Characterization of Weighted Function Spaces In Terms of Wavelet Transforms

In this section, we recall some notations and basic definitions, also mention certain weight functions and results given in [2], which we will invoke in the analysis. In Section 2, we define the spaces ∧p,q ω in terms of differences △x, and B p,q ω,ψ, 1 ≤ p, q <∞ by means of wavelet transforms. Furthermore, by using the techniques of Ansorena and Blasco [2], we show that the norms on these spaces are equivalent. Notations: Throughout the paper, R denote the set of positive real numbers, S denote the Schwartz class of test functions on R, S ′ the space of tempered distributions, S0 the set of functions in S with mean zero and S ′ 0 its topological dual.

In this section, we recall some notations and basic definitions, also mention certain weight functions and results given in [2], which we will invoke in the analysis.In Section 2, we define the spaces p,q ω in terms of differences △ x , and B p,q ω,ψ , 1 ≤ p, q < ∞ by means of wavelet transforms.Furthermore, by using the techniques of Ansorena and Blasco [2], we show that the norms on these spaces are equivalent.
Notations: Throughout the paper, R + denote the set of positive real numbers, S denote the Schwartz class of test functions on R n , S ′ the space of tempered distributions, S 0 the set of functions in S with mean zero and S ′ 0 its topological dual.
Definition 1.1.The Fourier transform of a function f is denoted by f and defined as provided the integral exists.
Definition 1.2.The wavelet transform W ψ of a function f with respect to a wavelet ψ is defined as where a ∈ R + , b ∈ R n , ψ a,b (x) = 1 a n ψ x−b a and h(x) = ψ(−x), provided the integral exists.Definition 1.5.Let ǫ ≥ 0, δ ≥ 0 and ω be a weight function.Then ω is said to be a d ǫ -weight if there exists C ≥ 0 such that Some important properties:

For any δ
Definition 1.7 (Radial function).A function defined on Euclidean space R n whose values at each point depends only on the distance between that points and the origin is called a radial function.For example a radial function Φ in two dimensional space has the form Φ(x, y) = φ(r), r = x 2 + y 2 where φ is a function of a single non-negative real variable.
Definition 1.8.In this paper, A and A 1 denote the space of the functions defined by A 1 = ψ ∈ A : ψ radial and real, and supp ψ ⊆ {|x| ≤ 1}, Characterization of Function Spaces In Terms of Wavelet Transforms 71 Definition 1.9 (Calderón Reproducing Formula [2]).Let ψ ∈ A and f ∈ S .For ξ ∈ R n \ {0}, the Fourier transform of f is given by Then f ǫ,δ (x) converges to f in S ′ 0 as ǫ → 0 and δ → ∞.

Characterization of Function Spaces by Using the Wavelet Transform
Definition 2.1 (The space p,q ω ).Given a weight function ω and 1 ≤ p, q ≤ ∞, the space p,q ω denotes the space of measurable functions f : R n → C such that Now, we define a new function space B p,q ω,ψ by means of the wavelet transform.
Definition 2.2 (The space B p,q ω,ψ ).For 1 ≤ p, q ≤ ∞, ψ ǫ S 0 and a weight ω, the space B p,q ω,ψ denotes the space of functions f : and for q = ∞, where P is the parity operator defined by ) n+1 , we have and where C > 0, is a constant.
Proof.Since ψ is a wavelet, therefore R n ψ(x)dx = 0 and hence the wavelet transform of f with respect to P ψ may be written as Using L p norm and Minkowski's inequality [7, p-41], we get and hence we get the following inequality

6)
Characterization of Function Spaces In Terms of Wavelet Transforms 73 Suppose ψ satisfies the following estimates Then by using (2.7) in (2.6), we get Now we prove the second part.For 0 < ǫ < δ, we have Using Minkowski's inequality [7, p-41] we get the following estimate (2.8) and where ▽ denotes the gradient n j=1 e j ∂ ∂xj , where e j is the unit vectors.Hence where Hence from (2.8), we have Then from Lemma 2.4, it follows that Using (1.3) and (1.4) we get 3), (1.4) and (2.2) in (2.4), we get Proof.Let us assume that f ∈ p,1 ω .We have to prove that From (2.10) that is,
Proof.Let f ∈ p,q ω .Let us first show that (2.17) Since λ ∈ W 0,1 = (d 0 ) ∩ (b 1 ) then right hand side of above inequality is bounded a.e., and hence Now, using Holder's inequality, we have and hence by lemma (2.4) the result (2.17) is proved.

Definition 1 . 3 .
A non-negative bounded measurable function ω : R + → R + is referred to as a weight function or simply a weight.Definition 1.4.A weight function ω is said to satisfy Dini's condition if there exists a constant C > 0 such that s 0 ω(t) t dt ≤ C ω(s) a.e.s > 0.

. 11 )
Characterization of Function Spaces In Terms of Wavelet Transforms 77 Again C |x| n min |x| n ,
2.15) and (2.16) gives (2.13) and (2.14) in lemma(2.7).Hence T k define Characterization of Function Spaces In Terms of Wavelet Transforms 81 in (2.12) is bounded operator from L q R n , dx