A critical topology for $L^p$-Carleman classes with $0

In this paper, we explain a sharp phase transition phenomenon which occurs for $L^p$-Carleman classes with exponents $0<p<1$. In principle, these classes are defined as usual, only the traditional $L^\infty$-bounds are replaced by corresponding $L^p$-bounds. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the $L^p$-Carleman class. A particular degenerate instance is when we obtain the $L^p$-Sobolev spaces, analyzed previously by Peetre, following an initial insight by Douady. Peetre found that these $L^p$-Sobolev spaces are highly degenerate for $0<p<1$. Essentially, the contact is lost between the function and its derivatives. Here, we analyze this degeneracy for the more general $L^p$-Carleman classes defined by a weight sequence. Under some reasonable growth and regularity properties, and a condition on the collection of test functions, we find that there is a sharp boundary, defined in terms of the weight sequence: on the one side, we get Douady-Peetre's phenomenon of"disconnexion"between the function and its derivatives, while on the other, we obtain a collection of highly smooth functions. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the $L^p$ setting, with $0<p<1$.

Introduction 1.1. The Sobolev spaces for 0 < p < 1. We first survey the properties of the Sobolev spaces with exponents in the range 0 < p < 1. These were considered by Jaak Peetre [18] after Adrien Douady suggested that they behave very differently from the standard 1 ≤ p ≤ +∞ case. For an exponent p with 0 < p < 1, and an integer k ≥ 0, we define the Sobolev space W k,p (R) on the real line R to be the abstract completion of the space C k 0 of compactly supported k times continuously differentiable functions, with respect to the quasinorm Here, as a matter of notation, · p denotes the quasinorm of L p . Defined in this manner, W k,p becomes a quasi-Banach space whose elements consist of equivalence classes of Cauchy sequences of test functions. At first glance, this definition is very natural, and other approaches to define the same object in the classical case p ≥ 1 do not generalize. For instance, we cannot use the usual notion of weak derivatives to define these spaces, as functions in L p need not be locally L 1 for 0 < p < 1. Some observations suggest that things are not the way we might expect them to be. For instance, as a consequence of the failure of local integrability, for a given point a ∈ R, the formula for the primitive cannot be expected to make sense. A related difficulty is the invisibility of Dirac "point masses" in the quasinorm of L p . Indeed, if 0 < p < 1 and u := −1 1 [0, ] , then as → 0 + , we have the convergence u → 0 in L p while u → δ 0 in the sense of distribution theory. Here, we use standard notational conventions: δ 0 denotes the unit point mass at 0, and 1 E is the characteristic function of the subset E, which equals 1 on E and vanishes off E.

1.2.
Independence of the derivatives in Sobolev spaces. For 1 ≤ p ≤ +∞, we think of the Sobolev space W k,p as a subspace of L p , consisting of functions of a specified degree of smoothness. As such, the identity mapping α : W k,p → L p defines a canonical injection. Douady observed that we cannot have this picture in mind when 0 < p < 1, as the corresponding canonical map α fails to be injective. Peetre built on Douady's observation and showed that this uncoupling (or disconnexion) between the derivatives goes even deeper. In fact, the standard map f → ( f, f , . . . , f (k) ) on test functions f defines a topological isomorphism of the completion W k,p onto the direct sum of k + 1 copies of L p . For convenience, we state the theorem here. Theorem 1.2.1 (Peetre, 1975). Let 0 < p < 1 and k = 1, 2, 3, . . .. Then W k,p is isometrically isomorphic to k + 1 copies of L p : (1.2.1) W k,p L p ⊕ L p ⊕ . . . ⊕ L p .
This decoupling occurs as a result of the availability of approximate point masses which are barely visible in the quasinorm. As a consequence, if we define Sobolev spaces as completions with respect to the Sobolev quasinorm, we obtain highly pathological (and rather useless) objects.
1.3. A bootstrap argument to control the L ∞ -norm in terms the L p -quasinorms of the higher derivatives. From the Douady-Peetre analysis of the Sobolev spaces W k,p for exponents 0 < p < 1, we might be inclined to believe that for such small p, we always run into pathology. However, there is in fact an argument of bootstrap type which can save the situation if we simultaneously control the derivatives of all orders. To explain the bootstrap argument, we take a (weight) sequence M = {M k } +∞ k=0 of positive reals, and define the quasinorm on some appropriate linear space S ⊂ C ∞ (R) of test functions that we choose to begin with.
In the analogous setting of periodic functions (i.e., with the unit circle T R/Z in place of R), it would be natural to work with the linear span of the periodic complex exponentials as S .
In the present setting of the line R, there is no such canonical choice. Short of a natural explicit linear space of functions, we ask instead that S should satisfy a property. Moreover, for subsets S ⊂ C ∞ (R), we say that S is (p, θ)-tame if every element f ∈ S is (p, θ)-tame.
Remark 1.3.2. (a) For θ < 0, only the trivial function f = 0 is (p, θ)-tame, unless if p = 1, in which no (p, θ)-tame function exists. For this reason, in the sequel, we shall restrict our attention to θ ≥ 0. (b) Loosely speaking, for θ ≥ 0, the requirement (1.3.2) asks that the L ∞ -norms of the higher order derivatives do not grow too wildly. We note that for p close to one, (p, θ)-tameness is a very weak requirement; indeed, at the endpoint value p = 1, it is void.
A natural suitable choice of a (p, θ)-tame collection of test functions might be the following Hermite class: Indeed, a rather elementary argument shows that (1.3.2) holds with θ = 0 for S = S Her (for the details, see Lemma 4.1.1 below). However, it might be the case that not all f ∈ S Her have finite quasinorm f p,M < +∞ (this depends on the choice of weight sequence M). In that case, we would then replace S Her by its linear subspace S Her p,M := f ∈ S Her : f p,M < +∞ , and hope that this collection of test functions is not too small.
The bootstrap argument. We proceed with the bootstrap argument. We assume that our parameters are confined to the intervals 0 < p ≤ 1 and 0 ≤ θ < +∞. Moreover, we assume that the collection of test functions S is (p, θ)-tame and that f p,M < +∞ holds for each f ∈ S . We pick a normalized element f ∈ S with f p,M = 1. Since 1 = p + (1 − p), it follows from the fundamental theorem of Calculus that for x ≤ y, As f ∈ L p (R), the function f must assume values arbitrarily close to 0 on rather big subsets of R. By taking the limit of such y in (1.3.3), we arrive at which gives that By iteration of the estimate (1.3.4), we obtain that for n = 1, 2, 3, . . ., As it is given that f p,M = 1, we have that f ( j) p ≤ M j , which we readily implement into (1.3.5): Finally, we let n approach infinity, so that in view of the (p, θ)-tameness assumption (1.3.2) and homogeneity, we obtain that Next, we let n tend to infinity in (1.3.9) and use homogeneity and (p, θ)-tameness as in (1.3.7), and arrive at it follows from (1.3.10) and (1.3.8) that It is immediate from (1.3.11) that under the summability condition (1.3.8), we may in fact control the sup-norm of all the higher order derivatives, which guarantees that the elements of the completion of the test function class S under the quasinorm · p,M consists of C ∞ functions, and the failure of the Douady-Peetre disconnexion phenomenon is complete. We refer to the argument leading up to (1.3.11) as a "bootstrap" because we were able to rid ourselves of the sup-norm control on the right-hand side by diminishing its contribution in the preceding estimate and taking the limit.
Remark 1.3.3. The above argument is inspired by an argument which goes back to work of Hardy and Littlewood on Hardy spaces of harmonic functions. The phenomenon is coined Hardy-Littlewood ellipticity in [7]. To explain the background, we recall that for a function u(z) harmonic in the unit disk D, the function z → |u(z)| p is subharmonic if p ≥ 1. As such, it enjoys the mean value estimate A remarkable fact is that this inequality survives even for 0 < p < 1 (with a different constant) even though subharmonicity fails. See, e.g., [7,Lemma 4.2], [13,Lemma 3.7], and the original work of Hardy and Littlewood [14]. The similarity with the bootstrap argument used here is striking if we compare with e.g. [13,Lemma 3.7].
1.4. The L p -Carleman spaces and classes. The Carleman class C M associated with the weight sequence M is defined as the linear subspace of f ∈ C ∞ (R) for which for some positive constant A = A f (which may depend on f ). The theory for Carleman classes was developed in order to understand for which classes of functions the (formal) Taylor series at a point uniquely determines the function. Denjoy [12] provided an answer under regularity assumptions on the weight sequence, and Carleman [10,8] is an upper bound for the number of zeros of f on the interval [0, 1]. As this result was contained in Bang's thesis [2], written in Danish, the result appears not to be known to a wider audience. For an account of several of the interesting results in [2], as well as of some further developments in the theory of quasianalytic functions, we refer to the work of Borichev, Nazarov, Sodin, and Volberg [17,6]. The present work is devoted to the study of the analogous classes defined in terms of L pnorms, mainly for 0 < p < 1. In view of the preceding subsection, it is natural to select the biggest possible collection of test functions so that the bootstrap argument has a chance to apply under the assumption (1.3.8): Here, we keep our standing assumptions that 0 < p < 1 and θ ≥ 0 (many assertion will hold also at the endpoint value p = 1). Then S p,θ,M automatically meets the asymptotic growth condition (1.3.2), but for (p, θ)-tameness to hold we also need for each individual derivative to be bounded. However, the condition lim sup guarantees that that f (n) ∈ L ∞ at least for big positive integers n, say for n ≥ n 0 . But then, since f p,M < +∞ and in particular f (k) ∈ L p for all k = 0, 1, 2, . . ., the estimate (1.3.4) gives that f (n−1) ∈ L ∞ as well. Proceeding iteratively, we find that f (n) ∈ L ∞ for all n = 0, 1, 2, . . . if f ∈ S p,θ,M . This means that all the estimates of the preceding subsection are sound for f ∈ S p,θ,M . Of course, for very degenerate weight sequences M, it might unfortunately happen that S p,θ,M = {0}. We proceed to define the L p -Carleman spaces.
M is the completion of the test function class S p,θ,M with respect to the quasinorm · p,M .
In the standard textbook presentations, the Carleman classes are defined for a regular weight sequence M in the same way, only the L p quasinorm is replaced by the L ∞ norm, and the class is to be minimal given the following two requirements: (i) the space is contained in the class, and (ii) the class is invariant with respect to dilatation [10,15,16]. It is easy to see that claiming that f ∈ C M is the same as saying that f (t) = g(at) for some positive real a, where for some constant C, which we understand as the requirement g ∞,M < +∞ (with exponent p = +∞). This allows us to extend the notion of the Carleman classes for exponents 0 < p < 1 as follows.
Here, to avoid unnecessary abstraction, we need to understand that each element in W p,θ M gives rise to an element of the Cartesian product space L p × L p × · · · via the lift of the map f → ( f, f , f , . . .) initially defined on test functions f ∈ S p,M (for more details, see the next subsection). Moreover, it is easy to see that f ∈ W p,θ M is uniquely determined by the corresponding element in L p × L p × . . .. It is important to note that in the Cartesian product space, the dilatation operation is well-defined, so that the L p -Carleman class C p,θ M can be understood as a submanifold of L p × L p × · · · in the above sense. Moreover, C p,θ M is actually a linear subspace, as the quasinorm criterion for being in the class is (analogously) for some positive constants C and A, and this kind of bound is closed under linear combination.

Classes of weight sequences.
From this point onward, we will restrict attention to positive, logarithmically convex weight sequences, i.e. sequences M = {M j } j of positive numbers such that the function j → log M j is convex. We will consider the following notions of regularity for weight sequences.
It is a simple observation that p-regular sequences are stable under the process of shifts and under replacing M n for a finite number of indices, as long as log-convexity is kept.
We note in the context of Definition 1.5.1 that if M grows so fast that κ(p, M) = +∞ holds, then in particular the asymptotic estimate log M n = O(q −n ) fails as n → +∞ for any given q with 1 − p < q < 1. The second condition in (ii), which says that n log n ≤ (δ + o(1)) log M n , is a rather mild lower bound on log M n compared with this exponential growth along subsequences.
(b) For logarithmically convex sequences M, the sum is actually convergent to an extended real number in R ∪ {+∞}.
We conclude with a notion of regularity that applies in the regime with finite p-characteristic. 1.6. The three phases. We mentioned already the phenomenon that the Carleman classes exhibit the phase transition associated with quasianalyticity. Here, the concept of quasianalyticity is usually defined in terms of the unique continuation property that the (formal) Taylor series at any given point determines the function uniquely. Under the regularity condition that the sequence M = {M n } n is logarithmically convex, it is known classically that the Carleman class In the small exponent range 0 < p < 1 considered here, it turns out that we have actually two phase transitions: (i) the Douady-Peetre disconnexion barrier, and (ii) the quasianalyticity barrier.
Here, we shall attempt to explore both phenomena.
In the degenerate case when M n = 1 for n = 0, . . . , k and M n = +∞ for n > k, the Carleman space W p,θ M does not depend on the parameter θ ≥ 0, and is the same as the Sobolev space W p,k , except that it is equipped with another (but equivalent) quasinorm. So in this instance, we get the Douady-Peetre disconnexion phenomenon (1.2.1) for W p,θ M , while under the bounded p-characteristic condition (1.3.8), the following result shows that it does not happen.
To properly formulate the result, we consider the canonical mapping π : W p,θ It is natural to replace here the product space L p × L p × . . . by its linear subspace ∞ (L p , M) supplied with the standard quasinorm: Indeed, the linear mapping π : W p,θ M → ∞ (L p , M) then becomes an isometry. This is obvious for test functions in S p,θ,M , and, then automatically holds for elements of the abstract completion as well. We denote the n-th component projection of π by π n : π n ( f 0 , f 1 , f 2 , . . .) := f n . Theorem 1.6.1. (0 < p < 1) Suppose that the weight sequence M is logarithmically convex and meets the finite p-characteristic condition (1.3.8). Then, for each n = 0, 1, 2 . . ., π n maps W p,θ where ∂ stands for the differentiation operation. Moreover, the projection π 0 is injective, and, in the natural sense, the space equals the collection of test functions: W p,θ The proof of this theorem is supplied in Subsection 4.2. The conclusion of Theorem 1.6.1 is that under the the strong finite p-characteristic condition, W p,θ M is a space of smooth functions, and, indeed, it is identical with the test function class S p,θ,M . The situation is drastically different when κ(p, M) = +∞.
A sketch of the proof of Theorem 1.6.2 is supplied in Subsection 5.3. As for the formulation, the summand W p,θ 1 M 1 on the right-hand side arises as the space of "derivatives" of functions in W p,θ M . Here, the reason why M gets replaced by M 1 is due to a one-unit shift in the sequence space ∞ (L p , M). Moreover, the reason why θ gets replaced by θ 1 = θ/(1 − p) is the corresponding shift in the space of test functions S p,θ,M when we take the derivative. Remark 1.6.3. Since M 1 inherits all relevant properties from M, the theorem can be applied iteratively to obtain an isomorphism We briefly comment on the remaining transition, between non-quasianalyticity and quasianalyticity. For θ = 0, we characterize quasianalyticity for C p,θ M in terms of the quasianalyticity of the standard Carleman class C N for a certain related sequence N. Moreover, the classical Denjoy-Carleman theorem supplies criteria for when the class C N is quasianalytic or non-quasianalytic. The associated sequence N = {N n } n is given by which we recognize as coming from the L ∞ -bound of the higher order derivatives in (1.3.11).
The result runs as follows.
Theorem 1.6.4. (0 < p < 1) Assume that M is logarithmically convex and bounded away from zero. If κ(p, M) < +∞, the following holds: Finally, we comment on the dependence of the classes on the parameter θ in the smooth context of Theorem 1.6.1.
We do not know whether such a strict inclusion holds in the uncoupled regime when κ(p, M) = +∞. It remains a possibility that the spaces are then so large that the parameter θ is not felt. In any case, we are able to show that (Proposition 5.4.1) Remark 1.6.6. (a) In view of the Douady-Peetre disconnexion phenomenon, the fact that for 0 < p < 1, L p functions fail to define distributions is a serious obstruction. An alternative approach is to consider the real Hardy spaces H p in place of L p , since H p functions automatically define distributions. The drawback of that approach is that for p = 1, H 1 is substantially smaller than L 1 . Our theme here is to keep L p and to let a bootstrap argument (involving infinitely many higher order derivatives) take care of the smoothness, and to explore what happens when the bootstrap argument fails to supply appropriate bounds.
(b) A word on the title. The term a critical topology used here is borrowed from Beurling's work [5], where another phase transition is the object of study.
2. Sobolev spaces: Peetre's proof and failure of embedding 2.1. Sobolev spaces for 0 < p < 1. We fix a number p with 0 < p < 1 and an integer k ≥ 0. Following Peetre [18] we consider the Sobolev space W k,p = W k,p (R), defined as the abstract completion of C k 0 (R) with respect to the quasinorm The resulting space W k,p is then a quasi-Banach space. Here, C k 0 (R) denotes the space of compactly supported functions in C k (R).
Remark 2.1.1. In this paper we shall mostly work on the entire line. If at some place we consider spaces on bounded intervals, this will be explicitly mentioned. The definition of W k,p (I) for a general interval I is entirely analogous.
The space W k,p comes with two canonical mappings, α = α k : W k,p → L p , and δ = δ k : W k,p → W k−1,p . These are both initially defined for test functions f ∈ C k 0 (R) by α f = f, and δ f = f .
The mappings α and δ are bounded and densely defined, and hence extend to bounded operators on the entire space W k,p .
2.2. Douady-Peetre disconnexion. We begin this section by considering a simple example which explains how a crucial feature differs in the setting of 0 < p < 1 as compared to the classical Sobolev space case.
We are used to thinking of W k,p as being a certain subspace of L p , consisting of functions that are sufficiently smooth. As mentioned in the introduction, the first observation, made by Douady, is that this is not the right way to think when 0 < p < 1. Indeed, the canonical map α is not injective.
This would suggest that there ought to exist functions that vanish identically but nevertheless the derivative equals the nonzero constant 1. This is of course absurd, and the right way to think about it is to realize that in the completion, the function and its derivative lose contact, they disconnect.
Proof. A small argument (see [18,Lemma 2.1]) shows that we are allowed to work with functions whose derivatives have jumps. We let { j } j be a sequence of positive reals, such that j j → 0 as j → +∞, and define f j on the interval [0, 1 and extend it periodically to R with period 1 j + . The resulting function f j will be a skewed saw-tooth function that rises slowly with slope 1 and then drops steeply. By differentiating f j , we have that Since f j assumes values between 0 and 1 j , it is clear that f j → 0 as j → +∞ in L ∞ and hence in L p . Within the interval [0, 1], there are at most j full periods of the function f j , which allows us to estimate In view of the above observations, f j → 0 while f j → 1, both in the quasinorm of L p , as j → +∞. In particular, { f j } j is a Cauchy sequence, and if we let f denote the abstract limit in the completion, we find that α f = 0 while δ f = 1.

The isomorphism and construction of the canonical lifts.
We fix an integer k = 1, 2, 3, . . . and an exponent 0 < p < 1. The space L p ⊕ W k−1,p consists of pairs (g, h), where g ∈ L p and h ∈ W k−1,p , and we equip it with the quasinorm From the definition of the norm (2.1.1), we see that the operator A : W k,p → L p ⊕ W k−1,p given by A f := (α f, δ f ) is an isometry. Indeed, for f ∈ C k 0 (I), we have that and this property survives the completion process. If A can be shown to be surjective, then it is an isometric isomorphism A : W k,p → L p ⊕ W k−1,p . Proceeding iteratively with W k−1,p , we obtain the desired decomposition, since clearly W 0,p = L p .
To obtain the surjectivity of A, we shall construct two canonical lifts, β : L p → W k,p and γ : W k−1,p → W k,p of α and δ, respectively. These are injective mappings, from L p and W k−1,p to W k,p , respectively, satisfying certain relations with α and δ. The properties of these are summarized in the following lemma (the notation id X stands for the identity mapping on the space X). The details of the construction are postponed until Section 3.2.
With this result at hand, the proof of the main theorem about the W k,p -spaces becomes a simple exercise.
Proof of Theorem 1.2.1. As noted above, it will be enough to show that the isometry A : W k,p → L p ⊕ W k−1,p given by A f := (α f, δ f ) is surjective. To this end, we pick (g, h) ∈ L p ⊕ W k−1,p . Then βg and γh are both elements of W k,p , and so is their sum It now follows from Lemma 2.3.1 that A f = (α(βg + γh), δ(βg + γh)) = (αβg + αγh, δβg + δγh) = (g, h).
As a consequence, A is surjective, and hence A induces an isometric isomorphism W k,p L p ⊕ W k−1,p .
By iteration of the same argument, the claimed decomposition of W k,p follows. Remark 2.3.2. The lift γ does not appear in Peetre's work [18]. Reading between the lines one can discern its role, but here we fill in the blanks and treat it explicitly.

A collection of smooth functions by iterated convolution.
For an integer k ≥ 0 and a real 0 ≤ α ≤ 1 let C k,α denote the class of k times continuously differentiable functions, whose derivative of order k is Hölder continuous with exponent α. Given two functions f, g ∈ L 1 (R), their convolution f * g ∈ L 1 (R) is as usual given by For a > 0, we let the function H a denote the normalized characteristic function H a = a −1 1 [0,a] . For a decreasing sequence of positive reals a 1 , a 2 , a 3 , . . ., consider the associated repeated convolutions (for j ≤ k) The function Φ j,k then has compact support [0, a j + · · · a k ] and belongs to the smoothness class C k− j,1 which means that the derivative of order k − j is Lipschitz continuous. We will assume that the sequence a 1 , a 2 , a 3 , . . . decreases to 0 at least fast enough for (a j ) j≥1 ∈ 1 to hold. Then we may form the limits Φ j,∞ := lim k→+∞ Φ j,k , j = 1, 2, 3, . . . , and see that each such limit Φ j,∞ is C ∞ -smooth with support [0, a j + a j+1 + . . . ]. Moreover, we have the sup-norm controls we may calculate the higher order derivatives by the formula interpreted when needed in the sense of distribution theory. Here, we should ask that j+n ≤ k+1 in the first formula. By calculation, which when expanded out is the sum of delta masses at 2 n (generically distinct) points, each with mass (a j · · · a j+n−1 ) −1 . By the convolution norm inequality f * g ∞ ≤ f 1 g ∞ , where the L 1 norm may be extended to the finite Borel measures, we have that where we used the estimate (3.1.2). The analogous estimate holds for k = ∞ as well: j,∞ ∞ ≤ 2 n a j · · · a j+n .
We need to estimate the L p -norm of the function Φ (n) j,∞ as well. The standard norm estimate for convolutions is f * g q ≤ f 1 g q which holds provided that 1 ≤ q ≤ +∞. For our small exponents 0 < p < 1 this is no longer true. However, there is a substitute, provided f is a finite sum of point masses: for some finite collection of reals x j . This follows immediately from the p-triangle inequality and the translation invariance of the L p -norm. In our present context we see that where n + j ≤ k + 1. Correspondingly for k = +∞ we have that

Existence of invisible mollifiers in W k,p .
We now employ the repeated convolution procedure of Section 3.1, to exhibit mollifiers with L p -vanishing properties. Lemma 3.2.1 (invisibility lemma). Let k ≥ 0 be an integer. Then for any given > 0, there exists a non-negative function Φ ∈ C k,1 (R) such that Proof. For positive decreasing numbers {a j } k+1 j=1 (to be determined), let Φ = Φ 1,k+1 be given by (3.1.1). Then clearly Φ ∈ C k,1 (R) with Φ ≥ 0 and R Φ dx = 1. Moreover, the support of Φ equals [0, a 1 + · · · + a k+1 ]. By the estimate (3.1.4), it follows that We need to show that the finite sequence {a l } k+1 l=1 may be chosen such that the sum of the righthand side in (3.2.1) over 0 ≤ n ≤ k is bounded by p , while at the same time k+1 l=1 a l ≤ . As a first step, we assume that a j+1 ≤ 1 2 a j for integers n ≥ 0, and observe that it then follows that k+1 l= j a l ≤ 2a j . Consequently, we get that supp Φ ⊂ [0, 2a 0 ] and We put It then follows that 2 n+1 (a 0 · · · a n ) p ≤ p k + 1 , n = 0, . . . , k.
3.3. The definition of the lift β for 0 < p < 1. The lift β maps boundedly L p → W k,p , and we need to explain how it gets to be defined. Let F denote the collection of step functions, which we take to be the finite linear combination of characteristic functions of bounded intervals, and when also equip it with the quasinorm of L p , we denote it by F p := (F , · p ). We note that F p is quasinorm dense in L p . We first define βg for g ∈ F p . For g ∈ F p , we will write down a W k,p -Cauchy sequence {g j } j of test functions g j ∈ C k 0 (R), and declare βg ∈ W k,p to be the abstract limit of the Cauchy sequence g j as j → +∞.
We will require the following properties of the test functions g j : These properties uniquely determine βg for g ∈ F . Indeed, ifg j were another Cauchy sequence satisfying (3.3), then {g j } j and {g j } j are equivalent as Cauchy sequences, in light of In particular, β : F p → W k,p is then a well-defined bounded operator, and since F p is dense in L p it extends uniquely to a bounded operator β : L p → W k,p which is actually an isometry.
In view of the above, it will be enough to define βg when g is the characteristic function of an interval g = 1 [a,b] and to check (3.3.1) for it, since general step functions in F are obtained using finite linear combinations.
j=0 be a sequence of numbers tending to zero with 0 < j < 1 2 (b − a). By Lemma 3.2.1 applied to W k−1,p , there exists non-negative functions Φ j ∈ C k−1 , such that We define g j by convolution: g j := Φ j * 1 [a,b] . It is then clear that g j − g has support on [a, a + j ] ∪ [b, b + j ], and there, it is bounded by 1 in modulus. As a consequence, g j − g p p ≤ 2 j , so g j → g in L p . Next, we consider the derivative g j , which we may express as g j = (τ a − τ b )Φ j , where we recall that τ with subscript is a translation operator. It is clear that Consequently, g j p k−1,p also tends to zero, as needed. This establishes (3.3.1). Proof of Lemma, part 1. We show that αβ = id L p and δβ = 0. Since βg ∈ W k,p is the abstract limit of the Cauchy sequence g j with (3.3.1), and by definition αg j = g j and δg j = g j , it follows from (3.3.1) that αβg = g and δβg = 0 for every g ∈ L p . The assertion follows.

The lift γ.
To construct γ, we let g ∈ W k−1,p be an arbitrary element, which is by definition the abstract limit of some Cauchy sequence {g j } j , where g j ∈ C k−1 0 (R). For any given > 0, Lemma 3.2.1 provides a function Φ ∈ C k−1 0 (R) with Φ ≥ 0, Φ R := R Φ (t)dt = 1, supported in [0, ], while at the same time, Φ k−1,p < . We use the functions Φ j to modify each g j (x) to have vanishing zeroth moment, by defining where the j are chosen to tend to zero so quickly that Next, we define the functions u j as primitives: Then asg j has integral 0, we see that u j ∈ C k 0 (R). We put f j := u j − βu j ∈ W k,p , and observe that from the known properties of β, it follows that Then, from the isometry of A : W k,p → L p ⊕ W k−1,p (see (2.3.1)), we have that where in the last step, we applied the p-triangle inequality. A similar verification shows that { f j } j is a Cauchy sequence, so that it has a limit γg := lim j→+∞ f j in W k,p . Moreover, in view of (3.4.1), it follows that in L p and W k−1,p , respectively. In the construction of the sequence of functions f j , there is some arbitrariness e.g. in the choice of the sequence of the j (they were just asked to tend to 0 sufficiently quickly). To investigate whether this matters, we suppose another Cauchy sequence {F j } j in W k,p is given, with properties that mimic (3.4.1): that αF j = 0 in L p , and that for some Cauchy sequence {G j } j in W k−1,p converging to g ∈ W k−1,p , we know that δF j = G j , then by the isometric properties of A. If we let F denote the abstract limit of the Cauchy sequence F j in W k,p , we conclude that F = f in W k,p . In conclusion, it did not matter whether we used the prescribed Cauchy sequence or a competitor, and hence γg ∈ W k,p is well-defined for g ∈ W k−1,p . Finally, we observe from (3.4.2) that γg k,p ≤ g k−1,p , g ∈ W k−1,p , which makes γ : W k−1,p → W k,p a linear contraction.
Proof of Lemma, part 2. We show that the remaining properties, those involving γ: αγ = 0 and δγ = id W k−1,p . We read off from (3.4.1) that αγg = 0 and δγg = g for g ∈ W k−1,p , which does it.  [11] states that the L p -spaces for 0 < p < 1 have trivial dual. It follows from Douady-Peetre's isomorphism W k,p L p ⊕ · · · ⊕ L p that the same is true for the Sobolev spaces W k,p . We note here that any space that could be realised as a space of distributions would necessarily admit non-trivial bounded functionals (the test functions, for instance). Hence it follows that the elements of W k,p cannot even be interpreted as distributions.

Classes of test functions.
Although we mainly focus on the classes S p,θ,M , we also mention the Hermite class S Her of weighted polynomials where Pol(R) denotes the linear space of all polynomials of a real variable. We show that S Her consists of (p, 0)-tame functions. To get a class which fits into the associated L p -Carleman class, we also intersect the Hermite class with the space { f ∈ C ∞ (R) : f p,M < +∞} to obtain the S Her p,M . Proof. Let f ∈ S Her . Then f (x) = q(x)e −x 2 for some polynomial q ∈ Pol(R). Let d be the degree of q, and assume that all coefficients of the polynomial q(x) are bounded by some number m = m q . Let L be the operator on Pol(R) defined by the relation It follows that the Taylor coefficients Lq(j) of Lq can be estimated rather crudely in terms of d and m q : and the coefficients vanish for j > d + 1, so that the degree of Lq is at most d + 1. By repeating the same argument, with Lq in place of q, we obtain and, by iteration of (4.1.1), it follows more generally that for j = 0, 1, 2, . . ., where we use the the standard Pochhammer notation (x) k = x(x + 1) · · · (x + k − 1) for x ∈ R. We proceed to estimate the L ∞ -norm of f (n) . By the way L was defined, we may estimate By trivial calculus, the supremum on the right-hand side is attained at |x| = j/2, and this we may implement in the above estimate while we recall the coefficient estimate (4.1.2): where in the last step we just estimated by the number of terms multiplied by the largest term. Next, we take logarithms and apply elementary estimates to arrive at As the expression in brackets is is of growth order O (n + d) log(n + d) , it follows that the right-hand side expression tends to 0 as n → +∞. It is then immediate that f is (p, 0)-tame, and since f was an arbitrary element f ∈ S Her , the claim follows.
Next, we turn to the question of what is required of the weight sequence M in order for S Her p,M to contain non-trivial functions. We thus need to estimate the L p -norms of f (n) for f ∈ S Her . By performing the same estimates as in the above proof and by appealing to the p-triangle inequality (which says that (a + b) p ≤ a p + b p for 0 < p ≤ 1 and positive a and b), we see that The integrals on the right-hand side are easily evaluated: R |x| p j e −px 2 dx = p −(p j+1)/2 Γ((p j + 1)/2), so that the above gives the estimate Ignoring the specific constants, we find that where α > 0 and β > 1 are some constants, and hence it follows that f (n) p = O (n!) 3/2 . We do not proceed to analyse in full detail what M = {M n } should have to fulfill in order for S Her p,M to be a meaningful test class, but we note that the above implies that S Her p,M = S Her if e.g. M = {M n } n meets M n ≥ (n!) σ for any σ ≥ 3/2, which we understand as a Gevrey class condition.
4.2. The smooth regime: Theorem 1.6.1. We have already presented the bootstrap argument, which is the main ingredient in the proof of Theorem 1.6.1, in the introduction. We proceed to fill in the remaining details. and passing to the completion the mapping extends to an isometry π : W p,θ M → ∞ (L p , M). In particular, the mapping π is injective. We recall that we think of an element f ∈ W p,θ M as an abstract limit of a Cauchy sequence { f j } j in the norm · p,M , with f j ∈ S p,θ,M . For such an element, the mapping is obtained by taking the L p -limit in all coordinates (that is, the sequence of higher order derivatives). This is well-defined, since if we were to take two Cauchy sequences { f j } and {f j } with the same abstract limit f ∈ W where the right hand side tends to zero as j, k → +∞. It is now pretty obvious that under the finite p-characteristic condition κ(p, M) < +∞, each function f (n) j has a limit g n ∈ C(R) as j → +∞. Moreover, as all the derivatives converge uniformly, we conclude that g n = g n−1 = . . . = g (n) 0 for each n = 0, 1, 2, . . ., and hence that ∂π n f = π n+1 f, n = 0, 1, 2, . . . .
Finally, we show that the given test class is stable under the completion. We already saw that the limiting functions are of class C ∞ , and naturally f p,M < +∞ for each f ∈ W p,θ M . What remains is to show that lim sup However, the bound (1.3.11) tells us that By Remark 1.5.2 (b), it follows that and hence lim sup as claimed.
5. Independence of derivatives 5.1. Some notation and preparatory material. We previously considered the mapping π : f → ( f, f , f , . . .). To conform with notation introduced earlier in this paper as well as in the work of Peetre, we will use the letter α instead of π 0 to refer to the first coordinate map α : W p,θ M → L p . Likewise, we write δ for the mapping W As we saw in connection with the Sobolev W k,p -spaces for 0 < p < 1, the key step was the construction of the two lifts β and γ. If we may find the analogous lifts in the present setting, the rest of the proof will carry over almost word for word from the Sobolev space case. It turns out that the lifts β and γ are built in the same manner as before, using the existence of invisible mollifiers Φ analogous to the case of W k,p (compare with Lemma 3.2.1), with slight technical obstacles in the construction of γ, related to the unbounded support of test functions.
Before turning to the mollifiers, we make an observation regarding the weight sequence. We will make a dichotomy between the cases  For more complete details of this proof, see [4], as well as Behm's thesis [3], where the former paper is discussed.
Proof sketch. Let c 0 = lim inf n→+∞ (1 − p) n log M n > 0. Then We now put where the positive constant C 1 is chosen such that N n ≤ M n for all integers n ≥ 0. It follows that N = {N n } n is p-regular with κ(p, N) = +∞, and that (1 − p) n log N n = o(1), as needed.

The construction of mollifiers.
The invisibility lemma runs as follows.
Lemma 5.2.1 (invisibility lemma for L p -Carleman spaces). Assume that M is p-regular and has infinite p-characteristic κ(p, M) = +∞. For any > 0 there exists a non-negative function Φ ∈ S p,0,M (with θ = 0) such that The tools needed to prove the lemma were developed back in Section 3.1, in connection with the Invisibility Lemma for W k,p . For a sequence {a l } ∞ l=1 we form the associated convolution product Φ 1,∞ . We recall that Φ 1,∞ has support [0, l≥1 a l ], and enjoys the estimates (3.1.3) and (3.1.5), provided that (a j ) j≥1 is a decreasing 1 -sequence.
In order to bound the support of Φ (n) , will need to estimate the sums j≥n+1 a j . If we write a j = c j α j and ask for {α j } j to be a decreasing sequence of positive numbers, and that {c j } j ∈ 1 , we obtain the bound where c = {c j } j 1 . Hence the right-hand side of (3.1.5) may be estimated further: We summarise what we ask for a sequence a = {a l } l = {c l α l } l to satisfy in to guarantee that Φ 1,∞ = Φ 1,∞,a satisfies the conclusion of the lemma. First, for the estimates (3.1.3) and (3.1.5) to come into play, we need (5.2.1), which is satisfies as soon as {c j } j ∈ 1 and {α j } j is decreasing. The (p, 0)-tameness is, in view of (3.1.3), ensured by (5.2.2) lim sup n→∞ (1 − p) n log 1 a 1 · · · a n ≤ 0.
In order to verify the norm control Φ 1,∞ p,M < , we shall require Moreover, the assertion that supp Φ 1,∞ ⊂ [0, ] is equivalent to having a 1 ≤ , which follows from the requirement α 1 ≤ /c.
Let us briefly discuss the structural consequences for sequences {α l } l satisfying a requirement and recall that we want a l,k = c l α l,k . Since M is assumed to be logarithmically convex, It follows that {a l } l is a decreasing sequence, so since {c l } l ∈ 1 , the above discussion is applicable.
In particular, the support of Φ is included in [0, cα 1,k ], which we may calculate as Since κ(p, M) = +∞ and since M k+1 ≥ 1 for large enough k, this tends to zero as k → +∞.
To verify that Φ is (p, 0)-tame, we observe that Proof. It is enough to demonstrate the result for θ = 0, since the general case then follows by inclusion. The definition procedure remains the same as that in Section 3.3. We denote by F p the space of test functions, and to each g ∈ F p we aim to associate a Cauchy sequence {g j } j ⊂ S p,0,M such that We proceed to define a partition of R into intervals I n := [η n , η n+1 ] with . . . ≤ η −1 ≤ η 0 ≤ η 1 ≤ . . ., such that n∈Z |I n |( f osc I n ) p .
To see how this can be done, observe that f = g is uniformly bounded, so f is uniformly continuous. Starting with with the partition induced by Z, i.e. I 1 m,1 = [m, m + 1], where m ∈ Z, we aim to repeatedly cut intervals in half until the requirement is satisfied. For any interval I 1 m,1 , if ( f osc we do nothing. If this inequality fails, consider the dyadic children {I k m, j } 2 k j=1 of generation k of |I k m m, j | ≤ 3 .
We may then order the intervals and index them by the single index n ∈ Z, so that the right end-points are increasing. This is our partition I n .
Indeed, choosing first the sequence {δ n } n small enough in 1 to ensure the first summability property while requiring δ n < 1, and then choosing n ≤ δ p −1 n , we may then use Lemma 5.2.1 and a simple translation to find functions Φ n,1 and Φ n,2 supported on the right intervals with the right norm bounds. Let ϕ(x) denote the function gives by where λ n = 1 |I n | I n f (t)dt, i = 1, 2.
Define u ∈ C ∞ (R) by This proves our assertion.
6. The quasianalyticity transition and the parameter θ 6.1. A remark on the case 1 ≤ p < +∞. It is of course a basic question is if the Denjoy-Carleman theorem remains valid in the setting of the L p -Carleman Classes in the parameter range 1 ≤ p < +∞ (without any tameness requirement of course). This is of course true and should be well-known, but we have unfortunately not been able to find a suitable references for this fact. For this reason, we supply a short self-contained presentation. We begin with the following lemma. it follows that N n,θ N n+1,θ ≤ e −p 2 (1−p) −n−1 N n−1,θ N n,θ .
S Her p,M = f (x) = e −x 2 p(x) : p a polynomial such that f p,M < +∞ which was mentioned earlier, or, in the case of the unit circle, the class P trig of trigonometric polynomials, and ask whether the corresponding L p -Carleman spaces undergo the same phase transitions.