Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

Continuity envelopes for the spaces of generalised smoothness Bpq(s,Ψ)(ℝn) and Fpq(s,Ψ)(ℝn) are studied in the so-called supercritical s=1


Introduction
This paper continues the study of continuity envelopes in spaces of generalised smoothness begun in [21].Moreover, as we shall immediately explain, there also appear close connections to [6] as we concentrate on some 'limiting' situation now : In [21] Haroske and Moura considered spaces of type B (s,Ψ) pq (R n ), F (s,Ψ) pq (R n ), with 0 < p, q ≤ ∞ (p < ∞ for F -spaces), n p < s < 1 + n p ; now we turn to the situation s = 1 + n p .Here Ψ is a so-called admissible function, typically of log-type near 0. The study of spaces of generalised smoothness has a long history, resulting on one hand from the interpolation side (with a function parameter), see the results by Merucci [29] and Cobos, Fernandez [8], whereas the rather abstract approach (approximation by series of entire analytic functions and coverings) was independently developed by Gol'dman and Kalyabin in the late 70's and early 80's of the last century; we refer to the survey [24] by Kalyabin and Lizorkin and the appendix [28] which cover the extensive (Russian) literature at that time.More recent contributions are due to Gol'dman, Netrusov and Burenkov, see [21] for further details.The notion was revived and extended in the way we shall use it in connection with limiting embeddings and spaces on fractals by Edmunds and Triebel in [14,15], Leopold in [26,27] and Moura in [30,31].Closely linked, but slightly different is the approach to more general Lipschitz spaces as developed by Edmunds and Haroske in [11,12,18].The present state of the art is reviewed and covered in [16] by Farkas and Leopold linking function spaces of generalised smoothness with negative definite functions -and thus referring to applications for pseudo-differential operators (as generators of sub-Markovian semi-groups).Plainly these latter applications and also the topic in its full generality are out of the scope of the present paper; it explains, however, the increased interest on function spaces of generalised smoothness quite recently.As a prototype one can think of spaces of Besov type B (s,Ψ) p,q (R n ), where the function Ψ might behave like (1 + | log x|) b , x ∈ (0, 1], b ∈ R; for example, we have for 1 < p ≤ ∞, 0 < q ≤ ∞, 0 < s < 1, an easy characterisation by differences, (with the usual modification if q = ∞).
In contrast to the notion of spaces of generalised smoothness, the study of continuity envelopes has a rather short history; this new tool was developed only recently in [19,20,35], initially intended for a more precise characterisation of function spaces.It turned out, however, that it leads not only to surprisingly sharp results based on classical concepts, but allows a lot of applications, too, e.g. to the study of compact embeddings.We return to this point later.Roughly speaking, a continuity envelope E C (X) of a function space X consists of a so-called continuity envelope function together with some 'fine index' u X ; here ω(f, t) stands for the modulus of continuity, as usual.Forerunners of continuity envelopes in a wider sense are well-known for decades: dealing with spaces of type B s pq , F s pq we refer to the result of Sickel and Triebel [33,Thm. 3.3.1](also for further historical comments), the paper [22] by Kalyabin, which concerns the question of embeddings into C in the special context of spaces of generalised smoothness mentioned above, whereas the famous result [3] of Brezis and Wainger can be regarded as some origin of the idea of continuity envelopes at all.It states that any function g ∈ H 1+n/p p (R n ), 1 < p < ∞, is 'almost' Lipschitz-continuous in the sense that for all x, y ∈ R n , 0 < |x − y| < 1  2 , |g(x) − g(y)| ≤ c |x − y| log |x − y| Here c is a constant independent of x, y and g, and 1 p + 1 p = 1.The sharpness of this assertion and parallel questions for more general spaces were studied in [11,12].These considerations led finally to the introduction of continuity envelopes : obviously (2) results after some reformulation in In this paper we extend this result as follows.Denoting by A either B or F , the main objective is to characterise the above continuity envelopes (1) of spaces A (s,Ψ) pq (R n ) when these spaces are not continuously embedded into the Lipschitz space Lip 1 (R n ); for that reason we shall first prove a criterion for this embedding to hold.Moreover, when Ψ ≡ 1 and, consequently, we are dealing with Besov or Triebel-Lizorkin spaces A s pq (R n ), Haroske [19] and Triebel [35] proved that E = 1 (thus also extending the inequality (3) to equivalence).In [21] it was shown that E Hence, regarding these two forerunners, the question arises which might be the correct guess for our situation.Fortunately we gain here from a similar consideration in [6], resulting in the introduction of some auxiliary function Φ r,u : (0, 2 ] and Remark 3.2 below.We shall prove in this paper that with 0 < σ = s − n p ≤ 1 and u A as above, thus containing both previous results and generalising it in a natural way.In fact, we can give even more precise characterisations leading to the concept of so-called continuity envelopes, where the continuity envelope function E X C is complemented by that fine index u X .Moreover, we obtain two sharp embedding results, namely -as already announced above -that using the above abbreviation for u A again, and a criterion for the embedding where Finally, turning to spaces defined on bounded domains, say the unit ball U ⊂ R n for simplicity, it is reasonable to consider compact embedding operators, id : B (s,Ψ) pq (U ) −→ C(U ), where C stands for the space of complex-valued bounded uniformly continuous functions.More precisely, we shall further inquire into the nature of this compactness and characterise the asymptotic behaviour of the corresponding approximation numbers; we obtain two-sided estimates for , and Ψ as above.
The paper is organised as follows.We collect the necessary background material in Section 1; in Section 2 we obtain criteria for sharp embeddings as briefly sketched above, and construct extremal functions.This is not only needed afterwards, but also of some interest of its own.Our main result on continuity envelopes in spaces of generalised smoothness can be found in Section 3, whereas Section 4 contains some applications : Hardy-type inequalities, further sharp embedding results and, finally, (mainly two-sided) asymptotic estimates for approximation numbers of related compact embeddings.For a better illustration of our results we give a typical example at the beginning and relate all essential outcomes to it afterwards.

General notation
As usual, R n denotes the n-dimensional real Euclidean space, N the collection of all natural numbers and N 0 = N ∪ {0}.We use the equivalence '∼' in always to mean that there are two positive numbers c 1 and c 2 such that for all admitted values of the discrete variable k or the continuous variable x, where (a k ) k , (b k ) k are nonnegative sequences and ϕ, ψ are non-negative functions.If a ∈ R then a + := max(a, 0) and [a] denotes the integer part of a.If 0 < u ≤ ∞, the number u is given by 1/u := (1 − 1/u) + .Given two quasi-Banach spaces X and Y , we write X → Y if X ⊂ Y and the natural embedding of X into Y is continuous.All unimportant positive constants will be denoted by c, occasionally with additional subscripts within the same formula.If not otherwise indicated, log is always taken with respect to base 2. Apart from the last section we shall always deal with function spaces on R n ; so for convenience we shall usually omit the 'R n ' from their notation.

Function spaces of generalised smoothness
Recall our introductory remarks on spaces of generalised smoothness, relating this topic with some historical background as well as the present state of the art.In our context, we shall be concerned with function spaces of generalised smoothness of Besov and Triebel-Lizorkin type, where the usual main smoothness parameter s is replaced by a couple (s, Ψ) and Ψ is an admissible function according to the following definition.
Definition 1.1 A positive monotone function Ψ on the interval (0, 1] is called admissible if Example 1.2 If c ∈ (0, 1) and b ∈ R, then is an admissible function.In particular, is admissible; we return to this particular choice for illustration in the sequel.
The proposition below gives some properties of admissible functions that will be useful in what follows.We refer to Lemma 2.3 of [5], where a simple proof can be found.

Proposition 1.3 Let Ψ be an admissible function.
(i) There exist constants b ≥ 0 and c 1 , c 2 > 0 such that for any t ∈ (0, 1]. (ii) For any a, d > 0, there is some δ > 0 such that Before introducing the function spaces under consideration we need to recall some notation.By S we denote the Schwartz space of all complex-valued, infinitely differentiable and rapidly decreasing functions on R n and by S the dual space of all tempered distributions on R n .Furthermore, L p with 0 < p ≤ ∞, stands for the usual quasi-Banach space of p-integrable (measurable, essentially bounded if p = ∞) functions with respect to the Lebesgue measure, quasi-normed by denotes the Fourier transform of ϕ.As usual, F −1 ϕ or ϕ ∨ stands for the inverse Fourier transform, given by the right-hand side of (7) with i in place of −i.Here xξ denotes the scalar product in R n .Both F and F −1 are extended to S in the standard way.Let ϕ 0 ∈ S be such that and for each j ∈ N let Then the sequence (ϕ j ) ∞ j=0 forms a dyadic resolution of unity.
(with the usual modification if q = ∞) is finite.
is the collection of all f ∈ S such that (with the usual modification if q = ∞) is finite.
Remark 1.5 The above spaces were introduced by Edmunds and Triebel in [14,15] and also considered by Moura in [30,31].If Ψ ≡ 1 then the spaces B (s,Ψ) pq and F (s,Ψ) pq coincide with the usual Besov and Triebel-Lizorkin spaces, B s pq and F s pq , respectively, and the following elementary embeddings hold: for all ε > 0 and A ∈ {B, F }.For convenience, we shall continue writing A s pq or A (s,Ψ) pq , respectively, when both B-and F -spaces are concerned and no distinction is needed.In a more general setting such spaces were also studied in [22], [23], [24] and, quite recently, in [16].
Example 1.6 With the particular choice of Ψ b given by ( 6) we obtain spaces B s,b pq consisting of those f ∈ S for which is finite (usual modification for q = ∞); similarly for F s,b pq .These spaces were studied by Leopold in [26,27].
For later use we also recall a special lift property for spaces A (s,Ψ) pq obtained in [5].Let Ψ be an admissible function and (ϕ j ) j∈N 0 a smooth dyadic partition of unity according to (8), ( 9), where we additionally assume ϕ 0 to be non-negative and radially monotonically decreasing.Denote and Proposition 1.7 Let 0 < p, q ≤ ∞ (with p < ∞ in F -case), s ∈ R and Ψ be an admissible function. Then A proof is given in [5,Prop. 3.2].The essential advantage of this result is that it enables us to gain from the wider knowledge concerning embeddings and spaces of type A s pq .
An important tool in our later considerations is the characterisation of spaces of generalised smoothness by means of atomic decompositions.We state this here only for the B-spaces and with some restriction to the parameters which will be sufficient for us.We refer to [30,31] for a complete description.We need some preparation.
As for Z n , it stands for the lattice of all points in R n with integer-valued components, Q νm denotes a cube in R n with sides parallel to the axes of coordinates, centred at 2 if, and only if, it can be represented as with (λ νm ) ν,m a sequence of complex numbers for which (appropriately modified if p or/and q are ∞) is finite and a νm certain kind of atoms (see definition below) localised on Q νm .Furthermore, the infimum of the expressions (14) taken over all admissible representations (13) is an equivalent quasi-norm in B (s,Ψ) pq .
The atoms referred to above have the following properties: for some given numbers c > 1 and k > s, with k ∈ N, they are k-times differentiable complex-valued functions on R n such that Next we recall the definition of differences of functions.If f is an arbitrary function on Note that ∆ k h can also be defined iteratively via For convenience we may write ∆ h instead of ∆ 1 h .Furthermore, the k-th modulus of smoothness of a function We shall write simply ω(f, t) We recall here Marchaud's inequality, which will be crucial in our estimates: The following result, obtained in [21], gives an equivalent expression for the quasi-norm in some B-spaces and will also play a key role later on.
Theorem 1.9 Let 0 < p, q ≤ ∞, s > σ p and Ψ be an admissible function.If k is an integer such that k > s, then Example 1.10 We return to our example Ψ b given by (6).
(with the usual modification if q = ∞), see Example 1.6.

Continuity envelopes
The concept of continuity envelopes was introduced by Haroske in [19] and Triebel in [35].
Here we quote the basic definitions and results concerning continuity envelopes.However, we shall be rather concise and we mainly refer to [19,20,35] for heuristics, motivations and details on this subject.
Let C be the space of all complex-valued bounded uniformly continuous functions on R n , equipped with the sup-norm as usual.Recall that the classical Lipschitz space Lip 1 is defined as the space of all functions f ∈ C such that is finite, the expression (18) defining its norm, where ω(f, t) stands for the modulus of continuity, The proposition below gives some properties of continuity envelope functions which will be useful in the sequel.A proof can be seen in [19,Prop. 4.3].
Proposition 1.12 Let X → C be some quasi-normed function space on R n .
there is some positive constant c such that, for all t > 0, where c can be chosen equal to id : where h is a continuous monotonically decreasing function equivalent to E X C in (0, ε], and let µ H be the associated Borel measure.The number

then called the continuity envelope for the function space X.
There are different ways to define the number u X in the literature.So let us point out explicitly that when writing results of the type E C (X) = E X C , u X in the sequel (for particular choices of X) we shall always mean that the infimum of v, 0 < v ≤ ∞, such that (19) holds for some c = c(v) > 0 and all f ∈ X, is in fact a minimum which we are denoting by u X .
We briefly recall some properties of continuity envelopes.In view of Definition 1.11 we obtain -strictly speaking -equivalence classes of continuity envelope functions when working with equivalent (quasi-) norms in X as we shall do in the sequel.However, for convenience we do not want to distinguish between representative and equivalence class in what follows and thus stick at the notation introduced.Concerning Definition 1.13 it is obvious that (19) holds for v = ∞.Moreover, one verifies that  19) true.Note that we are not saying that such a smallest v always exist.However, if it exists, together with a continuous h as assumed above, the same happens to the continuity envelope of X, and the given definition can be seen to be independent of the chosen number ε and function h (in this respect, it is useful to look at [35, Prop.12.2 (ii)]).Clearly, for the continuity envelope to be well-defined, the E X C in the pair should be understood in the sense of an equivalence class, of functions equivalent to the actual E X C for small positive values of the argument, containing at least one continuous decreasing function (this requirement will always hold in the cases for which we are going to apply Definition 1.13 later on).In the important case when H = − log h happens to be continuously differentiable in (0, ε], we have µ H (dt) = H dt, and for the functions we want to integrate we can calculate the left-hand side of (19) as an improper Riemannian integral: We finish this preliminary section with one more result, very easy to prove (on the basis of [35, Prop.12.2 (ii)]) but which turns out very useful in what follows.
Proposition 1.14 Let X i → C, i = 1, 2, be some quasi-normed function spaces on R n with X 1 → X 2 .Assume for their continuity envelope functions that for some ε ∈ (0, 1).If (19) holds for some v for X 2 , it also holds for the same v for X 1 .In particular, in the case the indices u X i , i = 1, 2, exist we have that

Embeddings and extremal functions
Recall that we shall write A as long as no distinction is needed.In view of Proposition 1.12 (ii) we are only interested in spaces X → Lip 1 in the sequel.Note that A (s,Ψ) pq → C for 0 < p, q ≤ ∞ (with p < ∞ in the F -case), s > n p , and Ψ an admissible function; this follows immediately from (10) together with the corresponding well-known results for spaces A s pq .By an analogous argument it turns out that A (s,Ψ) pq → Lip 1 for s > n p + 1, while there is not such an embedding if s < n p + 1; thus, by Definition 1.13, it is reasonable to study continuity envelopes at least in the situation X = A (s,Ψ) pq , n p < s < n p + 1.This case was dealt with in [21], where the tricky limiting situations s = n p and s = n p + 1 were postponed.Proposition 2.1 Let 0 < p, q ≤ ∞ and Ψ be an admissible function.
We come to the second limiting case s = n p +1 now and study the conditions such that A (1+n/p,Ψ) pq → Lip 1 .In fact it turns out that this characterisation resembles the one mentioned above.Proposition 2.2 Let 0 < p, q ≤ ∞ and Ψ be an admissible function.
For the estimates to be derived later we need some results related to so-called extremal functions.Concerning this one can essentially follow the reasoning in [35, pp. 220-221], adapted to our situation.We have also made some modifications to the setting considered in [35] and concentrated only in what is needed here.But, as mentioned, the ideas are contained in [35, pp. 220-221] and therefore here we just write the assertion we shall need later on and briefly refer to the proof of ( 27) below where the atomic decomposition of B-spaces is used.
Proposition 2.4 Let Ψ be an admissible function and h and h 0 be the functions in C ∞ 0 (R) defined, respectively, by h(y) = e Then the function f b given by the uniformly convergent series belongs to B (1+n/p,Ψ) pq and there exists c 1 > 0 (independent of b) such that If, moreover, b j ≥ 0, j ∈ N, then there also exists c 2 > 0 (depending only on h and h 0 ) such that Remark 2.5 Observe that if, in (28), one uses a special sequence b in such a way that there is a strictly increasing sequence (j k ) k∈N of natural numbers such that b j = 0 whenever j = j k , then instead of (28) we can also write

Continuity envelopes : the main result
We shall use a convenient notation introduced in [6].For 0 < r, u ≤ ∞ and a continuous admissible function Ψ, define the function Φ r,u : (0, 2 with the usual modification for u = ∞: Φ r,∞ (t) := sup In particular, Φ r,u is positive, monotonically decreasing and continuous, being also differentiable when u = ∞, cf.[6, Prop.2.5].We refer to [6] for a detailed discussion on Φ r,u , though, for later use, we also mention here the following discretisation result [6, Prop.2.7]: (with the right-hand side modified to sup j=1,...,[| log t|/n] 2 j n r Ψ(2 −j ) −1 if u = ∞).Though not explicitly mentioned in the assertion that follows -which is our main result -, when convenient we assume in its proof that the admissible function Ψ satisfies the condition Ψ(1) = 1.There is no loss of generality in doing this.Theorem 3.1 Let 0 < p, q ≤ ∞ (with p < ∞ in the F -case), 0 < σ ≤ 1, s = n p + σ and Ψ be a continuous admissible function.
Proof.Note that in case of 0 < σ < 1 and any number u, 0 < u ≤ ∞, the admissibility of Ψ leads to ∈ p for the F -spaces in the sequel.Actually, since the result for the latter follows immediately from the results for the former, due to (25), Proposition 1.12 (iii) and 1.14, in what follows we concentrate on the B-spaces only.
Step 2.Here we prove that, for small t > 0, (modification if q = ∞, that is, if 0 < q ≤ 1).We take advantage of some choices made in the proof of [6,Prop. 4.2], where a corresponding result for so-called local growth envelope functions also in a 'critical' situation was obtained.For each J ∈ N denote by f b J the function f b in ( 26) with b = (b j ) j∈N being the sequence defined by Notice that b j ≥ 0, j ∈ N, b ∈ q and, moreover, b| q = 1, so that Proposition 2.4 allows us to write, with constants independent of J, if 0 < q ≤ 1.So we finally have, for 0 < t ≤ 1/2, by means of choosing J ∈ N such that 2 −(J+1) < t ≤ 2 −J and with the help of ( 30) and the properties of E X C and Ψ, that Step 3. We want to show now that u B (1+n/p,Ψ) pq ≤ q, in case this index exists.The case q = ∞ follows immediately from the previous two steps, as they imply that, for some chosen ε ∈ (0, 2 −1 ] and c > 0, and Consider now 0 < q < ∞.It is enough to prove that for some ε ∈ (0, 2 −1 ] and c > 0, where µ n ∞,q stands for the Borel measure associated with − log Φ ∞,q (t n ) in (0, ε].Actually, in view of the continuous embedding B and the fact that Φ ∞,q does not depend on p, we need only prove (35) for p = ∞, which we shall do next, taking advantage of some ideas used in Steps 2 and 3 of the proof of [6,Prop. 4.1], where a corresponding result for so-called local growth envelope functions also in a critical situation was obtained.Consider first the case 1 < q < ∞.We benefit from the fact that Φ ∞,q is continuously differentiable in this case and µ n ∞,q (dt) can thus be easily described, see Remark 4.2 below.We use Marchaud's inequality (17) and a generalised Hardy's inequality (cf.[17, p. 247]), as well as properties of Ψ, in order to write, Clearly, the last inequality follows from Theorem 1.9.Finally, consider the case 0 < q ≤ 1.We observe that, because of the hypothesis (Ψ(2 −j ) −1 ) j / ∈ q , we have in this case that Ψ must be increasing and such that lim t→0+ Ψ(t) = 0.In particular, Φ ∞,∞ (t n ) = Ψ(t) −1 .On the other hand, those hypotheses guarantee that we can construct a strictly increasing sequence (t k ) k∈N 0 of natural numbers in the following way: (i) t 0 is a given natural number; We can thus write, with ε = 2 −t0 , and again with the help of Marchaud's inequality, where the last inequality follows again from Theorem 1.9.
Step 4. It finally remains to show that the index u B (1+n/p,Ψ) pq exists and is equal to q.
Let ε ∈ (0, 2 −1 ].From what we have already proved in the preceding steps (and also by the considerations after the general definition of the index), it suffices now to show that when v < q it is not possible to find c > 0 such that holds for all f ∈ B (1+n/p,Ψ) pq . We assume, on the contrary, that for some v ∈ (0, q) it was possible to find c > 0 such that (38) holds for all f ∈ B (1+n/p,Ψ) pq and take advantage of some ideas used in the proof of [6,Thm. 4.4], where a corresponding result for so-called local growth envelope functions also in a critical situation was obtained.We start with the case 1 < q ≤ ∞.The hypothesis (Ψ(2 −j ) −1 ) j / ∈ q guarantees that we can construct a strictly increasing sequence (t k ) k∈N0 of natural numbers in the following way: where 0 < c 1 ≤ c 2 < ∞ are the equivalence constants implied in (30) (with r = ∞ and u = q ).Notice that Given such a sequence (t k ) k∈N 0 , denote by f b J , for each J ∈ N, the function f b in ( 26) with b = (b j ) j∈N being the sequence defined by Notice that b j ≥ 0, j ∈ N, b ∈ q and, moreover, b| q ≤ J 1/q , so that (38), Proposition 2.4, (39) and (40), together with the properties of ω(f, t)/t and Φ ∞,q , allow us to write, with constants independent of J, and we get a contradiction if we let J → ∞.Now for the case 0 < q ≤ 1.We shall explain how to modify the proof of the preceding case.The sequence (t k ) k∈N0 shall obey (i) above, but instead of (39) we shall now assume that (36) holds, so that instead of (40) we now have (37).The sequence (b j ) j∈N shall be defined by Clearly, b j ≥ 0, j ∈ N, b ∈ q and, moreover, b| q = J 1/q , so that (38), Proposition 2.4, (36) and (37), together with the properties of ω(f, t)/t and Φ ∞,∞ , allow us to write, .Actually, we can give up the continuity assumptions for Ψ in Theorem 3.1, keeping the result, as long as the function Φ r,u (t n ) appearing in the continuity envelope -and, especially, the corresponding measure µ n r,u -is built by means of an equivalent continuous admissible function Ψ.
Example 3.3 Using our particular choice Ψ b given by ( 6) we obtain for b < (appropriately interpreted for u = ∞ ) and Theorem 3.1 (i) thus reads as , q , where 0 < p, q ≤ ∞ and b < 1 q .When 1 < q ≤ ∞ it also makes sense to consider b = 1/q (cf.Example 2.3), and in this case Theorem 3.1 (i) gives us where here t should be considered in (0, 2 −2 ], for example.

Applications
As a first application we can conclude some Hardy-type inequalities.This follows immediately from our above assertions together with the monotonicity (20), see [35,Prop. 12.2,, the properties of ω(f, t)/t and the fact that, given κ non-negative on (0, ε], holds for some c > 0 and all f ∈ X, f |X ≤ 1, if, and only if, κ is bounded (adapted from [20,Prop. 4.3.3 (iv)]).We only state the version for σ = 1, i.e. s = n p + 1, as the remaining case is already contained in [21,Cor. 3.5].
and Ψ be a continuous admissible function.
(i) Let κ(t) be a positive monotonically decreasing function on (0, ε] and let 0 < u ≤ ∞.Then for some c > 0 and all f ∈ B (1+n/p,Ψ) pq if, and only if, κ is bounded and q ≤ u ≤ ∞, with the modification Moreover, if κ is an arbitrary non-negative function on (0, ε], then (42) holds if, and only if, κ is bounded.
Step 3. We turn to the F -case in (ii).Note first that the necessity of (48) can be shown completely parallel to the B-case in Step 1, applying now Theorem 3.1 (ii) with 0 < σ < 1 in (51).Conversely we gain from the B-case and ( 25), where the assumption p 1 < p 2 is now essentially involved.In particular, choose numbers σ i ∈ R, r i ∈ (0, ∞), i = 1, 2, additionally satisfying Thus we conclude from (48), p 1 < p 2 , and (i) that which together with (25) leads to (47) and finishes the proof of (ii).
Remark 4.5 Note that the result (i), assuming additionally p 2 > p 1 ≥ 1, can be found in [28,D 1.7,Thm. 4,p. 396].Using the Ψ-lift J e Ψ given by (12) together with Proposition 1.7 one might be tempted to reduce the problem to the question whether B (s1, b Ψ) p1q1 → B s2 p 2 q 2 , however this neglects the fact that Ψ = Ψ 2 Ψ 1 is not necessarily an admissible function in general unlike in case of our standard example (6).Comparison of Proposition 4.3 (i) with Propositions 2.1 (i), 2.2 (i), together with the known inclusions , respectively, and (25) shows that the sufficiency of the assertions in Propositions 2.1, 2.2 can also be regarded as a consequence of our above result; however -due to the special role played by C or Lip 1 within the scale of Besov spaces -it also shows that for an embedding within the Besov scale one gains nothing extending B 0 ∞,1 to C or B 1 ∞,1 to Lip 1 , respectively.
Remark 4.6 In case of the F -spaces the assumption s 1 > s 2 in (ii) (implying also p 1 < p 2 ) is essentially used in our argument.Unlike in the B-case there is no interplay between the functions Ψ i and the fine indices q i , i = 1, 2; for Ψ 1 ≡ Ψ 2 ≡ 1 this was already known [34, Thm.2.7.1 (ii)].The situation changes essentially when approaching the case s 1 = s 2 studied in (iii).
Our last application presented here concerns compact embeddings of function spaces of generalised smoothness.We shall qualify their compactness further by means of approximation numbers essentially gaining from our above envelope results.First, we briefly recall this notion.For details and properties of approximation numbers we refer to [7], [10], [25] and [32] (restricted to the case of Banach spaces), and [13] for some extensions to quasi-Banach spaces.
Remark 4.8 A strong motivation to study approximation numbers comes from spectral theory, in particular, the investigation of eigenvalues of compact operators.For instance, let H be a complex Hilbert space and T ∈ L(H) compact, the non-zero eigenvalues of which are denoted by {µ k (T )} k∈N ; then T * T has a non-negative, self-adjoint, compact square root |T |, and for all k ∈ N, see [10, Thm.II.5.10, p. 91].Hence, if in addition T is non-negative and self-adjoint, then the approximation numbers of T coincide with its eigenvalues.Outside Hilbert spaces the results are less good but still very interesting, cf.[7], [10], [25] and [32] for further details.
The interplay between continuity envelopes and approximation numbers relies on the following outcome.
Proposition 4.9 Let X be some Banach space of functions defined on the unit ball U in R n with X(U ) → C(U ).Then there is some c > 0 such that for all k ∈ N , a k (id : complementing an earlier outcome of Moura in [31, Prop.1.1.13(iv)-(vi)].Here assertion (4) is meant for parameters u A ∈ (1, ∞) only, but we shall prove the corresponding statement for all u A , 0 < u A ≤ ∞, (suitably interpreted) in Proposition 2.2 below.

Definition 4 . 7
Let A 1 and A 2 be two complex (quasi-) Banach spaces, T a linear and continuous operator from A 1 into A 2 , and let k ∈ N. The k th approximation number a k of T is the infimum of all numbers T − S where S runs through the collection of all continuous linear maps from A 1 to A 2 with rank S < k, a k (T ) = inf{ T − S : S ∈ L(A 1 , A 2 ), rank S < k}.
with constants independent of J; we get a contradiction if we let J → ∞.2