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

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

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

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

and

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 ℝ 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:

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:

where n ∈ N, n > k (the definition itself is independent of n), and

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:

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:

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:

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

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

For N > 0, we set

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

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,

Furthermore, if there exists a pair of element x 0 ∈ Q and operator S 0 ∈ L such that

then Ω ( ∥ x 0 ∥ X , A ; Q ) = ∥ A x 0 ∥ Y and

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

for every δ ⩾ 0, we have

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,

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 ).

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 ℛ we denote either the set ℒ 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

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 ℛ on elements of the set Q with given error δ consists in finding the quantity

called the best recovery of the operator A with the help of mappings from ℛ 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,

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,

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

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 ℒ) on elements of the class W p , s r ( G ) given with error δ.

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.

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

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

and

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 ),

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.

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

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,

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

Using (15), changing variables, altering the order of integration and applying the Hölder inequality we obtain

It is easy to show that N r ( x ) ⩽ x r − 1, for every x ∈ [ 0 , r ]. Hence, for every v ∈ ( 0 , r h ),

From the latter and estimate (16) we conclude that

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

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

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:

Proof of Proposition 2 First we note that

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

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

is convergent. Hence, it is sufficient to prove the convergence of the second integral. The latter is obvious because

and, for m = 1 , 2 , … , r,

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. □

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 ),

Then, for every f ∈ L ∞ , ∞ r ( G ) and h > 0, we have the inequality

Furthermore, if a function Φ ∈ W ∞ , ∞ r ( G ) satisfies the equalities

and

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

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:

We remark that the following results on sharp inequalities of the form (7) concretize Corollary 1.

where p = 1 − 2 k / ( 1 − k ).

For integer values of k, the extremal function Φ on ℝ 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.