\begin{matrix}c \\ ◯ \end{matrix} Revista de Matemática: Teoría y Aplicaciones 2019 26(1) : 21–44 cimpa – ucr issn: 1409-2433 (Print), 2215-3373 (Online)

doi: https:// doi. org/ 10.15517/ rmta. v26i1. 35516

characterization of bmo using wavelets through triebel-lizorkin spaces

caracterización de bmo usando ondículas por medio del espacio de

triebel-lizorkin

Jorge Eliécer Hernández Hernández^{∗}

Received: 9/Nov/2017; Revised:2/Jul/2018; Accepted: 7/Nov/2018

∗Departamento de Técnicas Cuantitativas, Decanato de Ciencias Económicas y Empresariales, Universidad Centroccidental Lisandro Alvarado. Barquisimeto, Venezuela. E-Mail: jorgehernandez@ucla.edu.ve

Abstract

In the present article it is presented a characterization of all those functions in the space of bounded mean oscillation functions, BMO, in terms of an appropriate wavelet, using an isomorphism between the aforementioned space and the homogeneous space of Triebel-Lizorkin Ḟ_{∞}^{0,2}. In addition, a new inequality that involves the vector inequality of the maximal function of Hardy-Littlewood is proved.

Keywords: BMO function space; Triebel-Lizorkin’s homogeneous space; wavelets.

Resumen

En el presente artículo se presenta una caracterización de todas aquellas funciones pertenecientes al espacio de oscilación media acotada, BMO, en términos de una apropiada ondícula, usando un isomorfismo entre el mencionado espacio de funciones y el espacio homogéneo de Triebel-Lizorkin Ḟ_{∞}^{0,2}. Además, se prueba una versión nueva que involucra la desigualdad vectorial de la función maximal de Hardy-Littlewood.

Palabras clave: espacio de funciones de oscilación media acotada; espacio homogéneo de Triebel-Lizorkin; ondículas.

Mathematics Subject Classification: Primary 22E46, 53C35, Secondary 57S20.

1 Introduction

Wavelets were introduced in early’s 80, they have been of interest for the mathematical scientific community and other disciplines. The wavelet analysis has been used as an alternative for the windowed Fourier analysis, that is, for the case in which the objective is to measure the frequency content of a signal, while the case of wavelets is to compare several sizes of this signal with different resolutions. To fix a clear definition: a function ψ ∈ L_{2}(ℝ) is an orthonormal wavelet provided the system {ψ_{j,k} : j, k ∈ ℤ} is an orthonormal basis for L_{2}(ℝ) where

ψ_{j,k}(x) = 2^{\frac{j}{2}} ψ(2^{j}x − k), for all j, k ∈ ℤ.

Besides providing us with orthogonal bases for the Hilbert space L_{2}(ℝ), some wavelets gives us natural basis for other topological spaces.

Recently, in 2008, Triebel [13] presented some results that associate the use of wavelets with the Besov function spaces and certain estimates for local means. Also, in 2010, Han and Lu [7] made use of the Littlewood-Paley theory and the multiparameter Hardy space theory.

Using the Littlewood-Paley theory, and the wavelet representation of functions in a fixed function space, it is possible to obtain characterization of such function space. To cite a few instances, in [8] it is the characterization of the functions in the Lebesgue space L_{p}(ℝ), (1 < p < ∞), Hardy space H_{1}(ℝ) and Sobolev space L^{p,s}(ℝ), 1 < p < ∞, s = 1, 2, 3, … Below it is showed these characterizations.

Theorem 1. Let ψ be an orthonormal wavelet such that ψ ∈ R^{0}. If 1 < p < ∞, there exist constants A_{p} and B_{p}, 0 < A_{p} < B_{p} < ∞ such that

A_{p} ∥f∥_{L_{p}(ℝ)} ≤ ∥(∑_{j∈ℤ} ∑_{k∈ℤ} |⟨f, ψ_{j,k}⟩|^{2} 2^{\frac{j}{2}} χ_{[2^{−j}k,2^{−j}(k+1)]}(·))^{1/2}∥_{L_{p}(ℝ)}

≤ B_{p} ∥f∥_{L_{p}(ℝ)} ,

for all f ∈ L_{p}(ℝ).

Theorem 2. Let ψ be an orthonormal wavelet such that ψ ∈ R^{0}. If 1 < p < ∞, there exist constants A_{p} and B_{p}, 0 < A_{p} < B_{p} < ∞ such that

A_{p} ∥f∥_{H_{1}(ℝ)} ≤ ∥(∑_{j∈ℤ} ∑_{k∈ℤ} |⟨f, ψ_{j,k}⟩|^{2} 2^{\frac{j}{2}} χ_{[2^{−j}k,2^{−j}(k+1)]}(·))^{1/2}∥_{L_{p}(ℝ)}

≤ B_{p} ∥f∥_{H_{1}(ℝ)} ,

for all f ∈ H_{1}(ℝ).

Theorem 3. Let ψ in the Schwartz class, S, be a limited band orthonormal wavelet. For 1 < p < ∞ and s = 1, 2, 3, …; there exist constants A_{p,s} and B_{p,s}, 0 < A_{p_{s}} < B_{p,s} < ∞ such that

A_{p} ∥f∥_{L^{p,s}(ℝ)} ≤ ∥(∑_{j∈ℤ} ∑_{k∈ℤ}|⟨f, ψ_{j,k}⟩|^{2}(1 + 2^{js})2^{j}χ_{[2^{−j}k,2^{−j}(k+1)]}(·))^{1/2}∥_{L_{p}(ℝ)}

≤ B_{p} ∥f∥_{L^{p,s}(ℝ)} ,

for all f ∈ L^{p,s}(ℝ).

Frazier, Jawerth and Weiss showed, in [5], that the homogeneous space of Triebel-Lizorkin Ḟ_{1}^{0,2} is isomorphic to the Hardy space H^{1}; also, Fefferman in [1], proved that BMO space is the dual space of H^{1}; and Frazier and Jawerth in [3], proved that Ḟ_{∞}^{0,2} is the dual space of Ḟ_{1}^{0,2}. We use these results to achieve a characterization of BMO, with wavelets coefficients, through functions in Ḟ_{∞}^{0,2}.

2 Preliminares

By L_{p}(ℝ), 1 ≤ p ≤ ∞ we denote the Banach space of equivalence classes of measurable functions on ℝ whose p’th power is integrable (respectively, which are essentially bounded if p = ∞). The norm in L_{p}(ℝ) is defined by

∥f∥_{p} = (∫_{ℝ} |f(x)|^{p} dx)^{1/p} , (1 ≤ p < ∞),

in the case of p = ∞

∥f∥_{∞} = ess sup |f(x)| .

Similarly, by l_{p}(ℝ), 1 < p < ∞ we denote the Banach space of sequences {a_{k}}_{k∈ℤ} of real (complex) numbers whose p’th power is summable (respectively, which are bounded if p = ∞). The norm in l_{p}(ℝ) is defined by

∥{a_{k}}_{k∈ℤ}∥_{p} = (∑_{k∈ℤ} |a_{k}|^{p} dx)^{1/p} ,

in the case of p = ∞

∥{a_{k}}_{k∈ℤ}∥_{∞} = sup_{k∈ℤ} |a_{k}| .

When p = 2 the inner product of functions f, g in L_{2}(ℝ) is defined by

⟨f, g⟩ = ∫_{ℝ} f(x)\overline{g(x)}dx,

so that we shall say that two functions f, g are orthonormal if

⟨f, g⟩ = ∫_{ℝ} f(x)\overline{g(x)}dx = 0.

We will say that the sequence {f_{n}}_{n∈ℤ} is orthogonal if ⟨f_{n}, f_{m}⟩ = δ_{n,m}, where δ is the Dirac delta function

δ_{n,m} = \{ \begin{matrix}1, if n = m, \\ 0, if n ≠ m. \end{matrix}

The convolution f ∗ g is defined by

(f ∗ g) (x) = ∫_{ℝ} f(t)g(x − t)dt.

The Schwartz class S, consist of those functions φ satisfying

i) φ ∈ C_{∞}(ℝ),

ii) sup_{m≤k} sup_{x∈ℝ} (1 + x^{2})^{k} |φ^{(m)}(x)| < ∞.

This kind of functions are known by rapidly decreasing functions.

We shall say that a distribution φ is a tempered distribution if φ ∈ S^{∗}, where S^{∗} is the dual space of S. Also, a tempered distribution φ is modulus polynomial if φ ∈ S^{∗}/P where P is the set of polynomials over ℝ. All these basic concepts can be found in [11].

The following lemma shows how a tempered distribution can be written in series using the Schwartz class of functions S, which have Fourier transform with compact support, and its proof is outlined in [8].

Lemma 1. Let ψ ∈ L^{2}(ℝ) be a function such that supp(ψ̂) ⊆ [−π, π], and

∑_{j∈ℤ} |ψ̂(2^{j}ξ)|^{2} = 1, for a.e. ξ ∈ ℝ − {0} .

If f is a tempered distribution then

f = ∑_{j∈ℤ} ∑_{k∈ℤ} ⟨f, ψ_{j,k}⟩ ψ_{j,k}.

The following result is mentioned in several publications, it will be useful for the present work, and next a proof of the same one is presented.

Proposition 1. If f is a tempered distribution and φ ∈ S then

|⟨f, φ_{j,k}⟩| ≤ 2^{\frac{−j}{2}} sup_{y∈I_{j,k}} |(φ̃_{2^{−j}} ∗ f) (y)| , for all j, k ∈ ℤ,

where

φ_{j,k}(x) = 2^{\frac{j}{2}} φ (2^{j}x − k) , φ̃(x) = \overline{φ(−x)}, φ̃_{2^{−j}}(x) = \frac{1}{2^{−j}} \overline{φ(\frac{−x}{2^{−j}})}

and

I_{j,k} = [2^{−j}k, 2^{−j}(k + 1)].

Proof. Let φ ∈ S. Since |\overline{φ}^{(m)}(x)| = |φ^{(m)}(x)| we have that \overline{φ} ∈ S. In other hand, if f is tempered distribution, we can observe that

⟨f, φ⟩ = ∫_{ℝ} f(t)\overline{φ}(t)dt = f(\overline{φ}),

so, treating f as a bounded linear operator, we get that ⟨f, φ⟩ is finite. Also we get

⟨f, φ_{j,k}⟩ = ∫_{ℝ} f(t) · \overline{φ_{j,k}} (t) dt
= ∫_{ℝ} f(t) · \overline{2^{\frac{j}{2}} φ(2^{j}t − k)} dt
= 2^{\frac{−j}{2}} ∫_{ℝ} f(t) · \frac{2^{j}}{2^{j}} \overline{φ(−2^{j}(−t + 2^{−j}k))} dt
= 2^{\frac{−j}{2}} ∫_{ℝ} f(t) · φ_{2^{−j}}(2^{−j}k − t)dt
= 2^{\frac{−j}{2}} (φ̃_{2^{−j}} ∗ f) (2^{j}k),

for all j, k ∈ ℤ. In consequence, if I_{j,k} = [2^{−j}k, 2^{−j}(k + 1)] then

|⟨f, φ_{j,k}⟩| ≤ 2^{−j/2} sup_{y∈I_{j,k}} |(φ̃_{2^{−j}} ∗ f) (y)| .

Given an orthonormal system {f_{n}}_{n∈ℤ} and a function f, we define the Fourier coefficients respect to the system {f_{n}}_{n∈ℤ} as

c_{k} = ⟨f, f_{k}⟩, k ∈ ℤ,

and the Fourier transform of a function f ∈ L_{1}(ℝ) ∩ L_{2}(ℝ) is defined by

f̂(ξ) = ∫_{ℝ} f(x)e^{−iξx}dx,

and the inverse transform by

f(x) = \frac{1}{2π} ∫_{ℝ} f̂(ξ)e^{iξx}dξ,

in the case of a tempered distribution f, the Fourier transform is also a tempered distribution defined by

f̂(φ) = f \binom{}{φ}, (φ ∈ S),

moreover

(f ∗ φ) ˆ = f̂φ̂.

It is known the following result showed by Hernandez and Weis in [8].

Lemma 2. Let ε > 0. Let f, g be functions and C_{1}, C_{2} > 0 such that

|f(x)| ≤ \frac{C_{1}}{(1 + |x|)^{1+ε}} and |g(x)| ≤ \frac{C_{2}}{(1 + |x|)^{1+ε}} ,

for all x ∈ ℝ. Then there exist C > 0 such that for all j, k, l, m ∈ ℤ, j ≥ l we have

|(f_{j,k} ∗ g_{l,m}) (x)| ≤ \frac{C2^{\frac{j−1}{2}}}{(1 + 2^{j} |x − 2^{−j}k − 2^{−l}m|)^{1+ε}} ,

for all x ∈ ℝ.

We shall say that ψ ∈ L_{2}(ℝ) is a wavelet if {ψ_{j,k}}_{i,k∈ℤ} is an orthonormal

basis for L_{2}(ℝ) where ψ_{j,k}(x) = 2^{\frac{−j}{2}} ψ(2^{j}x − k).

The next Theorem gives us a characterization of the wavelets in terms of properties of their Fourier transform (see [8]).

Theorem 4. A function ψ ∈ L_{2}(ℝ), with ∥ψ∥_{L_{2}(ℝ)} = 1, is an orthonormal wavelet if and only if

∑_{j∈ℤ} |ψ̂(2^{j}ξ)|^{2} = 1, for a.e. ξ ∈ ℝ,

and

∑_{j=0}^{∞} ψ̂(2^{j}ξ) \overline{ψ̂(2^{j} (ξ + 2mπ))} = 0, for a.e. ξ ∈ ℝ, m ∈ 2ℤ + 1.

Definition 1. A wavelet ψ is called limited band wavelet if supp(ψ̂) ⊆ I for some I ⊂ ℝ with l(I) < ∞.

Remark 1. The Lemarié-Meyer wavelet is in the Schwartz class with Fourier transform given by

ψ̂(ξ) = b(ξ)e^{i \frac{ξ}{2}} ,

where

b(ξ) = \begin{cases}sin (\frac{3}{4} (|ξ| − \frac{2}{3}π)) & if \frac{2}{3} π < |ξ| ≤ \frac{4}{3} π, \\ sin (\frac{3}{8} (\frac{8}{3} π − |ξ|)) & if \frac{4}{3}π < |ξ| ≤ \frac{8}{4} π, \\ 0 & in other case. \end{cases}

2.1 Maximals functions

Let f be a locally integrable function over ℝ, we define the maximal function of Hardy-Littlewood by

Mf(x) = sup_{r>0} \frac{1}{2r} ∫_{|y−x|≤r} |f(y)| dy.

An important result regarding this maximal function, and related to L_{p}(ℝ) functions, was established by Fefferman and Stein in [2].

Theorem 5. Let 1 < p, q < ∞, then there exist C_{p,q} such that

∥{∑_{i=1}(Mf_{i})^{q}}^{1/q}∥_{L^{p}(ℝ)} ≤ C_{p,q} ∥{∑_{i=1} |f_{i}|^{q}}^{1/q}∥_{L^{p}(ℝ)} ,

for any sequence f_{i} : i = 1, 2, 3, …; of locally integrable functions.

As an immediately consequence, if f is a locally integrable function then the functions f_{1} = f, 0 = f_{2} = f_{3} = …; are locally integrable functions, and an application of Theorem 5 with q = p shows that

∥M(f)∥_{p} ≤ C_{p} ∥f∥_{p} , p > 1.

Therefore, for any sequence f_{i} : i = 1, 2, 3, …; of locally integrable functions we have

∥M(f_{i})∥_{p} ≤ C_{p} ∥f_{i}∥_{p} , p > 1, i = 1, 2, 3, …  (1)

In the main results section it will be established a new similar result. The following result is proved by Hernández and Weiss in [8].

Lemma 3. Given ε > 0 and 1 ≤ r < 1 + ε there exist C such that for any sequence {s_{j,k} : j, k ∈ ℤ} of complex numbers and any x ∈ I_{j,k} we have

(a) if l ≤ j,

∑_{m∈ℤ} \frac{|s_{l,m}|}{(1 + 2^{l} |2^{−j}k − 2^{−l}m|)^{1+ε}} ≤ C [M (∑_{m∈ℤ} |s_{l,m}|^{1/r} χ_{I_{l,m}}) (x)]^{r} ,

(b) if l ≥ j,

∑_{m∈ℤ} \frac{|s_{l,m}|}{(1 + 2^{l} |2^{−l}k − 2^{−j}m|)^{1+ε}} ≤ 2^{(l−j)r}C [M (∑_{m∈ℤ} |s_{l,m}|^{1/r} χ_{I_{l,m}}) (x)]^{r} ,

where M is the maximal function of Hardy-Littlewood.

For a function g on ℝ and for a real number λ > 0, we consider the maximal function

g_{λ}^{∗}(x) = sup_{y∈ℝ} \frac{g (|x − y|)}{(1 + |y|)^{λ}}, x ∈ ℝ,

Hernández and Weiss show a relation between these maximal functions in [8].

Lemma 4. Let g be a limited band function defined over ℝ, such that g_{λ}^{∗}(x) < ∞, for all x ∈ ℝ and λ > 0. Then there exists a constant C_{λ} > 0 such that

g_{λ}^{∗}(x) ≤ C_{λ} [M (|g|^{1/λ}) (x)]^{λ} , x ∈ ℝ.

Also, the same authors in [8], proved the following Lemma.

Lemma 5. Let φ be a limited band function, f ∈ S′, 0 < p ≤ ∞ such that φ_{2^{−j}} ∗ f ∈ L_{p}(ℝ), for all j ∈ ℤ. Then, for all λ > 0 there exists a constant C_{λ} such that

(φ_{j,λ}^{∗∗}f) (x) ≤ C_{λ} [M (|φ_{2^{−j}} ∗ f|^{1/λ}) (x)]^{λ} , x ∈ ℝ,

where

(φ_{j,λ}^{∗∗}f) (x) = sup_{y∈ℝ} \frac{|(φ_{2^{−j}} ∗ f) (x − y)|}{(1 + 2^{j} |y|)^{λ}}, x ∈ ℝ,

and φ_{t}(x) = (1/t)φ(x/t).

2.2 Hardy, BMO and Triebel-Lizorkin function spaces

Garcia and Rubio de Francia in [6] wrote about the following function space. Given the functions

P_{t}(x) = \frac{1}{π} \frac{t}{x^{2} + t^{2}} ,

Q_{t}(x) = \frac{1}{π} \frac{x}{x^{2} + t^{2}} ,

and a function f ∈ L_{p}(ℝ), 1 ≤ p < ∞, we define the functions

u(x, t) = (P_{t} ∗ f) (x) = \frac{1}{π} ∫_{ℝ} \frac{t}{(x − y)^{2} + t^{2}} f(y)dy,

v(x, t) = (Q_{t} ∗ f) (x) = \frac{1}{π} ∫_{ℝ} \frac{x − y}{(x − y)^{2} + t^{2}} f(y)dy,

both are harmonic functions and

lim_{t→0^{+}}u(x, t) = f(x) a.e in ℝ.

We define F : ℂ → ℂ by

F (z) = u(x, t) + iv(x, t) = \frac{i}{π} ∫_{ℝ} \frac{f(y)}{x − y} dy,

then F is an analytic function.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 26(1): 21–44, Jan–Jun 2019

The set

Re(H)^{1} = {f ∈ L^{1}(ℝ) : sup_{t>0} (∫_{ℝ} |F (x + it)| dx) < ∞} ,

equipped with

∥f∥ = sup_{t>0} (∫_{ℝ} |F (x + it)| dx) ,

is a Banach space. When Re(H)^{1} is "complexified" then it is named Hardy space and denoted by H^{1}(ℝ), that is to say

f ∈ H^{1}(ℝ) if and only if f = g + ih, g, h ∈ Re(H)^{1}.

The sharp maximal function of a locally integrable function f over ℝ is defined by

f^{#}(x) ≡ sup \frac{1}{I} ∫_{I} |f(y) − f_{I}| dy,

where the supremum is taking over all intervals I = (a, b) such that x ∈ I and

f_{I} ≡ \frac{1}{|I|} ∫_{I} f(y)dy.

Let B be the set of all locally integrable functions f such that f^{#} ∈ L^{∞}(ℝ), and

∥f∥_{∗} = ∥f^{#}∥_{∞} .

So we obtain a seminorm ∥·∥_{∗} such that ∥f∥_{∗} = 0 if and only if f is a constant function almost everywhere.

In [9] we can find the following definition.

Definition 2. The quotient space of B modulus the constant functions is called Bounded Mean Oscillation space functions and is denoted by BMO ≡ BMO(ℝ) and ∥·∥_{∗} is a norm over this space, making (BMO, ∥·∥_{∗}) a Banach space. Also, f ∈ BMO when it determines one of the equivalence classes in this quotient space.

This function space was introduced by John and Niremberg in [9], contains all bounded functions, that is to say, L^{∞} ⊂ BMO, and the unbounded functions for which holds

|\{x ∈ I : |f(x) − f_{I}| > t\}| < C_{1}e^{( \frac{−C_{2}t}{∥f∗∥|I|} )},

for some C_{1}, C_{2} > 0. This last inequality is known as John-Niremberg inequality.

C. Fefferman in [1] established that BMO is the dual space of H^{1}(ℝ), also A. Torchinsky enunciated and proved, in [12], this same result.

Theorem 6. BMO(ℝ^{n}) is the dual space of the Hardy space H^{1}(ℝ^{n}). The inner product ⟨f, g⟩ = ∫_{ℝ^{n}} f(x)g(x)dx for f ∈ BMO and g belonging to the dense subspace of C^{∞} rapidly decreasing functions in H^{1}.

Other function spaces that will be of interest in this work are defined below. (See [10]).

Definition 3. For 0 < p < ∞, 0 < q ≤ ∞ and s ∈ ℝ, the Homogeneus Triebel Lizorkin Ḟ_{p}^{s,q} space function is defined as the space consisting of all distributions (modulus polinomia) such that

(∫_{ℝ} (∑_{j∈ℤ} 2^{js} |φ_{2^{−j}} ∗ f(x)|^{q})^{p/q} dx)^{1/p} < ∞,

where φ is a function that satisfy the following conditions

(i) φ ∈ S,

(ii) supp \binom{}{φ̂} ⊆ \{ξ ∈ ℝ : \frac{1}{2} ≤ |ξ| ≤ 2\},

(iii) ∑_{j∈ℤ} |φ̂ (2^{−j}x)|^{2} = 1.

For p = ∞, s ∈ ℝ and 0 < q ≤ ∞ the space Ḟ_{∞}^{sq} is the set consisting of all tempered distributions (modulus polinomial) such that

sup_{l,m∈ℤ} (2^{l} ∫_{ℝ} 2^{js} ∑_{j∈ℤ} |φ_{2^{−j}} ∗ f(x)|^{q} dx)^{1/q} < ∞,

where φ is a function that satisfy the conditions (i), (ii) and (iii).

In the following, ℜ_{∞}^{0,2} will denote the set of all functions that satisfy (i), (ii), (iii) conditions in Definition 3.

For the purpose of this work is important next theorem, which is enunciated and proved by Jawerth and Frazier in [3].

Theorem 7. Suppose that s ∈ ℝ and 0 < q < ∞. Then \binom{}{Ḟ_{1}^{s,q}}^{∗} ≈ Ḟ_{∞}^{−s,q′}.

That is, if g ∈ Ḟ_{∞}^{−s,q′}, the mapping l_{g}, defined by l_{g}(f) = ⟨f, g⟩, for f ∈ S_{0}, where

S_{0} = {f : f̂ = 0 in some open set containing zero} ,

can be extended to bounded linear functional over Ḟ_{1}^{s,q} with ∥l_{g}∥ ≈ ∥g∥Ḟ_{∞}^{−s,q′}.

Reciprocally, if l ∈ \binom{}{Ḟ_{1}^{s,q}}^{∗} then there exist g ∈ Ḟ_{∞}^{−s,q′} such that l = l_{g}.

Also, Frazier et. al. in [5] proved the following result.

Theorem 8. The Hardy space H^{1} is isomorphic to the Triebel Lizorkin space Ḟ_{1}^{0,2}.

Remark 2. The BMO space is isomorphic to the Triebel Lizorkin space Ḟ_{∞}^{0,2} due to Theorems 6, 7 and 8.

3 Main results

The following result is about Hardy-Littlewood maximal function.

Theorem 9. Let {f_{n}}_{n∈ℤ} be a sequence of locally integrable functions. Then there exist C > 0 such that

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{i≥l} |Mf_{i}(x)|^{2} dx)^{1/2} ≤ C sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{i≥l} |f_{i}(x)|^{2} dx)^{1/2} ,

where I_{l,m} = [2^{−l}m, 2^{−l} (m + 1)] for l, m ∈ ℤ.

Proof. Let f_{i} : i = 1, 2, 3, …; be a sequence of locally integrable functions. From inequality (1) it is obtained that

∥M(f_{i})∥_{p} ≤ C_{p} ∥f_{i}∥_{p} , p > 1, i = 1, 2, …

34 j.e. hernández In particular if p = 2 then

∫_{ℝ} M(f_{i}(x))^{2}dx ≤ C_{2}^{2} ∫_{ℝ} |f_{i}(x)|^{2} dx.

Let I ⊂ ℝ be an interval. Let g_{i} : i = 1, 2, 3, …; be the sequence of locally integrable functions defined by g_{i} = f_{i} · χ_{I}, i = 1, 2, 3, … From (1) it is had

∫_{ℝ}(Mg_{i}(x))^{2}dx ≤ C_{2}^{2} ∫_{ℝ} |g_{i}(x)|^{2} dx,

that is

∫_{ℝ}(M(f_{i} · χ_{I})(x))^{2}dx ≤ C_{2}^{2} ∫_{ℝ} |f_{i} · χ_{I}(x)|^{2} dx = C_{2}^{2} ∫_{I} |f_{i}(x)|^{2} dx,

therefore

∫_{I}(M(f_{i})(x))^{2}dx ≤ C_{2}^{2} ∫_{I} |f_{i}(x)|^{2} dx.

Let l, m ∈ ℤ arbitrary and fixed integers. We can observe that if I = I_{l,m} then

∫_{I_{l,m}} ∑_{i≥l}(Mf_{i}(x))^{2}dx = ∑_{i≥l} ∫_{I_{l,m}}(Mf_{i}(x))^{2}dx  (2)
≤ C_{2}^{2} ∑_{i≥l} ∫_{I_{l,m}} |f_{i}(x)|^{2} dx = C_{2}^{2} ∫_{I_{l,m}} ∑_{i≥l} |f_{i}(x)|^{2} dx.

So, multiplying both sides of (2) the inequality by 2^{l} and taking supreme over l, m ∈ ℤ it is obtained

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{i≥l} |Mf_{i}(x)|^{2} dx)^{1/2} ≤ C sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{i≥l} |f_{i}(x)|^{2} dx)^{1/2} ,

where C = C_{2}. The proof is complete. Theorem 10. Let ψ ∈ ℜ_{∞}^{0,2}. Then

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊂l_{l,m}} 2^{j} |⟨f, ψ_{j,k}⟩|^{2} χ_{I_{j,k}}(x)) ≤ C ∥f∥Ḟ_{∞}^{0,2} ,

for all distribution f ∈ Ḟ_{∞}^{0,2}.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 26(1): 21–44, Jan–Jun 2019

Before proceed with the proof, let us stablish the following observation about the notation; the symbol

∑_{I_{j,k}⊂l_{l,m}},

can be written as

∑_{j≥l} ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1},

since I_{j,k} and I_{l,m} are dyadic intervals, and I_{j,k} ⊆ I_{l,m} indicates that the sum is taken over those index j, k ∈ ℤ such that the inclusion condition hold.

Proof. Let us fix l, m ∈ ℤ. Let j ∈ ℤ such that j ≥ l and x ∈ I_{l,m}. Using Proposition 1 we find

∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x)  (3)

≤ ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} ( sup_{y∈I_{j,k}} |(ψ̃_{2^{−j}} ∗ f) (y)|)^{2} χ_{I_{j,k}}(x).

Note that if x ∈ I_{l,m} , then there exists an unique k′ ∈ {2^{j−l}m, …, 2^{j−l}(m + 1) − 1} such that x ∈ I_{j,k′}, because of I_{j,k} are disjoint intervals. Then the inequality (3) can be rewritten as

∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x) ≤ ( sup_{y∈I_{j,k}} |(ψ̃_{2^{−j}} ∗ f) (y)|)^{2} χ_{I_{j,k′}} (x).

(4)

But, if x ∈ I_{j,k′} then for all y ∈ I_{j,k′} there exists z ∈ ℝ such that y = x − z, and since

2^{−j}k′ ≤ x ≤ 2^{−j}(k′ + 1) and 2^{−j}k′ ≤ y ≤ 2^{−j}(k′ + 1),

then

2^{−j}k′ − 2^{−j}(k + 1) ≤ x − y ≤ 2^{−j}(k′ + 1) − 2^{−j}k′,

that is

−2^{−j} ≤ z ≤ 2^{−j},

which is equivalent to

|z| ≤ 2^{−j},

so, the inequality (4) can be written as

∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x) ≤ ( sup_{|z|≤2^{−j}} |(ψ̃_{2^{−j}} ∗ f) (x − z)|)^{2} . (5)

Let λ be an arbitrary positive real number. Multiplying and dividing by (1 + 2^{j} |z|)^{2λ} in inequality (5), it is had

∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x) ≤(\begin{matrix}sup_{|z|≤2^{−j}} & |(ψ̃_{2^{−j}} ∗ f)(x − z)| \\ (1 + 2^{j} |z|) & λ \end{matrix} )^{2}(1+2^{j}|z|)^{2λ}

≤(\begin{matrix}sup_{|z|≤2^{−j}} & |(ψ̃_{2^{−j}} ∗ f) (x − z)| \\ (1 + 2^{j} |z|) & λ \end{matrix} )^{2} 2^{2λ}

=(2^{λ} (ψ_{j,λ}^{∗∗}f) (x))^{2} ,

where ψ_{jλ}^{∗∗} f is the maximal function defined in Lemma 5.

Taking the sum respect to j it is obtained

∑_{j≥l} ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x) ≤ ∑_{j≥l} (2^{λ} (ψ_{j,λ}^{∗∗}f) (x))^{2} , (6)

in consequence, using the Lemma 5

∑_{j≥l} ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x) ≤ C_{λ} ∑_{j≥l} {M (|ψ_{2^{−j}} ∗ f|^{1/λ}) (x)}^{2λ} .

In particular, taking λ = 1 and using the Theorem 9 with f_{j} = ψ_{2^{−j}} ∗ f , it follows

C_{1} ∫_{I_{l,m}} ∑_{j≥l} {M (|ψ_{2^{−j}} ∗ f|) (x)}^{2} dx ≤ C_{1}C ∫_{I_{l,m}} ∑_{j≥l} |(ψ_{2^{−j}} ∗ f) (x)|^{2} dx,

so

∫_{I_{l,m}} ∑_{j≥l} ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1}|⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x) ≤ C_{1}C∫_{I_{l,m}} ∑_{j≥l} |(ψ_{2^{−j}} ∗ f) (x)|^{2} dx,

multiplying by 2^{l}, and taking supreme over l, m ∈ ℤ, it is attained

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{j≥l} ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x)) ≤ C ∥f∥Ḟ_{∞}^{0,2} .

Here, the capital letter C is used for denote the product C_{1}C. The proof is complete.

Next result establishes an opposite inequality to that found in Theorem 10.

Theorem 11. Let ψ be a function in ℜ_{∞}^{0,2}. If f ∈ Ḟ_{∞}^{0,2} then

∥f∥Ḟ_{∞}^{0,2} ≤ D sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{j≥l} ∑_{k=2^{j−l}m}^{2^{j−l}(m+1)−1} |⟨f, ψ_{j,k}⟩|^{2} 2^{j} χ_{I_{j,k}}(x)) .

Proof. Let ψ ∈ ℜ_{∞}^{0,2} and f ∈ Ḟ_{∞}^{0,2} be as in the statement. Since f is a tempered distribution (modulus polinomial), by Lemma 1, it can be written as

f = ∑_{r∈ℤ} ∑_{s∈ℤ} ⟨f, ψ_{r,s}⟩ ψ_{r,s}.

For a fixed and arbitrary j ∈ ℤ, it follows

(ψ_{2^{−j}} ∗ f) (x) = ∑_{r∈ℤ} ∑_{s∈ℤ} ⟨f, ψ_{r,s}⟩ (ψ_{2^{−j}} ∗ ψ_{r,s}) (x),

and taking Fourier transform in that expression, it is obtained

(ψ_{2^{−j}} ∗ f)^{^}(ξ) = ∑_{r∈ℤ} ∑_{s∈ℤ} ⟨f, ψ_{r,s}⟩ ψ̂_{2^{−j}}(ξ)ψ̂_{r,s}(ξ).  (7)

Now, it is known that supp \binom{}{ψ} ⊆ \{ξ ∈ ℝ : (\frac{1}{2}) ≤ |ξ| ≤ 2\} implies that

supp \binom{}{ψ_{2^{−j}}} ⊆ \{ξ ∈ ℝ : 2^{j−1} ≤ |ξ| ≤ 2^{j+1}\} ,

and

supp \binom{}{ψ_{r,s}} ⊆ \{ξ ∈ ℝ : 2^{r−1} ≤ |ξ| ≤ 2^{r+1}\} ,

therefore, if ξ ∉ supp \binom{}{ψ_{2^{−j}}} ⋂ supp \binom{}{ψ_{r,s}} then the terms ψ̂_{2^{−j}}(ξ)ψ̂_{r,s}(ξ) are zeroes for those r ∈ ℤ such that r ∉ {j − 1, j, j + 1}, so the series (7) can be written as

(ψ_{2^{−j}} ∗ f)^{^}(ξ) = ∑_{r=j−1}^{j+1} ∑_{s∈ℤ} ⟨f, ψ_{r,s}⟩ ψ̂_{2^{−j}}(ξ)ψ̂_{r,s}(ξ).

Applying Fourier inverse transform it is had

(ψ_{2^{−j}} ∗ f) (x) = ∑_{r=j−1}^{j+1} ∑_{s∈ℤ} ⟨f, ψ_{r,s}⟩ (ψ_{2^{−j}} ∗ ψ_{r,s}) (x).

Since ψ ∈ S then (ψ_{2^{−j}} ∗ ψ_{r,s}) ∈ S, and for ε > 0, an application of Lemma 2, it follows that

|ψ_{2^{−j}} ∗ ψ_{r,s}(x)| ≤ \frac{2^{\frac{j}{2}} C}{(1 + 2^{r} |x − 2^{−r}s|)^{1+ε}} .

Since g(x) = \frac{1}{(1 + |x − a|)^{1+ε}} has an absolute maximum in x = a, then, if

x ∈ I_{j−1,s_{1}} ⊆ I_{j,s_{2}} ⊆ I_{j+1,s_{3}} it is obtained

\frac{2^{\frac{j}{2}} C}{(1 + 2^{j−1} |x − 2^{−j+1}s|)^{1+ε}} ≤ \frac{2^{\frac{j}{2}} C}{(1 + 2^{j−1} |2^{j−1}s_{1} − 2^{−j+1}s|)^{1+ε}} ,

\frac{2^{\frac{j}{2}} C}{(1 + 2^{j} |x − 2^{−j}s|)^{1+ε}} ≤ \frac{2^{\frac{j}{2}} C}{(1 + 2^{j} |2^{j}s_{2} − 2^{−j}s|)^{1+ε}} ,

and

\frac{2^{\frac{j}{2}} C}{(1 + 2^{j} |x − 2^{−j}s|)^{1+ε}} ≤ \frac{2^{\frac{j}{2}} C}{(1 + 2^{j+1} |2^{j+1}s_{3} − 2^{−j−1}s|)^{1+ε}} ,

for all s ∈ ℤ. characterization of bmo using wavelets 39

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 26(1): 21–44, Jan–Jun 2019

In consequence,

|ψ_{2^{−j}} ∗ f(x)| |ψ_{2^{−j}} ∗ f(x)| ≤ ∑_{s∈ℤ} \frac{2^{\frac{j}{2}} C |⟨f, ψ_{j−1,s}⟩|}{(1 + 2^{j−1} |2^{j−1}s_{1} − 2^{−j+1}s|)^{1+ε}}

+ ∑_{s∈ℤ} \frac{2^{\frac{j}{2}} C |⟨f, ψ_{j,s}⟩|}{(1 + 2^{j} |2^{j}s_{2} − 2^{−j}s|)^{1+ε}}

+ ∑_{s∈ℤ} \frac{2^{\frac{j}{2}} C |⟨f, ψ_{j+1,s}⟩|}{(1 + 2^{j+1} |2^{j+1}s_{3} − 2^{−j−1}s|)^{1+ε}} .

Now, with an application of Lemma 3 to each sum, it follows estimate

|ψ_{2^{−j}} ∗ f(x)| ≤ 2^{\frac{j}{2}} C [M (∑_{s∈ℤ} |⟨f, ψ_{j−1,s}⟩| χ_{I_{j−1,s}}) (x)

+ M (∑_{s∈ℤ} |⟨f, ψ_{j,s}⟩| χ_{I_{j,s}}) (x)

+ M (∑_{s∈ℤ} |⟨f, ψ_{j+1,s}⟩| χ_{I_{j+1,s}}) (x)] .

Taking power operations, summing about j, integrating over I_{l,m} and taking supreme over l, m ∈ ℤ it is attained

∥f∥Ḟ_{∞}^{0,2} = sup_{l,m∈ℤ} (2^{l}∫_{I_{l,m}} ∑_{j≥l} |ψ_{2^{−j}} ∗ f(x)|^{2} dx)^{1/2}

≤ C [ sup_{l,m∈ℤ} (2^{l}∫_{I_{l,m}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j−1,s}⟩| χ_{I_{j−1,s}}) (x)dx)^{1/2}

+ sup_{l,m∈ℤ}(2^{l}∫_{I_{l,m}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j,s}⟩| χ_{I_{j,s}}) (x)dx)^{1/2}

+ sup_{l,m∈ℤ}(2^{l}∫_{I_{l,m}} ∑_{j≥l} M^{2}(∑_{s∈ℤ} |⟨f, ψ_{j+1,s}⟩| χ_{I_{j+1,s}})(x)dx)^{1/2}] . (8)

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 26(1): 21–44, Jan–Jun 2019

40 j.e. hernández Now, doing the index change k = j − 1 in the first term of the previous sum,

2^{l} ∫_{I_{l,m}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j−1,s}⟩| 2^{\frac{j}{2}} χ_{I_{j−1,s}})(x)dx
= 2^{l} ∫_{I_{l,m}} ∑_{k≥l−1} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{k,s}⟩| 2^{\frac{j}{2}} χ_{I_{k,s}})(x)dx,

since I_{l,m} ⊂ I_{l−1,m−1}, it is had

2^{l} ∫_{I_{l,m}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j−1,s}⟩| 2^{\frac{j}{2}} χ_{I_{j−1,s}})(x)dx
≤ 2^{l}C ∫_{I_{l−1,m−1}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{k,s}⟩| 2^{\frac{k}{2}} χ_{I_{k,s}})(x)dx,

so, renaming the index in the series and taking supreme over l, m ∈ ℤ

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j+1,s}⟩| 2^{\frac{j}{2}} χ_{I_{j+1,s}}) (x)dx)^{1/2}
≤ C sup_{l,m∈ℤ} (2^{l−1} ∫_{I_{l−1,m−1}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j,s}⟩| 2^{\frac{j}{2}} χ_{I_{j,s}}) (x)dx)^{1/2} .

Now, adding the positive terms of the form |⟨f, ψ_{j,s}⟩| with s ∈ ℤ in the third term of (8), it is seen that

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j−1,s}⟩| 2^{\frac{j}{2}} χ_{I_{j−1,s}}) (x)dx)^{1/2}
≤ C sup_{l,m∈ℤ} (2^{l} ∫_{I_{l−1,m−1}} ∑_{j≥l} M^{2} (∑_{s∈ℤ} |⟨f, ψ_{j,s}⟩| 2^{\frac{j}{2}} χ_{I_{j,s}}) (x)dx)^{1/2} .

Thus inequality (8) remains as

∥f∥Ḟ_{∞}^{0,2} ≤ C sup_{l,m∈ℤ} (2^{l}∫_{I_{l−1,m−1}} ∑_{j≥l} M^{2}(∑_{s∈ℤ} |⟨f, ψ_{j,s}⟩| 2^{\frac{j}{2}} χ_{I_{j,s}}) (x)dx)^{1/2} .

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 26(1): 21–44, Jan–Jun 2019

Applying Theorem 9 in the right side of the previous inequality, it is attained

∥f∥Ḟ_{∞}^{0,2} ≤ C sup_{l,m∈ℤ} (2^{l} ∫_{I_{l−1,m−1}} ∑_{j≥l} ∑_{s∈ℤ} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx)^{1/2} .

The proof is complete.

The following proposition shows another valid expression for the considered characterization. Proposition 2.

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx)^{1/2} < ∞,

if and only if there exist a C > 0 such that

2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} ≤ C,

for all l, m ∈ ℤ. Proof. If

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx)^{1/2} = A < ∞,

then

2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x) dx ≤ A^{2},

for all m, l ∈ ℤ. Since

∫_{I_{l,m}} χ_{I_{j,s}}(x)dx = 2^{.j},

it is obtained that

2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx

=2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} ∫_{I_{l,m}} χ_{I_{j,s}}(x)dx ≤ A^{2}.

That is

2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} ≤ C,

where C = A^{2}.

Reciprocally, if there exist C > 0 such that

2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} ≤ C,

for all l, m ∈ ℤ then

2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{j}2^{−j} = 2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} ∫_{I_{l,m}} χ_{I_{j,s}}(x)dx

= 2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx,

so

2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx ≤ C.

Operating with the \frac{1}{2} power and taking supreme over l, m ∈ ℤ, it is had

sup_{l,m∈ℤ} (2^{l} ∫_{I_{l,m}} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} 2^{\frac{j}{2}} χ_{I_{j,s}}(x)dx)^{1/2} ≤ C.

So, the proof is complete.

As immediate consequence from Proposition 2, Theorems 10and 11we have the following.

Corollary 1. Let ψ ∈ ℜ_{∞}^{0,2}. f ∈ BMO if and only if there exist C > 0 such that

2^{l} ∑_{I_{j,k}⊆I_{l,m}} |⟨f, ψ_{j,s}⟩|^{2} ≤ C,

for all l, m ∈ ℤ.

In particular, if ψ is a orthonormal wavelet in ℜ_{∞}^{0,2} we obtain a characterization of BMO using wavelets coefficients. The Lemorié - Meyer wavelet, showed Preliminaries section, is in the class ℜ_{∞}^{0,2}.

4 Conclusion

In the present study, results concerning the characterization of the space BMO functions are found by using a special class of wavelet and homogeneous spaces of Triebel Lizorkin Ḟ_{∞}^{0,2}. Also, a new version of a vector-valued inequality using the classic maximal function of Hardy-Littlewood was also found (Theorem 9). Finally, as a consequence of the characterization established for BMO (Theorems 10 and 11), a result was found (Corollary 1) which responses to a proposal made by Wojtaszczyk in [14].

This work is expected to serve as a useful and motivating tool to find new characterizations of function spaces using wavelets.

Acknowledgments

The author thanks to the Council for Scientific, Humanistic and Technological Development (CDCHT) of Universidad Centroccidental Lisandro Alvarado, and especially to Dr. Ventura Echandía Dean of the School of Sciences of the Faculty of Sciences of the Central University of Venezuela for his teachings and contributions.

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