Proyecciones Journal of Mathematics Vol. 33, No 1, pp. 43-59, March 2014. Universidad Cat´olica del Norte Antofagasta - Chile
Hardy-Type Spaces and its Dual Ren´e Erl´in Castillo Universidad Nacional de Colombia, Colombia and Julio C. Ramos Fern´ andez and Eduard Trousselot Universidad de Oriente, Venezuela Received : August 2012. Accepted : October 2013
Abstract In this paper we defined a new Hardy-type spaces using atoms on homogeneous spaces which we call H ϕ,q . Also we prove that under (p) certain conditions BM Oϕ is the dual of H ϕ,q . Keywords : BMO, Dual space, Hardy space, Space of homogeneous type. Subjclass [2010] : 32A37; 43A85.
44
Ren´e E. Castillo, Julio C. Ramos and Eduard Trousselot
1. 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 Mp (r, f ) =
µ
1 2π
Z 2π 0
iθ p
|f (re )| dθ
¶ p1
,
0 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 CM (Ct) ≤ ψ(t) ≤ M (t). 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 also that
ϕ(x) is a decreasing function of x, and suppose x
ϕ(x) is an increasing function for some 0 < < 1. Let x ψ(x) =
Z x ϕ(t)
t
0
dt.
Then ψ is concave, ϕ is quasi-concave, and xψ(x) is quasi-convex. Proof: The derivative
ϕ(x) x is decreasing by hypothesis. Therefore, ψ is concave. To show that ϕ is quasi-concave, first note that ϕ(x) ≤ ψ(x) since ψ 0 (x) =
ψ(x) =
Z x ϕ(t) 0
t
dt ≥
Z x ϕ(x) 0
x
dt = ϕ(x).
To show the other inequality, we estimate ψ(Cx), for C < 1 by ψ(Cx) =
Z Cx ϕ(t)
Z Cx ϕ(t)
t t t1− 0 C ϕ(x) (Cx) = ϕ(x) . x 0
=
dt =
dt ≤
ϕ(x) x
Z Cx 0
t
−1
dt
50
Ren´e E. Castillo, Julio C. Ramos and Eduard Trousselot Therefore, we have C
ψ(x) ≤ ϕ(x). 1
Next, we choose C by letting C = +1 . Since 0 < < 1, C also satisfies C < 1 and Cψ(Cx) ≤ ϕ(x). Thus, we have shown that ϕ is quasi-concave. g(x) = To show that xψ(x) is quasi-convex, let g(x) = xψ(x). Note that x ψ(x) is increases, so Z x g(u) du A(x) = u 0 is convex. Also, A(x) =
Z x g(u) 0
u
du ≤
g(x) x = g(x), x
so A(x) ≤ g(x). We also have A(2x) =
Z 2x g(u) 0
u
du ≥
Z 2x g(u) x
u
du ≥
g(x) x = g(x), x
thus g(x) ≤ A(2x) ≤ 2A(2x), and we have shown that A(x) ≤ g(x) ≤ 2A(2x). 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(t1 ) Cϕ(Ct2 ) ≥ t1 t2 for all 0 ≤ t1 ≤ t2 . 2. ψ is quasi-convex if and only if there exists a C > 1 such that ϕ(t1 ) Cϕ(Ct2 ) ≥ t1 t2 for all 0 < t1 ≤ t2 .
Hardy-Type Spaces and its Dual
51
Proof: of (1) (⇒) since ϕ is quasi-concave, we have M concave and C < 1 such that CM (Ct) ≤ ϕ(t) ≤ M (t). Now,
M (t) is a non-increasing function of t, so, for all 0 < t1 ≤ t2 , we have t M (t1 ) M (t2 ) ≥ . t1 t2
Thus, ϕ(t1 ) t1
CM (Ct1 ) C 2 M (Ct1 ) C 2 M (Ct2 ) ≥ ≥ t1 Ct1 Ct2 CM (Ct2 ) Cϕ(Ct2 ) ≥ . t2 t2
≥ =
(⇐) Let C < 1, t1 ≤ t2 , and suppose function
1 ψ(t) = C
Z
0
t C
ϕ(t1 ) Cϕ(Ct2 ) ≥ . Consider the t1 t2
inf
x 0. We obtain a function g˜j satisfying Lf
(7.1)
Z
=
f g˜j dµ
Bj
for each j. Now, let gj )B1 . gj = g˜j − (˜
R
Then, B1 gj dµ = 0. It remains to show that gj |Bk = gk for all k ≤ j. By the above remark, we know that on Bk ⊃ B1 , we have gj − gk = C. Now, integrate both sides over B1 to obtain Z
B1
Z
(gj − gk )dµ =
Cdµ,
B1
which implies that 0 = µ(B1 ). Therefore, C = 0, and we conclude that gj |Bk = gk for k ≤ j. We now have a unique function g such that if f ∈ Lq (B), then Lf =
Z
fgdµ,
B
which holds for any ball B. In particular, if a is a (ϕ, q) atom supported in B, we have ¯Z ¯
kLk ≥ |La| = ¯¯
(7.2)
B
¯ ¯
¯Z ¯
gadµ¯¯ = ¯¯
B
if f is supported in B and kf kLq = 1, then
¯ ¯
(g − gB )adµ¯¯ ,
µ
¶
1 [µ(B)]1/q −1 w a= (f − fB ) 2 µ(B) is a (ϕ, q) atom, since µ
1 µ(B)
Z
B
¶1
|a(x)|q dµ
q
= ≤
µ
1 [µ(B)]1/q −1 w µ(B) 2[µ(B)]1/q w−1
≤ w
³
1 µ(B)
2
−1
µ
´
2
¶
µZ
1 . µ(B)
B
¶ µZ
B
¶1
|f |q dµ
q
¶1
|f − fB |q dµ
q
Hardy-Type Spaces and its Dual
55
Now, using this atom in (7.2) above, and using the fact that g − gB has mean zero on B, we obtain ¯Z ¯ µ ¶ ¯ ¯ 1 [µ(B)]1/q −1 ¯ ¯ w (f − fB )dµ¯ ≤ kLk, ¯ (g − gB ) ¯ B ¯ 2 µ(B)
which implies that
¯Z ¯ µ ¶ ¯ ¯ 1 [µ(B)]1/q −1 ¯ ¯ w fdµ¯ ≤ kLk, ¯ (g − gB ) ¯ B ¯ 2 µ(B)
which in turn implies that
¯Z ¯ ¯ ¯ ¯ (g − gB )f dµ¯ ≤ ¯ ¯ B
2kLk [µ(B)]1/q w−1
³
1 µ(B)
´.
If we now take the supremum of all f supported in B such that kf kLq = 1, we obtain 2kLk
kg − gB kLp ≤
³
[µ(B)]1/q w−1
1 µ(B)
´
2kLk µ(B)ϕ(µ(B)) [µ(B)]1/q
=
= 2kLk[µ(B)]1/q ϕ(µ(B)). Rewriting this inequality, we obtain µ
1 µ(B)[ϕ(µ(B))]p
Z
p
B
¶1
|g − gB | dµ
p
≤ 2kLk,
(p)
so g ∈ BM Oϕ . By H¨ older’s inequality, we also have 1 ϕ(µ(B))µ(B)
Z
B
|g(x) − gB |dµ(x) ≤ ≤
[µ(B)]1/q µ(B)ϕ(µ(B)) µ
µZ
B
1 µ(B)[ϕ(µ(B))]p
So g ∈ BM Oϕ , also. We have now shown that (H ϕ,q )∗ ⊂ BM Oϕ(p) ⊂ BM Oϕ .
¶1
|g − gB |p dµ
Z
p
B
p
¶1
|g − gB | dµ
p
.
56
Ren´e E. Castillo, Julio C. Ramos and Eduard Trousselot (p)
Now, suppose that g ∈ BM Oϕ . We will show that g defines a bounded linear functional on H ϕ,q . Let a be a (ϕ, q) atom. Then ¯Z ¯ ¯Z ¯ Z ¯ ¯ ¯ ¯ ¯ gadµ¯ = ¯ (g − gB )a(u)dµ¯ ≤ |g − gB |a(u)dµ ¯ ¯ ¯ ¯ B B µZ ¶ 1 µZ ¶1
≤
B
|g − gB |p dµ
p
|g − gB |p dµ
p
≤
µZ
≤
[µ(B)]1/q µ(B)ϕ(µ(B))
B
B
¶1
µ
µZ
|a(u)|q dµ
[µ(B)]1/q w−1 ¶1
p
B
1 ≤ µ(B)[ϕ(µ(B))]p ≤ kgkBMO(p) .
|g − gB | dµ
Z
B
p
q
µ
1 µ(B)
¶
p
¶1
|g − gB | dµ
p
ϕ
Therefore, we have shown that ¯Z ¯ ¯ ¯ ¯ gadµ¯ ≤ kgk (p) . ¯ ¯ BMOϕ
(7.3)
B
Now, let h ∈ H ϕ,q and let h = (ϕ, q) atom such that w−1
³X
P∞
j=1 αj aj
be decomposition of h into
´
w(|αj |) ≤ (1 + C 4 )khkH ϕ,q ,
where C < 1 is the quasi-concavity constant for w. Since C < 1 then X
µ ¶¶ 1 X |αj | w C C µ X µ ¶¶ 1 C|αj | ≤ w−1 w . C C2
|αj | ≤ w−1
µ
|αj | Now, let t1 = |αj |, and let t2 = 2 . Since C < 1, we have t1 < t2 and C by Lemma 6.2 implies that C
w(Ct2 ) w(t1 ) ≤ . t2 t1
Therefore, we have w(Ct2 ) ≤
t2 w(t1 ) t1 C
Hardy-Type Spaces and its Dual which implies that µ
w C
|αj | C2
¶
≤
w(|αj |) |αj | w(|αj |) = . C 2 |αj | C C3
Thus X
¶ µ ¶ 1 X w(|αj |) 1 X −1 |αj | ≤ w =w w(|αj |) C C3 C4 µ ¶ 1 ≤ w−1 (1 + C 4 )khkH ϕ,q C4 µ ¶ 2 −1 ≤ w khkH ϕ,q . C4 −1
µ
(p)
Now, if g ∈ BM Oϕ , since C < 1, we have ¯Z ¯ ¯Z ¯ X ¯ ¯ ¯ ¯ ¯ ghdµ¯ ≤ ¯ |αj | ¯ gaj dµ¯¯ ¯ ¯
³
´
≤ kgkBMO(p) w−1 C24 khkH ϕ,q . ϕ Therefore, g defines a bounded linear functional L on H ϕ,q given by Lg (h) =
Z
ghdµ,
which satisfies
kLg k =
|Lg (h)| ≤ w−1
sup khkH ϕ,q =1
µ
Thus, BM Oϕ(p) ⊂ (H ϕ,q )∗ and the Theorem is proved.
¶
2 kgkBMO(p) . ϕ C4
57
58
Ren´e E. Castillo, Julio C. Ramos and Eduard Trousselot
References [1] Castillo, R. E., Ramos Fern´andez, J. C. and Trousselot, E. Functions of Bounded (ϕ, p) Mean Oscillation. Proyecciones Journal of Math. Vol 27, No. 2, pp. 163-177, August (2008). [2] Coifman, R. R, A real variable characterization of H p . Studia Math. 51, pp. 269-274, (1974). [3] Coifman, R. R and Weiss, G. Extensions of Hardy spaces and their use in Analysis. Bull. Amer. Math. Soc. 83, pp. 99-157, (1977). [4] Duren, P. L., Romberg, B. W. and Shields, A. L., Linear functionals on H p spaces with 0 < p < 1. J. Reine Angew, Math 238, pp. 32-60, (1969). [5] Janson, S. Generalization of Lipschitz spaces and an application to Hardy space and bounded mean oscillation. Duke Math. 47, pp. 959982, (1980). [6] Latter, R. A. A characterization of H p (Rn ) in terms of atoms. Shidia Math 62 (1978) pp. 93-101, (1978). [7] Macias, R. and Segovia, C. Lipschitz functions on space of homogeneous type, Avd. Math., 33, pp. 257-270, (1979). Ren´ e Erl´in Castillo Departamento de Matem´ aticas, Universidad Nacional de Colombia AP360354 Bogot´ a e-mail :
[email protected] Julio C. Ramos Fern´ andez Departamento de Matem´ aticas, Universidad de Oriente 6101 Cuman´ a, Edo. Sucre, Venezuela e-mail :
[email protected]
Hardy-Type Spaces and its Dual and Eduard Trousselot Departamento de Matem´ aticas, Universidad de Oriente 6101 Cuman´ a, Edo. Sucre, Venezuela e-mail :
[email protected]
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