Hardy-Type Spaces and its Dual

In this paper we defined a new Hardy-type spaces using atoms on homogeneous spaces which we call H. Also we prove that under certain conditions BMO (p) φ is the dual of H.


Introduction
The Hardy space H p were first studied on the unit disk in the complex plane.In their 1968 paper Duren, Romberg and Shield (see [4]) make the following definitions and comments about H p .For 0 < p ≤ ∞, H p is the linear space of functions f (z) analytic in |z| < 1 such that remains bounded as r → 1.If 1 ≤ p ≤ ∞, H p is a Banach space under the norm kf k p = lim r→1 M p (r, f ).
For 0 < p < 1, this is not a norm, but H p is still a complete metric space with a translation invariant metric d(f, g) = kf − gk p p .
A linear functional ϕ on H p is bounded (ϕ ∈ (H p ) * ) if It is easily verify that (H p ) * is a Banach space.Duren, Romberg and Shield (see [4]) were the first to study the linear space structure of the H p space with 0 < p < 1.These H p spaces are not Banach spaces and are not locally convex.
They may be regarded as closed subspaces of L p for 0 < p < 1; however,it is interesting to note that although there are no continuous linear functionals on L p for 0 < p < 1, there are many on H p .Duren, Romberg and Shield (see [4]) prove for 1/2 < p < 1, that (H p ) * = Λ α the Lipschitz space of order α = 1 p − 1.For p ≤ 1 2 , the results are similar.Even though H p is not locally convex, there are still enough linear functionals to distinguish elements.For example as noted in [4], Later, the study of H p spaces was extended to H p (R n ).The results were highly specialized to R n until Latter (see [5]), Coifman and Weiss (see [3]) defined H p (R n ) using the notion of an atom and proved that the atomic H p (R n ) space were equivalent to the original H p (R n ).Roughly speaking, an atom is a "building block" function which is supported on a ball, has zero integral and has a bounded average.
By thinking of the H p spaces in terms of atoms Coifman and Weiss (see [2]) were able to prove that the dual of H p is again a Lipschitz space of order α = 1 p − 1 not only in R n , but on any homogeneous space X .The H p space for 0 < p ≤ 1 on R n were first characterized in terms of atoms by Coifman (see [3]) and Latter (see [6]).Coifman and Weiss (see [2]) then used this characterization to define H p (X ), where X is a homogeneous space.
In this paper, we extend the work of Coifman and Weiss (see [3]) by defining new Hardy-type spaces using atoms on homogeneous space which we call H ϕ,q .The main result of this paper is the following.
Theorem 1.1.Suppose ϕ and w are related by .
Suppose also that ϕ(t) t is a decreasing function of t and that ϕ(t) t is an increasing function for some 0 < < 1.Let 1 ≤ q < ∞, and let p be conjugate of q.Then the dual of H ϕ,q is BMO p ϕ .

Atoms
We begin by defining atoms.The idea for the relationship between w and ϕ functions come from Janson's paper (see [5]).Throughout this paper, we will assume that the measure µ is a regular measure.
Definition 2.1.A measurable function a is said to be a (ϕ, q) atom if it satisfies: 1.The support of a is contained in a ball B(x 0 , r), ´, where w and ϕ are related by and that (3) can be written as , where B = B(x 0 , r).

Spaces of Homogeneous type
Let us begin by recalling the notion of space of homogeneous type.
with the following properties: 1. d(x, y) = 0 if and only if x = y.
3. There exists a constant K such that for all x, y, z ∈ X .
A quasimetric defines a topology in which the balls B(x, r) = {y ∈ X : d(x, y) < r} form a base.These balls may be not open in general; anyway, given a quasimetric d, is easy to construct an equivalent quasimetric d 0 such that the d 0 -quasimetric balls are open (the existence of d 0 has been proved by using topological arguments in [7]).So we can assume that the quasimetric balls are open.A general method of constructing families {B(x, δ)} is in terms of a quasimetric.
Example 3.1.A space of homogeneous type (X , d, µ) is a set X with a quasimetric d and a Borel measure µ finite on bounded sets such that, for some absolute positive constant A the following doubling property holds for all x ∈ X and r > 0.
Next, we are ready to give some example of a space of homogeneous type.
, put in X the euclidean distance and the following measure µ: µ is the usual surface measure on {x : |x| = 1} and µ ({0}) = 1.Then µ is doubling so that (X , d, µ) is a homogeneous space.
k=1 B k with the euclidean distance and the measure µ such that µ (B k ) = 2 k and on each ball B k , µ is uniformly distributed.Claim 1. µ satisfies the doubling condition.Let B r = B(P, r) with P = (P 1 , . . ., P n ) and r > 0. Case 1. Assume for some k, B k ⊂ B r and let k 0 = max {k : Hence the doubling condition holds with A = 4. Case 2. If for all k, B k B r , then r < 1 so that B r and B 2r intersect only on ball B k .Then the doubling condition holds.

ϕ-Lipschitz space
We define the ϕ-Lipschitz space and denoted it by L ϕ to be the space for all measurable functions f on X for which where B is any ball containing both x and y and C is a constant depending only of f .Let γ(f ) be the inf over all C for which the above inequality holds.Then if we define a straightforward argument shows that L ϕ , with this norm, is a Banach space.To simplify calculations, we assume that if µ(X ) is finite, then µ(X ) = 1.We now define H ϕ,q to be the subspace of (L ϕ ) * consisting of those linear functionals admitting an atomic decomposition as follows: h ∈ H ϕ,q if h can be written as a sum h = P j∈N λ j a j , where a j is a (ϕ, q) atom, and We denote by the symbol khk ϕ,q the quantity (which is not, in general a norm)

Functions of Bounded (ϕ, p) mean Oscillation
In this section, we recall the definition of the space of functions of bounded (ϕ, p) mean oscillation, BM O (p) ϕ (X ), where X is a space of homogeneous type.Let ϕ be a nonnegative function on [0, ∞).A locally µ-integrable Where the sup is taken over all balls B ⊂ X and For more detail on functions of bounded (ϕ, p) mean oscillation see Castillo, Ramos Fernández and Trousselot [1].

Quasi-Concavity
In this section, we study the notion of quasi-concavity, which is the condition that we will need to prove our main result.Definition 6.1.A non-negative function φ is said to be quasi-convex if there exists a convex function A and a constant C > 1 such that A(t) ≤ φ(t) ≤ CA(t).Definition 6.2.A function ψ is said to be quasi-concave if there exists a constant C > 1 and a concave function M such that We will use the following Lemmas to prove that the function W as introduced in the definition of a (ϕ, q) atom is quasi-concave under appropriate conditions on ϕ.Lemma 6.1.Suppose that ϕ(x) x is a decreasing function of x, and suppose also that ϕ(x) x is an increasing function for some 0 < < 1.Let Then ψ is concave, ϕ is quasi-concave, and xψ(x) is quasi-convex.
Proof: The derivative x is decreasing by hypothesis.Therefore, ψ is concave.To show that ϕ is quasi-concave, first note that ϕ(x) ≤ ψ(x) since To show the other inequality, we estimate ψ(Cx), for C < 1 by Therefore, we have Next, we choose C by letting C = 1 +1 .Since 0 < < 1, C also satisfies C < 1 and Cψ(Cx) ≤ ϕ(x).Thus, we have shown that ϕ is quasi-concave.
To show that xψ(x) is quasi-convex, let g(x) = xψ(x).Note that g(x) x = ψ(x) is increases, so We also have and we have shown that Therefore, g is quasi-convex, which completes the proof.Lemma 6.2.
1. ϕ is quasi-concave if and only if there exists a constant C < 1 such that ϕ(t 1 ) 2. ψ is quasi-convex if and only if there exists a C > 1 such that for all 0 < t 1 ≤ t 2 .

Now, M (t)
t is a non-increasing function of t, so, for all 0 < t 1 ≤ t 2 , we have Thus, Then ψ is concave by Lemma 6.1.Also, as in the proof of Lemma 1, we have Thus, Cψ(Ct) ≤ ϕ(t), which gives us the first inequality in the quasiconcavity definition.For the other inequality, note that since ϕ(t) t decreases, and C < 1, Thus, we have shown that Cψ(Ct) ≤ ϕ(t) ≤ ψ(t), where ψ is concave, proving that ϕ is quasi-concave.
The proof of ( 2) is similar to the above proof of (1).

Duality
Theorem 1.1 and its proof are modeled on H ϕ,q , where ϕ(t) = t 1/p−1 and w(t) = t p .Clearly, in this case, ϕ(t) t decreases and ϕ(t) t increases for some 0 < < 1.To prove Theorem 1.1, we let L be a bounded linear functional on H ϕ,q , and we fix a ball B in X .We show first that L is a bounded linear functional on the subspace Then, using the Hahn-Banach Theorem and the Riesz Representation Theorem, we extend L to L q (B) with the same norm, and we uniquely represent L by an integral with L p function g.Using an increasing sequence of balls converging to X , we then find a unique function g such that if f ∈ L q (B), for any ball B. Finally, by making a (ϕ, q) atom from ϕ , and we note that by Hölder Inequality, BMO ϕ ⊂ BM O ϕ .To show that any g ∈ BMO (p) ϕ defines a bounded linear functional on H ϕ,q , we first show for an atom a ∈ H ϕ,q , supported on a ball B, ¯ZB gadµ ¯≤ kgk BM (p) ϕ for h ∈ H ϕ,q , we decompose h into a sum of (ϕ, q)-atoms and we use the quasi-concavity of w to show that Therefore, g defines a bounded linear functional on H ϕ,q given by This shows that L g is a bounded linear functional on H ϕ,q .
Proof of Theorem 1.1 The proof of this Theorem follows along the same lines as the proof of [2].Let L be a bounded linear functional on H ϕ,q , and let kLk be the norm of L. Fix a ball B in X .Let is a (ϕ, q) atom, since, by (2) of the atomic definition We also have Hence, Lf is defined and That is, L is a bounded linear functional on L q 0 (B).By the Hahn-Banach Theorem, we can extend L to L q (B) with the same norm and by the Riesz Representation Theorem, we can conclude that there exists g ∈ L p (B) such that Lf = R B fgdµ for all f ∈ L q 0 (B).The function g is uniquely determined up to a constant , or, equivalently if R B fgdµ = 0 for all f ∈ L q 0 (B), then it follows that g is a constant.To see this, suppose Since this equality holds for all h ∈ L q (B), it must be true that g(x) = g B a.e.x in B.
Let {B j } ∞ j=1 be an increasing sequence of balls converging to X , such that µ(B 1 ) > 0. We obtain a function gj satisfying for each j.Now, let g j = gj − (g j ) B 1 .
Then, R B 1 g j dµ = 0.It remains to show that g j | B k = g k for all k ≤ j.By the above remark, we know that on B k ⊃ B 1 , we have g j − g k = C. Now, integrate both sides over B 1 to obtain Cdµ, which implies that 0 = µ(B 1 ).Therefore, C = 0, and we conclude that We now have a unique function g such that if f ∈ L q (B), then which holds for any ball B.
In particular, if a is a (ϕ, q) atom supported in B, we have is a (ϕ, q) atom, since . Now, using this atom in (7.2) above, and using the fact that g − g B has mean zero on B, we obtain ´.
If we now take the supremum of all f supported in B such that kf k L q = 1, we obtain Rewriting this inequality, we obtain ϕ .By Hölder's inequality, we also have ϕ .We will show that g defines a bounded linear functional on H ϕ,q .Let a be a (ϕ, q) atom.Then Now, let h ∈ H ϕ,q and let h = P ∞ j=1 α j a j be decomposition of h into (ϕ, q) atom such that where C < 1 is the quasi-concavity constant for w.Since C < 1 then  ´.
Therefore, g defines a bounded linear functional L on H ϕ,q given by L g (h) = Z ghdµ, which satisfies Thus, BMO (p) ϕ ⊂ (H ϕ,q ) * and the Theorem is proved.