norway Full list of author information is available at the end of the article

Abstract The complete characterization of the weighted Lp – Lr inequalities of supremum operators on the cones of monotone functions for all 0 < p, r ≤ ∞ is given. MSC: Primary 26D15; secondary 47G10 Keywords: Hardy type inequality; weight; supremum operator; Lebesgue space; monotone function

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 define  Lpv

:= f ∈ M : f Lpv :=



∞

 f (x)p v(x) dx

 p

 <∞ ,



     ∞ L∞ v := f ∈ M : f Lv := ess sup v(x) f (x) < ∞ . x≥

Let w ∈ M+ and k(x, y) ≥  is a Borel function on [, ∞) satisfying Oinarov’s condition: k(x, y) =  if x < y, and there is a constant D ≥  independent of x ≥ z ≥ y ≥  such that 

 

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

(.)

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, provided 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.

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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 f ∈ M↑ ,

(Tf )(x) = ess sup k(y, x)w(y)f (y), y≥x

f ∈ M↓ ,

(Sf )(x) = ess sup k(y, x)w(y)f (y), y≥x

(T f )(x) = ess sup k(x, y)w(y)f (y),

f ∈ M↓ ,

(S f )(x) = ess sup k(x, y)w(y)f (y),

f ∈ M↑ .

≤y≤x

≤y≤x

Let  < p, r ≤ ∞ and u, v ∈ M+ . The paper is devoted to the necessary and sufficient conditions for the inequalities Tf Lru ≤ CT f Lpv ,

f ∈ M↑ ,

(.)

Sf Lru ≤ CS f Lpv ,

f ∈ M↓ ,

(.)

T f Lru ≤ CT f Lpv ,

f ∈ M↓ ,

(.)

S f Lru ≤ CS f Lpv ,

f ∈ M↑ ,

(.)

where the constants CT and others are taken as the least possible. This problem was first 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 different supremum operators some similar problems were studied in [–]. This area is currently developing intensively and finds 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  G, if F ≤ cG with some positive constant c, which depends only on irrelevant parameters. F ≈ G means F  G  F or F = cG. χE 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 



t

V (t) :=

v,

∞

V∗ (t) :=



v. t

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

∞ 

p ess sup k(y, x)w(y) y≥x

 

y

pr  pr  p h u(x) dx ≤ CT

∞

hV∗ , 

h ∈ M+ .

(.)

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If 

p Tp h(x) := ess sup k(y, x)w(y)

y

h,

y≥x



then (.) is equivalent to Tp h

r p

Lu

p

≤ CT hL , V∗

h ∈ M+ .

(.)

Analogously, if

p Sp h(x) := ess sup k(y, x)w(y)



∞

h,

y≥x

y

p Tp h(x) := ess sup k(x, y)w(y) ≤y≤x



∞

h, y

p Sp h(x) := ess sup k(x, y)w(y)



y

h,

≤y≤x



then (.), (.), and (.) are equivalent to Sp h

p

≤ CS hL ,

r p

V

Lu

Tp h

r p Lu

p

≤ CT hL , V

h ∈ M+ ,

(.)

h ∈ M+ ,

(.)

and Sp h

r p

Lu

p

≤ CS hL , V∗

h ∈ M+ ,

(.)

respectively. For the border cases  < p < r = ∞,  < r < p = ∞, and r = p = ∞ we have the following four groups of inequalities:  p p ess sup u(x) Tp h(x) ≤ CT,p



∞

h ∈ M+ ,

hV∗ ,

x≥



∞

(.)



 r r ess sup k(y, x)w(y)f (y) u(x) dx ≤ CT,r f L∞ , v y≥x



  , ess sup u(x) ess sup k(y, x)w(y)f (y) ≤ CT,∞ f L∞ v y≥x

x≥

f ∈ M↑ ,

f ∈ M↑ ,

(.) (.)

for the operator T;  p p ess sup u(x) Sp h(x) ≤ CS,p x≥





∞

hV , r

∞

ess sup k(y, x)w(y)f (y) u(x) dx 

y≥x

h ∈ M+ ,  r

≤ CS,r f L∞ , v

  ess sup u(x) ess sup k(y, x)w(y)f (y) ≤ CS,∞ f L∞ , v x≥

y≥x

(.)



f ∈ M↓ ,

f ∈ M↓ ,

(.) (.)

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for the operator S;  p p ess sup u(x) Tp h(x) ≤ CT ,p x≥



∞



∞

hV ,

h ∈ M+ ,

(.)



 r r ess sup k(x, y)w(y)f (y) u(x) dx ≤ CT ,r f L∞ , v ≤y≤x



  , ess sup u(x) ess sup k(x, y)w(y)f (y) ≤ CT ,∞ f L∞ v ≤y≤x

x≥

f ∈ M↓ ,

f ∈ M↓ ,

(.) (.)

for the operator T , and  p p ess sup u(x) Sp h(x) ≤ CS ,p



hV∗ ,

x≥



∞ 

∞

h ∈ M+ ,

(.)



 r r ess sup k(x, y)w(y)f (y) u(x) dx ≤ CS ,r f L∞ , v ≤y≤x

  , ess sup u(x) ess sup k(x, y)w(y)f (y) ≤ CS ,∞ f L∞ v ≤y≤x

x≥

f ∈ M↑ ,

f ∈ M↑ ,

(.) (.)

for the operator S . We characterize the inequalities (.)-(.) in the next section. To deal with the inequalities (.)-(.) we study first the case k(x, y) =  and then a general case.

3 Border cases of summation parameters For a measurable function v ∈ M+ we define monotone envelopes (see [], Section ) as follows: v↓ (x) := ess sup v(y), y≥x

v↑ (x) := ess sup v(y). ≤y≤x

Theorem . For the best possible constants of the inequalities (.)-(.) we have CT,p ≈ sup u↑ (x) ess sup

k(y, x)w(y)

, /p V∗ (y)

 ∞  r k(y, x)w(y) r CT,r = u(x) dx , ess sup v↓ (y) y≥x 

k(y, x)w(y) . CT,∞ = ess sup u(x) ess sup v↓ (y) y≥x x≥

(.)

y≥x

x≥

(.) (.)

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

p Tp h(t) = ess sup k(y, t)w(y) y≥t



y

h 

p ≤ Dp ess sup k(y, x)w(y) y≥t

p ≤ Dp ess sup k(y, x)w(y) y≥x



y

h 

 y

h = Dp Tp h(x). 

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Hence, Tp h(x) ≈ sup Tp h(t) := ϕ(x) ∈ M↓ . t≥x

It implies (see [], Proposition .)  p  p ess sup u(x) Tp h(x) ≈ ess sup u(x) ϕ(x) x≥

x≥

 p  p = ess sup u(x) sup ϕ(t) = sup ϕ(t) u↑ (t) t≥x

x≥

t≥

 p ≈ sup u↑ (x) Tp h(x), x≥

and (.) is equivalent to  p sup u↑ (x) Hx hL∞

p

(k(·,x)w(·))p

x≥



 CT,p

∞

hV∗ ,

h ∈ M+ ,

(.)



where 

y

Hx h(y) := χ[x,∞) (y)

h. 

Thus,  p p CT,p ≈ sup u↑ (x) Hx L

∞ V∗ →L(k(·,x)w(·))p

x≥

.

Since by a well-known theorem ([], Theorem .) Hx L

∞ V∗ →L(k(·,x)w(·))p

= ess sup y≥x

(k(y, x)w(y))p , V∗ (y)

we obtain (.). Now, (.) is equivalent to the inequality 

∞

r

 r

ess sup k(y, x)w(y)f (y) u(x) dx 

y≥x

≤ CT,r f L∞↓ ,

The lower bound of (.) follows from (.) with f = f L∞

v

 v↓

f ∈ M↑ .

(.)

and the upper bound from the

↓

estimate f (y) ≤ v↓ (y)v . 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 u↑ (x) ess sup x≥

y≥x

k(y, x)w(y) , V /p (y)

(.)

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

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∞

 r k(y, x)w(y) r ess sup CS,r = u(x) dx , v↑ (y) y≥x 

k(y, x)w(y) CS,∞ = ess sup u(x) ess sup . v↑ (y) y≥x x≥ 

CT ,p ≈ sup u↓ (x) ess sup ≤y≤x

x≥

(.) (.)

k(x, y)w(y) , V /p (y)

(.)

∞

 r k(x, y)w(y) r ess sup u(x) dx , CT ,r = v↑ (y) ≤y≤x 

k(x, y)w(y) , CT ,∞ = ess sup u(x) ess sup v↑ (y) x≥ ≤y≤x 

CS ,p ≈ sup u↓ (x) ess sup

(.) (.)

k(y, x)w(y)

, /p V∗ (y)  ∞  r

k(y, x)w(y) r CS ,r = u(x) dx , ess sup v↓ (y) ≤y≤x 

k(y, x)w(y) . CS ,∞ = ess sup u(x) ess sup v↓ (y) x≥ ≤y≤x

(.)

≤y≤x

x≥

(.) (.)

4 The case k(x, y) = 1 t Let u, v , w ∈ M+ be weights. We suppose for simplicity that  <  u < ∞, for all t > , ∞ – : [; ∞) → [; ∞), by  u = ∞ and define the functions σ : [; ∞) → [; ∞), σ   x   y u≥ u , σ (x) := inf y >  : 



   y   x u≥ u . σ – (x) := inf y >  :    Let σ  := σ (σ ). For  ≤ c < d < ∞ and h ∈ M+ we put 

x

Hc h(x) := χ[c,∞) (x)

h, 

 Hc,d h(x) := χ[c,d) (x) Hc∗ h(x) := χ[c,∞) (x) ∗ h(x) := χ[c,d) (x) Hc,d

x

h, 

σ – (c) ∞

h, x



σ (d)

h. x

We need the following partial cases of [], Theorems . and . (see also [, ]). Theorem . Let  < r < ∞. Then: (a) For validity of the inequality 

∞

 ess sup w (y)



y≥x



y

r  r h u(x) dx ≤ C hLv , 

h ∈ M+ ,

(.)

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

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it is necessary and sufficient that the inequality 

∞





 ↓ r u(x) w (x)

x



r  r h dx ≤ A hLv , 

h ∈ M+ ,

holds and the constant ⎧  ⎨supt> ( t u) r Ht L →L∞ , r ≥ ,  v w r A :=  ∞ x r –r –r ⎩( –r H – [σ (x),σ (x)] L →L∞ dx) r ,  < r < ,  u(x)(  u) v

w

is finite. Moreover, C ≈ A + A . (b) For validity of the inequality 

∞

 ess sup w (y) y≥x



y

∞

r  r h u(x) dx ≤ C hLv , 

h ∈ M+ ,

(.)

it is necessary and sufficient that the inequality 

r 



∞

u(x) ess sup w (y)

σ  (x)

x≤y≤σ  (x)



∞

r  r h dx ≤ B hLv , 

h ∈ M+ ,

holds and the constant ⎧  ⎨supt> ( t u) r Ht∗ L →L∞ , r ≥ ,  v w r B :=  ∞ x r –r ⎩( –r H ∗  –r dx) r ,  < r < ,  u(x)(  u) [σ – (x),σ (x)] L →L∞ v

w

is finite. 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 ,

C S ≈ B  + B ,

(.)

where  –  A = sup V∗ (t) p



t>

A =

V∗ (x)



 r u w↓

 r ,

r ≥ p,

t ∞ 



∞

–



∞

 r u w↓

r  p–r

 r u(x) w↓ (x) dx

 p–r pr ,

 < r < p,

x

 t  r w↓ (y) A = sup u sup r ≥ p,  , y≥t [V (y)] p t>  ∗ r   ∞  x  p–r  p–r  r pr [w(y)]p p–r A = u(x) u dx , ess sup   σ – (x)≤y≤σ (x) V∗ (y)

 < r < p,

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



–  

B = sup V σ  (t) p



 r

r

V σ (z)

–



,

r ≥ p,

x≤y≤σ  (x)

 ∞ 

B =



t

u(x) ess sup w(y) dx

t>



Page 8 of 18





z

r

r  p–r

u(x) ess sup w(y) dx x≤y≤σ  (x)



 r  × u(z) ess sup w(y) dz

p–r pr

,

 < r < p,

z≤y≤σ  (z)

 t  r w(y) B = sup u ess sup r ≥ p,  , y≥t t>  [V (y)] p r   r  ∞  x  p–r  p–r pr [w(y)]p p–r B = u(x) u dx , ess sup   σ – (x)≤y≤σ (x) V (y)

 < r < p.

Proof Since (.) ⇔ (.) and (.) ⇔ (.), the proof follows by applying Theorem . with r replaced by pr , w = wp , v = V∗ in (.) and v = V in (.). Thus, CT ≈ A + A , where A is the best constant in the inequality 

∞

 r u(x) w↓ (x)





x 

 pr  pr  p h dx ≤ A hL , V∗

h ∈ M+ ,

(.)

and 

  p

A

⎧  p ⎨supt> ( t u) r Ht L →L∞ , r ≥ p, V∗ wp r = ∞ x r p–r p–r ⎩( p–r H – [σ (x),σ (x)] L →L∞ dx) r ,  < r < p.  u(x)(  u) V∗

wp

If k(x, y) ≥  is a measurable kernel on R+ × R+ and 

∞

Kf (x) :=

k(x, y)f (y) dy, 

then by well-known results ([], Chapter XI, Section ., Theorem , see also [], Theorem .)   KL →Lq = ess supk(·, s)Lq ,

 ≤ q ≤ ∞.

(.)

s≥

If k(x, y) = w(x)χ[,x] (y)u(y) and  < q < , then ([], Theorem .)  KL →Lq ≈

∞

ess sup u(y) 

≤y≤x

q   –q

∞

q

w

q  –q



q

w(x) dx

 –q q .

x

Applying (.) and (.) to (.) we find that A ≈ A . Again, applying (.), when k(x, y) = wp (x)χ[t,∞) (x)

χ[,x] (y) V∗ (y)

(.)

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

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we obtain Ht L

∞ V∗ →Lwp

  = ess supk(·, s)L∞ = ess sup s≥

s≥

 ess sup wp (x) V∗ (s) {x≥t}∩{x≥s}

p   ↓

= sup w max(t, s) s≥ V∗ (s)   [w↓ (t)]p [w↓ (s)]p [w↓ (s)]p , sup = sup . = max sup s≥t V∗ (s) s≥t V∗ (s) ≤s≤t V∗ (s) Similarly, using the monotonicity of V∗ , we find r p–r

H[σ – (x),σ (x)] L

∞ V∗ →Lwp

 ess sup wp (y) s≥σ – (x) V∗ (s) {σ – (x)≤y≤σ (x)}∩{y≥s}

= ess sup =

 sup ess sup wp (y) σ – (x)≤s≤σ (x) V∗ (s) σ – (x)≤y≤σ (x) s≤y≤σ (x)

=

wp (y) σ – (x)≤y≤σ (x) V∗ (y)

sup

ess sup

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 

∞

r 



∞

u(x) ess sup w(y) σ  (x)

x≤y≤σ  (x)



 pr  pr  p h dx ≤ B hL ,

h ∈ M+ ,

V

(.)

and 

  p

B

⎧  p ⎨supt> ( t u) r Ht∗ L →L∞ , r ≥ p, V wp r = ∞ x r p–r p–r ⎩( p–r H ∗  dx) r ,  < r < p.  u(x)(  u) [σ – (x),σ (x)] L →L∞ V

wp

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

∞

r 



∞

u(x) ess sup w(y) 

x≤y≤σ  (x)

 pr  pr  p h dx ≤ B hL

V  σ

x

,

h ∈ M+ ,

(.)

where Vσ  (y) := V (σ  (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  < t u < ∞ for all t > ,  u = ∞ and define the functions ζ : [; ∞) → [; ∞), ζ – : [; ∞) → [; ∞), by   ζ (x) := sup y >  :

∞

u≥

y

  ζ – (x) := sup y >  :

∞ y

 



∞

 u ,

x



u≥ x

∞

 u .

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

Page 10 of 18

Let ζ  := ζ (ζ ). For  ≤ c < d < ∞ and h ∈ M+ we put 

∞

Hd h(x) := χ(,d] (x)

h, x



Hc,d h(x) := χ(c,d] (x)

h, 

∗

Hc h(x) := χ(,d] (x) ∗ Hc,d h(x) := χ(c,d] (x)

ζ (d)

x x

h, 



x

h. ζ – (c)

We need the following partial cases of [], Theorems . and .. Theorem . Let  < r < ∞. Then: (a) For validity of the inequality 

∞



∞

ess sup w (y) ≤y≤x



r  r h u(x) dx ≤ C hLv , 

y

h ∈ M+ ,

it is necessary and sufficient that the inequality 

∞



 r u(x) w↑ (x)



∞

x

r  r h dx ≤ D hLv ,

h ∈ M+ ,



holds and the constant ⎧  ⎨supt> ( ∞ u) r Ht L →L∞ , r ≥ , t w v r D :=  ∞ ∞ r –r –r ⎩( –r H – [ζ (x),ζ (x)] L →L∞ dx) r ,  < r < ,  u(x)( x u) v

w

is finite. Moreover, C ≈ D + D . (b) For validity of the inequality 

∞

 ess sup w (y) ≤y≤x





y

r  r h u(x) dx ≤ C hLv , 

h ∈ M+ ,

it is necessary and sufficient that the inequality 

∞



r 

ζ – (x)

u(x) ess sup w (y) 

ζ – (x)≤y≤x



r  r h dx ≤ E hLv , 

h ∈ M+ ,

holds and the constant ⎧  ⎨supt> ( ∞ u) r Ht ∗ L →L∞ , r ≥ , t w v r E :=  ∞ ∞ r –r ⎩( –r H ∗  –r dx) r ,  < r < ,  u(x)( x u) [ζ – (x),ζ (x)] L →L∞ v

is finite. Moreover, C ≈ E + E .

w

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 V (t) p



t>

t

 r u w↑

 r

r ≥ p,

,

 ∞ 



D =

V (x)

–





x

 r u w↑

r  p–r

 r u(x) w↑ (x) dx

 p–r pr ,

 < r < p,





 r w↑ (y) u sup r ≥ p,  , t> 
D = sup

∞



–  E = sup V∗ ζ – (t) p



t>



E =



∞

u(x) ess sup w(y) dx

– V∗ ζ – (z)



,

r ≥ p,

ζ – (x)≤y≤x

t ∞ 

 r

r

 < r < p,





∞

r

r  p–r

u(x) ess sup w(y) dx ζ – (x)≤y≤x

z

 r  p–r pr × u(z) ess sup w(y) dz ,

 < r < p,

ζ – (z)≤y≤z

 E = sup

∞

 r u ess sup

w(y)

r ≥ p,  , [V∗ (y)] p r   r  ∞  ∞  p–r  p–r pr [w(y)]p p–r ess sup E = u(x) u dx ,  x ζ – (x)≤y≤ζ (x) V∗ (y) t>

≤y≤t

t

 < 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) satisfies Oinarov’s condition (.). Then: (a) For validity of the inequality 

∞



y

ess sup k (y, x)w (y) y≥x





r  r h u(x) dx ≤ C hLv , 

h ∈ M+ ,

it is necessary and sufficient that the inequalities  

∞

 r  x r  r u(x) ess sup k (y, x)w (y) h dx ≤ A hLv , y≥x





h ∈ M+ ,

(.)

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

Page 12 of 18

and 

∞

  y r  r   r ess sup w (y) u(x) k σ (x), x h dx ≤ A hLv , y≥σ  (x)



h ∈ M+ ,





hold and the constant ⎧ t  ⎪ , ⎨supt> (  u) r Ht Lv →L∞ w (·)k (·,t)  r A :=  ∞ x r –r ⎪ ⎩(  u(x)(  u) –r H[σ – (x),σ  (x)] Lv →L∞ 

r ≥ , dx)

–r r

, r < ,

w (·)k (·,σ – (x))

is finite. Moreover, C ≈ A + A + A . (b) For validity of the inequality 

∞



∞

ess sup k (y, x)w (y) y≥x



y

r  r h u(x) dx ≤ C hLv , 

h ∈ M+ ,

(.)

it is necessary and sufficient that the inequalities 

r 



∞

u(x) ess sup k (y, x)w (y) x≤y≤σ  (x)



∞

σ  (x)

r  r h dx ≤ B hLv ,

h ∈ M+ ,



and 

∞

    r ess sup w (y) u(x) k σ (x), x y≥σ  (x)



∞ y

r  r h dx ≤ B hLv , 

h ∈ M+ ,

hold and the constant ⎧ t  ⎪ , ⎨supt> (  u) r Ht∗ Lv →L∞ w (·)k (·,t)  r B :=  ∞ x r –r ∗ ⎪ ⎩(  u(x)(  u) –r H[σ – (x),σ  (x)] L →L∞ v

r ≥ , dx)

–r r

,

r < ,

w (·)k (·,σ – (x))

is finite. Moreover, C ≈ B + B + B . Using Theorem . we obtain the characterization of (.) and (.) for  < p, r < ∞. Denote Wk (x) := ess sup k(y, x)w(y), y≥x

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

wσ (w)(x) := ess sup w(y), x≤y≤σ  (x)



 gσ k (y) := g σ k (y) , kσ (x) := k σ  (x), x ,    y  x r r σ (x) := inf y >  : u[kσ ] ≥  u[kσ ] , 



    y  x – r r σ (x) := inf y >  : u[kσ ] ≥ u[kσ ] .   

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 V∗ (t) p



t>

∞

u[Wk ]r

 r

r ≥ p,

,

t



∞ 

A =

V∗ (x)

–



∞

u[Wk ]



r

r  p–r



r

,

r ≥ p,

 p–r pr

u(x) Wk (x) dx

,

 < r < p,

x

 –  A, = sup (V∗ )σ  (t) p



t>

t ∞ 

 A, =

∞

(V∗ )σ  (x)

–

 ↓ r u kσ wσ 





∞

x

 r ↓ × u(x) kσ (x)wσ  (x) dx

 ↓ r u kσ wσ 

r  p–r

 p–r pr ,

 t  r r A, = sup u[kσ ] ess sup t>

 r

 < r < p,

wσ  (y) 

y≥t

r ≥ p,

,

[(V∗ )σ  (y)] p r  x  p–r  ∞  r u(x) kσ (x) u[kσ ]r A, = 







[wσ  (y)]p × ess sup σ – (x)≤y≤σ (x) (V∗ )σ  (y)

r  p–r

 p–r pr ,

dx

 < r < p,



 t  r w(y)k(y, t) A = sup u ess sup  , y≥t t>  [V∗ (y)] p r  ∞  x  p–r A = u(x) u 

r ≥ p,

 r  p–r



[w(y)k(y, σ – (x))]p × ess sup V∗ (y) σ – (x)≤y≤σ  (x)  –  B = sup Vσ  (t) p



t>

 B =

t

u[Wk ]r

 r

 p–r pr ,

dx

 < r < p,

r ≥ p,

,

 ∞ 

Vσ  (z)

–





z

u[Wk ]

r

r  p–r



r

u(z) Wk (z) dz

 p–r pr





–  B, = sup Vσ  σ (t) p t>

∞ 

 B, = 



t 

 r u kσ wσ (wσ  )

 – Vσ  σ (t) (z)



z r





 r ,

u[kσ ] wσ (wσ  )

 r × u(z) kσ (z)wσ (wσ  )(z) dz

r ≥ p, r

 p–r pr ,

 < r < p,

r  p–r

,

 < r < p,

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

Page 14 of 18

 t  r wσ  (y) r B, = sup u[kσ ] ess sup  , y≥t t>  [Vσ  (y)] p r  x  p–r  ∞  r r u(x) kσ (x) u[kσ ] B, = 



 ×

r ≥ p,

[wσ  (y)]p σ – (x)≤y≤σ (x) Vσ  (y)

r  p–r

 p–r pr ,

dx

ess sup

 < r < p,



 t  r w(y)k(y, t) B = sup u ess sup  , y≥t t>  [V (y)] p r  ∞  x  p–r B = u(x) u 

r ≥ p,





[w(y)k(y, σ – (x))]p × ess sup V (y) σ – (x)≤y≤σ  (x)

r  p–r

 p–r pr ,

dx

 < 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 

∞

y≥x





 r  x  pr  pr   p u(x) ess sup k(y, x)w(y) h dx ≤ A hL ,

∞

V∗



   pr  pr  p y   r  p u(x) k σ (x), x h dx ≤ A hL , ess sup w(y) y≥σ  (x)



h ∈ M+ ,

V∗



(.) h∈M , +

and ⎧ t p ⎪ ,   p ⎨supt> (  u) r Ht LV∗ →L∞ [w(·)k(·,t)]p r A :=  ∞ x r p–r ⎪ ⎩(  u(x)(  u) p–r H[σ – (x),σ  (x)] L →L∞ V∗

r ≥ p, dx)

p–r r

,  < r < p.

[w(·)k(·,σ – (x))]p

Applying (.) and (.) we see that A ≈ A and A ≈ A . By a change of variable we find that (.) is equivalent to the inequality 

∞

   pr  pr   r p y u(x) kσ (x) ess sup wσ  (y) h dx y≥x





  p

≤ A hL

[V∗ ]  σ

,

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 

∞ 

 r  ↓ r u(x) kσ (x) wσ  (x)



x

 pr  pr p  h dx ≤ A, hL

[V∗ ]  σ



,

h ∈ M+ ,

and  

p A, p A,

 t  pr r := sup u[kσ ] Ht L t>

∞ [V∗ ]  →Lwp σ σ





∞

:=

 r u(x) kσ (x)





x

u[kσ ]

r

r  p–r

r ≥ p,

,

H



r p–r ∞ [σ– (x),σ (x)] L [V∗ ]  →Lwp σ σ



 p–r r dx

,

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 Oinarov’s condition (.). Then: (a) For validity of the inequality 

∞



∞

ess sup k (x, y)w (y) ≤y≤x



r  r h u(x) dx ≤ C hLv ,

h ∈ M+ ,



y

it is necessary and sufficient that the inequalities 

∞

 r  u(x) ess sup k (x, y)w (y) ≤y≤x



x

∞

r  r h dx ≤ A hLv , 

h ∈ M+ ,

and 

∞



r u(x) k x, ζ – (x)



 ess sup w (y)

≤y≤ζ – (x)



≤ A hLv , 

∞

r  r h dx

y

h ∈ M+ ,

hold and the constant ⎧ ∞  ⎪ , ⎨supt> ( t u) r Ht Lv →L∞ w (·)k (t,·)  r A :=  ∞ ∞ r –r ⎪ ⎩(  u(x)( x u) –r H[ζ – (x),ζ  (x)] L →L∞ v

r ≥ , dx)

–r r

, r < ,

w (·)k (ζ  (x),·)

is finite. Moreover, C ≈ A + A + A . (b) For validity of the inequality 

∞



y

ess sup k (y, x)w (y) 

≤y≤x



r  r h u(x) dx ≤ C hLv , 

h ∈ M+ ,

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

Page 16 of 18

it is necessary and sufficient that the inequalities 

∞

 r  u(x) ess sup k (x, y)w (y) ζ – (x)≤y≤x



ζ – (x)

r  r h dx ≤ B hLv , 



h ∈ M+ ,

and 

∞



r u(x) k x, ζ – (x)





y

ess sup w (y) ≤y≤ζ – (x)





r  r h dx ≤ B hLv , 

h ∈ M+ ,

hold and the constant ⎧ ∞  ⎪ , ⎨supt> ( t u) r Ht ∗ Lv →L∞ w (·)k (t,·)  r B :=  ∞ ∞ r –r ∗ ⎪ ⎩(  u(x)( x u) –r H[ζ – (x),ζ  (x)] Lv →L∞ 

r ≥ , dx)

–r r

, r < ,

w (·)k (ζ  (x),·)

is finite. Moreover, C ≈ B + B + B . Using Theorem . we obtain the characterization of (.) and (.) for  < p, r < ∞. Denote Wk∗ (x) := ess sup k(x, y)w(y),

Wk∗ (x) := ess sup k(x, y)w(y),

≤y≤x

ζ – (x)≤y≤x

ζ (w)(x) := ess sup w(y), ζ – (x)≤y≤x

 kζ (x) := k x, ζ – (x) ,   ζ (x) := sup y >  :

 gζ –k (y) := g ζ –k (y) ,   ∞  ∞ u[kζ ]r ≥ u[kζ ]r ,  x y    ∞  ∞ u[kζ ]r ≥  u[kζ ]r . ζ– (x) := sup y >  : y

x

Theorem . Let  < p, r < ∞. Then for the best possible constants of the inequalities (.) and (.) the following equivalences hold: CT ≈ D + D, + D, + D ,

CS ≈ E + E, + E, + E ,

(.)

where  –  D = sup V (t) p



t>

 D =

t

 ∞ 

V (x)

–

 r u Wk∗





x 

 –  D, = sup Vζ – (t) p



t>

 ∞ 

 D, =

Vζ – (x)



–

 r

 r u Wk∗ t

,

r ≥ p,

r  p–r

 ↑ r u kζ wζ –

 

x

 r u(x) Wk∗ (x) dx

 r

 ↑ r u kζ wζ –

,

r ≥ p,

r  p–r

 p–r pr ,

 < r < p,

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

× u(x)  D, = sup t>



r kζ (x)w↑ζ – (x)

∞

u[kζ ]r

Page 17 of 18

 p–r pr ,

dx

 r

 < r < p,

wζ – (y)

ess sup



≤y≤t

r ≥ p,

,

[Vζ – (y)] p r  ∞  p–r  ∞  r r D, = u(x) kζ (x) u[kζ ] t



x



[wζ – (y)]p × ess sup ζ – (x)≤y≤ζ (x) Vζ – (y)

r  p–r

 p–r pr ,

dx

 < r < p,



 r w(y)k(t, y) u ess sup  , t> ≤y≤t t [V (y)] p r  ∞  ∞  p–r D = u(x) u

 D = sup

∞



r ≥ p,

x r  p–r



[w(y)k(ζ  (x), y)]p × ess sup V (y) ζ – (x)≤y≤ζ  (x)  –  E = sup (V∗ )ζ – (t) p



t>

∞

t



∞ 

E =

(V∗ )ζ – (z)

–





r  u Wk∗ ∞

z



–  E, = sup (V∗ )ζ – ζ– (t) p



t>

∞ 

 E, = 

∞ t

 – (V∗ )ζ – ζ– (t) (z)

t>

∞

u[kζ ]r

 r ess sup

r  p–r

 r u(z) Wk∗ (z) dz

 r u[kζ ] ζ (wζ – ) 

∞

z

 < r < p,

r ≥ p,

,

r

 r  r × u(z) kζ (z) ζ (wζ – ) dz  E, = sup

,

dx

 r

r  u Wk∗

 p–r pr

 r ,

 r u[kζ ]r ζ (wζ – )

 p–r pr , r ≥ p,

r  p–r

 p–r pr ,

 < r < p,

wζ – (y)

,



≤y≤t

r ≥ p,

[(V∗ )ζ – (y)] p r  ∞  p–r  ∞  r r E, = u(x) kζ (x) u[kζ ] t



x

 ×

[wζ – (y)]p ζ – (x)≤y≤ζ (x) (V∗ )ζ – (y)

r  p–r

ess sup

 p–r pr ,

dx

 < r < p,



 r w(y)k(t, y) u ess sup  , t> ≤y≤t [V (y)] p t ∗ r  ∞  ∞  p–r E = u(x) u

 E = sup





∞

r ≥ p,

x

[w(y)k(ζ  (x), y)]p × ess sup V∗ (y) ζ – (x)≤y≤ζ  (x)

r  p–r

 p–r pr dx

,

 < r < p.

 < 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 significantly in writing this paper. All the authors read and approved the final manuscript. Author details 1 Department of Engineering Sciences and Mathematics, Lulea University of Technology, Lulea, 97187, Sweden. 2 UiT, The Arctic University of Norway, P.O. Box 385, Narvik, 8505, Norway. 3 Department of Mathematics, Financial University under the Government of the Russian Federation, Leningradsky Prospekt 49, Moscow, 125993, Russia. 4 Department 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 financially supported by the Russian Science Foundation (Project no. 16-41-02004). Received: 24 March 2016 Accepted: 7 September 2016 References 1. Kufner, A, Persson, LE: Weighted Inequalities of Hardy Type. World Scientific, River Edge (2003) 2. Kufner, A, Maligranda, L, Persson, LE: The Hardy Inequality: About Its History and Some Related Results. Vydavatelsky Servis Publishing House, Pilsen (2007) 3. Kokalishvili, V, Meskhi, A, Persson, LE: Weighted Norm Inequalities for Integral Transforms with Product Kernels. Nova Science Publishers, New York (2010) 4. Kalybay, A, Person, LE, Temirkhanova, A: A new discrete Hardy-type inequality with kernels and monotone functions. J. Inequal. Appl. 2015, 321 (2015) 5. Machihara, S, Ozawa, T, Wadade, H: Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space. J. Inequal. Appl. 2015, 281 (2015) 6. Mejjaoli, H: Hardy-type inequalities associated with the Weinstein operator. J. Inequal. Appl. 2015, 267 (2015) 7. Persson, L-E, Shambilova, GE, Stepanov, VD: Hardy-type inequalities on the weighted cones of quasi-concave functions. Banach J. Math. Anal. 9(2), 21-34 (2015) 8. Oguntuase, J, Fabelurin, O, Adeagbu-Sheikh, A, Persson, LE: Time scale Hardy inequalities with ‘broken’ exponent p. J. Inequal. Appl. 2015, 17 (2015) 9. Persson, LE, Shaimardan, S: Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator. J. Inequal. Appl. 2015, 296 (2015) 10. Gogatishvili, A, Opic, B, Pick, L: Weighted inequalities for Hardy-type operators involving suprema. Collect. Math. 57, 227-255 (2006) 11. Stepanov, VD: On a supremum operator. In: Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends. Operator Theory: Advances and Applications, vol. 219, pp. 233-242 (2012) 12. Gogatishvili, A, Pick, L: A reduction theorem for supremum operators. J. Comput. Appl. Math. 208, 270-279 (2007) 13. Cwikel, M, Pustylnik, E: Weak type interpolation near ‘endpoint’ spaces. J. Funct. Anal. 171, 235-277 (2000) 14. Evans, WD, Opic, B: Real interpolation with logarithmic functions and reiteration. Can. J. Math. 52, 920-960 (2000) 15. Pick, L: Optimal Sobolev embeddings. Rudolph-Lipshitz-Vorlesungsreihe, no. 43. Rheinische Friedrich-Wilhelms-Universität Bonn (2002) 16. Prokhorov, DV: Inequalities for Riemann-Liouville operator involving suprema. Collect. Math. 61, 263-276 (2010) 17. Prokhorov, DV: Lorentz norm inequalities for the Hardy operator involving suprema. Proc. Am. Math. Soc. 140, 1585-1592 (2012) 18. Prokhorov, DV: Boundedness and compactness of a supremum-involving integral operator. Proc. Steklov Inst. Math. 283, 136-148 (2013) 19. Prokhorov, DV, Stepanov, VD: On weighted Hardy inequalities in mixed norms. Proc. Steklov Inst. Math. 283, 149-164 (2013) 20. Prokhorov, DV, Stepanov, VD: Estimates for a class of sublinear integral operators. Dokl. Math. 89, 372-377 (2014) 21. Prokhorov, DV, Stepanov, VD: Weighted inequalities for quasilinear integral operators on the semiaxis and application to the Lorentz spaces. Sb. Math. 207(8), 135-162 (2016). doi:10.1070/SM8535 22. Krepela, M: Integral conditions for Hardy type operators involving suprema. Collect. Math. (2016). doi:10.1007/s13348-016-0170-6 23. Sinnamon, G: Transferring monotonicity in weighted norm inequalities. Collect. Math. 54, 181-216 (2003) 24. Gogatishvili, A, Stepanov, VD: Reduction theorems for weighted integral inequalities on the cone of monotone functions. Russ. Math. Surv. 68(4), 597-664 (2013) 25. Kantorovich, LV, Akilov, GP: Functional Analysis. Pergamon, Oxford (1982) 26. Sinnamon, G, Stepanov, VD: The weighted Hardy inequality: new proofs and the case p = 1. J. Lond. Math. Soc. 54, 89-101 (1996)