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Introduction
Since Konyagin and Temlyakov's seminal paper [22] was released, several types of bases that can be characterized by combining derived forms of unconditionality and democracy have appeared when studying the thresholding greedy algorithm from a functional-analytic point of view. Apart from greedy bases, the most relevant types of bases that have emerged within this area of research are almost greedy ones and quasi-greedy ones. Almost greediness is a weakened form of greediness and, in turn, quasi-greediness is a weakened form of unconditionality. In some sense, the role played by quasi-greediness within the study of almost greedy bases runs parallel to the role played by unconditionality within the study of greedy ones. In fact, while a basis is greedy if and only if it is unconditional and democratic [22], a basis is almost greedy if and only if it is quasi-greedy and democratic (see ([14], Theorem 3.3) and ( [6], Theorem 6.3)).
An intrinsic characteristic of democratic bases is its fundamental function. So, from a functional analytic point of view, it is very natural to ask in which way the geometry of the space affects the fundamental functions of almost greedy bases and greedy ones. As far as greedy bases are concerned, this topic connects with that of uniqueness of greedy bases in Banach spaces. Suppose that a Banach space has a greedy basis and that any greedy basis of is equivalent to a permutation of B . Then, obviously, all the greedy bases of have, up to equivalence, the same fundamental function. Within the class of Banach spaces with a unique (up to equivalence and permutation) greedy basis, we must differentiate two disjoint subclasses. On the one hand, if has a unique (semi-normalized) unconditional basis B that is democratic, then B also is the unique greedy basis of . On the other hand, there are Banach spaces with non-democratic semi-normalized unconditional bases and a unique greedy basis. To the former subclass belong the spaces 1 , 2 , c 0 , the Tsirelson space T , and the 2-convexified Tsirelson space T (2) (see [11,12,23,24,26]). To the latter subclass belong certain Orlicz sequence spaces near either to 1 or to 2 and the separable part of weak p for 1 < p < ∞ (see [3]). If we broaden the scope of the study to nonlocally quasi-Banach spaces, the former subclass enlarges considerably. In fact, given p ∈ (0, 1) , p , Hardy space H p ( ) , Lorentz sequence spaces p,q for 0 < q ≤ ∞ , and Orlicz sequence spaces F for a concave Orlicz function F belong to it (see [19,21,30]).
There also exist Banach spaces without a unique greedy basis in which all greedy bases have the same fundamental function. For instance, the sequence space p for p ∈ (1, 2) ∪ (2, ∞) has a continuum of mutually permutatively non-equivalent greedy bases (see [15,29]), and the fundamental function of all of them grows as More generally, it is known that given 0 < p ≤ ∞ , the fundamental function of any super-democratic (that is, democratic and unconditional for constant coefficients) basis of p (we replace p with c 0 if p = ∞ ) is equivalent to p (see ( [5], Proposition 4.21)). Therefore, the fundamental function of any almost greedy basis of p is equivalent to p . We draw reader's attention that any super-democratic basis whose fundamental function is equivalent to ∞ is equivalent to the unit vector system of c 0 . This observation gives that c 0 has a unique almost greedy basis. In contrast, 2 and p for 0 < p ≤ 1 , despite having a unique greedy basis, have a continuum of mutually permutatively non-equivalent almost greedy bases (see ( [15], Theorem 3.2) and ( [8], Corollary 6.2)). As far as quasi-greedy bases are concerned, the main structural difference between the spaces p for p ∈ (0, 1] ∪ {2, ∞} and the spaces p for p ∈ (1, 2) ∪ (2, ∞) is that, unlike the former, the latter spaces have non-democratic quasi-greedy bases (see [9,13,16,31]).
In p -spaces, the fundamental functions of greedy bases behave like those of almost greedy ones. However, a priori, the geometry of the space provides less information on the fundamental function of almost greedy bases than that it does on the fundamental function of greedy bases. An important example of this situation is the Lebesgue L p = L p ([0, 1]) for 1 < p < ∞ . Since any unconditional basis of L p possesses a subbasis equivalent to the unit vector system of p (see [18]), the fundamental function of any greedy basis of L p grows as p . In this sense, the behavior of greedy bases in L p , 1 < p < ∞ , runs parallel to that of greedy bases in p . This parallelism breaks down when dealing with almost greedy bases. To see this, we bring up Nielsen' paper [27], in which the author constructs a uniformly bounded orthogonal system of L 2 that is an almost greedy basis for L p for each 1 < p < ∞ . Although not explicitly stated by the author, the proof of ( [27], Theorem 1.4) gives that the fundamental function of the achieved basis of L p grows as 2 . This fact is not casual: by [4, Proposition 2.5], any quasi-greedy basis Ψ of L p with sup n ‖ n ‖ ∞ < ∞ is democratic with fundamental function of the same order as 2 . Hence, L p , 1 < p < ∞ and p ≠ 2 , has almost greedy bases whose fundamental functions grow differently. As for L 2 , we point out that any quasi-greedy basis of a Hilbert space is democratic with fundamental function equivalent to 2 (see ( [31], Proof of Theorem 3)). As the case p = 1 is concerned, we bring up [16,Theorem 4.2], which implies that all quasi-greedy bases of L 1 and 1 are democratic with fundamental function of the same order as 1 . Moreover, by ( [15], Theorem 3.2), there is a continuum of mutually non-permutatively equivalent quasi-greedy bases of L 1 . Thus, in some sense, almost greedy bases in L 1 and 1 behave similarly. We also notice that since L 1 has no unconditional basis [24], it has no greedy basis either.
Once realized that, for 1 < p < ∞ and p ≠ 2 , there are almost greedy bases of L p with essentially different fundamental functions, the question should be determine what functions are possible fundamental functions of an almost greedy basis of L p . The Radamacher type and cotype of the space shed some information in this respect. In fact, the fundamental function, say , of any unconditional for constant coefficients basis of any Banach space of type r and cotype s satisfies (see, e.g., ( [1], Proof of Lemma 2.5)). In particular, any almost greedy basis of L p satisfies (1) with r = min{2, p} and s = max{2, p} . As above explained, it is known that the ends of this range are possible fundamental functions. Oddly enough, as we shall prove, these extreme functions are, up to equivalence, all fundamental functions possible for almost greedy bases of L p .
If is the fundamental function of an almost greedy basis of L p , then there is r ∈ {2, p} such that ≈ r .
We close this introductory section by briefly describing the structure of the paper. Section 3 revolves around the proof of Theorem 1.1. In Sect. 2, we settle the terminology we will use, and we collect some auxiliary results that we will need.

Background and terminology
Although we are mainly interested in Banach spaces, as the theory of greedy-like bases can be carry out for (not necessarily locally convex) quasi-Banach spaces (see [6]), we will state the results we record in this section in this more general framework. All of them are essentially known. Nonetheless, for the reader's ease, we (1) s ≲ ≲ r will sketch some proofs. We use standard terminology on Functional Analysis and greedy-like bases as can be found in [2] and the aforementioned paper [6]. For clarity, however, we record the notation that is used most heavily.
The symbol j ≲ j for j ∈ J means that there is a positive constant C < ∞ such that the families of non-negative real numbers ( j ) j∈J and ( j ) j∈J are related by the inequality j ≤ C j for all j ∈ J . If j ≲ j and j ≲ j for j ∈ J we say ( j ) j∈J are ( j ) j∈J are equivalent, and we write j ≈ j for j ∈ J.
Let be a quasi-Banach space over the real or complex scalar field , and let X = (x n ) ∞ n=1 be a linearly independent sequence that generates the whole space . For a fixed sequence = ( n ) ∞ n=1 ∈ ℕ , let us consider the map The sequence X is an unconditional basis if and only if S is well defined on for all ∈ ∞ , and ≈ 1 for j ∈ J . For convenience, we will adopt a definition of basis that implies that only semi-normalized Schauder bases become bases. A basis of will be norm-bounded sequence X = (x n ) ∞ n=1 that generates whole space , and for which there is a (unique) normbounded sequence X * ∶= (x * n ) ∞ n=1 in the dual space * , called the dual basis of X , such that (x n , x * n ) ∞ n=1 is a biorthogonal system. A basic sequence will be a sequence in which is a basis of its closed linear span. Notice that, according to our terminology, any basic sequence is semi-normalized.
Let X = (x n ) ∞ n=1 and Y = (y n ) ∞ n=1 be bases of quasi-Banach spaces and , respectively. We say that X C-dominates Y if there is a linear map T ∶ → such that ||T|| ≤ C and T(y n ) = x n for all n ∈ ℕ . If the constant C is irrelevant, we simply n a n x n .
drop it from the notation. If X dominates Y and, in turn, Y dominates X , we say that the bases are equivalent. Given a basis X = (x n ) ∞ n=1 of a quasi-Banach space , the mapping is a bounded linear operator from into c 0 , hence for each m ∈ ℕ there is a unique and n < j . The mth greedy approximation to f ∈ with respect to the basis X is The basis X is said to be almost greedy if there is a constant C such that We say that X is greedy if it satisfies the more demanding condition The basis X is said to be quasi-greedy if the TGA with respect to it is uniformly bounded. Equivalently, X is quasi-greedy if and only if there is a constant C ≥ 1 such that In turn, we say that X is truncation quasi-greedy if the restricted truncation operators are uniformly bounded, i.e., there is a constant C such that Semi-normalized unconditional bases are a special kind of quasi-greedy bases, and although the converse is not true in general, quasi-greedy basis always retain in a certain sense a flavor of unconditionality. For example, any quasi-greedy basis is truncation quasi-greedy (see ( [6], Theorem 4.13)). In turn, if the basis X = (x n ) ∞

Lemma 2.1 Let s = (s m ) ∞ m=1 be an essentially increasing sequence in (0, ∞) . Then, s has the URP if and only if there is a constant C such that the weight (1∕s m ) ∞ m=1 satisfies the Dini condition
A weight will be a sequence w = (w n ) ∞ n=1 of non-negative scalars with w 1 > 0 . The primitive sequence (s m ) ∞ m=1 of the weight w is defined by s m = ∑ m n=1 w n for all m ∈ ℕ . Given 0 < q ≤ ∞ , we will denote by d 1,q (w) the Lorentz space consisting of all sequences f ∈ c 0 whose non-increasing rearrangement of (b n ) ∞ n=1 satisfies where u = (w n ∕s n ) ∞
Given  There is also a connection between bidemocratic bases and truncation quasi-greedy ones.

Proposition 2.3 ([6] [Proposition 5.7])
Let X be a bidemocratic basis of a quasi-Banach space. Then X and X * are truncation quasi-greedy, hence UCC and super-democratic.
Next, we bring up a partial converse of Proposition 2.3. We say that a quasi-Banach space has Rademacher type (respectively cotype) r, 0 < r < ∞ , if there is a constant C such that for any finite family (f j ) j∈A in , being we have A ≤ CS (resp., S ≤ CA ). Since the optimal type (resp., cotype) of the scalar field is 2, if a nonzero quasi-Banach space has type (resp., cotype) r, then r ≤ 2 (resp., r ≥ 2 ). Given 0 < p < ∞ and a measure space (Ω, Σ, ) such that the dimension of the vector space consisting of al integrable simple functions is infinite, the optimal type of L p ( ) is min{2, p} , and its optimal cotype is max{2, p}.
Since any quasi-Banach space with a Rademacher type larger than one is locally convex ( [20], Theorem 4.1), the following result lies within the theory of Banach spaces.
If we choose k large enough, so that CT r k 1∕r−1 ≤ 1∕2 , taking the supremum on A we Our approach also leads to settling the structure of subsymmetric basic sequences of L p , p > 2 . Although this result is probably well known to specialists, we make a detour on our route to record it. Recall that a sequence in a Banach space is said to be a subsymmetric basis if it is an unconditional basis and it is equivalent to all its subsequences. All we need to know about this important class of bases is the following. Lemma 3.10 (see [7] [Lemma 2.2]) Let X = (x n ) ∞ n=1 be a subsymmetric basis of a quasi-Banach space. Let (n j ) ∞ j=1 and (m j ) ∞ j=1 in ℕ be sequences in ℕ with n j < m j < n j+1 for all j ∈ ℕ . Then, (x n j − x m j ) ∞ n=1 is equivalent to X.

Theorem 3.11
Let Ψ be a subsymmetric basic sequence in L p , 2 ≤ p < ∞ . Then, Ψ is equivalent the unit vector system of either 2 or p .
Proof Just combine Lemma 3.10 with Lemma 3.

◻
A Banach space is said to be an L p -space, 1 ≤ p ≤ ∞ , if for every finite-dimensional subspace ⊆ there is a further finite-dimensional subspace ⊆ ⊆ whose Banach-Mazur distance to dim( ) p is uniformly bounded. It is known that, in the case when 1 < p < ∞ , is a separable L p -space if and only if it is isomorphic to a non-Hilbertian complemented subspace of L p (see [25]).
Given 1 ≤ p ≤ ∞ , we denote by p ′ its conjugate exponent defined by Proof If 2 ≤ p < ∞ the result follows from combining Proposition 2.3 with Corollary 3.9. Assume that 1 < p < 2 . Then, by Proposition 2.3, X * is a super-democratic basis of a Banach space isomorphic to a subspace of L * p . Since L * p is isometrically isomorphic to L p ′ , there is s ∈ {2, p � } such that u [X * , * ] ≈ s . Consequently, by Equation (5), u [X, ] ≈ r for some r ∈ {2, p � } � = {2, p} . ◻ Now, we obtain Theorem 1.1 as a consequence of the following more general result. Theorem 3.13 Let 1 < p < ∞ , and let be the fundamental function of a super-democratic basis of an L p -space. Then, ≈ r , where either r = p or r = 2.
Proof Just combine Corollary 2.6 with Corollary 3.12. ◻ We close the paper by exhibiting an application of Theorem 1.1. Let be a symmetric space, i.e., a quasi-Banach space ⊆ ℕ for which the unit vector system is a symmetric basis. We say that a quasi-Banach space with a basis X embeds in via X , and we write X ↪ , if X dominates unit vector system of . In the reverse direction, we say that embeds in via X , and we write X ↪ , if the unit vector system of dominates X . Let 1 and 2 be symmetric spaces whose unit vector systems have equivalent fundamental functions. Squeezing the Banach space as is a condition that guarantees in a certain sense the optimality of the compression algorithms with respect to the basis X . Besides, such embeddings serve in some situations as a tool to derive other properties of the basis X . We refer the reader to [6, Section 9] for details. Corollary 3.14 Let Ψ be an almost greedy basis of L p , 1 < p < ∞ . Then, there are r ∈ {2, p} and 1 < q < s < ∞ such that