Interpolation of Gentle Spaces

Abstract

The notion of gentle spaces, introduced by Jaffard, describes what would be an “ideal” function space to work with wavelet coefficients. It is based mainly on the separability, the existence of bases, the homogeneity, and the γ-stability. We prove that real and complex interpolation spaces between two gentle spaces are also gentle. This shows the relevance and the stability of this notion. We deduce that Lorentz spaces 𝐿^𝑝,𝑞 and 𝐻^𝑝,𝑞 spaces are gentle. Further, an application to nonlinear approximation is presented.

1. Introduction

Interpolation is a powerful technique for proving continuity of linear operators. Let us recall some basic notions concerning interpolation between Banach spaces. Let 𝑋₀ and 𝑋₁ be two Banach spaces. We say that (𝑋₀,𝑋₁) is a compatible couple if 𝑋₀ and 𝑋₁ are continuously embedded in a common Hausdorff topological vector space ℋ (we write 𝑋_𝑖↪𝐻 for 𝑖=0,1).

Let (𝑋₀,𝑋₁) be a compatible couple. Then 𝑋₀∩𝑋₁ and 𝑋₀+𝑋₁ are Banach spaces under the norms 𝑓_{𝑋0∩𝑋1}=max𝑓_𝑋0,𝑓_𝑋1,𝑓_𝑋0+𝑋1=inf𝑓_0𝑋0+𝑓_1𝑋1∶𝑓=𝑓₀+𝑓₁, (1) where the infimum extends over all representations 𝑓=𝑓₀+𝑓₁ of 𝑓 with 𝑓₀∈𝑋₀ and 𝑓₁∈𝑋₁.

If (𝑋₀,𝑋₁) is a compatible couple, then a Banach space 𝑋 is said to be an intermediate space between 𝑋₀ and 𝑋₁ if 𝑋 is continuously embedded between 𝑋₀∩𝑋₁ and 𝑋₀+𝑋₁.

Let (𝑌₀,𝑌₁) be a second compatible couple. A linear operator 𝑇 defined on 𝑋₀+𝑋₁ and taking values in 𝑌₀+𝑌₁ is said to be admissible with respect to couples (𝑋₀,𝑋₁) and (𝑌₀,𝑌₁) if, for each 𝑖=0,1, the restriction of 𝑇 to 𝑋_𝑖 is a linear continuous operator from 𝑋_𝑖 into 𝑌_𝑖 (if 𝑋_𝑖=𝑌_𝑖, then 𝑇 is said to be admissible with respect to (𝑋₀,𝑋₁)). One looks for intermediate spaces 𝑋 and 𝑌 of the couples (𝑋₀,𝑋₁) and (𝑌₀,𝑌₁), respectively, such that every admissible operator 𝑇 maps 𝑋 into 𝑌. The pair (𝑋,𝑌) is called interpolation pair relative to (𝑋₀,𝑋₁) and (𝑌₀,𝑌₁). If 𝑋_𝑖=𝑌_𝑖 for 𝑖=0,1 and 𝑋=𝑌, then 𝑋 is called an interpolation space between 𝑋₀ and 𝑋₁.

Note that 𝑋₀, 𝑋₁, 𝑋₀∩𝑋₁, and 𝑋₀+𝑋₁ are examples of interpolation spaces between 𝑋₀ and 𝑋₁. Other examples can be constructed by several methods.

In 1926, Riesz found the first interpolation method for (𝐿^𝑝(𝑑𝜇),𝐿^𝑞(𝑑𝜇)). A generalized version was given by Thorin in 1939/1948 and is known as the Convexity Theorem of Riesz and Thorin or the Riesz-Thorin interpolation theorem. There are many extensions of this theorem. In this connection, we mention the Marcinkiewicz interpolation theorem (in 1939) which extends the Riesz-Thorin interpolation theorem to couples of weak 𝐿^𝑝-spaces and which was proved by Zygmund in 1956. In 1958, Stein and Weiss generalized the method for couples (𝐿^𝑝(𝑑𝜇),𝐿^𝑞(𝑑𝜈)) with different measures 𝜇 and 𝜈. At the end of 1958, Lions gave the first proof of the interpolation theorem for quadratic interpolation between Hilbert spaces. Since then several authors have introduced and developed different interpolation methods for couples of general Banach spaces. We mention here essentially two methods: the real interpolation method introduced by Lions and Peetre and the complex method developed by Lions, Calderón, and Krejn. In general, these methods lead to different interpolation spaces. Let us quote the example of usual complex spaces 𝐿^𝑝=𝐿^𝑝(ℝ^𝑑) of 𝑝-integrable functions and homogeneous Bessel-potential spaces 𝑊̇^𝑠,𝑝=𝑊̇^𝑠,𝑝(ℝ^𝑑). If we denote by (𝑋₀,𝑋₁)_[𝜃] and (𝑋₀,𝑋₁)_𝜃,𝑞, with 0<𝜃<1, the interpolation spaces obtained, respectively, by the complex and the real method, then we have (see for instance [1])(i)(𝐿^𝑝0,𝐿^𝑝1)_[𝜃]=𝐿^𝑝, 1≤𝑝₀, 𝑝₁≤∞ and 1/𝑝=((1−𝜃)/𝑝₀)+(𝜃/𝑝₁).(ii)(𝐿^𝑝0,𝐿^𝑝1)_𝜃,𝑞=𝐿^𝑝,𝑞, 1≤𝑝₀, 𝑝₁≤∞, 1/𝑝=((1−𝜃)/𝑝₀)+(𝜃/𝑝₁) and 𝑝₀<𝑞≤∞.(iii)(𝑊̇^𝑠,𝑝0,𝑊̇^𝑠,𝑝1)_𝜃,𝑝=𝑊̇^𝑠,𝑝, 𝑠∈ℝ, 1≤𝑝₀,  𝑝₁≤∞ and (1/𝑝)=((1−𝜃)/𝑝₀)+(𝜃/𝑝₁).(iv)(𝑊̇^𝑠0,𝑝,𝑊̇^𝑠1,𝑝)_𝜃,𝑞=𝐵̇_𝑝^𝑠,𝑞, 𝑠₀≠𝑠₁, 𝑠=(1−𝜃)𝑠₀+𝜃𝑠₁ and 1≤𝑝,  𝑞≤∞.(v)(𝑊̇^𝑠0,𝑝0,𝑊̇^𝑠1,𝑝1)_[𝜃]=𝑊̇^𝑠,𝑝, 𝑠₀≠𝑠₁, 𝑠=(1−𝜃)𝑠₀+𝜃𝑠₁ and (1/𝑝)=((1−𝜃)/𝑝₀)+(𝜃/𝑝₁).

Here, 𝐿^𝑝,𝑞=𝐿^𝑝,𝑞(ℝ^𝑑) and 𝐵̇_𝑝^𝑠,𝑞=𝐵̇_𝑝^𝑠,𝑞(ℝ^𝑑) are, respectively, Lorentz and homogenous Besov spaces (see [1–3]). The spaces 𝐿^𝑝(ℝ^𝑑) for 1<𝑝<∞, 𝑊̇^𝑠,𝑝(ℝ^𝑑) where 𝑠∈ℝ and 1<𝑝<∞, and 𝐵̇_𝑝^𝑠,𝑞(ℝ^𝑑) with 𝑠∈ℝ and 0<𝑝, 𝑞≤∞ are gentle spaces (see [4]). Nonetheless, inhomogeneous Besov spaces 𝐵_𝑝^𝑠,𝑞(ℝ^𝑑) are not gentle because the homogeneity property (the second requirement in Definition 1) is not verified. Recall that 𝐵_𝑝^𝑠,𝑞ℝ^𝑑=𝐿^𝑝ℝ^𝑑+𝐵̇_𝑝^𝑠,𝑞ℝ^𝑑𝐵if𝑠<0,1≤𝑝,𝑞≤∞,_{𝑝𝑠,𝑞}ℝ^𝑑=𝐿^𝑝ℝ^𝑑∩𝐵̇_𝑝^𝑠,𝑞ℝ^𝑑if𝑠>0,1≤𝑝,𝑞≤∞. (2) This means that the interpolation space between two gentle spaces is not always gentle (recall that, above, we said that 𝑋₀, 𝑋₁, 𝑋₀∩𝑋₁, and 𝑋₀+𝑋₁ are examples of interpolation spaces between 𝑋₀ and 𝑋₁). However, in the third section of this paper, we will prove that gentleness is stable by real and complex interpolation methods. In the next section, we give all the necessary recalls concerning these methods.

The notion of gentleness was introduced by Jaffard [5]. It describes what would be an “ideal” function space to work with wavelet coefficients, in “any” wavelet basis. This is the case for Sobolev spaces in PDEs (see [6, 7] for instance), and for Besov spaces in statistics, see [8]. Moreover, many signals and images are stored, denoised, or transmitted by their wavelet coefficients (see [9]). One often needs to obtain local or global information on signals or images by conditions bearing on the moduli of their wavelet coefficients. These conditions should be independent of the chosen wavelet basis.

Gentleness is based mainly on separability, existence of bases, homogeneity, and 𝛾-stability; let 𝒮(ℝ^𝑑) be the Schwartz space of all complex valued rapidly decreasing 𝐶^∞ functions on ℝ^𝑑. Set 𝒮_∞ℝ^𝑑=𝑓∈𝒮ℝ^𝑑∶_ℝ𝑑𝑥^𝛽𝑓(𝑥)𝑑𝑥=0∀𝛽∈ℕ^𝑑, (3) (where 𝑥^𝛽=𝑥^𝛽1₁⋯𝑥^𝛽𝑑_𝑑 for all 𝑥=(𝑥₁,…,𝑥_𝑑)∈ℝ^𝑑 and all 𝛽=(𝛽₁,…,𝛽_𝑑)∈ℕ^𝑑). By 𝒮^′_∞(ℝ^𝑑) we denote its topological dual (also called the space of tempered distributions modulo polynomials).

Definition 1. Let 𝐻∈ℝ. A function space 𝐸 is gentle of order 𝐻 if we have the following.(i)𝐸 is a Banach or a quasi-Banach space of distributions.(ii)𝐸 is homogeneous of order 𝐻; that is, there exists a constant 𝐶>0 such that for all 𝑓∈𝐸, all 𝑎∈ℝ^𝑑 and all 𝑟>0𝜏_𝑎𝑓_𝐸≤𝐶𝑓_𝐸,ℎ_𝑟𝑓_𝐸≤𝐶𝑟^𝐻𝑓_𝐸, (4) where 𝜏_𝑎 and ℎ_𝑟 are the shift and the dilation operators defined by 𝜏_𝑎𝑓(𝑥)=𝑓(𝑥−𝑎),ℎ_𝑟𝑓(𝑥)=𝑓(𝑟𝑥). (5)(iii)𝒮_∞(ℝ^𝑑)↪𝐸↪𝒮^′_∞(ℝ^𝑑).(iv)If 𝐸 is separable, then 𝒮_∞(ℝ^𝑑) is dense in 𝐸, and if 𝐸 is the dual of a separable space 𝐹, then 𝒮_∞(ℝ^𝑑) is dense in 𝐹.(v)There exists 𝛾>0 such that 𝐸 is 𝛾-stable.

The first requirement is explained in the following definition introduced by Bourdaud in [10]. Denote by 𝒟^′(ℝ^𝑑) the dual space of the space of all complex valued compactly supported 𝐶^∞ functions on ℝ^𝑑.

Definition 2. A Banach (resp., quasi-Banach) space of distributions is a vector subspace 𝐸 of 𝒟^′(ℝ^𝑑) endowed with a complete norm (resp., quasi-norm) such that the embedding 𝐸↪𝒟^′(ℝ^𝑑) is continuous.

Recall that a quasi-Banach space is a complete topological vector space endowed with a quasi-norm. A quasi-norm (see [1] page 59) satisfies the requirements of a norm except for the triangular inequality which is replaced by the weaker condition ∃𝐶>0;∀𝑥,𝑦∈𝐸,𝑥+𝑦_𝐸≤𝐶‖𝑥‖_𝐸+𝑦_𝐸. (6) We say that ‖⋅‖_𝐸 is a 𝑝-norm where 0<𝑝≤1 if in addition ∀𝑥,𝑦∈𝐸,𝑥+𝑦^𝑝_𝐸≤‖𝑥‖^𝑝_𝐸+𝑦^𝑝_𝐸. (7) Note that a quasi-norm is always equivalent to a 𝑝-norm (see [11]). The real Hardy spaces 𝐻^𝑝(ℝ^𝑑) and Besov spaces 𝐵̇_𝑝^𝑠,𝑝(ℝ^𝑑), with 𝑠∈ℝ and 0<𝑝<1, are quasi-Banach spaces.

We should point out that the definitions of intermediate and interpolation spaces carry over without change for a compatible couple of quasi-Banach spaces (see [12–14]).

In the second requirement in Definition 1, the shift invariance implies that the definition of pointwise regularity introduced by Jaffard in [5] (which describes how the norm of 𝑓 (properly renormalized by substracting a polynomial) behaves in small neighbourhoods of a given point 𝑥₀) is the same at every point, and the dilation invariance is implicit in pointwise regularity through scaling invariance. This requirement and the third one imply that Meyer-Lemarié wavelets [15] belong to 𝐸 since they are obtained by translations and dyadic dilations of a basic function (mother wavelet) in 𝒮_∞(ℝ^𝑑).

The fourth requirement shows that wavelet bases are either unconditional bases or unconditional ∗-weak bases of gentle spaces. Examples of nonseparable spaces for which wavelets are ∗-weak bases include homogeneous Hölder spaces 𝐶̇^𝑠(ℝ^𝑑) and, more generally, homogeneous Besov spaces 𝐵̇_𝑝^𝑠,𝑞(ℝ^𝑑) with 𝑝=∞ or 𝑞=∞.

The last requirement implies that the characterization of gentle space by wavelet coefficients does not depend on the particular 𝑟-smooth wavelet basis (see [15]) which is chosen for 𝛾<𝑟; let ℳ_𝛾 be the space of infinite matrices 𝑀(𝜆,𝜆^′) indexed by dyadic cubes 𝜆=𝑘2^−𝑗+[0,2^−𝑗[^𝑑 and 𝜆^′=𝑘^′2^−𝑗′+[0,2^−𝑗′[^𝑑 (where 𝑗,𝑗^′∈ℤ and 𝑘,𝑘^′∈ℤ^𝑑) and satisfying ∃𝐶>0∶∀𝜆,𝜆^′𝑀𝜆,𝜆^′≤𝐶𝜔_𝛾𝜆,𝜆^′, (8) where 𝜔_𝛾𝜆,𝜆^′=2^{−((𝑑/2)+𝛾)|𝑗−𝑗′|}1+𝑗−𝑗^′21+2^{inf{𝑗,𝑗′}}𝑘2^−𝑗−𝑘^′2^{−𝑗′𝑑+𝛾} (9) (here |⋅| denotes the Euclidean norm). Meyer proved that ℳ_𝛾 is algebra. Besides, he defined 𝒪𝑝(ℳ_𝛾) as the algebra of bounded operators on 𝐿²(ℝ^𝑑) whose matrices on a 𝑟-smooth wavelet basis (for a 𝑟>𝛾) belong to ℳ_𝛾 and showed that this definition does not depend on the chosen wavelet basis. In particular, we can use compactly supported wavelet bases.

Definition 3. Let 𝛾>0. A Banach or a quasi-Banach space of distributions 𝐸 is 𝛾-stable if the operators of 𝒪𝑝(ℳ_𝛾) are continuous on 𝐸.

In [16], we extended the notion of gentle spaces to include anisotropic homogeneous Besov spaces.

In the fourth section of this paper, we will apply our results to exhibit new examples of gentle spaces, namely, Lorentz spaces 𝐿^𝑝,𝑞 (see [2]) and 𝐻^𝑝,𝑞 spaces (see [17]). We will also prove that if the Jackson and Bernstein inequalities are valid, then “nonlinear approximation space” as defined in [18] associated to a gentle space is gentle. Note that there are different types of nonlinear approximations. The 𝑛-term approximation is one of the dominant types. We can mention, for example, approximation by splines with 𝑛 free knots or by rational functions of degree 𝑛; see DeVore and Popov [19]. Approximation by a linear combination with 𝑛-term of 𝜑-function was developed by DeVore et al. in [20]. A generalization of 𝑛-term approximation (called restricted approximation) by a linear combination of compactly supported biorthogonal wavelets was presented by Cohen et al. in [21]. In this paper, we consider approximation by a linear combination of Lemarié-Meyer wavelets as was done by Kyriazis in [18]. This form of approximation occurs in several applications including image processing, statistical estimation, and numerical solutions of differential equations.

2. Real and Complex Interpolation Methods

Originally, real and complex interpolation methods were developed for Banach spaces. The extension of real interpolation for quasi-Banach spaces causes no serious problem (see [12, 13, 22]). However, for the complex method the situation is quite different (see [23–25]). Let us recall briefly some basic definitions and notations related to these two methods. For more details see [1, 12, 13, 22, 26–28].

Definition 4. Let (𝑋₀,𝑋₁) be a compatible couple of Banach or quasi-Banach spaces. (1)The 𝐾-functional is defined for each 𝑓∈𝑋₀+𝑋₁ and 𝑡>0 by 𝐾𝑓,𝑡=𝐾𝑓,𝑡,𝑋₀,𝑋₁=inf𝑓_0𝑋0+𝑡𝑓_1𝑋1∶𝑓=𝑓₀+𝑓₁, (10) where the infimum extends over all representations 𝑓=𝑓₀+𝑓₁ of 𝑓 with 𝑓₀∈𝑋₀ and 𝑓₁∈𝑋₁.(2)The 𝐽-functional is defined for each 𝑓∈𝑋₀∩𝑋₁ and 𝑡>0 by 𝐽𝑓,𝑡=𝐽𝑓,𝑡,𝑋₀,𝑋₁=max𝑓_𝑋0,𝑡𝑓_𝑋1. (11)

The 𝐾-functional and 𝐽-functional, introduced By Peetre, are nonnegative concave and increasing functions.

Definition 5. Let (𝑋₀,𝑋₁) be a compatible couple of Banach or quasi-Banach spaces. (1)Let 0<𝜃<1 and 0<𝑞<∞ or let 0≤𝜃≤1 and 𝑞=∞. The space (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐾} consists of all 𝑓∈𝑋₀+𝑋₁ such that 𝑓_{𝜃,𝑞,𝐾}∶=^∞0𝑡^−𝜃𝐾𝑓,𝑡^𝑞𝑑𝑡𝑡^1/𝑞,0<𝜃<1,0<𝑞<∞,sup_𝑡>0𝑡^−𝜃𝐾𝑓,𝑡,0≤𝜃≤1,𝑞=∞ (12) is finite.(2)Let 0<𝜃<1 and 0<𝑞≤∞. The space (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐽} consists of all 𝑓∈𝑋₀+𝑋₁ that are represented by Bochner-integral 𝑓=^∞₀𝑢(𝑠)𝑑𝑠𝑠, (13) where 𝑢 is measurable with values in 𝑋₀∩𝑋₁ and such that 𝑓_{𝜃,𝑞,𝐽}∶=inf^∞0𝑠^−𝜃𝐽(𝑢(𝑠),𝑠)^𝑞𝑑𝑠𝑠^1/𝑞,𝑞<∞,sup_𝑠>0𝑠^−𝜃𝐽(𝑢(𝑠),𝑠),𝑞=∞ (14) is finite, where the infimum is taken over all representations (13) of 𝑓.

Remark 6. There is a discrete representation of the space (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐽} (see [1]); in fact 𝑓∈(𝑋₀,𝑋₁)_{𝜃,𝑞,𝐽} if and only if there exists a sequence (𝑢_𝜈)_𝜈∈ℤ in 𝑋₀∩𝑋₁ such that 𝑓=_𝜈∈ℤ𝑢_𝜈,_𝜈∈ℤ2^−𝜈𝜃𝐽𝑢_𝜈,2^{𝜈𝑞1/𝑞}<∞,if𝑞<∞ (15)(16) or sup_𝜈∈ℤ2^−𝜈𝜃𝐽𝑢_𝜈,2^𝜈<∞,if𝑞=∞. (17) Moreover, 𝑓_{𝜃,𝑞,𝐽}isequivalenttoinf_𝜈∈ℤ2^−𝜈𝜃𝐽𝑢_𝜈,2^{𝜈𝑞1/𝑞},𝑞<∞,sup_𝜈∈ℤ2^−𝜈𝜃𝐽𝑢_𝜈,2^𝜈,𝑞=∞, (18) where the infimum extends over all sequences (𝑢_𝜈)_𝜈∈ℤ satisfying (15).

The following result is given in [1] (see also [29]).

Theorem 7. Let (𝑋₀,𝑋₁) be a compatible couple of Banach (resp., quasi-Banach) spaces. Then, spaces (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐾}, with 0<𝜃<1 and 0<𝑞<∞ or 0≤𝜃≤1 and 𝑞=∞, and (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐽}, with 0<𝜃<1 and 0<𝑞≤∞, equipped, respectively, by the norms (resp., quasi-norms) (12) and (14) are Banach (resp., quasi-Banach) spaces and are interpolation spaces between 𝑋₀ and 𝑋₁.
Furthermore, if 0<𝜃<1 and 0<𝑞≤∞, then (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐾}=(𝑋₀,𝑋₁)_{𝜃,𝑞,𝐽} with equivalence of norms (resp., quasi-norms).

Real interpolation method means either the 𝐾- or 𝐽-method. We will write (𝑋₀,𝑋₁)_𝜃,𝑞 instead of (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐾} or (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐽}, if 0<𝜃<1. If 𝜃=0 or 1 and 𝑞=∞, then (𝑋₀,𝑋₁)_𝜃,𝑞 denotes (𝑋₀,𝑋₁)_{𝜃,𝑞,𝐾}. By ‖⋅‖_𝜃,𝑞 we denote the norm or quasi-norm on (𝑋₀,𝑋₁)_𝜃,𝑞 depending whether this space is Banach or quasi-Banach.

In the complex case, we will first restrict ourselves to Banach spaces. There are two interpolation spaces whose norms are equivalent under some conditions. Set 𝑆={𝑧∈ℂ,0≤ℜ(𝑧)≤1},𝑆₀={𝑧∈ℂ,0<ℜ(𝑧)<1}. (19) Let (𝑋₀,𝑋₁) be a compatible couple of Banach spaces. We denote by ℱ the space of all functions 𝑓∶𝑆→𝑋₀+𝑋₁ that are bounded, continuous on 𝑆, and analytic on 𝑆₀, such that functions 𝑡↦𝑓(𝑘+𝑖𝑡), 𝑘=0,1, from ℝ into 𝑋_𝑘, are continuous and tend to zero as |𝑡|→∞.

By 𝒢 we denote the space of all functions 𝑔∶𝑆→𝑋₀+𝑋₁ that are continuous on 𝑆 and analytic on 𝑆₀, satisfying 𝑔(𝑧)_𝑋0+𝑋1≤𝐶(1+|𝑧|) (20) such that 𝑔(𝑘+𝑖𝑡₁)−𝑔(𝑘+𝑖𝑡₂) has values in 𝑋_𝑘, 𝑘=0,1, for any −∞<𝑡₁<𝑡₂<∞.

Spaces ℱ and 𝒢 provided, respectively, with 𝑓_ℱ∶=maxsup_𝑡∈ℝ𝑓(𝑖𝑡)_𝑋0,sup_𝑡∈ℝ𝑓(1+𝑖𝑡)_𝑋1,𝑔_𝒢∶=maxsup_{𝑡1,𝑡2∈ℝ}𝑔𝑖𝑡₁−𝑔𝑖𝑡₂𝑡₁−𝑡_2𝑋0,sup_{𝑡1,𝑡2∈ℝ}𝑔1+𝑖𝑡₁−𝑔1+𝑖𝑡₂𝑡₁−𝑡_2𝑋1 (21)(22) are Banach spaces.

Definition 8. Let (𝑋₀,𝑋₁) be a compatible couple of Banach spaces. For all 0≤𝜃≤1 (resp. 0<𝜃<1) we define (𝑋₀,𝑋₁)_[𝜃] (resp., (𝑋₀,𝑋₁)^[𝜃]) as the space of all 𝑥∈𝑋₀+𝑋₁ such that 𝑥=𝑓(𝜃) (resp., 𝑥=𝑔^′(𝜃)) for some 𝑓∈ℱ (resp., 𝑔∈𝒢).

Theorem 9. Let (𝑋₀,𝑋₁) be a compatible couple of Banach spaces. Then, spaces (𝑋₀,𝑋₁)_[𝜃], where 0≤𝜃≤1 and (𝑋₀,𝑋₁)^[𝜃], where 0<𝜃<1, equipped respectively with ‖𝑥‖_[𝜃]∶=inf𝑓_ℱ∶𝑓(𝜃)=𝑥,𝑓∈ℱ,‖𝑥‖^[𝜃]∶=inf𝑔_𝒢∶𝑔^′(𝜃)=𝑥,𝑔∈𝒢 (23)(24) are Banach spaces and are interpolation spaces between 𝑋₀ and 𝑋₁.

Remark 10. We have (𝑋₀,𝑋₁)_[𝜃]⊂(𝑋₀,𝑋₁)^[𝜃]. In general, (𝑋₀,𝑋₁)_[𝜃] and (𝑋₀,𝑋₁)^[𝜃] are not equal. However, if either 𝑋₀ or 𝑋₁ is reflexive and if 0<𝜃<1, then (𝑋₀,𝑋₁)_[𝜃]=(𝑋₀,𝑋₁)^[𝜃] and ‖𝑥‖^[𝜃]=‖𝑥‖_[𝜃], for all 𝑥∈𝑋₀+𝑋₁.

The extension of this method to quasi-Banach spaces is not routine; one cannot use duality as was done for Banach spaces. The duality theorem is not true in general in the quasi-Banach case, and the maximum principle fails for functions taking values in a quasi-Banach space (see [30]).

There are several possible ways to define complex interpolation spaces. For example, in [23] complex interpolation was defined in the framework of Fourier analysis, while in [24] complex interpolation was defined as in the Banach setting but by adding in ‖𝑓‖_ℱ (which was given in (21)) a third term sup_𝑧∈𝑆0‖𝑓(𝑧)‖_𝑋0+𝑋1. In [25], the authors described a new approach to interpolate by the complex method some quasi-Banach spaces; let 𝐴(𝑆) be the space of all scalar valued functions 𝑓 continuous and bounded on 𝑆 and analytic on 𝑆₀. Let (𝑋₀,𝑋₁) be a compatible couple of quasi-Banach spaces. Denote by 𝒜(𝑋₀,𝑋₁) the collection of all functions 𝑓 that can be written as a finite sum 𝑓(𝑧)=∑_𝑘𝑓_𝑘(𝑧)𝑎_𝑘 where 𝑓_𝑘∈𝐴(𝑆) and 𝑎_𝑘∈𝑋₀∩𝑋₁. We put 𝑓_{𝒜(𝑋0,𝑋1)}∶=maxsup_𝑡∈ℝ𝑓(𝑖𝑡)_𝑋0,sup_𝑡∈ℝ𝑓(1+𝑖𝑡)_𝑋1, (25) and for all 𝑥∈𝑋₀∩𝑋₁ and 0<𝜃<1 let ‖𝑥‖_𝜃∶=inf𝑓_{𝒜(𝑋0,𝑋1)}∶𝑓(𝜃)=𝑥,𝑓∈𝒜𝑋₀,𝑋₁. (26) This functional is a semi-quasi-norm. Let 𝑁=𝑥∈𝑋₀∩𝑋₁∶‖𝑥‖_𝜃=0; (27) then, ((𝑋₀∩𝑋₁)/𝑁,‖⋅‖_𝜃) is a quasi-normed space. We define [𝑋₀,𝑋₁]_𝜃 as the completion of ((𝑋₀∩𝑋₁)/𝑁,‖⋅‖_𝜃) (see [31]).

3. Real and Complex Interpolation between Gentle Spaces

Assume that 𝐸₀ and 𝐸₁ are two gentle spaces. Let 𝐸 be an interpolation space between them. Then, 𝐸 is a Banach or a quasi-Banach space of distributions and 𝒮_∞(ℝ^𝑑)↪𝐸↪𝒮^′_∞(ℝ^𝑑).

On the other hand, there exist two constants 𝛾₀>0 and 𝛾₁>0 such that 𝐸_𝑖 is 𝛾_𝑖-stable, 𝑖=0,1. Set 𝛾=min{𝛾₀,𝛾₁}. If 𝑇∈𝒪𝑝(ℳ_𝛾), then 𝑇∈𝒪𝑝(ℳ_𝛾𝑖). So 𝑇 is continuous on 𝐸_𝑖 and therefore continuous on 𝐸. Thus, 𝐸 is 𝛾-stable.

We will now prove the homogeneity property.

Proposition 11. Let 𝐸₀ and 𝐸₁ be two gentle spaces of order 𝐻₀ and 𝐻₁, respectively. Then, spaces (1)(𝐸₀,𝐸₁)_𝜃,𝑞, with 0<𝜃<1 and 1≤𝑞<∞ or 0≤𝜃≤1 and 𝑞=∞,(2)(𝐸₀,𝐸₁)_[𝜃], with 0≤𝜃≤1 (in the Banach setting),(3)(𝐸₀,𝐸₁)^[𝜃], with 0<𝜃<1 (in the Banach setting),(4)[𝐸₀,𝐸₁]_𝜃, with 0<𝜃<1 (in the quasi-Banach setting) are homogeneous spaces of order (1−𝜃)𝐻₀+𝜃𝐻₁.

Proof. (1) Let 0<𝜃<1 and 0<𝑞<∞ or let 0≤𝜃≤1 and 𝑞=∞. Let 𝑓∈(𝐸₀,𝐸₁)_𝜃,𝑞, 𝑓₀∈𝐸₀, and 𝑓₁∈𝐸₁ such that 𝑓=𝑓₀+𝑓₁. Let 𝑡>0. From the homogeneity of 𝐸₀ and 𝐸₁ we get 𝐾𝜏_𝑎𝑓,𝑡≤𝜏_𝑎𝑓_0𝐸0+𝑡𝜏_𝑎𝑓_1𝐸1≤𝐶𝑓_0𝐸0+𝑡𝑓_1𝐸1,𝐾ℎ_𝑟𝑓,𝑡≤ℎ_𝑟𝑓_0𝐸0+𝑡ℎ_𝑟𝑓_1𝐸1≤𝐶𝑟^𝐻0𝑓_0𝐸0+𝑡𝑟^{𝐻1−𝐻0}𝑓_1𝐸1. (28)
By taking the infimum over all such decompositions 𝑓=𝑓₀+𝑓₁ of 𝑓, we obtain 𝐾𝜏_𝑎𝑓,𝑡≤𝐶𝐾𝑓,𝑡so𝜏_𝑎𝑓_𝜃,𝑞≤𝐶𝑓_𝜃,𝑞,𝐾ℎ_𝑟𝑓,𝑡≤𝐶𝑟^𝐻0𝐾𝑓,𝑡𝑟^{𝐻1−𝐻0}. (29)(i)If 0<𝜃<1 and 0<𝑞<∞, then ℎ_𝑟𝑓_𝜃,𝑞=^∞₀𝑡^−𝜃𝐾ℎ_𝑟𝑓,𝑡^𝑞𝑑𝑡𝑡^1/𝑞≤𝐶𝑟^𝐻0∞₀𝑡^−𝜃𝐾𝑓,𝑡𝑟^{𝐻1−𝐻0𝑞}𝑑𝑡𝑡^1/𝑞. (30)
It follows, by a simple change of variable, that ℎ_𝑟𝑓_𝜃,𝑞≤𝐶𝑟^𝐻0𝑟^{−𝜃(𝐻0−𝐻1)∞}₀𝑡^−𝜃𝐾𝑓,𝑡^𝑞𝑑𝑡𝑡^1/𝑞=𝐶𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝑓_𝜃,𝑞. (31)(i)If 0≤𝜃≤1 and 𝑞=∞, we obtain ℎ_𝑟(𝑓)_𝜃,𝑞=sup_𝑡>0𝑡^−𝜃𝐾ℎ_𝑟𝑓,𝑡≤𝐶𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝑓_𝜃,𝑞. (32)
Therefore, the space (𝐸₀,𝐸₁)_𝜃,𝑞 is homogeneous of order (1−𝜃)𝐻₀+𝜃𝐻₁.
(2) Now, let 𝑓∈(𝐸₀,𝐸₁)_[𝜃]; there exists 𝑔∈ℱ such that 𝑓=𝑔(𝜃). Clearly 𝜏_𝑎𝑔(𝑧)(𝑥)=𝑔(𝑧)(𝑥−𝑎),ℎ_𝑟𝑔(𝑧)(𝑥)=𝑔(𝑧)(𝑟𝑥). (33)
Thus, 𝑧↦𝜏_𝑎(𝑔(𝑧)) and 𝑧↦ℎ_𝑟(𝑔(𝑧)) are analytic functions.
From (23), (21), and the homogeneity of 𝐸₀ and 𝐸₁, we can easily see that 𝜏_𝑎(𝑓)_[𝜃]≤𝐶𝑓_[𝜃]. (34)
Put 𝐹(𝑧)=𝑟^{(𝑧−1)𝐻0}𝑟^−𝑧𝐻1ℎ_𝑟𝑔(𝑧). (35)
Therefore, ℎ_𝑟𝑓=𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝐹(𝜃). (36)
Hence, ℎ_𝑟𝑓_[𝜃]≤𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}‖𝐹‖_ℱ=𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}maxsup_𝑡∈ℝ‖𝐹(𝑖𝑡)‖_𝐸0,sup_𝑡∈ℝ‖𝐹(1+𝑖𝑡)‖_𝐸1. (37) It follows from (35) and the homogeneity of 𝐸₀ and 𝐸₁ that ‖𝐹(𝑖𝑡)‖_𝐸0=𝑟^−𝐻0ℎ_𝑟𝑔(𝑖𝑡)_𝐸0≤𝐶𝑔(𝑖𝑡)_𝐸0,‖𝐹(1+𝑖𝑡)‖_𝐸1=𝑟^−𝐻1ℎ_𝑟𝑔(1+𝑖𝑡)_𝐸1≤𝐶𝑔(1+𝑖𝑡)_𝐸1. (38) This implies that ‖𝐹‖_ℱ≤𝐶𝑔_ℱ,ℎ_𝑟𝑓_[𝜃]≤𝐶𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝑔_ℱ. (39) Taking the infimum over all 𝑔∈ℱ such that 𝑔(𝜃)=𝑓, we obtain ℎ_𝑟(𝑓)_[𝜃]≤𝐶𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝑓_[𝜃]. (40)
(3) If 𝑓∈(𝐸₀,𝐸₁)^[𝜃], then there exists 𝑔∈𝒢 such that 𝑓=𝑔^′(𝜃). So as previously 𝑧→𝜏_𝑎(𝑔(𝑧)) and 𝑧→ℎ_𝑟(𝑔(𝑧)) are analytic functions. Clearly, from (24), (22), and the homogeneity of 𝐸₀ and 𝐸₁, we have 𝜏_𝑎𝑓^[𝜃]≤𝐶𝑓^[𝜃]. (41) Set 𝐺(𝑧)=^𝑧₀𝑟^{(𝜂−1)𝐻0}𝑟^−𝜂𝐻1ℎ_𝑟𝑔^′𝜂𝑑𝜂. (42) Hence, 𝐺^′(𝜃)=𝑟^{(𝜃−1)𝐻0}𝑟^−𝜃𝐻1ℎ_𝑟𝑓. (43) Thus, ℎ_𝑟𝑓^[𝜃]≤𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}‖𝐺‖_𝒢. (44) Let −∞<𝑡₁<𝑡₂<∞. From (42) and the fact that 𝐸₀ is homogeneous of order 𝐻₀, we get 𝐺𝑖𝑡₁−𝐺𝑖𝑡_2𝐸0≤𝑟^{−𝐻0𝑡2}_𝑡1ℎ_𝑟𝑔^′(𝑖𝑡)_𝐸0𝑑𝑡≤𝐶^𝑡2_𝑡1𝑔^′(𝑖𝑡)_𝐸0𝑔𝑑𝑡≤𝐶𝑖𝑡₂−𝑔𝑖𝑡_1𝐸0. (45) Similarly 𝐺1+𝑖𝑡₁−𝐺1+𝑖𝑡_2𝐸1≤𝐶𝑔1+𝑖𝑡₂−𝑔1+𝑖𝑡_1𝐸1. (46) Therefore, ‖𝐺‖_𝒢≤𝐶𝑔_𝒢,soℎ_𝑟𝑓^[𝜃]≤𝐶𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝑔_𝒢. (47) By taking the infimum over all 𝑔∈𝒢 such that 𝑔^′(𝜃)=𝑓, we obtain ℎ_𝑟(𝑓)^[𝜃]≤𝐶𝑟^{(1−𝜃)𝐻0+𝜃𝐻1}𝑓^[𝜃]. (48)
(4) Now, let 𝑓∈[𝐸₀,𝐸₁]_𝜃. There exists a sequence (𝑓_𝑛)_𝑛∈(𝐸₀∩𝐸₁)/𝑁 such that ‖𝑓−𝑓_𝑛‖_𝜃→0. Let 𝑔∈𝒜(𝐸₀,𝐸₁) such that 𝑔(𝜃)=𝑓−𝑓_𝑛 and 𝑔 is written as a finite sum 𝑔(𝑧)=∑_𝑘𝑔_𝑘(𝑧)𝑎_𝑘 where 𝑔_𝑘∈𝐴(𝑆) and 𝑎_𝑘∈𝐸₀∩𝐸₁. We define 𝜏_𝑎𝑔(𝑧)=_𝑘𝜏_𝑎𝑔_𝑘(𝑧)𝑎_𝑘,ℎ_𝑟𝑔(𝑧)=_𝑘ℎ_𝑟𝑔_𝑘(𝑧)𝑎_𝑘, (49) where 𝜏_𝑎𝑔_𝑘(𝑧)(𝑥)=𝑔_𝑘(𝑧),ℎ)(𝑥−𝑎_𝑟𝑔_𝑘(𝑧)(𝑥)=𝑔_𝑘(𝑧)(𝑟𝑥), (50) and 𝑧↦𝜏_𝑎(𝑔_𝑘(𝑧)) and 𝑧↦ℎ_𝑟(𝑔_𝑘(𝑧)) are analytic functions. By arguing similarly as in (𝐸₀,𝐸₁)_[𝜃], by taking 𝑓−𝑓_𝑛 instead of 𝑓, we get the desired result.

Let us now prove the density property.

Proposition 12. If 𝑞<∞, then 𝒮_∞(ℝ^𝑑) is dense in (𝐸₀,𝐸₁)_𝜃,𝑞.

Proof. Since 𝑞<∞, then 𝐸₀∩𝐸₁ is dense in (𝐸₀,𝐸₁)_𝜃,𝑞 (see e.g., [1]). Therefore it suffices to prove that every function 𝑓∈𝐸₀∩𝐸₁ can be approached by a sequence (𝑓_𝑛)_𝑛∈ℕ∈𝒮_∞(ℝ^𝑑) in (𝐸₀,𝐸₁)_𝜃,𝑞. The assumption 𝑞<∞ implies that 0<𝜃<1, so by the equivalence between real interpolation spaces we will prove this approach in (𝐸₀,𝐸₁)_{𝜃,𝑞,𝐽}.
On the other hand, since 𝐸₀ and 𝐸₁ are gentle spaces, then wavelets are either unconditional bases or unconditional ∗-weak bases, depending on the separability of these spaces. Using the 𝛾-stability, we can take Lemarié-Meyer wavelet basis; let 𝜓⁰ and 𝜓¹ be, respectively, the Lemarié-Meyer [32] father and mother wavelets; that is, 𝜓⁰ and 𝜓¹ are in the Schwartz class such that all moments of 𝜓¹ vanish, ∫_ℝ𝜓⁰(𝑥)𝑑𝑥=1, and the collection of the union of (𝜓⁰(⋅−𝑘))_𝑘∈ℤ and (2^𝑗/2𝜓¹(2^𝑗⋅−𝑘))_{𝑗∈ℕ,𝑘∈ℤ} is an orthonormal basis of 𝐿²(ℝ). Let 𝑉 be the set of nonzero vertices of the unit cube in ℝ^𝑑, for each vertex 𝑣=(𝑣₁,…,𝑣_𝑑)∈𝑉 we set 𝜓^𝑣(𝑥)∶=𝜓^𝑣1𝑥₁⋯𝜓^𝑣𝑑𝑥_𝑑,𝑥=𝑥₁,…,𝑥_𝑑∈ℝ^𝑑, (51) and we define 𝜓^𝑣_𝜆(𝑥)∶=2^𝑑𝑗/2𝜓^𝑣2^𝑗𝑥−𝑘, (52) indexed by dyadic cube 𝜆=2^−𝑗[𝑘,𝑘+1], where 𝑘∈ℤ^𝑑 and 𝑗∈ℤ. Denote by Λ (resp., Λ_𝑗) the set of all dyadic cubes (resp., dyadic cubes at scale 𝑗). Then, the collection 𝜓^𝑣_𝜆∶𝜆∈Λ,𝑣∈𝑉 (53) is an orthonormal basis of 𝐿²(ℝ^𝑑). Thus any function 𝑓∈𝐿²(ℝ^𝑑) can be written as 𝑓=_{𝜆∈Λ𝑣∈𝑉}𝑐^𝑣_𝜆𝜓^𝑣_𝜆,where𝑐^𝑣_𝜆=𝑓,𝜓^𝑣_𝜆. (54) Let 𝑓∈𝐸₀∩𝐸₁. Put 𝑓_𝑗=∑_{𝜆∈Λ𝑗}∑_𝑣∈𝑉𝑐^𝑣_𝜆𝜓^𝑣_𝜆; then, (𝑓_𝑗)_𝑗∈ℤ belongs to 𝒮_∞(ℝ^𝑑), so to 𝐸₀∩𝐸₁ and 𝑓=∑_𝑗∈ℤ𝑓_𝑗. Since 𝐸₀∩𝐸₁⊂(𝐸₀,𝐸₁)_{𝜃,𝑞,𝐽}, we therefore obtain from (15) and (16) that _𝑗∈ℤ2^−𝑗𝜃𝐽𝑓_𝑗,2^{𝑗𝑞1/𝑞}<∞. (55) Set now 𝑓_𝑛=^𝑛_𝑗=−𝑛𝑓_𝑗. (56) It follows from (18) that 𝑓−𝑓_{𝑛𝜃,𝑞,𝐽}≤_𝑗>𝑛2^−𝑗𝜃𝐽𝑓_𝑗,2^{𝑗𝑞1/𝑞}, (57) which tends to 0 as 𝑛 tends to infinity.

Proposition 13. Suppose that 𝒮_∞(ℝ^𝑑) is dense in 𝐸₀ and in 𝐸₁. (1)Assume that 𝐸₀ and 𝐸₁ are Banach spaces. Then(a)If 0≤𝜃≤1, then 𝒮_∞(ℝ^𝑑) is dense in (𝐸₀,𝐸₁)_[𝜃].(b)If 0<𝜃<1, then 𝒮_∞(ℝ^𝑑) dense in (𝐸₀,𝐸₁)^[𝜃].(2)Assume that 𝐸₀ and 𝐸₁ are quasi-Banach spaces. Then 𝒮_∞(ℝ^𝑑) is dense in [𝐸₀,𝐸₁]_𝜃, where 0<𝜃<1.

Proof . Let 𝑓∈(𝐸₀,𝐸₁)_[𝜃]. There exists 𝑔∈ℱ such that 𝑓=𝑔(𝜃).
Since 𝒮_∞(ℝ^𝑑) is dense in both 𝐸₀ and 𝐸₁, then, for all 𝜀>0, there exist two functions 𝑔₀ and 𝑔₁ in 𝒮_∞(ℝ^𝑑) such that for all 𝑡∈ℝ, 𝑔(𝑖𝑡)−𝑔_0𝐸0≤𝜀,𝑔(1+𝑖𝑡)−𝑔_1𝐸1≤𝜀. (58) Let 𝐻∈ℱ such that 𝐻 takes values in 𝒮_∞(ℝ^𝑑), 𝐻(𝑖𝑡)=𝑔₀, and 𝐻(1+𝑖𝑡)=𝑔₁ for all 𝑡∈ℝ. Put ℎ=𝐻(𝜃). Thus, from (23), (21), and (58), we obtain 𝑓−ℎ_[𝜃]≤𝑔−𝐻_ℱ≤𝜀. (59) Since ℎ∈𝒮_∞(ℝ^𝑑), then this implies the density of 𝒮_∞(ℝ^𝑑) in (𝐸₀,𝐸₁)_[𝜃].
Now let 𝑓∈(𝐸₀,𝐸₁)^[𝜃]. Arguing similarly as previously but by replacing ℱ by 𝒢, 𝑔 by 𝑔^′, and (58) by 𝑔𝑖𝑡₁−𝑔𝑖𝑡₂𝑡₁−𝑡₂−𝑔_0𝐸0𝑔≤𝜀,1+𝑖𝑡₁−𝑔1+𝑖𝑡₂𝑡₁−𝑡₂−𝑔_1𝐸1≤𝜀. (60) Taking 𝐻∈𝒢 such that 𝐻 takes values in 𝒮_∞(ℝ^𝑑), 𝐻(𝑖𝑡)=𝑡𝑔₀ and 𝐻(1+𝑖𝑡)=𝑡𝑔₁ for all 𝑡∈ℝ. To finish the proof, it suffices to put ℎ=𝐻^′(𝜃).
For the proof of the density of 𝒮_∞(ℝ^𝑑) in [𝐸₀,𝐸₁]_𝜃, we argue similarly as in (𝐸₀,𝐸₁)_[𝜃] by taking 𝑓−𝑓_𝑛 instead of 𝑓, where (𝑓_𝑛)_𝑛 is a sequence in (𝐸₀∩𝐸₁)/𝑁 such that ‖𝑓−𝑓_𝑛‖_𝜃→0.

Consequence. Suppose that 𝒮_∞(ℝ^𝑑) is dense in both 𝑋₀ and 𝑋₁, where 𝑋^′_𝑖=𝐸_𝑖, with 𝑖=0,1. Let 0<𝜃<1. It follows from the previous proposition that 𝒮_∞(ℝ^𝑑) is dense in both (𝑋₀,𝑋₁)_[𝜃] and (𝑋₀,𝑋₁)^[𝜃]. On the other hand, since 𝒮_∞ℝ^𝑑↪𝑋₀∩𝑋₁↪𝑋_𝑖,𝑖=0,1, (61) then 𝑋₀∩𝑋₁ is dense in 𝑋_𝑖, 𝑖=0,1. From the duality theorem (see [1]), it follows that 𝑋₀,𝑋₁^′_[𝜃]=𝑋^′₀,𝑋^′₁^[𝜃]=𝐸₀,𝐸₁^[𝜃],𝑋₀,𝑋₁^[𝜃]′=𝑋₀,𝑋₁^′_[𝜃]. (62) Furthermore, if either 𝐸₀ or 𝐸₁ is reflexive (then either 𝑋₀ or 𝑋₁ is reflexive), then 𝑋₀,𝑋₁^′_[𝜃]=𝐸₀,𝐸_1[𝜃],𝑋₀,𝑋₁^[𝜃]′=𝑋^′₀,𝑋^′₁^[𝜃]=𝐸₀,𝐸₁^[𝜃]. (63)

4. Applications

As a first application we show that Lorentz spaces 𝐿^𝑝,𝑞=𝐿^𝑝,𝑞(ℝ^𝑑) are gentle. This follows from the fact that 𝐿^𝑝,𝑞=(𝐿^𝑝0,𝐿^𝑝1)_𝜃,𝑞 (see [1]), where 0<𝜃<1, 1<𝑝₀,𝑝₁<∞, (1/𝑝)=((1−𝜃)/𝑝₀)+(𝜃/𝑝₁) and 𝑝₀<𝑞≤∞ and 𝐿^𝑝0=𝐿^𝑝0(ℝ^𝑑) and 𝐿^𝑝1=𝐿^𝑝1(ℝ^𝑑) (where 1<𝑝₀,𝑝₁<∞) are gentle spaces (see [4]).

In the second application, we consider the real interpolation between Hardy spaces 𝐻^𝑝0=𝐻^𝑝0(ℝ^𝑑) and BMO=BMO(ℝ^𝑑) spaces. In [4], we investigated the gentleness property of some functional spaces and we proved the stability of this property by duality. We also proved that homogeneous Lizorkin-Triebel spaces 𝐹̇_𝑝^𝑠,𝑞(ℝ^𝑑) (see [33]), where 𝑠∈ℝ, 0<𝑝<∞ and 0<𝑞≤∞ are gentle spaces. In particular, 𝐻^𝑝(ℝ^𝑑)=𝐹̇_𝑝^0,2(ℝ^𝑑), where 0<𝑝≤1 are gentle. As a consequence, since BMO is the topological dual of 𝐻¹, then BMO is gentle. In [34], it was shown that, for 0<𝜃<1, 0<𝑞≤∞, 0<𝑝₀<1, and 𝑝=𝑝₀/1−𝜃, (𝐻^𝑝0,BMO)_𝜃,𝑞=𝐻^𝑝,𝑞, where 𝐻^𝑝,𝑞=𝐻^𝑝,𝑞(ℝ^𝑑) is the space of all tempered distributions 𝑓 on ℝ^𝑑 such that sup_𝑡>0𝑡^−𝑑𝜙_𝑡∗𝑓∈𝐿^𝑝,𝑞, (64) where 𝜙_𝑡(𝑥)=𝜙(𝑥/𝑡) and 𝜙 is a smooth function with ∫_ℝ𝑑𝜙(𝑥)𝑑𝑥≠0 (see [17]). Consequently 𝐻^𝑝,𝑞 are gentle spaces. Note that 𝐻^𝑝,𝑞=𝐻^𝑝 for 𝑝=𝑞.

Our last application refers to some nonlinear approximation spaces introduced in [18]. Take the Lemarié-Meyer orthonormal basis (53). Let Σ_𝑛 be the set of all functions 𝑆=_{𝜆∈Λ(𝑛)}𝐴_𝜆(𝑆), (65) where 𝐴_𝜆(𝑆)=∑_𝑣∈𝑉𝑐^𝑣_𝜆𝜓^𝑣_𝜆 and Λ(𝑛) is a subset of Λ with cardinality ♯Λ(𝑛)≤𝑛. For a given distribution 𝑓 and any quasi-normed subspace 𝑋⊂𝒮^′_∞(ℝ^𝑑), we define 𝜎_𝑛𝑓,𝑋∶=inf_{𝑆∈Σ𝑛}𝑓−𝑆_𝑋. (66) For 0<𝑞≤∞ and 𝛼>0, we define the approximation class 𝒜^𝛼_𝑞(𝑋) as the set of all 𝑓∈𝒮^′_∞(ℝ^𝑑) such that 𝑓_{𝒜𝛼𝑞(𝑋)}=_𝑗∈ℤ2^𝑗𝛼𝜎₂^𝑗𝑓,𝑋^𝑞1/𝑞,0<𝑞<∞,sup_𝑗∈ℤ2^𝑗𝛼𝜎₂^𝑗𝑓,𝑋,𝑞=∞, (67) is finite.

In [18], Kyriazis proved that if 𝑋 and 𝑌 are quasi-normed spaces that verify, for some 𝑟>0, Jackson inequality (forall𝑓∈𝑌,𝜎_𝑛(𝑓,𝑋)≤𝐶𝑛^−𝑟/𝑑‖𝑓‖_𝑌) and Bernstein inequality (forall𝑆∈Σ_𝑛,‖𝑆‖_𝑌≤𝐶𝑛^𝑟/𝑑‖𝑆‖_𝑋), then for each 0<𝛼<𝑟 and 0<𝑞≤∞𝒜_𝑞^𝛼/𝑑(𝑋)=(𝑋,𝑌)_{𝛼/𝑟,𝑞}. (68) This result was also proved in [21], where Cohen et al. have used compactly supported biorthogonal wavelets, and in [35], by DeVore and Lorentz, where 𝑌 is embedded in 𝑋.

In the literature complete characterizations are known for the cases 𝑋=𝐻^𝑝, 0<𝑝<∞; 𝑋=𝐵̇_𝑝^0,𝑝, 0<𝑝<∞; and 𝑋=𝐹̇_𝑝^𝛽,𝑡, 𝛽∈ℝ, 0<𝑝<∞, and 0<𝑡≤∞. DeVore et al. have shown, in [20], that for any 0<𝛼<𝑠 and any 0<𝑞≤∞𝒜_𝑞^𝛼/𝑑𝐻^𝑝=𝐻^𝑝,𝐵̇_𝜏^𝑠,𝜏_{𝛼/𝑠,𝑞},1𝜏=𝑠𝑑+1𝑝. (69) This result was proved by simpler techniques in [21], where the authors proved also that for any 0<𝛼<𝑠 and any 0<𝑞≤∞𝒜_𝑞^𝛼/𝑑𝐵̇_𝑝^0,𝑝=𝐵̇_𝑝^0,𝑝,𝐵̇_𝜏^𝑠,𝜏_{𝛼/𝑠,𝑞} (70) and for any 𝛼>0 and any 0<𝑞≤∞𝒜_𝑞^𝛼/𝑑𝐻^𝑝=𝒜_𝑞^𝛼/𝑑𝐵̇_𝑝^0,𝑝. (71) Approximation in 𝐹̇_𝑝^𝛽,𝑡 was investigated in [18]. More precisely, it has been shown that, for every 0<𝑝<∞, 0<𝑞,  𝑡≤∞, 𝛼>0, and 𝛽∈ℝ, 𝒜_𝑞^𝛼/𝑑𝐹̇_𝑝^𝛽,𝑡=𝐹̇_𝑝^𝛽,𝑡,𝐵̇_𝜏^𝛾,𝜏_{𝛼/(𝛾−𝛽),𝑞}, (72) where 𝛼<𝛾−𝛽 and 1/𝜏=𝛾−𝛽/𝑑+1/𝑝.

Since 𝐻^𝑝, 𝐵̇_𝑝^0,𝑝, and 𝐹̇_𝑝^𝛽,𝑡 are gentle spaces, then their approximation spaces (which are real interpolation spaces) are also gentle. Moreover, the pairs (𝐻^𝑝,𝐵̇_𝜏^𝑠,𝜏), (𝐵̇_𝑝^0,𝑝,𝐵̇_𝜏^𝑠,𝜏), and (𝐹̇_𝑝^𝛽,𝑡,𝐵̇_𝜏^𝛾,𝜏), under the above conditions on all parameters 𝑝, 𝑞, 𝑠, 𝑡, 𝜏, and 𝛾, verify the Jackson and Bernstein inequalities. More generally, we deduce that if 𝑋 and 𝑌 are gentle spaces and if the Jackson and Bernstein inequalities are valid, then the approximation space is also gentle.

In [18, 21], the authors restricted themselves to particular wavelets. They remarked that all of their theorems hold in more generality. Using our result, this remark is now confirmed using the gentleness stability by real interpolation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group No. RGP-350.