Open Access

Kolmogorov type inequalities for the Marchaud fractional derivatives on the real line and the half-line

  • Vladislav F Babenko1,
  • Mariya S Churilova1,
  • Nataliia V Parfinovych1 and
  • Dmytro S Skorokhodov1Email author
Journal of Inequalities and Applications20142014:504

https://doi.org/10.1186/1029-242X-2014-504

Received: 11 August 2014

Accepted: 2 December 2014

Published: 17 December 2014

Abstract

In this paper we establish some new Kolmogorov type inequalities for the Marchaud and Hadamard fractional derivatives of the functions defined on a real axis or semi-axis. Simultaneously we solve two related problems: the Stechkin problem on the best approximation of unbounded operators by bounded ones on a given class of elements and the problem of optimal recovery of an operator on elements from some class given with prescribed error.

Keywords

inequalities for derivativesfractional derivativesapproximation of unbounded operators by boundedoptimal recovery of operatorsideal lattice

1 Introduction

Inequalities estimating the norm of an intermediate derivative of a function in terms of the norm of the function itself and the norm of its higher-order derivative (inequalities of Kolmogorov type) are important in many areas of mathematics and its applications. Due to the efforts of many mathematicians, nowadays, a great number of sharp Kolmogorov type inequalities are known (see, for instance, surveys [13] and monographs [46]). In many questions of analysis and its applications the study of fractional-order derivatives is also important (see, for instance, [7]). For some known results on the Kolmogorov type inequalities for derivatives of fractional order we refer the reader to [815], the book [[16], Chapter 2] and references therein.

In this paper we shall obtain some new Kolmogorov type inequalities for fractional derivatives. Simultaneously, we consider two closely related problems: the Stechkin problem on approximation of unbounded operators by bounded ones on a given class of elements Q, and the problem of optimal recovery of unbounded operator on the class Q under assumption that elements in Q are given with known error (for more information see [1, 2] and [[4], Section 7.1]).

1.1 The Kolmogorov type inequalities

Let G be the real line R = ( , + ) or half-line R + = [ 0 , + ) . By L p ( G ) , 1 p , we denote the space of measurable functions f : G R whose modulus to the p th power is integrable on G (essentially bounded on G if p = ), endowed with the standard norm
f L p ( G ) : = { ( G | f ( t ) | p d t ) 1 / p , if  1 p < , ess sup { | f ( t ) | : t G } , if  p = .

For r N and 1 s , by L p , s r ( G ) we denote the space of the functions f L p ( G ) having locally absolutely continuous on G derivative f ( r 1 ) and such that f ( r ) L s ( G ) .

Let 1 q , k N { 0 } , 0 k r 1 , and λ , μ R . Inequalities of the form
f ( k ) L q ( G ) K f L p ( G ) μ f ( r ) L s ( G ) λ
(1)
holding true, for every function f L p , s r ( G ) with some constant K independent of f, are called Kolmogorov type inequalities (Kolmogorov-Nagy type inequalities when k = 0 ). It is well known (see [17]) that the constant K in inequality (1) is finite if and only if
λ = k 1 / q + 1 / p r 1 / s + 1 / p , μ = 1 λ ,
(2)
and
r q r k p + k s .
(3)

Naturally, inequalities with the lowest possible (sharp) constant K are of the most interest. We refer the reader to [1, 2] and the books [4, 6] for the detailed survey on the Kolmogorov type inequalities and discussion of related questions.

Together with inequalities (1) the study of inequalities between the norms of intermediate function derivative, the function itself and its higher-order derivative in spaces more general than L p are also important. In Sections 2-4 we shall obtain several inequalities between the norms of derivatives in ideal lattices (see [[18], Chapter 2, Section 2]).

In this paper we focus on the study of the Kolmogorov type inequalities for non-integer (fractional) values of k. There are many ways to give a sense to the fractional derivative of a function defined on or R + . Among the first ones was the fractional derivative in the Riemann-Liouville sense (see [[7], Section 5.1]) that is defined for a function f : R R and x R , as follows:
D ± k f ( x ) : = ( ± 1 ) n Γ ( n k ) d n d x n 0 + t n k 1 f ( x t ) d t , n = [ k ] + 1 ,
(4)
where Γ ( z ) is the Euler gamma function and [ z ] stands for the integer part of real number z. We shall mostly consider fractional derivatives in the Marchaud sense (see [19] or [[7], Section 5.6]) that are defined for a function f : R R and x R , as follows:
D ± k f ( x ) = 1 ϰ ( k , n ) 0 + ( Δ ± t n f ) ( x ) t 1 + k d t ,
(5)
where n N , n > k (the definition itself is independent of n), and
( Δ ± t n f ) ( x ) : = m = 0 n ( 1 ) m ( n m ) f ( x m t ) , ϰ ( k , n ) : = Γ ( k ) m = 0 n ( 1 ) m ( n m ) m k .
(6)

For a function f : R + R , the right hand sided derivatives D k f and D k f are defined by formulas (4) and (5), respectively. The left hand sided derivatives D + k f and D + k f are defined with the help of slightly different constructions (see [[7], Sections 5.1, 5.5]), and we shall not study these derivatives here.

It is well known (see [7]) that D ± k f = D ± k f for ‘good’ functions f : G R . However, construction (5) is also suitable for a wider class of the functions, e.g. constant functions or functions whose power growth at infinity has order lower than k.

Let us consider Kolmogorov type inequalities of the form (1) with the term f ( k ) L q ( G ) being replaced by D k f L q ( G ) in the left hand side:
D k f L q ( G ) K f L p ( G ) μ f ( r ) L s ( G ) λ , f L p , s r ( G ) .
(7)

Similarly to inequalities (1) for derivatives of integer order, it is easy to see that the constant K in (7) is finite only if parameters λ and μ satisfy equalities (2).

Together with the Riemann-Liouville and the Marchaud fractional derivatives, Kolmogorov type inequalities were also studied [15, 20, 21] for other fractional derivatives, e.g. the Riesz fractional derivative, the Weyl fractional derivative, etc. To the best of our knowledge, the sharp constant in inequality (7) was found in the following situations:
  1. 1.

    G = R , p = q = s = , r = 2 , and k ( 0 , 1 ) , - Geĭsberg [8];

     
  2. 2.

    G = R + , p = q = s = , r = 2 , and k ( 0 , 2 ) { 1 } , - Arestov [10] (for D ± k );

     
  3. 3.

    G = R + , p = q = s = , 0 < k 1 , and k < r 2 , - Arestov [10] (for D ± k );

     
  4. 4.

    G = R or G = R + , p = s = 2 , q = , k < r , - Buslaev and Tihomirov [21] (for the Weyl derivative);

     
  5. 5.

    G = R or G = R + , p = q = , 1 s , r = 1 , k ( 0 , 1 1 / s ) , - Babenko and Churilova [22];

     
  6. 6.

    G = R , p = q = , 1 s , r = 1 , 2 , k ( 0 , r 1 / s ) { 1 } , - Babenko and Parfinovych [20] and Babenko, Parfinovych and Pichugov [15] (for the Riesz derivative).

     
Here we establish sufficient conditions which allow writing sharp Kolmogorov type inequalities. Specifically, we focus on inequalities between the uniform norms of the function and its derivatives; the uniform norms of the function and its intermediate derivative and the norm of its higher-order derivative in the ideal lattice; the norms of the function and its derivatives in the ideal lattice. As a consequence in Section 4 we obtain several new sharp Kolmogorov type inequalities in the following cases:
  1. 1.

    G = R or G = R + , p = q = , r = 1 , and k ( 0 , 1 ) , the norm of f is considered in an ideal lattice;

     
  2. 2.

    G = R , p = q = s = 1 , r = 1 , and k ( 0 , 1 ) ;

     
  3. 3.

    G = R + , p = q = , r = 2 , and k ( 0 , 1 ) , the norm of f is considered in an ideal lattice;

     
  4. 4.

    G = R , p = q = , 1 s , r = 2 , and k ( 0 , 1 ) ;

     
  5. 5.

    G = R or G = R + , p = q = , 1 s , r = 2 , and k ( 1 , 2 1 / s ) .

     

1.2 The Stechkin problem

The problem of the best approximation of unbounded operators by linear bounded ones is close to the problem of finding sharp constants in inequalities (1), (7) and, furthermore, presents an independent interest. We follow [23] (see also surveys [1, 2]) to set the problem rigorously.

Let X and Y be the Banach spaces; A : X Y be an operator (not necessarily linear) with domain of definition D A X ; Q D A be some set. The function
Ω ( δ ) = Ω ( δ , A , Q ) : = sup { A f Y : f Q , f X δ } , δ 0 ,
(8)

is called the modulus of continuity of the operator A on the set Q.

By L = L ( X , Y ) we denote the space of all linear bounded operators S : X Y . The error of approximation of the operator A by linear bounded operator S L on the set Q is defined by
U ( A , S ; Q ) : = sup x Q A x S x Y .
For N > 0 , we set
E N ( A ; Q ) : = inf S L , S N U ( A , S ; Q ) .
(9)

The Stechkin problem on the best approximation of the operator A by linear bounded operators on set Q consists in evaluating quantity (9) and finding extremal operators (if any exists) delivering an infimum in the right hand part of (9).

Now, we let
( δ , A ; Q ) : = inf N 0 ( E N ( A ; Q ) + N δ ) .

The following theorem by Stechkin [23] (see also [2, 4]) provides a simple but nevertheless effective lower estimate of quantity (9) in terms of the modulus of continuity Ω.

Theorem A If A is a homogeneous (in particular, linear) operator, Q D A is centrally symmetric convex set, then, for every N 0 and δ 0 ,
E N ( A ; Q ) sup δ 0 { Ω ( δ , A ; Q ) N δ } = sup x Q { A x Y N x X } , Ω ( δ , A ; Q ) ( δ , A ; Q ) .
(10)
Furthermore, if there exists a pair of element x 0 Q and operator S 0 L such that
A x 0 Y = U ( A , S 0 ; Q ) + S 0 x 0 X
(11)
then Ω ( x 0 X , A ; Q ) = A x 0 Y and
E S 0 ( A ; Q ) = U ( A , S 0 ; Q ) = A x 0 Y S 0 x 0 X .

Consequently, the operator S 0 is extremal in problem (9) for N = S 0 , and the element x 0 in problem (8) for δ = x 0 X .

We refer the reader to [1, 2] for a survey of other known results on the Stechkin problem and a discussion of related questions.

In the particular case when G = R or G = R + , X = L p ( G ) , Y = L ( G ) , A = D k , and
Q = W p , s r ( G ) : = { f L p , s r ( G ) : f ( r ) L s ( G ) 1 } ,
for every δ 0 , we have
Ω ( δ , D k ; W p , s r ( G ) ) = K δ 1 λ , λ = k + 1 / p r 1 / s + 1 / p ,
where K = Ω ( 1 , D k ; W p , s r ( G ) ) is the sharp constant in inequality (7) with q = . From the result of Gabushin [[24], Lemma 1] it can easily be shown that estimate (10) is sharp when Y = L ( G ) . Hence, for every N 0 ,
E N ( D k ; W p , s r ( G ) ) = sup δ 0 { K δ 1 λ N δ } = λ ( 1 λ ) 1 λ 1 K 1 λ N 1 1 λ .

Therefore, in all cases when the sharp constant K in inequality (7) for q = is found, we immediately know the exact value of the quantity of the best approximation of the operator D k by linear bounded operators on the class W p , s r ( G ) .

1.3 The problem of optimal recovery of operators on elements given with an error

Another problem that is closely related to the Stechkin problem and sharp Kolmogorov type inequalities is the problem of optimal recovery of an operator with the help of the set of linear operators (or mappings in general) on elements of some set that are given with an error. We follow [2] to set the problem rigorously.

Let X and Y be the Banach spaces; A : X Y be an operator (not necessarily linear) with domain of definition D A Q ; Q D A be some set. By we denote either the set of all linear operators acting from X to Y, or the set of all mappings O from X to Y. For an arbitrary δ 0 and S R , we set
U δ ( A , S ; Q ) : = sup { A f S f Y : f Q , g X , f g X δ } .
It is clear that U 0 ( A , S ; Q ) = U ( A , S ; Q ) . The problem of optimal recovery of the operator A with the help of the set of operators on elements of the set Q with given error δ consists in finding the quantity
E δ ( R , A ; Q ) : = inf S R U δ ( A , S ; Q ) ,

called the best recovery of the operator A with the help of mappings from on elements Q given with prescribed error δ. The detailed survey of existing results and further references can be found, for instance, in [2]. The following statement is a corollary of the result by Arestov [[2], Theorem 2.1] that indicates the close relations between this problem and the Stechkin problem.

Theorem B If A is a homogeneous (in particular, linear) operator, Q D A is a centrally symmetric convex set, then, for every N 0 and δ 0 ,
Ω ( δ , A ; Q ) E δ ( O , A ; Q ) E δ ( L , A ; Q ) ( δ , A ; Q ) .
Moreover, if there exist an element x 0 Q and an operator S 0 L ( X , Y ) satisfying (11) from Theorem  A then for δ = x 0 X ,
A x 0 Y = Ω ( δ , A ; Q ) = E δ ( O , A ; Q ) = E δ ( L , A ; Q ) .
Similarly to the Stechkin problem, in the case G = R or G = R + , X = L p ( G ) , and Y = L ( G ) , for every δ > 0 , we have
E δ ( O , D k ; W p , s r ( G ) ) = E δ ( L , D k ; W p , s r ( G ) ) = Ω ( δ , D k ; W p , s r ( G ) ) .

So once the sharp constant in inequality (7) is found, we immediately know the value of the error of optimal recovery of the operator D k by operators from O (or ) on elements of the class W p , s r ( G ) given with error δ.

1.4 Organization of the paper

The paper is organized in the following way. Section 2 is devoted to auxiliary results concerning properties of the Marchaud fractional derivatives: existence, continuity, and integral representation in terms of the higher-order function derivative. Then we establish some sufficient conditions when sharp Kolmogorov type inequalities (7) can be written and derive some consequences from these conditions for r = 1 , 2 in Section 4. Finally, in Section 5 we present applications of the main results: the Kolmogorov problem for three numbers consisting in finding necessary and sufficient conditions on the triple of real positive numbers that guarantee the existence of a function attaining these numbers as the norms of its three consecutive derivatives, and sharp Kolmogorov type inequalities for the weighted norms of the Hadamard fractional derivatives.

2 Auxiliary results

In this section we formulate auxiliary propositions on the existence and continuity of the Marchaud fractional derivative and its integral representation in terms of the higher-order derivative. These and similar questions were studied by many mathematicians. For an overview of known results we refer the reader to the books [7, 25] and references therein.

2.1 Definitions and results

Let G = R or G = R + . By S ( G ) we denote the space of measurable functions f : G R . The linear space E S ( G ) endowed with the norm E is called the ideal lattice on G (see [[18], Chapter 2, Section 2]) if, for every f E and g S ( G ) such that | g ( x ) | | f ( x ) | a.e. on G, it follows that g E and g E f E . The set A ( E ) G is called the support of the ideal lattice E if f ( x ) = 0 for every f E and x A ( E ) . By E 1 we denote the associated space to E (see [[18], Chapter 2, Section 3]), i.e. the space of the functions g S ( G ) such that supp g A ( E ) and
g E 1 : = sup f E , f E 1 G f ( x ) g ( x ) d x < .

It is clear that E 1 is the ideal lattice on G and is a subspace in the space dual to E. Ideal lattices generalize many important spaces e.g. spaces L p ( G ) , 1 p , the Orlicz spaces [26], the Lorentz spaces [18], the Marcinkiewicz spaces [18], etc.

In what follows we would also say that an ideal lattice E is semi shift invariant if, for every f E and x G , we have f ( + x ) E and either f ( + x ) E = f E if G = R or f ( + x ) E f E if G = R + .

Let r N , k ( 0 , r ) N , and F let be an ideal lattice. By L , E r ( G ) and L F , E r ( G ) we denote the spaces of the functions f L ( G ) and f F , respectively, such that f ( r 1 ) is locally absolutely continuous on G and f ( r ) E . In addition, let χ B stand for the characteristic (indicator) function of a measurable set B R .

Proposition 1 Let G = R or G = R + , r N , k ( 0 , r ) N , E be a semi shift-invariant lattice on G such that
( ) r k 1 χ ( 0 , 1 ) ( ) E 1
(12)
and
lim h 0 + ( ) r 1 k χ ( 0 , h ) ( ) E 1 = 0 ,
(13)

where E 1 is the associated space to E. Then D k f exists and is continuous on G, for every function f L , E r ( G ) .

Proposition 2 Let G = R or G = R + , r N , k ( 0 , r ) N , E be a semi shift-invariant lattice on G satisfying condition (12). Then, for every f L , E r ( G ) ,
D k f ( ) = ( 1 ) r Γ ( r k ) 0 + t r 1 k f ( r ) ( + t ) d u .
(14)

Proposition 3 Let G = R or G = R + , r N , k ( 0 , r ) N , E be a semi-shift invariant lattice on G satisfying condition (12) and F be a semi-shift invariant lattice on G such that χ ( 0 , 1 ) F 1 . Then D k f ( x ) exists, for every f L F , E r and x G , and the integral representation (14) for D k f holds true.

In particular, when E = L s ( G ) , 1 s , both conditions (12) and (13) are equivalent to the inequality k < r 1 / s . So the following corollaries hold true.

Proposition 4 Let G = R or G = R + , r N , 1 p , s , and k ( 0 , r 1 / s ) N . Then, for every f L p , s r ( G ) , D k f exists and is continuous on G, and (14) holds true.

Proposition 5 Let G = R or G = R + , r N , 1 s , and k ( 0 , r ) N . Then D k f ( x ) exists, for every f L s , s r ( G ) and x G , and (14) holds true.

2.2 The proofs of auxiliary results

For the sake of completeness, we prove Propositions 1-3. Preliminarily, we recall the definition of the B-splines and some of their properties (see, e.g., [[27], Section 4.2]). The first-order B-spline N 1 is the function χ ( 0 , 1 ) . For r 2 , the r-order B-spline N r is defined by
N r ( x ) = R N r 1 ( x t ) N 1 ( t ) d t = 0 1 N r 1 ( x t ) d t , x R .
It is well known that N r is continuous and positive on ( 0 , r ) function, compactly supported on [ 0 , r ] . Moreover (see Theorem 4.3 in [27]), for every r-times differentiable function f : G R and every t > 0 ,
( Δ t r f ) ( x ) = ( 1 ) r t r 0 r N r ( u ) f ( r ) ( x + u t ) d u , x G .
(15)
Proof of Proposition 1 Let a function f L , E r ( G ) and a point x G be arbitrary. We observe that | ( Δ t r f ) ( x ) | 2 r f L ( G ) , for every t > 0 . Hence, by definition (5), for every h > 0 , we have
ϰ ( k , r ) | D k f ( x ) | = | 0 + ( Δ t r f ) ( x ) t 1 + k d t | | 0 h ( Δ t r f ) ( x ) t 1 + k d t | + | h + ( Δ t r f ) ( x ) t 1 + k d t | | 0 h ( Δ t r f ) ( x ) t 1 + k d t | + 2 r f L ( G ) k h k .
(16)
Using (15), changing variables, altering the order of integration and applying the Hölder inequality we obtain
| 0 h ( Δ t r f ) ( x ) t 1 + k d t | = | 0 h 0 r N r ( u ) f ( r ) ( x + u t ) t k + 1 r d u d t | = | 0 h 0 r t N r ( v / t ) f ( r ) ( x + v ) t k + 2 r d v d t | = | 0 r h f ( r ) ( x + v ) v / r h N r ( v / t ) t k + 2 r d t d v | f ( r ) E χ ( 0 , r h ) ( ) ( ) / r h N r ( ( ) / t ) t k + 2 r d t E 1 .
It is easy to show that N r ( x ) x r 1 , for every x [ 0 , r ] . Hence, for every v ( 0 , r h ) ,
v / r h N r ( v / t ) t k + 2 r d t v r 1 v / r h d t t k + 1 v r k 1 k r k .
From the latter and estimate (16) we conclude that
ϰ ( k , r ) | D k f ( x ) | 2 r f L ( G ) k h k + ( ) r 1 k χ ( 0 , r h ) ( ) E 1 f ( r ) E k r k ,

which proves the existence and uniform boundedness of derivative D k f ( x ) at an arbitrary point x G .

Now, we turn to the proof of continuity of D k f on G. Let x G be an arbitrary point. For every ε > 0 , there exist numbers h > 0 and H > 0 such that
( ) r k 1 χ ( 0 , r h ) ( ) E 1 f ( r ) E k r k ϰ ( k , r ) < ε 6 and 2 r + 1 f L ( G ) k H k ϰ ( k , r ) < ε 3 .
The function f is continuous on G and is uniformly continuous on [ h / 2 , r H + r h / 2 ] . Hence, there exists δ ( 0 , h / 2 ) such that, for every y , y [ h / 2 , r H + r h / 2 ] , | y y | < δ , we have | f ( y ) f ( y ) | < 2 r k h k | ϰ ( k , r ) | ε . Then using similar arguments to the proof of the existence of D k f , we see that, for every x , y G , | x y | < δ , for the function g ( t ) = f ( x + t ) f ( y + t ) we have
| D k f ( x ) D k f ( y ) | 1 ϰ ( k , r ) | 0 + ( Δ t r g ) ( 0 ) t 1 + k d t | 1 ϰ ( k , r ) ( | 0 h ( Δ t r g ) ( 0 ) t 1 + k d t | + | h H ( Δ t r g ) ( 0 ) t 1 + k d t | + | H + ( Δ t r g ) ( 0 ) t 1 + k d t | ) g ( r ) E ( ) r k 1 χ ( 0 , r h ) ( ) E 1 k r k ϰ ( k , r ) + 2 r g L ( [ h , r H ] ) k h k ϰ ( k , r ) + 2 r g L ( G ) k H k ϰ ( k , r ) 2 f ( r ) E ( ) r k 1 χ ( 0 , r h ) ( ) E 1 k r k ϰ ( k , r ) + 2 r f ( x + ) f ( y + ) L ( [ h , r H ] ) k h k ϰ ( k , r ) + 2 r + 1 f L ( G ) k H k ϰ ( k , r ) < ε 3 + ε 3 + ε 3 = ε .

Therefore, the D k f is continuous on G. □

Remark 1 During the proof of Proposition 1 we have established the Kolmogorov type inequality between the uniform norm of fractional derivative of the function, the function itself and the norm of its higher-order derivative in the ideal lattice:
D k f L ( G ) 2 r f L ( G ) k h k ϰ ( k , r ) + χ ( 0 , r h ) ( ) ( ) / r h N r ( ( ) / t ) t k + 2 r d t E 1 f ( r ) E .
Proof of Proposition 2 First we note that
0 r N r ( u ) u r k d u = ( 1 ) r k ( k 1 ) ( k r + 1 ) ( Δ 1 r ( ) k ) ( 0 ) = ( 1 ) r k ( k 1 ) ( k r + 1 ) m = 0 r ( 1 ) m ( r m ) m k = ϰ ( k , r ) Γ ( k ) ( k ) ( k + 1 ) ( k + r 1 ) = ϰ ( k , r ) Γ ( r k ) .
Let f L , E r ( G ) and x G . The derivative D k f ( x ) exists due to Proposition 1. Altering the order of integration and applying the Tonelli theorem we obtain
D k f ( x ) = ( 1 ) r ϰ ( k , r ) 0 + 0 r t r 1 k N r ( u ) f ( r ) ( x + u t ) d u d t = ( 1 ) r ϰ ( k , r ) 0 + w r k 1 f ( r ) ( x + w ) ( 0 r N r ( u ) u r k d u ) d w = ( 1 ) r Γ ( r k ) 0 + w r 1 k f ( r ) ( x + w ) d w ,

which finishes the proof. □

Proof of Proposition 3 Let f L F , E r ( G ) and x G . Using the same arguments as in the proof of Proposition 1 we can prove that the first of two integrals
0 1 ( Δ t r f ) ( x ) t 1 + k d t and 1 + ( Δ t r f ) ( x ) t 1 + k d t
is convergent. Hence, it is sufficient to prove the convergence of the second integral. The latter is obvious because
| 1 + ( Δ t r f ) ( x ) t 1 + k d t | m = 0 r ( r m ) 1 + | f ( x + m t ) | t 1 + k d t = | f ( x ) | k + m = 1 r m k ( r m ) m + | f ( x + u ) | u 1 + k d u | f ( x ) | k + m = 1 r m k ( r m ) f E ( ) 1 k χ ( m , + ) ( ) F 1 ,
and, for m = 1 , 2 , , r ,
( ) 1 k χ ( m , + ) ( ) F 1 j = m χ ( 0 , 1 ) F 1 j k + 1 < m + k m k + 1 k χ ( 0 , 1 ) F 1 .

Hence, D k f ( x ) exists, for every x G . Finally, we remark that equality (14) immediately holds true if D k f ( x ) exists. The proof is finished. □

3 Main results

Let us present results on some general sufficient conditions allowing one to write a sharp Kolmogorov type inequality in various situations. We start with the Kolmogorov type inequality between the uniform norms of the Marchaud fractional derivative of a function, the function itself and its higher-order derivative. In Section 3.2 we extend this result on the case of inequalities between the norms of the function and its derivatives in an ideal lattice. Then in Section 3.3 we give another extension of results of Section 3.1 on the case of inequalities between the uniform norms of the Marchaud fractional derivative of a function, the uniform norm of the function itself, and the norm of the higher-order derivative in an ideal lattice.

The idea used to prove main results of the paper consists in approximating fractional differentiation operator with the help of linear bounded operators. More precisely, let G = R or G = R + , X, Y, Z be the linear normed spaces of measurable on G functions f : G R , L be the space of the functions f X having locally absolutely continuous on G derivative of order ( r 1 ) , and such that f ( r ) Z . Let also T : X Y and R : Z Y be linear bounded operators such that, for every f L , D k f = T f + R f ( r ) . Then, for every f L ,
D k f Y T f Y + R f ( r ) Y T X Y f X + R Z Y f ( r ) Z ,

which is an additive form of the Kolmogorov type inequality (7). If for some operators T and R there exists a function f W turning the above inequality into an equality then the corresponding Kolmogorov type inequality is sharp.

We remark that this idea is not new and is already contained in [23] by Stechkin. Besides, some similar ideas were even in the papers by Landau and Hadamard. The corresponding operators T and R as well as the extremal function f were found in many cases (see [2830], and surveys [1, 2, 4] for more details).

Let G = R or G = R + . By V ( G ) we denote the space of the functions f L 1 ( G ) with bounded on G variation. Also, we set x + : = max { x ; 0 } , for every x G , and for f L 1 ( G ) and m N , we denote by f [ m ] the m th-order integral of the function f:
f [ m ] ( x ) : = 1 ( m 1 ) ! G ( x t ) + m 1 f ( t ) d t , x G .
Finally, for τ > 0 , we define the function R τ : G R as follows:
R τ ( x ) = { x τ 1 Γ ( τ ) , x > 0 , 0 , x 0 .

3.1 The Kolmogorov type inequalities for the Marchaud fractional derivatives: case of uniform norms

The following results hold true.

Theorem 1 Let G = R + or G = R , r N , k ( 0 , r ) N , and a function Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) L 1 ( G ) and, for every f L , r ( G ) ,
D k f ( 0 ) G f ( x ) d Ω ( x ) = ( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) f ( r ) ( x ) d x .
(17)
Then, for every f L , r ( G ) and h > 0 , we have the inequality
D k f L ( G ) h k G Ω f L ( G ) + h r k R r k Ω [ r 1 ] L 1 ( G ) f ( r ) L ( G ) .
(18)
Furthermore, if a function Φ W , r ( G ) satisfies the equalities
G Φ ( t ) d Ω ( t ) = G Ω Φ L ( G )
(19)
and
( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ ( r ) ( x ) d x = R r k Ω [ r 1 ] L 1 ( G )
(20)

then (18) is sharp and the function Φ h ( ) : = Φ ( ( ) / h ) turns (18) into an equality.

Minimizing the right hand part of (18) by h we obtain the next consequence.

Corollary 1 Let G = R + or G = R , r N , k ( 0 , r ) N , and assume that the functions Ω and Φ satisfy assumptions of Theorem  1. Then, for every f L , r ( G ) , the following sharp inequality holds true:
D k f L ( G ) D k Φ L ( G ) Φ L ( G ) 1 k / r f L ( G ) 1 k / r f ( r ) L ( G ) k / r .
(21)
We remark that the following results on sharp inequalities of the form (7) concretize Corollary 1.
  1. 1.
    For G = R , r = 2 , and k ( 0 , 1 ) , the extremal function Φ in inequality (18) and the corresponding function Ω that satisfy the conditions of Corollary 1 were found by Geĭsberg [8] and Arestov [10], respectively:
    Ω ( x ) : = { 0 , x 0 , 1 Γ ( 2 k ) , x ( 0 , 1 ) , x k Γ ( 1 k ) , x 1 , Φ ( x ) : = { ( 1 + p ) 2 8 , x p , ( x + p ) 2 ( 1 + p ) 2 8 , x [ p , 1 p 2 ] , ( 1 + p ) 2 ( 1 x ) 2 8 , x [ 1 p 2 , 1 ] , ( 1 + p ) 2 8 , x 1 ,
     
where p = 1 2 k / ( 1 k ) .
  1. 2.
    For G = R + , r = 2 , and k ( 0 , 1 ) , the extremal function Φ, and the corresponding function Ω that satisfy the conditions of Corollary 1 were found by Arestov [[10], Theorem 3]:
    Ω ( x ) : = { 0 , x = 0 , 1 Γ ( 2 k ) , x ( 0 , 1 ) , x k Γ ( 1 k ) , x 1 , Φ ( x ) : = { 1 4 x + x 2 2 , x [ 0 , 1 ] , 1 4 , x 1 .
     
  2. 3.
    For G = R + , r = 2 , and k ( 1 , 2 ) , the extremal function Φ and the corresponding function Ω that satisfy the conditions of Corollary 1 were also found by Arestov [[10], p.32]:
    Ω ( x ) : = { 0 , x = 0 , 3 2 ( k + 1 ) / 2 Γ ( 2 k ) , x ( 0 , 2 1 ] , 2 k / 2 2 Γ ( 2 k ) ( 2 1 ) , x ( 2 1 , 1 ) , x k Γ ( 1 k ) , x 1 , Φ ( x ) : = { 3 2 2 4 ( 2 1 ) x + 2 x 2 4 , x [ 0 , 1 2 ] , 1 2 2 + 4 x 2 x 2 4 , x ( 1 2 , 1 ) , 3 2 2 4 , x 1 .
     

For integer values of k, the extremal function Φ on in inequality (21) was found by Kolmogorov [31] (see also [3]), for every r = 2 , 3 ,  . In the surveys [1, 2, 4] the reader could find more references and a detailed history of the subject and overview of cases when the extremal function Φ in inequality (21) on R + is known.

In addition, for integer values of k, the function Ω on R + for which inequality (18) is sharp was explicitly constructed by Stechkin [23] in the case r = 2 , 3 . In the case G = R the existence of such a function Ω was proved by Domar [32] and explicitly it was constructed by Stechkin [23] for r = 2 , 3 , Arestov [33] for r = 4 , 5 , and Buslaev [34] for r > 5 .

Proof of Theorem 1 First, we let h = 1 and define the linear operator T : L ( G ) L ( G ) as follows:
T g ( ) : = G g ( + t ) d Ω ( t ) , g L ( G ) .
Clearly, T is bounded and T = G Ω . Next, let a function f L , r ( G ) and a point x G be arbitrary. Then from Proposition 2 and (17) we deduce
| D k f ( x ) | = | T f ( x ) + ( ( 1 ) r G R r k ( t ) f ( r ) ( x + t ) d t T f ( x ) ) | | T f ( x ) | + | ( 1 ) r G ( R r k ( t ) Ω [ r 1 ] ( t ) ) f ( r ) ( x + t ) d t | G Ω f L ( G ) + R r k Ω [ r 1 ] L 1 ( G ) f ( r ) L ( G ) ,
which implies the desired inequality (18) in the case h = 1 :
D k f L ( G ) G Ω f L ( G ) + R r k Ω [ r 1 ] L 1 ( G ) f ( r ) L ( G ) .
(22)
Next, we assume that there exists a function Φ W , r ( G ) satisfying equalities (19) and (20). Due to Proposition 1 the derivative D k Φ is continuous on G. Hence, taking into account equalities (19) and (20) we have
D k Φ L ( G ) | D k Φ ( 0 ) | = | G Φ ( x ) d Ω ( x ) + ( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ ( r ) ( x ) d x | G Ω Φ L ( G ) + R r k Ω [ r 1 ] L 1 ( G ) Φ ( r ) L ( G ) .

Therefore, the statement of the theorem is proved in the case h = 1 .

Now, we let h > 0 and f L , r ( G ) be arbitrary, and consider the function f h ( x ) : = f ( x / h ) , x G . Evidently, f h L , r ( G ) and by substituting f h into (22) we derive inequality (18). Clearly, Φ h turns (18) into an equality. □

3.2 The Kolmogorov type inequalities for the Marchaud fractional derivatives: case of norms in an ideal lattice

Let us generalize Theorem 1 to the case of Kolmogorov type inequalities between the norms of the Marchaud fractional derivative of a function, the function itself, and its higher-order derivative in an ideal lattice.

Theorem 2 Let G = R or G = R + , E be a semi shift-invariant lattice on G, r N , and k ( 0 , r ) N . Let also Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) L 1 ( G ) and (17) hold true, for every f L E , E r ( G ) . Then, for every f L E , E r ( G ) ,
D k f E G Ω f E + R r k Ω [ r 1 ] L 1 ( G ) f ( r ) E .

An immediate consequence of Theorem 2 is the following.

Corollary 2 Let G = R or G = R + , 1 s , r N , and k ( 0 , r ) N . Let also a function Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) L 1 ( G ) and (17) holds true, for every f L s , s r ( G ) . Then, for every f L s , s r ( G ) and h > 0 ,
D k f L s ( G ) h k G Ω f L s ( G ) + h r k R r k Ω [ r 1 ] L 1 ( G ) f ( r ) L s ( G ) .
Moreover, if a function Φ W , r ( G ) satisfies (19) and (20) then, for f L s , s r ( G ) ,
D k f L s ( G ) D k Φ L ( G ) Φ L ( G ) 1 k / r f L s ( G ) 1 k / r f ( r ) L s ( G ) k / r .
(23)

Evidently, inequality (23) is sharp for s = . In Section 4.1 we shall show that this inequality is also sharp when s = 1 , r = 1 , and G = R . For integer values of k and G = R , inequality (23) is known as the Stein inequality [35] (see also [36, 37]).

Proof of Theorem 2 Using Proposition 3 and the generalized Minkowskii inequality (see [18]), for every function f L E , E r ( G ) , we have
D k f E G f ( x ) d Ω ( x ) E + G ( R r k ( x ) Ω [ r 1 ] ( x ) ) f ( r ) ( x ) d x E G Ω f E + R r k Ω [ r 1 ] L 1 ( G ) f ( r ) E .

The proof is finished. □

3.3 The Kolmogorov type inequalities for the Marchaud fractional derivatives: case when the norm of the higher-order derivative is considered in an ideal lattice

In this subsection we generalize the results of Section 3.1 on the case when the norm of the higher-order derivative is taken in an ideal lattice. For convenience, we split the subsection into two parts: first we present results concerning the case when extremal function in the Kolmogorov type inequality (i.e. turning it into an equality) exists and then we present results concerning the case when the extremal function in the Kolmogorov type inequality does not exist. For integral-order derivatives, the existence of an extremal function in the Kolmogorov type inequalities (1) was proved in the case when 1 p , q , 1 < s , and inequality (3) is strict. For the corresponding results, we refer the reader to [24, 38, 39].

3.3.1 Case of existence of extremal function in the Kolmogorov type inequality

For an ideal lattice E on G and r N , we set
W , E r ( G ) = { f L , E r ( G ) : f ( r ) E 1 } .
Theorem 3 Let G = R + or G = R , r N , k ( 0 , r ) N , E be an ideal semi shift-invariant lattice on G satisfying conditions (12) and (13), and E 1 be the associated space to E. Also, let a function Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) E 1 and (17) hold true, for every f L , E r ( G ) . Then, for every f L , E r ( G ) ,
D k f L ( G ) G Ω f L ( G ) + R r k Ω [ r 1 ] E 1 f ( r ) E .
(24)
Furthermore, if a function Φ W , E r ( G ) satisfies equalities (19) and
( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ ( r ) ( x ) d x = R r k Ω [ r 1 ] E 1 ,
(25)

then inequality (24) is sharp and Φ turns (24) into an equality.

We remark that Theorem 3 can be generalized as follows.

Theorem 4 Let G = R or G = R + , r N , k ( 0 , r ) N , E be a semi-shift invariant lattice on G that satisfy conditions (13), E 1 be the associated space to E, F be an ideal lattice such that its associated space F 1 contains the function χ ( 0 , 1 ) . Let also a locally absolutely continuous function Ω V ( G ) on G be such that ( R r k Ω [ r 1 ] ) E 1 and (17) holds true, for every f L F , E r ( G ) . Then, for every f L F , E r ( G ) ,
D k f L ( G ) Ω F 1 f F + R r k Ω [ r 1 ] E 1 f ( r ) E .

For the spaces L s ( G ) , 1 < s , we obtain the following consequence.

Corollary 3 Let G = R or G = R + , 1 < s , s = s / ( s 1 ) , r N , and k ( 0 , r 1 / s ) N . Let also a function Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) L s ( G ) and (17) holds true, for every f L , s r ( G ) . If a function Φ W , s r ( G ) satisfies equality (19) and the relation
( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ ( r ) ( x ) d x = R r k Ω [ r 1 ] L s ( G )
then, for every f L , s r ( G ) and h > 0 , the sharp inequalities
D k f L ( G ) h k G Ω f L ( G ) + h r k 1 / s R r k Ω [ r 1 ] L s ( G ) f ( r ) L s ( G )
(26)
and
D k f L ( G ) D k Φ L ( G ) Φ L ( G ) 1 λ f L ( G ) 1 λ f ( r ) L s ( G ) λ , λ = k r 1 / s ,
(27)

hold true. Moreover, the function Φ h ( ) : = h r 1 / s Φ ( ( ) / h ) turns (26) and (27) into equalities.

We remark that Theorems 3.1.2 and 3.2.2 [16] are concretizations of Corollary 3. In addition, in the case of integer values of k and r = 2 , 3 and 1 < s < , the functions Φ and Ω satisfying conditions of Corollary 26 were explicitly constructed by Arestov in [28].

Proof of Theorem 3 The proof is similar to the proof of Theorem 1 in the case h = 1 . The difference is that for a function f L , E r ( G ) and x G , we need to use the inequality
| G ( R r k ( t ) Ω [ r 1 ] ( t ) ) f ( r ) ( x + t ) d t | R r k Ω [ r 1 ] E 1 f ( r ) ( x + ) E R r k Ω [ r 1 ] E 1 f ( r ) E .

The extremity of the function Φ can be proved in a similar way to Theorem 1. □

Proof of Corollary 3 For every h > 0 , we observe that the functions Ω h ( x ) : = h k Ω ( x / h ) , x G , and Φ h satisfy conditions (17), (19), and (25). Moreover,
G Ω h = h k G Ω , R r k Ω h [ r 1 ] L s ( G ) = h r k 1 / s R r k Ω [ r 1 ] L s ( G ) .

Hence, by Theorem 3 the desired inequality (26) holds true and the function Φ h turns (26) into an equality. Finally, minimizing the right hand part of (26) by the variable h, we arrive at inequality (27). The proof is finished. □

3.3.2 Case of non-existence of extremal function in the Kolmogorov type inequality

Let us present two results showing when conditions (19) and (25) can be relaxed.

Theorem 5 Let G = R + or G = R , and numbers k, r, an ideal semi shift-invariant lattice E on G and a function Ω V ( G ) satisfy assumptions of Theorem  3. Also, assume that there is a family of the functions { Φ ε } ε > 0 W , E r ( G ) satisfying the equality (19) and, for ε > 0 , the inequality
( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ ε ( r ) ( x ) d x > R r k Ω [ r 1 ] E 1 ε .
(28)
Then inequality (24) holds true and is sharp in the sense that, for every sufficiently small ε > 0 , there exists a function f ε L , E r ( G ) such that
D k f ε L ( G ) > G Ω f ε L ( G ) + ( R r k Ω [ r 1 ] E 1 ε ) f ε ( r ) E .
Theorem 6 Let G = R + or G = R , r N , k ( 0 , r 1 ) N , E be an ideal semi shift-invariant lattice on G such that lim inf h 0 + ( h 1 χ ( 0 , h ) E ) = : μ ( 0 , + ) . Also, let a function Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) E 1 and (17) holds true, for every f L , E r ( G ) . Assume that there exists a function Φ L ( G ) such that its derivative Φ ( r 1 ) is piecewise constant on G, G Φ ( r 1 ) = μ 1 , there exists h 0 > 0 such that the distance between each pair of discontinuity points of Φ ( r 1 ) is bounded below by h 0 , and equalities (19) and
( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) d Φ ( r 1 ) ( x ) = R r k Ω [ r 1 ] E 1

are valid. Then inequality (24) holds true and is sharp.

In the case E = L 1 ( G ) we obtain the following.

Corollary 4 Let G = R or G = R + , r N and k ( 0 , r 1 ) N . Let also a function Ω V ( G ) be such that ( R r k Ω [ r 1 ] ) L ( G ) and (17) holds true, for every f L , 1 r ( G ) . If an ( r 1 ) -times differentiable function Φ with piecewise constant derivative Φ ( r 1 ) satisfies equalities (19), G Φ ( r 1 ) = 1 , there exists h 0 > 0 such that the distance between each pair of discontinuity points of Φ ( r 1 ) is bounded below by h 0 , and
( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) d Φ ( r 1 ) ( x ) = R r k Ω [ r 1 ] L ( G )
then, for every f L , 1 r ( G ) and h > 0 , the sharp inequalities
D k f L ( G ) h k G Ω f L ( G ) + h r k 1 R r k Ω [ r 1 ] L ( G ) f ( r ) L 1 ( G )
and
D k f L ( G ) D k Φ L ( G ) Φ L ( G ) 1 λ f L ( G ) 1 λ f ( r ) L 1 ( G ) λ , λ = k r 1 ,

hold true.

We remark that for integer values of k and r = 2 , 3 , the functions Φ and Ω satisfying conditions of Corollary 4 were constructed by Arestov in [28].

Proof of Theorem 5 We observe that inequality (24) holds true, for every f L , E r ( G ) . Let us prove that (24) is sharp. Let ε > 0 be arbitrary and sufficiently small. Due to Proposition 1 the fractional derivative D k Φ ε is continuous on G. Hence, taking into account equalities (19) and (28) we obtain
D k Φ ε L ( G ) | D k Φ ε ( 0 ) | = | G Φ ε ( x ) d Ω ( x ) + ( 1 ) r G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ ε ( r ) ( x ) d x | G Ω Φ ε L ( G ) + R r k Ω [ r 1 ] E 1 ε G Ω Φ ε L ( G ) + ( R r k Ω [ r 1 ] E 1 ε ) Φ ε ( r ) E .

The proof is finished. □

Proof of Theorem 6 Since k < r 1 we see that ( ) r k 1 χ ( 0 , 1 ) ( ) E 1 and condition (13) is also fulfilled. Hence, by Theorem 3, inequality (24) holds true. Let us prove that inequality (24) is sharp. To this end by B = { β j } j J (J is a finite or countable set of indices) we denote the discontinuity points of Φ ( r 1 ) and set α j : = Φ ( β j + ) Φ ( β j ) , j J . Due to the assumption there exists h 0 > 0 such that | β j β i | h 0 for all distinct indices i , j J . Now, for every h ( 0 , h 0 ) , we define the function
Φ h ( x ) : = 1 h 0 h Φ ( x + t ) d t , x G .
It is easy to show that as h 0 + , we have
G Φ h ( x ) d Ω ( x ) G Φ ( x ) d Ω ( x ) = G Ω Φ L ( G ) , G ( R r k ( x ) Ω [ r 1 ] ( x ) ) Φ h ( r ) ( x ) d x G ( R r k ( x ) Ω [ r 1 ] ( x ) ) d Φ ( r 1 ) ( x ) = ( 1 ) r R r k Ω [ r 1 ] E 1 ,
and
lim inf h 0 + Φ h ( r ) E lim inf h 0 + j J | α j | h χ ( 0 , h ) E = 1 μ lim inf h 0 + χ ( 0 , h ) E h = 1 .
Due to the continuity of D k Φ h , for every ε and every sufficiently small h > 0 ,
D k Φ h L ( G ) D k Φ h ( 0 ) > G Ω Φ h L ( G ) + R r k Ω [ r 1 ] E 1 ε G Ω Φ h L ( G ) + ( R r k Ω [ r 1 ] E 1 ε ) Φ h ( r ) E .

The proof is finished. □

4 Consequences of main results

In this section we deduce new sharp Kolmogorov type inequalities from the results of the previous section when the order of the higher-order derivative is 1 or 2.

4.1 Case r = 1 and k ( 0 , 1 )

Let G = R or G = R + . For k ( 0 , 1 ) and h > 0 , we set
τ h ( x ) : = { 0 , x G ( 0 , h ) , x k h k , x ( 0 , h ) .

The following proposition is the consequence of Theorem 5.

Corollary 5 Let G = R or G = R + , k ( 0 , 1 ) , E be an ideal semi shift-invariant lattice on G satisfying conditions (12) and (13), E 1 be the associated space to E. Then, for every f L , E 1 ( G ) and h > 0 , the sharp inequality
D k f L ( G ) 2 h k Γ ( 1 k ) f L ( G ) + τ h E 1 Γ ( 1 k ) f E
(29)

holds true.

Proof For every h > 0 , we define the function
Ω ( x ) : = R 1 k ( x ) τ h ( x ) Γ ( 1 k ) = { 0 , x G ( 0 , + ) , h k Γ ( 1 k ) , x ( 0 , h ) , x k Γ ( 1 k ) , x h .
(30)

It is easy to check that (17) holds true, for every f L , E 1 ( G ) , G Ω = 2 h k Γ ( 1 k ) , and R 1 k Ω = τ h Γ ( 1 k ) E 1 .

Let us construct a family of functions { Φ ε } ε > 0 W , E 1 ( G ) satisfying conditions (19) and (28). For every ε > 0 , there exists a function g ε E , g ε E 1 , such that
0 h ( R 1 k ( x ) Ω ( x ) ) g ε ( x ) d x > R 1 k Ω E 1 ε = τ h E 1 Γ ( 1 k ) ε .
Without loss of generality we may assume that g ε is non-negative on G and supp g ε = [ 0 , h ] . Next, we define the function Φ ε as the first integral of ( g ε ) :
Φ ε ( x ) : = 0 x g ε ( t ) d t + 1 2 0 h g ε ( t ) d t , x G .
Clearly, Φ ε W , E 1 ( G ) and Φ ε ( x ) = Φ ε ( 0 ) = Φ ε L ( G ) , x h . As a result,
G Φ ε ( x ) d Ω ( x ) = h k Φ ε ( 0 ) Γ ( 1 k ) + h Φ ε ( x ) d ( x k ) Γ ( 1 k ) = G Ω Φ ε L ( G )
and
G ( R 1 k ( x ) Ω ( x ) ) Φ ε ( x ) d x = 0 h ( R 1 k ( x ) Ω ( x ) ) g ε ( x ) d x > R 1 k Ω E 1 ε .

Therefore, the function Ω and the family of the functions { Φ ε } ε > 0 satisfy the assumptions of Theorem 5. Hence, inequality (29) holds true and is sharp. □

Next, we formulate the following Stein type inequality.

Corollary 6 For k ( 0 , 1 ) , h > 0 , and f L , 1 1 ( R ) , the sharp inequalities
D k f L 1 ( R ) 2 h k f L 1 ( R ) Γ ( 1 k ) + k h 1 k f L 1 ( R ) Γ ( 2 k ) , D k f L 1 ( R ) 2 1 k Γ ( 2 k ) f L 1 ( R ) 1 k f L 1 ( R ) k
(31)

hold true.

Proof For h > 0 , let Ω be defined by (30). Then both desired inequalities follow from Corollary 2. Let us prove that inequalities (31) are sharp. To this end, for every ε ( 0 , h ) , we consider the Steklov averaging operator S ε : L ( G ) L ( G )
S ε f ( ) = 1 ε 0 ε f ( + t ) d t , f L ( G ) ,
and define the function Φ ε : = S ε χ ( 0 , h ) . Clearly, Φ ε L 1 ( R ) = h , Φ ε L 1 ( R ) = 2 , and
lim ε 0 + D k Φ ε L 1 ( R ) = D k χ ( 0 , h ) L 1 ( R ) = 2 h 1 k Γ ( 2 k ) .

Plugging the latter relations into the first of inequality (31) we turn it into an equality. The proof is finished. □

4.2 Case G = R + , r = 2 , and k ( 0 , 1 )

For k ( 0 , 1 ) and h > 0 , we define
τ h ( x ) : = { x 1 k h k x , x [ 0 , h ) , 0 , x h .

The following consequence of Theorem 5 holds true.

Corollary 7 Let k ( 0 , 1 ) , E be an ideal semi shift-invariant lattice on R + satisfying conditions (12) and (13), E 1 be the associated space to E. Then, for every f L , E 2 ( R + ) and h > 0 , the sharp inequality
D k f L ( R + ) 2 h k Γ ( 2 k ) f L ( R + ) + τ h E 1 Γ ( 2 k ) f E
(32)

holds true.

Proof For every h > 0 , we define the function
Ω ( x ) : = { 0 , x = 0 , h k Γ ( 2 k ) , x ( 0 , h ) , x k Γ ( 1 k ) , x h .
Evidently, G Ω = 2 h k Γ ( 2 k ) and R 2 k Ω [ 1 ] = τ h Γ ( 2 k ) E 1 . Moreover, for every f L , E 2 ( R + ) , we have (17). Indeed,
D k f ( 0 ) 0 + f ( x ) d Ω ( x ) = D k f ( 0 ) h k f ( 0 ) Γ ( 2 k ) + k h k f ( h ) Γ ( 2 k ) + k Γ ( 1 k ) h f ( x ) d x x 1 + k = k Γ ( 1 k ) 0 h f ( 0 ) f ( x ) x 1 + k d x + k h k f ( h ) Γ ( 2 k ) = k Γ ( 1 k ) 0 h 0 x ( x t ) f ( t ) x 1 + k d t d x + k h k Γ ( 2 k ) 0 h ( h t ) f ( t ) d t = 0 + ( R 2 k ( t ) Ω [ 1 ] ( t ) ) f ( t ) d t .
Let us construct a family of functions { Φ ε } ε > 0 W , E 2 ( G ) satisfying conditions (19) and (28). For every ε > 0 , there exists a function g ε E , g ε E 1 , such that
0 h ( R 2 k ( x ) Ω [ 1 ] ( x ) ) g ε ( x ) d x > R 2 k Ω [ 1 ] E 1 ε = τ h E 1 Γ ( 2 k ) ε .
Without loss of generality we may assume that g ε is non-negative on G and supp g ε = [ 0 , h ] . Next, we define the function Φ ε as the second integral of g ε :
Φ ε ( x ) : = 0 h ( x + t / 2 ) g ε ( t ) d t + 0 x ( x t ) g ε ( t ) d t , x R + .
Clearly, Φ ε W , E 2 ( G ) , Φ ε ( x ) = Φ ε ( 0 ) = Φ ε L ( R + ) , x h . As a result,
0 + Φ ε ( x ) d Ω ( x ) = h k [ Φ ε ( 0 ) k Φ ε ( h ) ] Γ ( 2 k ) k ( 1 k ) Γ ( 2 k ) h + Φ ε ( x ) x 1 + k d x = 0 + Ω Φ ε L ( R + )
and
0 + ( R 2 k ( x ) Ω [ 1 ] ( x ) ) Φ ε ( x ) d x = 0 h ( R 2 k ( x ) Ω [ 1 ] ( x ) ) g ε ( x ) d x > R 2 k Ω [ 1 ] E 1 ε .

Therefore, the function Ω and the family of the functions { Φ ε } ε > 0 satisfy assumptions of Theorem 5. Hence, inequality (32) holds true and is sharp. □

Let us formulate the consequence of Corollaries 3, 4, and 7. For s > 1 , we set
φ k , s ( x ) : = 0 h ( x + t / 2 ) τ 1 s 1 ( t ) d t + 0 x ( x t ) τ 1 s 1 ( t ) d t , x R + ,
and Φ k , s : = φ k , s L s ( R + ) 1 φ k , s . Also, we define
Φ k , 1 ( x ) = 1 2 max { ( 1 k ) 1 / k 2 x ; ( 1 k ) 1 / k } , x R + .
Corollary 8 Let k ( 0 , 1 ) , 1 s , and s = s / ( s 1 ) . Then, for every f L , s 2 ( R + ) , the sharp inequality
D k f L ( R + ) D k Φ k , s L ( R + ) Φ k , s L ( R + ) 1 λ f L ( R + ) 1 λ f L s ( R + ) λ , λ = k 2 1 / s ,

holds true.

4.3 Case G = R + , 1 < s , and k ( 1 , 2 1 / s )

Let 1 < s , s = s / ( s 1 ) , and k ( 1 , 2 1 / s ) . Consider the set
M : = { ( a , b ) ( 0 , 1 ) 2 : a b }
and, for every ( a , b ) M , we define the function
ω ( a , b ; x ) = { 0 , x = 0 , a 1 ( 1 b ) 1 ( 1 b ( 1 b 1 k ) ( 1 a ) ) , x ( 0 , a ] , ( 1 b ) 1 ( 1 b 1 k ) , x ( a , 1 ) , ( 1 k ) x k , x 1 .
For x R + , we consider the functions τ ( a , b ; x ) : = Γ ( 2 k ) R 2 k ( x ) ω [ 1 ] ( a , b ; x ) and
φ ( a , b ; x ) : = 0 a ( x + t / 2 ) τ ( s ) ( a , b ; t ) d t + 0 x ( x t ) τ ( s ) ( a , b ; t ) d t ,
where g ( s ) : = | g | s 1 sign g . Below in Lemma 1 we shall show that system (33) has a unique solution ( a k , s , b k , s ) on M. For convenience, we denote the functions ω ( a k , s , b k , s ; ) , τ ( a k , s , b k , s ; ) , φ ( a k , s , b k , s ; ) by ω k , s , τ k , s , φ k , s , respectively. The graphs of the functions ω [ 1 ] ( a , b ; ) , τ ( a , b ; ) , and φ k , s are shown in Figure 1.
Figure 1

Functions from left to right: Γ ( 2 k ) R 2 k and ω [ 1 ] ( a , b ; ) ; τ ( a , b ; ) ; φ k , s .

The next proposition is a consequence of Corollary 3.

Corollary 9 Let 1 < s , s = s / ( s 1 ) , k ( 1 , 2 1 / s ) , and Φ k , s : = φ k , s L s ( R + ) 1 φ k , s . Then, for every f L , s 2 ( R + ) , the sharp inequality
D k f L ( R + ) D k Φ k , s L ( R + ) Φ k , s L ( R + ) 1 λ f L ( R + ) 1 λ f L s ( R + ) λ , λ = k 2 1 / s ,

holds true.

We start with the proof of an auxiliary lemma.

Lemma 1 The system of the following equations has a unique solution on M:
{ F 1 ( a , b ) : = a 1 τ ( s ) ( a , b ; t ) d t = 0 , F 2 ( a , b ) : = 0 1 t τ ( s ) ( a , b ; t ) d t = 0 .
(33)

We remark that in some cases the pair ( a k , s , b k , s ) can be found explicitly, e.g. a k , = 2 1 and b k , = 1 / 2 .

Proof First, we observe that, for every ( a , b ) M , the function τ ( a , b ; ) is positive on ( 0 , b ) , is negative on ( b , 1 ) , and supp τ ( a , b ; ) = [ 0 , 1 ] . Next, the functions F 1 and F 2 are continuous on M, and can be continuously extended on a wider set M ˜ : = { ( a , b ) [ 0 , 1 ) 2 : b > 0 , a b } . Let us prove that system (33) has a unique solution on M ˜ . First, we note that F 2 strictly increases in the variables a and b, while F 1 strictly increases in b and strictly decreases in a. Hence, system (33) could have only one solution. Next, we have
lim b 0 + F 2 ( b , b ) lim b 0 + { 0 b t ( 1 k ) ( s 1 ) + 1 d t b 1 t τ ( s ) ( b , b ; t ) d t } = , lim b 1 F 2 ( b , b ) = 0 1 t ( t 1 k t ) s 1 d t > 0 , lim b 1 F 2 ( 0 , b ) = 0 1 t ( t 1 k 1 + ( 1 k ) ( 1 t ) ) s 1 d t > 0 .
Hence, there exist points a , b ( 0 , 1 ) such that F 2 ( 0 , b ) = F 2 ( a , a ) = 0 . Taking into account continuity of the function F 2 and its monotonicity in both variables we conclude that, for every a [ 0 , a ] , there exists b = ϱ ( a ) [ a , b ] such that F 2 ( a , b ) = 0 . Moreover, the function ϱ is continuous and is decreasing on the interval [ 0 , a ] because it has an inverse function. Finally, we observe that
F 1 ( 0 , b ) = 1 b { b 0 1 τ ( s ) ( 0 , b ; t ) d t } > 1 b { 0 b t τ ( s ) ( 0 , b ; t ) d t } = F 2 ( 0 , b ) b = 0
and
F 1 ( a , a ) = a 1 τ ( s ) ( a , a ; t ) d t < 0 .

Hence, there exists a 0 ( 0 , a ) such that F 1 ( a 0 , ϱ ( a 0 ) ) = 0 . The latter implies that ( a 0 , ϱ ( a 0 ) ) M and satisfies system (33). □

Proof of Corollary 9 We set Ω : = ω k , s Γ ( 2 k ) . It is easy to check that for f L , s 2 ( R + ) ,
D k f ( 0 ) 0 + f ( t ) d Ω ( t ) = 0 + ( R 2 k ( t ) Ω [ 1 ] ( t ) ) f ( t ) d t .

Moreover, Φ k , s W , s 2 ( R + ) , Φ k , s ( 0 ) = Φ k , s ( a k , s ) = Φ k , s L ( R + ) , Φ k , s ( x ) = Φ k , s ( 0 ) , for every x h , Φ k , s decreases on [ 0 , a k , s ] and increases on [ a k , s , + ) . Hence, the functions Ω and Φ k , s satisfy the conditions of Corollary 3. □

4.4 Case G = R , 1 s , and k ( 0 , 1 )

Let k ( 0 , 1 ) , 1 s , and s = s / ( s 1 ) . For p [ 0 , k / ( 1 k ) ] , we consider the function
ω ( p ; x ) = { 0 , x p , ( 1 + p ) 1 , x ( p , 1 ] , ( 1 k ) x k , x 1 .
For x R , we consider the function τ ( p ; x ) : = Γ ( 2 k ) R 2 k ( x ) ω [ 1 ] ( p ; x ) and define
φ s ( p ; x ) : = 1 2 p 1 t τ ( s ) ( p ; t ) d t + p x ( x t ) τ ( s ) ( p ; t ) d t , when  s > 1 , φ 1 ( p ; x ) : = { ( 1 + p ) / 4 , x p , ( 1 + p 2 x ) / 4 , x ( p ,