A unified approach to inequalities for K-functionals and moduli of smoothness

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.


Introduction
Some, nowadays well-known, inequalities between moduli of continuity, or more general, between moduli of smoothness are attached to the names of Marchaud, Ul'yanov, and Kolyada.These inequalities play an important role in approximation theory as well as in the theory of function spaces, in particular, they can be used to derive embedding properties of function spaces with fixed degree of smoothness, see, e.g., [6,Section 5.4], [8], [15].
The purpose of this paper is to consider crucial inequalities (Marchaud, Ul'yanov, etc.) from an abstract point of view.To this end, in Section 4 we assume suitable embeddings between interpolation and potential spaces (the interpolation spaces may be interpreted as abstract Besov spaces).Simultaneously, abstract versions of the Holmstedt formulas are developed, which allow also to cover limiting cases.In Section 5 applications are given in the case of general weighted Lorentz spaces.Finally, Section 6 deals with applications to Lorentz-Karamata spaces.
To illustrate our results, we start in Subsection 1.1 with the formulation of the aforesaid basic inequalities adapted to Lebesgue spaces L p (R n ), 1 < p < ∞.Their improvements and extensions in the framework of Lorentz spaces L p,r (R n ) (note that L p,p = L p ) are described in Subsection 1.2, proofs are given in Section 2.
1.1.Some basic results.A detailed study of inequalities between different moduli of smoothness on L p (R n ), 1 p ∞, can be naturally divided into two parts: inequalities for moduli of smoothness of different orders in L p and inequalities in different metrics (L p , L p * ).In the paper a modulus of smoothness of order κ > 0 on an r.i.function space X (defined in Section 3, e.g., X = L p ) is given by (1.1) ω κ (f, t) X = sup Let us begin with the key inequalities on L p (R n ).Trivially, if k, m, n ∈ N and 1 p ∞, then (1.2) ω k+m (f, t) Lp ω k (f, t) Lp for all t > 0 and f ∈ L p (R n ).
Observe again the natural formal passage from (1.5) to (1.2), this time when p → ∞.We call (1.5) a reverse Marchaud inequality (in [18] it is called a sharp Jackson inequality).Consider now inequalities for moduli of smoothness in different Lebesgue metrics.In 1968 P.L. Ul'yanov [66] proved such an inequality for periodic functions in L p (T).Its R ncounterpart reads as follows (see, e.g., [9]): If k, n ∈ N, 1 p < ∞, 0 < δ < min{n/p, k}, and 1/p * = 1/p − δ/n, then (1.6) holds for all f ∈ L p (R n ) (for which the right-hand side of(1.6) is finite). 1   1 One can show that if f ∈ L p (R n ) and the right-hand side of (1.6) is finite for some t > 0, then f ∈ L p * (R n ) and so the modulus of smoothness appearing on the left-hand side of (1.6) is well defined.Note that we always look at inequalities involving moduli of smoothness in different metrics at this way.One can also show that if f ∈ L p (R n ) and the right-hand side of (1.6) is finite for some t > 0, then it is finite for all t > 0 -cf.Remark 6.8 mentioned below).
1.2.Inequalities for moduli of smoothness on Lorentz spaces.We say that a measurable function f belongs to the Lorentz space L p,r = L p,r (R n ), 1 p, r ∞, if (see, e.g., [6,Section 4.4]) where f * denotes the non-increasing rearrangement of f.Thus L p = L p,p and f p = f p,p .

Denote by
f and all its (weak) derivatives up to the order k belong to L p (R n ).It is well known that, by Taylor's formula, Here we use the multi-index notation |µ| := n j=1 µ j , D µ = n j=1 (∂/∂x j ) µ j .We want to state an improvement and some type of reverse of this inequality in the case 1 < p < ∞.To this end, we need Besov spaces and Riesz potential spaces, both modelled upon Lorentz spaces.
We make use of the Fourier analytical approach in S ′ (cf.[7]): Take a C ∞ -function ϕ such that For j ∈ Z and x ∈ R n , let (1.12) The sequence {ϕ j } j∈Z is a smooth dyadic resolution of unity, i.e., 1 = ∞ j=−∞ ϕ j (x) for all x ∈ R n , x = 0. Let 1 p, q, r ∞ and σ > 0. The Besov space (the sum should be replaced by the supremum if q = ∞).Here the symbol F −1 is used for the inverse Fourier transform.An equivalent characterization of this semi-norm in terms of moduli of smoothness is given by under the assumptions on the parameters q 0 and r of Proposition 1.1 (A), for all t > 0, (1.16) then, under the assumptions on the parameters q 1 and r of Proposition 1.1 (B), for all t > 0, (1.18) .

Remarks and proofs in outlines
Peetre's K-functional K 0 for the compatible couple (L p,r , H σ p,r ) plays a decisive role in the proofs of Propositions 1.1-1.3.It is defined by We also need the characterization, for 1 < p < ∞, σ > 0, 1 r ∞, [67] and its extension in [30, (1.13)]) and the identification of the interpolation space given by where (•, •) θ,q denotes Peetre's real interpolation method.The improvements and extensions of inequalities (1.3)-(1.8)can be easily proved via the Holmstedt formulas [6, Section 5.2].One only needs to exchange in [63] the embeddings between Besov and potential spaces modelled on Lebesgue spaces by the corresponding ones modelled on Lorentz spaces.Therefore, we only sketch the proofs of the propositions stated in Subsection 1.2.Concerning (1.9) and (1.10), note that, under the restrictions on q 0 and q 1 given in Proposition 1.1, the following embeddings are true: [57]) and [57]).
Remark 2.1.In parts (i) and (ii) of this remark we assume the same restrictions on the parameters under which (1.9) and (1.10) hold, respectively.
For the proof of Proposition 1.3, suppose that β, δ > 0 and that p, p * and δ satisfy the assumptions.By Theorem 1.1 (iii) in [57], Moreover, Theorem 1.6 (i, iii) in [57] contains a version of the Hardy-Littlewood-Sobolev theorem on fractional integration, which states that (2.6) The use of Holmstedt's formula completes the proof of (1.19).
Concerning the proof of (1.20), we need the embedding (2.7) H σ+δ p,r 0 ֒→ B σ (p * ,r 1 ),q 0 , if 1 r 0 q 0 ∞, 1 r 1 ∞, which holds by [57, Theorem 1.2 (iii)], and also embedding (2.5), which requires the additional restriction q 1 r 1 . Remark This follows on applying to the left-hand side of (1.20) Marchaud inequality (1.9), where we replace p by p * .(b) Similarly, if 1 r := r 0 = r 1 = q 1 ∞, r max{p * , 2}, and γ > 0, then the combination of Ul'yanov inequality (1.19) and reverse Marchaud inequality (1.10) (where p is replaced by p * ) yields a special case of the Kolyada inequality, namely, for all Note that in the case 0 < γ < δ the order of the modulus of smoothness on the left-hand side is smaller than the one on the right-hand side.
2.1.Sharp Ul'yanov and Kolyada inequalities for p = 1.As it was mentioned above both (1.19) and (1.20) do not hold in general when p = 1.However, under some additional conditions on parameters both results are still valid even in the Lorentz space setting.
, then in light of (2.1), for all f ∈ L p * ,r 1 and all positive t, .
Now we take into account the following result by Alvino [3] (appeared in 1977, rediscovered by Poornima [54] in 1983 and by Tartar [59] in 1998) Together with Hörmander's multiplier criterion and [57, Theorem 1.6 (iii)], this yields and for the corresponding seminorms we have, for all Note that using Alvino's result, we necessarily assume δ 1.
By [57, Theorem 1.1 (iii)], the first embedding below is valid, the second one is elementary and, therefore, (2.15), and (2.11), we arrive at Together with Holmstedt's formula, this yields , where the condition β + δ ∈ N allows us to identify the resulting K 0 -functional with the classical modulus of smoothness in L 1 .
Proof of Proposition 1.3 ′ (B).Following the proof of (1.20), we need analogues of (2.5) and (2.7) for p = 1.In fact, in this case (2.5) holds whenever 1 q 1 r 1 ∞ (see (2.15)).Concerning (2.7), we modify it by repeating the argument in (2.13) to get Hence, applying (2.7) upon H β+δ−1 n/(n−1),1 , under our assumptions, we arrive at By the Holmstedt formula, Since this estimate holds for all g ∈ W β+δ 1 , we have Now simple substitutions, the characterizations of the K 0 -functionals via moduli of smoothness of integer order give the assertion.

Notation and preliminaries
Throughout the paper, we write A B (or for some positive constant c, which depends only on nonessential variables involved in the expressions A and B, and In the whole paper the symbol (R, µ) denotes a totally σ-finite measurable space with a non-atomic measure µ, and M(R, µ) is the set of all extended complex-valued µ-measurable functions on R. By M + (R, µ) we mean the family of all non-negative functions from M(R, µ).When R is an interval (a, b) ⊆ R and µ is the Lebesgue measure on (a, b), we denote these sets by M(a, b) and M + (a, b), respectively.Moreover, by M + (a, b; ↓) (and M + (a, b; ↑)) we mean the subset of M + (a, b) consisting of all non-increasing (non-decreasing) functions on (a, b).We denote by λ n the n-dimensional Lebesgue measure on R n .
For two normed spaces X and Y, we will use the notation Y ֒→ X if Y ⊂ X and f X f Y for all f ∈ Y.A normed linear space X of functions from M(R, µ), equipped with the norm • X , is said to be a Banach function space if the following four axioms hold: (1) 0 g f µ-a.e.implies g X f X ; (2) we obtain a rearrangement-invariant Banach function space (shortly r.i.space).Note that, by [6, Chapter 2, Theorem 6.6] and [6, Chapter 2, Theorem 2 Given a Banach function space X on (R, µ), the set equipped with the norm is called the associate space of X.It turns out that X ′ is again a Banach function space and that X ′′ = X.Furthermore, the Hölder inequality holds for every f and g in M(R, µ).It will be useful to note that For every r.i.space X on (R, µ), there exists an r.i.space X over ((0, ∞), dt) such that f X = f * X for every f ∈ X (cf.[6, Chapter 2, Theorem 4.10]).This space, equipped with the norm A Banach space F of real valued measurable functions defined on the measurable space (R, µ) is called a Banach function lattice if its norm has the following property: In this paper we will consider a Banach lattice F over a measurable space ((0, ∞), dt/t), satisfying the condition where Φ(x) := min(x, •) F for all x ∈ (0, ∞).(The function Φ is sometimes called the fundamental function of the lattice F .)Note that Φ is a quasiconcave function on (0, ∞), which means that Φ ∈ M + ((0, ∞); ↑) and Φ Id ∈ M + ((0, ∞); ↓) (here Id stands for the identity map on (0, ∞)).
Let (X, Y ) be a compatible couple of Banach spaces (cf., [6, p. 310]).The Kfunctional is defined for each f ∈ X + Y and t > 0 by where the infimum extends over all representation Similarly, we define, for each f ∈ X + Y and t > 0, and If (X, Y ) is a compatible couple of Banach spaces and F is a Banach lattice, then we define the space (X, Y ) F to be the set of all f ∈ X + Y for which the norm ) and the Banach lattice F is the set of all functions h ∈ M(0, ∞) such that then the space (X, Y ) F coincides with the classical space (X, Y ) θ,r defined, e.g., in [6, p. 299].
We will also work with more general classes of functions, which are not linear.Let ρ be a functional on ) and α 0. Such a functional is called a gage and the collection [17]).Moreover, we put An associate space of a gaged cone X is defined in the same way as for Banach function spaces.
If X is a gaged cone, then the functional | • | X : X → R is called a semi-gage on X provided that the functional | • | X is non-negative and positively homogeneous on X.
Given two function gaged cones X and Y , the embedding Y ֒→ X means that Y ⊂ X and f X f Y for all f ∈ Y.A pair of function gaged cones (X, Y ) is said a compatible couple of function gaged cones if there is some Hausdorff topological vector space, say Z, in which each of X and Y is continuously embedded.Given a compatible couple (X, Y ) of function gaged cones, the K-functionals K(f, t; X, Y ), K 0 (f, t; X, Y ), and K 1 (f, t; X, Y ) are defined analogously to (3.3)-(3.5).Moreover, if F is a Banach lattice over a measure space ((0, ∞), dt/t) satisfying (3.2), then the space (X, Y ) F is defined analogously to the case when (X, Y ) is a compatible couple of Banach spaces.
In this paper we work with function gaged cones being the subsets of Given k ∈ N and a Banach function space X = X(R n ), we denote by W k X the corresponding Sobolev space, that is, the space of all functions on R n whose distributional derivatives D α f , |α| k, belong to X.This space is equipped with the norm where A is the Sobolev integral operator; see, for example, the representation theorem in [12,Section 3.4].If X is a function gaged cone, then the Sobolev class W k X is defined similarly.
We are going to use the classical equivalence between the K-functional K 0 and modulus of smoothness: for any k ∈ N and an r.i.Banach function space X, one has The following properties of the generalized reverse function can be easily verified.
We note that Lemma 3.1 does not hold without the assumption that ξ is left continuous.Moreover, an analogue of ξ (Rξ)(t) = t, namely (Rξ)(ξ(t)) = t for any t ∈

General inequalities for K-functionals
4.1.Holmstedt-type formulas.The next theorem is a folklore in some way and it can be considered as an abstract form of the limiting cases of the Holmstedt-type formulas (see, e.g., [6, Corollary 2.3, p. 310 and p. 430] and [11, p. 466]).Since we have not been able to find an explicit reference of the needed general form (cf. [2,48]), we prove it below.The importance of this result can be seen in, e.g., [55].
Here A τ X, τ ∈ M, is the range of A τ equipped with the gage (or norm) A σ+τ X ֒→ Y, A σ X ֒→ Z, for some τ > 0 and σ 0.
In part (B) embedding (4.11) is reverse to (4.9), therefore the above inequality sign is also reverse.
Combining parts (A) and (B) of Theorem 4.1, we obtain the following result.
To prove (4.16), first, we apply Sobolev's embedding Ẇ k L p ֒→ L c p with 1 p < n/k and 1/p = 1/p − k/n (here, as usual, Ẇ k L p is the homogeneous Sobolev space and L c p = L p/{constants} is the factor space with the norm f L c p = inf c∈R 1 f − c p). See the book [45, 1.77, 1.78] for the case k = 1.For k > 1, it follows from the Poincaré inequality, namely, where c = lim t→∞ f * (t) and f # k is the maximal function given by f For p = 1 we obtain by same way By truncated method ([1, Theorem 7.2.1]),we can obtain By interpolation (see [53]), (4.17On the other hand, since where the equality follows from [51] and [50]. Embeddings (4.17), (4.18), Theorem 4.6 (with σ = 0 and K 0 instead of K), and the known relation Note that in the previous example, we did not use optimal Sobolev embeddings and thus did not obtain the sharp Ul'yanov inequality (1.19).The optimal embeddings require to use Lorentz spaces.
We will also need weighted Lorentz spaces defined as follows (cf., e.g., [16]) We will use the following conditions on weights: In general, Λ r (w)(R n , λ n ) and S r (w)(R n , λ n ) are not r.i.spaces, they are not even linear.On the other hand, Γ r (w)(R n , λ n ) is always an r.i.space for 1 r < ∞ and in this case the representation space of Γ r (w If Λ r (w)(R n , λ n ) is an r.i.space (e.g., if 1 < r < ∞ and w ∈ B r , see Lemma 5.1 below), then the representation space of Λ r (w)(R n , λ n ) is the space Λ r (w)((0, ∞), dt).
Similarly, if S r (w)(R n , λ n ) is an r.i.space (e.g., if 1 < r < ∞ and w ∈ RB r , i.e. w(1/t)t r−2 ∈ B r ; see [16,Theorem 3.3]), then the representation space of S r (w The dilation operator E t , t ∈ (0, ∞), is defined on M + (0, ∞) by Given an r.i.space X and t ∈ (0, ∞), the operator E t is bounded from X to X (cf.[6, p. 148]).If h X denotes the dilation function, i.e., h X (t) := E 1/t X→X for all t ∈ (0, ∞), then the lower and upper Boyd index of the space X is given by respectively.The Boyd indices satisfy (cf.[6, p. 149]) The Hardy averaging operator P and its dual Q are defined on M + (0, ∞), for each t ∈ (0, ∞), by respectively.Recall that (cf.[6, p. 150]) given an r.i.space X, the operator P is bounded on X if and only if α X < 1, while the operator Q is bounded on X if and only if 0 < α X .We will need the following result, which is partially known but the present formulation seems to be new.Lemma 5.1.Let w be a weight, 1 < r < ∞, and X := Λ r (w)(R n , λ n ).

The following conditions are equivalent:
(a) w ∈ B r , (b) X is an r.i.space, (c) the operator P is bounded on X, and η ∈ (0, 1), then the following conditions are equivalent: (a) w ∈ B * q with q = ηr, (b) the operator 3. If w ∈ B r , then the following conditions are equivalent: ) to the fact that the operator Q η is bounded in It remains to show that the operator Q η is bounded on L ↓ r (w) if and only if it is bounded in Λ r (w).Part "if" is clear.To prove the part "only if", we first note that, by Fubini's theorem and the Hardy-Littlewood rearrangement inequality (see [6, p. 44]), (5.1) Therefore, the fact that Q η f ∈ M + (0, ∞; ↓), the B r condition, the first part of this lemma, inequality (5.1), and the boundedness of .
The proof of part 3 is similar, one makes use of the fact that the condition w ∈ B * ∞ is equivalent to the boundedness of the operator Q on the space L ↓ r (w) (cf.[49, Theorem 3.3]).
In the rest of this section we work with spaces over (R n , λ n ) and sometimes we omit the symbol (R n , λ n ) from the notation of spaces in question.
for all t > 0, and since we obtain that Together with the condition w ∈ B r and the first part of Lemma 5.1 (recall that in our case X = Λ r (w)((0, ∞), dt)), this implies that In what follows, given γ 0 and n ∈ N, we define the weight v γ,n by The next lemma represents a key step in the proof of Proposition 5.4 below.It was proved in [32,Theorem 1.1] for k = 1, the proof for k ∈ N is analogous.
Let f ∈ M + (R n , λ n ) and g(s) := f * * (s) − f * (s) for all s > 0. Making use of (5.2) and the estimate t Together with the fact that the function t → t kr n is non-decreasing on (0, ∞), this implies that RHS(5.5) Given t > 0, we define the non-increasing function h t by ds for all t > 0 and, on applying the Hardy-Littlewood-Pólya rearrangement inequality, we arrive at Consequently, RHS(5.5) Making use of the assumption w ∈ B r ∩ B * ∞ and Lemma 5.1, the fact that the function t → g * (t)t − m n is non-increasing on (0, ∞) and the definition of g, we get Proof.Put X := Λ r (wv mr,n ), Y := Λ r (w).Embedding (5.6) means that, for all This can be proved quite analogously as the estimate RHS(5.5)RHS(5.4).
To prove the needed embeddings for Sobolev spaces modelled upon weighted Lorentz spaces given in Proposition 5.7 below, we make use of the following lemma, which is closely related to the results from [47] and can be seen as a Sobolev-Gagliardo-Nirenberg type inequality.
Lemma 5.6.Suppose that X(R n ) is an r.i.space such that k−1 n < α X , k ∈ N, k < n, and the set of bounded functions is dense in X.Then where Proof.First, from Theorem 2 in [47], we have Further, we show that the condition k−1 n < α X implies lim t→0+ ϕ X (t) = 0, where ϕ X is the fundamental function of X.Indeed, for t ∈ (0, 1  2 ), it follows that t Thus, lim t→0+ ϕ X (t) = 0. Using [6, Theorem 5.5, Chapter 2, p. 67], we obtain that X a = X b and X b is separable, where X a is the subset of functions f ∈ X which have absolutely continuous norms and X b is the closure in X of the set of simple functions.By our assumption X = X b .Thus X = X a = X b .Then in light of Semenov's theorem (see [44,Theorem 8, Chapter II]), it follows that continuous functions are dense in X b .Further, by standard density argument, one can see that (For another proof see Remark 3.13 in [25].Somewhat similar argument can be found in [39].) By Lorentz-Shimogaki result [6, Theorem 7.4, p. 169] and [6, Theorem 4.6, p. 61], if Thus, using a limiting argument, we may extend the validity of (5.7) from functions in C ∞ 0 (R n ) to all functions in W k X.Since t(f * * (t) − f * (t)) is an increasing function, we have Taking into account the condition k−1 n < α X , the operator Qk−1 n is bounded on X and therefore and embeddings (5.8) and (5.9) follow.
5.2.The Ul'yanov inequality between weighted Lorentz spaces.The next theorem provides an estimate of the K-functional K(f, t; S r (w), W k S r (w)).Note that in general, the function gaged cone S r (w) is not linear (cf., e.g., [16]).For the definition of the K-functional for the couple (S r (w), W k S r (w)) see the discussion in Section 3.
r ds s for all f ∈ Z and all t > 0.
To estimate RHS (5.11), we are going to apply Theorem 4.3 (A), with X := Λ r (w), Y := S Λr(w) (v k+m,n ), the function gaged cone Z mentioned above, with the Sobolev integral operator as the potential operator A, and the Banach lattice F 0 defined as the set of all functions h ∈ M(0, ∞) such that Note also that the assumption w ∈ B r ∩ B * (5.12) .
Remark 5.9.Note that Theorem 5.8 remains true if the K-functionals K are replaced by the K-functionals K 0 (cf.Remark 4.7).
Since S r (w) is not a linear space, the calculation of the K-functional K(f, t; S r (wv mr,n ), W k S r (wv mr,n )) may cause additional difficulties.In order to use the previous theorem, we would like to find a Banach function space Y such that S r (wv mr,n ) ֒→ Y .The smallest such space Y is the second associate space (S r (wv mr,n )) ′′ .
By [16,Theorem 4.1], if (5.13) Consequently, S r (wv mr,n ) ֒→ Γ r (ν).Hence, for all t > 0 and f ∈ S r (wv mr,n ), which, together with (5.15), implies that Therefore, by Corollary 5.10, for all t > 0 and f ∈ L p,r;b , , where Using the estimate ω k+m (f, u) L p,r;b ω k+m (f, u) L p,r;b with r r, we immediately obtain the following corollary.
for all t > 0 and f ∈ L p,r;b (for which RHS(5.17) is finite).
In particular, if b ≡ 1, then (5.17) yields the known estimate (1.19) for integer parameters k and m satisfying k + m < n.Note that the restriction r r is natural since (5.17) does not hold in general for r > r, see [30,Theorem 1.1(iii)].
In the next section we will investigate inequalities of type (5.17) in more details.

Sharp Ul'yanov inequality between the Lorentz-Karamata spaces
In the previous section we obtained the Ul'yanov-type inequalities for K-functionals and moduli of smoothness between the general weighted Lorentz spaces, which causes restrictions on the parameters.In particular, we assumed that k, m ∈ N. On the other hand, it is clear that, when dealing with more specific Lorentz spaces, one could get better results, i.e., sharp Ul'yanov inequalities for a wider range of parameters.
Our main goal in this section is to establish new sharp Ul'yanov inequalities between the Lorentz-Karamata spaces introduced in the previous subsection.
First we mention some simple properties of slowly varying functions (recall that slowly varying functions have been introduced in Definition 5.11 at the end of Subsection 5.
If b ∈ SV , then also b −1 := 1/b ∈ SV .We will show that these functions have comparable, sufficiently smooth regularizations: (a) Given N ∈ N, following [35, Lemma 6.3], we set Then, by direct computation, we obtain, for all t > 0 and ℓ ∈ N, 1 ℓ N, that (6.2) and hence, for all t > 0 and j, ℓ ∈ N, 1 j ℓ N, to obtain, for all t > 0 and ℓ ∈ N, 1 ℓ N, and hence, for all t > 0 and j, ℓ ∈ N, 1 j ℓ N, (with some constants D j,k ).Now we introduce the subclass SV ↑ of non-decreasing slowly varying functions by (6.7) and extend the classical Riesz potential to a fractional integration with slowly varying component b −1 , where b ∈ SV ↑ .To this end, if the slowly varying function a N , N ∈ N, is given by (6.1), set When we choose N > (n + 1)/2, we can apply the formula contained in [62], with m(t) = t −σ/2 a N ( √ t).To this end, observe that N ( √ t)/(t ℓ/2 t (ℓ−k)/2 ) (with some constants c k,ℓ,N ) which, by (6.3), implies that |m (N ) (t)| t −N −σ/2 b −1 ( √ t), and hence for all x = 0, ) for all t > 0 .Therefore, the proof of [26,Theorem 4.6] can be taken over to get the following analog of a fractional integration theorem.where the slowly varying function c N is given by (6.4).
Proof.Take R > 0, N ∈ N, N > n/2, and define Proof.Let N ∈ N, N > (n + 1)/2 and let the slowly varying functions a N , c N be given by (6.1) and (6.4).Then 1 = a N c N /(a N c N ) on the interval (0, ∞).Therefore, supposing that the Fourier symbol 1/(a N (|ξ|)c N (|ξ|)) generates a bounded operator on L p (R n ), 1 < p < ∞, then, by Lemma 6.2 and by Lemma 6.3, we obtain that Therefore, (6.12) D j 1 a N (t) c N (t) j k=1 M j,k (t) (a N (t) c N (t)) k+1 , 1 j N, for all t > 0, where the numerators M j,k (t) are appropriate linear combinations of terms of the type j i=1 D α k,j i (a k (t) c ℓ (t)) β k,j i , α k,j , β k,j ∈ N j 0 , j i=1 α k,j i β k,j i = j.
In view of (6.1) -(6.6), it is clear that the denominators on the right-hand side of (6.12) satisfy (a N (t) c N (t)) k+1 ≈ 1 for all t > 0, and that, on account of (6.11), |M j,k (t)| 1 for all t > 0 if 1 j N and 1 k j.Therefore, 1/(a N c N ) satisfies (6.10) and the proof is complete.

Remark 4 . 7 .
(i) Note that Theorems 4.3, 4.5, and 4.6 remain true if the K-functional K is replaced by the K-functional K 0 or by the K-functional K 1 given by(3.4)  or by(3.5).(ii)Theorems 4.1, 4.3, 4.5, and 4.6 are true if the Banach function spaces are replaced by function gaged cones.To give a flavor of how to use Theorems 4.3, 4.5, and 4.6, we present the following examples on the classical Ul'yanov inequality (1.6) and sharp Ul'yanov inequality in the Lorentz setting, cf.Proposition 1.3.Example 4.8.We obtain the following extension of the classical Ul'yanov inequality (1.6):If 1 p < ∞, k, n ∈ N, 0 < δ < min(k, n/p), and 1/p * = 1/p − δ/n, then (4.16) a linear projection mapping L 1 onto the space of polynomials of degree at most k, and f # = f # 0 .The first estimate follows from [5, Corollary 4.3] and Hardy type inequalities, the second and third estimates from [19, Theorem 9.3, Theorem 5.6, and Corollary 2.2].

I(− 1 )
σ g p * ,s;B F −1 [|ξ| −σ a N (|ξ|)c N (|ξ|) g(ξ)] p * ,s;B F −1 [c N (|ξ|) g(ξ)] p,r;bnB b(R) g p,r;bnBfor all R > 0 and for all entire functions g ∈ L p,r;bnB (R n ) with supp g ⊂ B R (0).Thus, by [10, Theorem 0.2], it remains to show that 1/(a N c N ) satisfies the condition (6.10) (with the function m R;N replaced by 1/(a N c N )).Introduce the differential operator D = t(d/dt), define D 0 to be the identity operator andD j = D D j−1 , j ∈ N. Now note that t N (d/dt) N can be expressed as a linear combination of D j , 1 j N, that D [a N (t)c N (t)] = a N −1 (t)c N (t) − a N (t)c N −1 (t)and, by induction, that (6.11) D j (a N (t) c N (t)) = j k=0 k+1 j k a N −k (t) c N −j+k (t) , 1 j N.
w satisfies the B r condition) if there is c > 0 such that t 0 w(s) ds for every t > 0;• w ∈ B * ∞ (i.e., w satisfies the B * ∞ condition) if there is c > 0 such that t 0 w(s) ds for every t > 0.