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68
UDC 517.98 Вестник СПбГУ. Математика. Механика. Астрономия. Т. 4 (62). 2017. Вып. 1
MSC 42B35, 26A33
CHARACTERIZATION OF
BMO VIA BALL BANACH FUNCTION SPACES
Mitsuo Izuki1, Yoshihiro Sawano2
1 Okayama University,
3-1-1 Tsushima-naka, Okayama, 700-8530 Japan
2 Tokyo Metropolitan University,
1-1 Minami-Ohsawa, Tokyo, Hachioji, 192-0397 Japan
The aim of this paper is to characterize the BMO norm via ball Banach function spaces based on the Rubio de Francia algorithm. The method in this paper can be applicable to the Campanato spaces. Refs 28.
Keywords: BMO norm, ball Banach function spaces, Rubio de Francia algorithm, Campanato spaces.
1. Introduction. The BMO space is known as the dual space of the Hardy space H 1(Rn) and plays an important role in real analysis due to many important characterizations. The space BMO(Rn) consists of all locally integrable functions b satisfying that the semi-norm
IHIbmo := sup f \b(x)-bQ\dx
Q:cube |Q| JQ
is finite, where for each cube Q C Rn, |Q| is the Lebesgue measure, and bQ is the mean value of the function b on Q, namely
6q :=mJQb{y)dy•
The semi-norm ||b||BMO is called the BMO norm. If b G BMO(Rn), then there exist positive constants Ci and C2 such that for all cubes Q and A > 0,
\{x (E Q : \b(x) - bQ\ > A}| < Ci |Q|exp f-^M • (1-1)
V ||b||BMO/
The inequality (1.1) is proved by John and Nirenberg [1] and implies that for any constant 1 < p < to there exists a constant C > 1 such that
l|b||BMOLi < ihibmolp < C||b||BMOLi, (1.2)
where xq is the characteristic function for Q and
IHIbmo^p := sup ---\\{b-bQ)xQ\\Lv{^)- (1-3)
Q:cube HxqHlpcRn)
Mitsuo Izuki was partially supported by Grand-in-Aid for Scientific Research (C), No. 15K04928, for Japan Society for the Promotion of Science.
Yoshihiro Sawano was partially supported by Grand-in-Aid for Scientific Research (C), No. 16K05209, for Japan Society for the Promotion of Science.
© Санкт-Петербургский государственный университет, 2017
The aim of this paper is to replace Lp(Rn) by general function spaces having similar properties.
We work on ball Banach function spaces, whose definition we present now.
Definition 1.1. [2, Definition 2] Let M be the set of all complex-valued measurable functions defined on Rn. A mapping p : M ^ [0, to] is called a ball Banach function norm if, for all f, g, fk, (k = 1, 2,3,...), in M, for all constants a > 0 and for all cubes Q in Rn, the following properties hold:
(P1) p(f) = 0 if and only if f = 0 a. e.; p(af) = ap(f); p(f + g) < p(f) + p(g); (P2) If 0 < g < f a. e., then p(g) < p(f); (P3) If 0 < fk f f a. e. then pf) f p(f); (P4)' If |Q| < to, then p(xq) < to;
(P5)' If f > 0 a.e. and |Q| < to, then JQ f (x)dx < Cqp(J) for some constant CQ, 0 < Cq < to, depending on Q and p but independent of f.
The definition remains unchanged if we replace "cube" by "ball" in the above. So this definition deserves this name.
Accrodingly, the space generated by such p is called the ball Banach function space.
Suppose that X is a ball Banach function space equipped with a norm || • ||X. The associate space X' is defined by
X' := {f GM : |f ||x' < to},
where
||x' := sup
/ f (x)g(x) dx
IIX
^ 1
By using a similar technique in [3], we see that X' is a ball Banach function space as well. We also recall that the Hardy—Littlewood maximal operator M is given by
Mf(x):= sup f \f(y)\dy.
Q:cube |Q| JQ
In this paper we aim to provide a sufficient condition to characterize the BMO norm in terms of X, X' and M.
Theorem 1.1. Let X be a ball Banach function space. If the Hardy—Littlewood maximal operator M is bounded on the associate space X', then there exist positive constants Ci < C2 such that for all b e BMO(Rn),
Ci||6||bmo< sup -l—\\(b-bQ)XQ\\x <С2||Ь||вмо. (1-4)
Q:cube HxQHx
The quantity
||Ь||вмох := sup --¡r\\(b-bQ)xQ\\x
Q:cube ^XQ^X
is the X-based generalized BMO, which is one of our targets in this paper. Ho's proof [4] is based on the theory of Hardy spaces. We will give another proof of Theorem 1.1 using the Rubio de Francia algorithm. Our result is based on the following inequalities.
Theorem 1.2. Let X be a ball Banach function space such that the Hardy—Littlewood maximal operator M is bounded on the associate space X'. Then we have
for some 1 < p < to.
Note that (1.4) is a consequence of Theorem 1.2 and (1.2).
Let 0 € (0,1) and 1 ^ p < to. As an application of Theorem 1.2, we can characterize the Campanato space Lp'0(Rn) as well. Recall that the Campanato space Lp'0(Rn) is the set of all f € Lfoc(R") for which the quantity
\\f\\cP,°(R~) ■■= sup £(Q)-S(-L [ \f(x)-fQ\Pdx)P
Q:cube V M Q J
is finite, where l(Q) is the side length of Q. We know that there exists a constant c0,p such that
llfWl1.8^") ^ llfIIlp.8(Rn) ^ cp\\fIIl1,8(R")- (1.5)
This equivalence dates back to the works by Campanato and Meyer (see [5, Theorem, p. 183] and [6, Theorem, p. 718]), where both authors showed that these norms are equivalent to the Lip0 norm. See [7, p. 72] for an account of these facts. See also [8, Theorem 3.1].
Let X be a ball Banach function space. We consider the quantity:
nfn — «„r, p(n\-e Hxqlf ~
ll/ll£^.8(R") •— sup t{Q) -----.
Q:cube WXqWx
Theorem 1.3. Let 1 ^ p < to, 0 € (0,1) and X be a ball Banach function space. If the Hardy—Littlewood maximal operator M is bounded on the associate space X', then there exist positive constants C\ ^ C2 such that for all f € L1'0(Rn),
C1llf wl^.^R") < llf ¡£1>e(R») < C2 Wf y£X.e(Rn)- (1.6)
Note that (1.6) is a consequence of Theorem 1.2 and (1.5). Also, from the general pointwise estimate in Theorem 1.2, we learn that a passage to generalized Campanato spaces and to higher order Campanato spaces are also possible.
We work on ball Banach function spaces instead of Banach function spaces. We recall the definition of Banach function spaces to explain that Morrey spaces do not fall under the scope of Banach function spaces.
Definition 1.2. Let X be a linear subspace of M.
1. The space X is said to be a Banach function space if there exists a functional || • ||X : M ^ [0,to] satisfying the following properties. Let f, g, fj € M (j = 1, 2, •••), then
(a) f € X holds if and only if ||f ||X < to;
(b) norm property:
(i) positivity: ||f ||x > 0;
(ii) strict positivity: ||f ||x = 0 holds if and only if f (x) = 0 for almost every x G Rn;
(iii) homogeneity: ||Af ||x = |A| • ||f ||x holds for all complex numbers A;
(iv) triangle inequality: ||f + g||x < If ||x + llgllx;
(c) symmetry: ||f||x = |||f|||x;
(d) lattice property: if 0 < g(x) < f (x) for almost every x G Rn, then ||g||X < ||f ||x;
(e) fatou property: if 0 < f (x) < fj+1 (x) for all j and fj (x) ^ f (x) as j ^ to for almost every x G Rn, then lim 11fj||x = ||f ||x;
(f) for every measurable set F C Rn such that |F| < to, ||xf||x is finite. Additionally there exists a constant CF > 0 depending only on F such that for all h G X,
[ |h(x)| dx < CF||h||X.
Jf
Remark 1.1. In other literatures (for example [9]) the Banach function spaces and the associate space are called the Kothe space and the Kothe dual respectively.
The usual Lebesgue space Lp(Rn) with constant exponent 1 < p < to is an example of Banach function spaces. However, Morrey spaces are not Banach function spaces in general. When 1 < q < p < to, then Mp(Rn) is a ball Banach function space trivially but is not a Banach function space [10, Example 3.3]. In [10, Theorem 4.1] the second author and Tanaka showed that the associate space of the ball Banach space Mp(Rn) is Hp (Rn), where Hp (Rn) is the block space defined by Zorko [11]. According to [12, Theorem 4.1], the
Hardy—Littlewood maximal operator is bounded on Hp (Rn) as long as 1 < q < p < to.
We organize the remaining part of this paper as follows: In Section 2, we review preliminary facts on ball Banach function spaces and on the Muckenhoupt weights. We prove Theorem 1.2 in Section 3. In Section 4, we consider some examples of X together with related results.
2. Preliminaries. We describe some of fundamental facts of ball Banach function spaces, whose proof is similar to the one corresponding to Banach function spaces; see Bennett and Sharpley [3]. For further informations on the theory of Banach function spaces including the proof of Lemma 2.1 below we refer to the book [3].
Lemma 2.1. Let X be a ball Banach function space. Then the following hold:
1) (The Lorentz—Luxemburg theorem) (X')' = X holds, in particular, the norms II • II(X')' and || • ||x are equivalent;
2) (The generalized Holder inequality) If f G X and g G X', then we have
f |f (x)g(x)| dx < If llxllgllx'.
' R'
Under a certain condition on the boundedness of the Hardy—Littlewood maximal operator M on X, the norm || • ||X enjoys properties similar to the Muckenhoupt weights.
Lemma 2.2. Let X be a ball Banach function space. Suppose that the Hardy— Littlewood maximal operator M is weakly bounded on X, that is, there exists a positive constant C such that
HX{m/>a}||x < CA-1||/||x (2.1)
is true for all / € X and A > 0. Then we have
SUP -AJWXQWXWXQWX' < co. (2.2)
Q:cube |Q|
Proof. The proof is similar to the first author's papers [13, Lemmas 2.4 and 2.5] and [14, Lemmas G' and H]. For readers' convenience we give the self-contained proof. Take a cube Q and a function / € L11oc(Rn). Suppose that |/|q > 0. Because |/|qXq(x) ^ M(/xq)(x) holds for almost every x € Rn, we obtain M(/xq)(x) > A for almost every x € Q, where A := |/|q/2. Hence by assumption (2.1) we get
|/|q||xq||x < |/|q||X{m(/xq)>a}!x < |/|q • CA-1||/Xq||x = 2C ||/Xq||x. Therefore we have
l^jllXQllxllXQlIx' = ^\\xq\\x • sup \g{x)\xQ{x) dx : g £ X, \\g\\x < 1
= sup {|g|Q ||xq ||x : g € X, ||g||x < 1} <
< sup {2C |gxQ|x : g € X, ||g||x < 1} < 2C.
□
Remark 2.1. If M is bounded on X, that is, there exists a positive constant C such
that
||M/||x < C ||/||x
holds for all / € X, then one can easily check that (2.1) holds. On the other hand, if M is bounded on the associate space X', then Lemma 2.1 shows that (2.2) is true.
Next, we recall the notion of weights. Let w be a locally integrable and positive function on Rn. The function w is said to be a Muckenhoupt A1 weight if there exists a positive constant C1 such that Mw(x) < C1w(x) holds for almost every x € Rn. The set A1 consists of all Muckenhoupt A1 weights. For every w € A1, the finite value
[w]Al := sup j-^- f w(x) dx • ||w_1||Lc„(q) 1
Q:cube I |Q| JQ J
is said to be a Muckenhoupt A1 constant. We remark that if w € A1, then
1 f
t— / w(x) dx < \w\a, inf w(x)
|Q| ./Q
for all cubes Q. We will use a classical result on the Muckenhoupt weights.
Lemma 2.3. [15, Chapter 7; 16, Chapter 9] Let w € Ai. We write w(Q) := Jq w(x) dx for a cube Q. Then the reverse Holder inequality holds, that is, there exist positive constants q > 1 and C depending on n and [w]ax such that for all cubes Q,
1 r \1/q
— / w(x)qdx s^Cwq. (2.3)
IQI JQ J
Actually, we know that the pair
c = 2, fl=l + ——!—-
does the job, where [w]A^ is the smallest number B > 0a for which
IQ
wq ^ Bexp [ ——■ f logw(x)dx]
V IQI JQ J
for every cube Q. Note that [w]A^ < [w]Al for all weights w. This result can be found in [17, Theorem 2.3]; see [18, Theorem 4.2] for a generalization to spaces of homogeneous type.
3. Proof of Theorem 1.2.
Proof. We first prove the left-hand side inequality. Using Lemma 2.2 and Remark 2.1, we get for all cubes Q,
1 [ 1 1
where C > 0 is a constant independent of f and Q. This shows the left-hand side inequality.
Next we prove the right-hand side inequality. Our idea is based on [19, Proof of Lemma 3.3]. Take g e X' with ||g||x' < 1. Let B := ||M||xч-x' and define a function
(i^'), (3.2)
fc=0 ( )
where
f|g| (k = 0), Mkg := [ Mg (k =1),
[M(Mk-1g) (k > 2).
For every g e X' with ||g||X' < 1, the function Rg satisfies the following properties:
1) |g(x)| ^ Rg(x) for almost every x e Rn;
2) ||Rg||x' < 2|g|x' < 2;
3) M(Rg)(x) < 2BRg(x), that is, Rg is a Muckenhoupt A1 weight with the A1 constant less than or equal to 2B.
By Lemma 2.3, there exist positive constants q > 1 and C independent of g such that for all cubes Q,
1 f , V/q C
'Q
By virtue of the generalized Holder inequality, we obtain
\\(Rg)xQ\\L^) = \Q\1/q J Rg(x)*cb) /q < |Q|1/9 • щЫЯ) <
< C |Q|-1/(q'>HRffHx'IIxqIIx < C |Q|-1/(q'^IxqIIx.
Thus we have
/ |f (x)g(x)| dx |f (x)|Rg(x) < ll/XQ^Lq'(Rn)y(Rg)XQyLq(Rn) <
1
<C^\JQlnX)lq dXl IIXqI|x'
1/(9')
By Lemma 2.1 we get
ll/XQilx = l/XQlx'' < C sup
/(x)g(x) dx
: g G X', llglx' < 1 <
<
IIxqIIX-
Consequently, the right-hand side inequality follows with p = q'.
□
Q
4. Examples. The authors have considered generalization of the equivalent BMO norm and proved the following statements.
1. (Izuki [20]) The variable Lebesgue norm Ц/llLP(-)(Rn) is defined by
( Г Лх)
lLp( )(Rn) := inM A > 0 :
A
dx < 1
Kovacik and Rakosnik [21] have proved that the generalized Lebesgue space Lp()(Rn) with variable exponent p(-) is a Banach function space and the associate space is Lp ( )(Rn) with norm equivalence, where p'(-) is the conjugate exponent given by
-J- + - 1
p(-) + P'(-) ~
The generalized Lebesgue space Lp()(Rn) consists of all measurable functions f such that the norm ||f ||Lp(-)(R") is finite.
By using a bounded measurable function p(-) : Rn ^ [1, то) we generalize the semi-norm (1.3) to
IHIbmo (.) := sup --—^-\\(b-bQ)xQ\\Lp()(R^- (4-1)
L Q:cube У XQ У Lp( ) (R")
If p(-) satisfies p- > 1 and the Hardy—Littlewood maximal operator M is bounded on Lp()(Rn), then the generalized BMO norm ||b||BMO ^ is equivalent to the classical one ||b||BMO.
2. (Izuki and Sawano [22]) If a bounded measurable functionp(-) : Rn ^ [1, to) satisfies 1 ^ inf p(x) and the log-Holder conditions:
С
- log(|x - y |) C
Ip(x) -p{y)I < —-—-.-Г- for x, y G К™, \x - y\ < 1/2,
\p{x) - pcok, , | for ier, log(e + |x|)
for some constants C and independent of x, y, then ||b||BMO is equivalent to ||&||bmo.
3. (Izuki, Sawano and Tsutsui [14]) If a variable exponent p(-) : Rn ^ [1, to) is bounded and M is of weak type (p(-),p(-)), that is, there exists a constant C > 0 such that for all f € Lp()(Rn) and all A > 0,
lX{Mf>\}lLp()(Rn) < CA 1|f|Lp(•)(Rn),
then ||6|bmolp(.) is equivalent to ||6|bmo.
4. (Ho [4]) Ho obtained a characterization in the context of general function space including Lebesgue spaces. Given a ball Banach function space X equipped with a norm || • ||x, we define the X-based generalized BMO norm
IHIbmo* := sup 1 W(b-bQ)xQ\\x-
Q:cube ^XQ^X
If M is bounded on the associate space X', then ||b||BMOx is equivalent to ||b||BMO. We remark that Ho's results [4, 23] have included the authors' one [22, 24]. The statements in [22, 24] are deeply depending on Diening's work [25] on variable exponent analysis. On the other hand, Ho's proof is self-contained and obtained as a by-product of the new results about atomic decomposition introduced in [4]. Our proof of the result, initially proved by Ho, is new in the sense that we use the Rubio de Francia algorithm [26-28].
* * *
The authors are thankful to the anonymous referee for his/her careful reading of this paper. The authors are appreicate to Professor Eiichi Nakai for his introducing the papers [5-8]. Finally, the authors thank Professor Tuomas Hytonen for his information on the paper [17] as well as (2.3).
Received: March 28, 2016; accepted: October 6, 2016.
Authors information
Mitsuo Izuki — Senior Assistant Professor; izuki@okayama-u.ac.jp Yoshihiro Sawano — Associate Professor; ysawano@tmu.ac.jp
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For citation: Mitsuo Izuki, Yoshihiro Sawano. Characterization of BMO via ball Banach function spaces. Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 2017, vol. 4(62), issue 1, pp. 78-86. DOI: 10.21638/11701/spbu01.2017.110
