1

数理解析研究所講究録
第2041巻 2017年 1-10

Generalized Hardy spaces
based on Banach function spaces
Yoshihiro Sawano

Department of Mathematics and Information Science,

Tokyo Metropolitan University
Abstract
In this note, we recall a result in [11] together with some examples.

1

Example of function spaces

The following function spaces are fundamental in harmonic analysis and we wantto understarnd them inaunified manner.

Here are some example of function

spaces that we envisage.

Lebesgue spaces for 0 < p \leq \infty One of our staring points is the BanachL^{p}(\mathbb{R}^{n}) for 1 \leq p\leq \infty Although it is not a Banach space, we can defineƯ (\mathbb{R}^{n}) for 0<p<1.

space

.

Weighted Lebesgue spaces

By a weight we mean a measurable function
which satisfies 0<w(x) <\infty for almost all x\in \mathbb{R}^{n} Let 0<p<\infty and w be aweight. One defines.

\displaystyle \Vert f\Vert_{Lp(w)}\equiv (\int_{\mathb {R}^{n} |f(x)|^{p}w(x)dx)^{\frac{1}{p}Morrey spaces

Let0<q\leq p<\infty Define the Morrey norm \Vert\star\Vert_{\mathcal{M}_{\mathrm{q} ^{p} by.

{\displaystyle \Vert f\Vert_{ $\Lambda$ 4_{q}^{p} \equiv\sup |B|^{\frac{1}{p}-\frac{1}{q}for a measurable function fable functions f for which.

(\displaystyle \int_{B}|f(x)|^{q}dx)^{\frac{1}{q}:The Morrey space\mathcal{M}_{q}^{p}(\mathbb{R}^{n}) is the set of all measur‐

B is a ball in \mathbb{R}^{n}

}

(1.1)

\Vert f\Vert_{\mathcal{M}_{\mathrm{q} ^{p} is finite. Among of them, the author would like2

to understand the property of Morrey spaces.

L^{1}(\mathbb{R}^{n})+L^{\infty}(\mathbb{R}^{n})Morrey spaces grasp more thanin general; see [10].

If we start Morrey spaces, we are led to Hardy‐Morrey spaces. See [19]Homogeneous and non‐homogeneous Herz spaces
and C_{j}\equiv[-2^{j}, 2^{j}]^{n}\backslash [-2^{j-1}, 2^{j-1}]^{n} for j\in \mathbb{Z}.

Write Q_{0}\equiv[-1, 1]^{n}Let 0 <p, q \leq \infty and $\alpha$\in \mathbb{R} The non‐homogeneous Herz space
K_{pq}^{ $\alpha$}(\mathbb{R}^{n}) is
the set of all measurable functions f for which the norm.

\displayst le\Vertf\Vert_{K_{pq}^{$\alpha$\equiv}\Vert$\chi$_{Q_{0}\cdotf\Vert_{p}+(\sum_{j=1}^{\infty}(2^{j$\alpha$}\Vert$\chi$_{C_{j}\cdotf\Vert_{p})^{q})^{\frac{1}q}
is finite. The homogeneous Herz spacetions f for which the norm

\Vert f\Vert_{\dot{K}_{pq}^{ $\alpha$}\equiv\dot{K}_{pq}^{ $\alpha$}(\mathb {R}^{n}) is the set of all measurable func‐(\displaystyle \sum_{j=-\infty}^{\infty} (2^{j $\alpha$}\Vert$\chi$_{C_{j} . f\Vert_{p})^{q})^{\frac{1}{q}is finite.

Hardy‐Herz spaces, made from Herz spaces, are studied in [13, 17].
Orlicz spaces, Musielak‐Orlicz spaces
spaces, we remark that

2

Although we do not define these

General function spaces

Let L^{0}(\mathbb{R}^{n}) be the space of all measurable functions defined on \mathbb{R}^{n}.
Definition 2.1. A linear space X=

X(\mathbb{R}^{n}) \subset L^{0}(\mathbb{R}^{n}) is said to be a quasi‐

Banach function space if X is equipped with a functional \Vert\cdot\Vert_{X} : L^{0}(\mathbb{R}^{n})\rightarrow[0, \infty]

enjoying the following properties:
Let f, g, f_{j}\in L^{0}(\mathbb{R}^{n}) (j=1,2, \ldots) and $\lambda$\in \mathbb{C}.

(1) f\in X holds if and only if \Vert f\Vert_{X} <\infty.
(2) (Norm property):(A1) (Positivity): \Vert f\Vert_{X}\geq 0.
(A2) (Strict positivity) \Vert f\Vert_{X}=0 if and only if f=0 a.e..
(B) (Homogeneity): \Vert $\lambda$ f\Vert_{X}=| $\lambda$|\cdot\Vert f| _{X}.
(C) (Triangle inequality): For some $\alpha$\geq 1, \Vert f+g\Vert_{X}\leq $\alpha$(\Vert f\Vert_{X}+\Vert g\Vert_{X}).

3(3) (Symmetry): \Vert f\Vert_{X}=\Vert|f|\Vert_{X}.
(4) (Lattice property): If 0\leq g\leq f a.e., then \Vert g\Vert_{X}\leq\Vert f\Vert_{X}.
(5) (Fatou property): If 0\leq f_{1} \leq f_{2} \leq.

.

.

and

\Vert f\Vert_{X}.

\displaystyle \lim_{j\rightar ow\infty}f_{j}=f then \displaystyle \lim_{j\rightar ow\infty}\Vert f_{j}\Vert_{X}=,(6) For all measurable sets E with |E| <\infty we have \Vert$\chi$_{E}\Vert_{X}<\infty.
,This framework is not enough in view of the following facts:

Remark 2.2. If $\alpha$=1 and the following condition:
For all measurable sets E with |E| <\infty andf\in X

holds, we have f\cdot$\chi$_{E} \in L^{1}(\mathbb{R}^{n}) ; holds, then X is said to be a Banach function
space; see [2]. Note that, we do not postulate this condition in the definition ofquasi‐Banach function space. As we have seen in [15], the Morrey space \mathcal{M}_{q}^{p}(\mathbb{R}^{n})with 1\leq q<p<\infty violates this additional condition.
For this reason, we need to introduce the following notion:

Definition 2.3. A linear space X \subset

L^{0}(\mathbb{R}^{n}) is said to be a ball quasi‐Uanach\Vert \Vert_{X} : L^{0}(\mathbb{R}^{n}) \rightarrow [0, \infty]

function space if X is equipped with a functional

enjoying (1)-(5) as well as the following properties (6) and (7):Let f, g, f_{j}\in L^{0}(\mathbb{R}^{n}) (j=1,2, \ldots) and $\lambda$\in \mathbb{C}.

(6) For all balls B we have \Vert$\chi$_{B}\Vert_{X}<\infty.
,(7) For all balls B and f\in X we have f\cdot$\chi$_{B}\in L^{1}(\mathbb{R}^{n}),.

If $\alpha$=1 , then X is said to be a ball Banach function space.
We recall the notion of the Köthe dual of a ball Banach function space X.

If\Vert\cdot\Vert_{X} is a ball function norm, its associate norm \Vert\cdot\Vert_{X'} is defined on L^{0}(\mathbb{R}^{n}) by\displaystyle \Vert g\Vert_{X'}\equiv\sup\{\Vert f . g\Vert_{L^{1}} : f\in L^{0}(\mathbb{R}^{n}), \Vert f\Vert_{X}\leq 1\},

(g\in L^{0}(\mathbb{R}^{n}))

.

(2.1)

The space X' collects all measurable functions f\in L^{0}(\mathbb{R}^{n}) for which the quantity\Vert f\Vert_{X'} is finite. The space X' is called the Köthe dual of X or the associated space

of X.

4

Hardy spaces

3
3.1

Classical definitions

Suppose that $\psi$\in S(\mathbb{R}^{n}) satisfies the non‐degenerate condition\displaystyle \int_{\mathb {R}^{n} $\psi$(x)dx\neq 0.
Using this function, the Hardy norm is defined by;

for f\in S'(\mathbb{R}^{n})for j \in \mathb {Z}

.

(3.1)

Here

$\psi$^{j}\equiv 2^{jn} $\psi$(2^{j}\cdot)

The space H^{p}(\mathbb{R}^{n}) is defined uniquely despite the ambiguity of the.

choice of $\psi$\displaystyle \Vert f\Vert_{Hp}^{ $\psi$}\equiv \Vert\sup_{j\in \mathb {Z} |$\psi$^{j}*f|\Vert_{p}, 0<p<\infty, f\in S'(\mathb {R}^{n}).

This fact justifies that we can omit $\psi$ in the notation\Vert\cdot\Vert_{H^{p} ^{ $\psi$}.

We have a couple of motivations of investigating Hardy spaces.
\bulletThe singular integral operators, which are represented by the j‐th Riesz

transform given by

R_{j}f(x)\displaystyle \equiv\lim_{\in\downar ow 0}\int_{\mathb {R}^{n}\backslash B(x, $\epsilon$)}\frac{x_{j}-y_{j} {|x-y|^{n+1} f(y)dy

(3.2)

,

integral operators with singularity (mainly at the origin). The bounded‐ness of such operators can be characterized by Hardy spaces. For example,
arelet f\in L^{1}(\mathbb{R}^{n})

Then the estimate.

\displaystyle\Vertf\Vert_{L^{1} +\sum_{j=1}^{n}\VertR_{j}f\Vert_{L^{1} <\infty
holds if and only if f\in H^{1}(\mathbb{R}^{n})also characterizes Ư (\mathbb{R}^{n})

.

.

(3.3)

The Hardy space H^{p}(\mathbb{R}^{n}) with 0<p<1

But the matters are subtler.

So we do not go

into the details.
\bulletLet 1 < p <\infty.

The Hardy space

H^{p}(\mathbb{R}^{n}) is isomorphic to L^{p}(\mathbb{R}^{n}),sodifferent expression of Ư (\mathbb{R}^{n}) , which in turm yields theresults
for the Lebesgue space L^{p}(\mathbb{R}^{n}) , for example.
decomposition

that we have\bullet

A spirit similar to above is that the Hardy space H^{p}(\mathbb{R}^{n}) and the Triebel‐
Lizorkin space

H^{p}(\mathbb{R}^{n})\bulletacan\dot{F}_{p2}^{0}(\mathb {R}^{n})areisomorphic for 0<p<\infty So the Hardy space.

play the model role of Triebel‐Lizorkin spaces.

An experience shows that many other operators
H^{p}(\mathbb{R}^{n}) to L^{p}(\mathbb{R}^{n}) but are not bounded on L^{p}(\mathbb{R}^{n})can.

be bounded from

5

3.2

Our main results

Let us go back to our fundamental setting: Let X be a quasi‐ball Banach function
space.

So we are going to define HX(\mathbb{R}^{n}).

Let $\psi$ be a function non‐degenerate

in the above sense. We want to define\displaystyle \Vert f\Vert_{HX}^{ $\psi$}\equiv \Vert\sup_{j\in \mathb {Z} |$\psi$^{j}*f|\Vert_{X}, 0<p<\infty, f\in S'(\mathb {R}^{n})(3.4)

S'(\mathbb{R}^{n}) Once we can show that different choices of admissible $\psi$ yield
equivalent norms, we can define HX(\mathbb{R}^{n}) to be the set of all f\in \mathcal{S}'(\mathbb{R}^{n}) for which
for f \in.

\Vert f\Vert_{HX}^{ $\psi$} is defined.

The powered Hardy‐Littlewood maximal operator M^{( $\eta$)} is defined by:M^{( $\eta$)}f\equiv[M[|f|^{ $\eta$}] ^{\frac{1}{ $\eta$}}, $\eta$>0
We write M=M^{(1)}

.

.

(3.5)

In [11], we proposed that the following condition to develop

the theory of HX(\mathbb{R}^{n}) :\displayst le\Vert(\sum_{j=1}^{\infty}M^{($\eta$)}f_{j}^{Q})^{\frac{1}Q}\Vert_{X}\sim<\Vert(\sum_{j=1}^{\infty}|f_{j}|^{Q})^{\frac{1}Q}\Vert_{X}(3.6)

We formulate the atomic decomposition, the main result in this paper after

giving the definition.
Definition 3.1. Let X be a ball quasi‐Banach function space and q \inAssume that d\in \mathbb{Z}_{+} satisfies d\geq d_{X}atom if there exists.

Q\in \mathcal{Q} such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(a)\subset Q,\displaystyle\Verta\Vert_{L^{q}\leq\frac{|Q^{\frac{1}{q} {\Vert$\chi$_{Q}|_{X},\int_{\mathb {R}^{n}x^{$\alpha$}a(x)dx=0
as[1, \infty].

Then the function a is called an (X, q, d)-(3.7)

long as | $\alpha$| \leq d.
We also let d_{X}=For a > 0,

we[\displaystyle\frac{n}{$\eta$}-n]define\Vert f\Vert_{X^{a}=that X^{a} is a ball quasi‐Banach space.
Theorem 3.2|f|^{a}\Vert_{X}]^{1/a} for a measurable function f so,

(Reconstruction, [11]). Let s \in (0,1], q \in (1, \infty ] and d_{X} be as

above. Assume that X is a ball quasi‐Banach function space such that the Köthe

dual of X^{1/s} is isomorphic to a Banach function space Y such that

\Vert M^{((q/Q)')}f\Vert_{Y}\sim<\Vert f\Vert_{Y}

,

(3.8)

6

and, for any f \in L^{0}(\mathbb{R}^{n}) Let \{a_{j}\}_{j=1}^{\infty} be a sequence of (X, q, d_{X}) ‐atoms, sup‐ported on the cubes \{Q_{j}\}_{j=1}^{\infty} \subset \mathcal{Q} and \{$\lambda$_{j}\}_{j=1}^{\infty} \subset[0, \infty ) satisfy that.

,\displaystle\Vert\{ sum_{j=1}^{\infty}(\frac{$\lambda$_{j} \Vert$\chi$_{Q j}\Vert_{X})^{\mathrm{s}$\chi$_{Q j}\^{ frac{1}s \Vert_{X}<\infty
Then the series

(3.9)

.

f :=\displaystyle \sum_{j=1}^{\infty}$\lambda$_{j}a_{j} converges in \mathcal{S}'(\mathbb{R}^{n}) f\in HX(\mathbb{R}^{n}) and,\displaystle\Vertf\Vert_{HX_{\sim}<\Vert\{ sum_{j=1}^{\infty}(\frac{$\lambda$_{j} |$\chi$_{Q j}\Vert_{X})^{s}$\chi$_{Q j}\^{\frac{1}\mathrm{s} \Vert_{X}
where the implicit positive constant is independent of f.
Theorem 3.3(Decomposition, [11]). Let X be a ball quasi‐Banach function
a fixed integer and f\in
HX(\mathbb{R}^{n}) Then there exist a

space satisfying d\geq d_{X} be

sequencea.

\{a_{j}\}_{j=1}^{\infty} of (X, \infty, d) ‐atoms, supported\{$\lambda$_{j}\}_{j=1}^{\infty} \subset [0, \infty ) such that f=\displaystyle \sum_{j=1}^{\infty}$\lambda$_{j}a_{j} in S'(\mathbb{R}^{n}) andon

the cubes \{Q_{j}\}_{j=1}^{\infty} \subset \mathcal{Q} , andsequence\displayst le\Vert\{ sum_{j=1}^{\infty}(\frac{$\lambda$_{j} \Vert$\chi$_{Q j}\Vert_{X})^{s}$\chi$_{Q j}\^{\frac{1}s \Vert_{X}\sim<_{s}\Vertf\Vert_{HX},
where the implicit positive constant is independent of f , but depends on s.

3.3

A reduction from HX to X

By mimicking the proof of H^{p}(\mathbb{R}^{n})L^{p}(\mathbb{R}^{n}) [16]=,we canprove thefollowingproposition.

Proposition 3.4. If X is a ball Banach function space that admits a predual andthat \Vert Mf\Vert_{X_{\sim}}<\Vert f\Vert_{X} for all f\in X(\mathbb{R}^{n}) then HX(\mathbb{R}^{n})=X(\mathbb{R}^{n}) with coincidenceof norms.
,

Proof. We generalized Proposition 3.4 in [11]. Here for the sake of convenience
for readers we supply a proof. Let f\in X Since.

\displaystyle \int_{B(x,1)}|f(z)|dz\leq(1+|x|)^{n}Mf(y)for all y\in B(1)=\{|z| <1\} , we obtain\displaystyle \int B(x,1)^{|f(z)|dz_{\sim}(1+|x|)^{n}\Vert f\Vert_{X} <.

7Thus, f \in S'(\mathbb{R}^{n})2.7]. Thus,

Let f\in HX(\mathbb{R}^{n}).

.

For each j \in \mathb {Z} , we have |$\psi$^{j}*f|\sim<Mf; see [7, Proposition\displaystyle \Vert f\Vert_{HX}= \Vert\sup_{j\in \mathb {Z} |$\psi$^{j}*f|\Vert_{X}\sim<\Vert Mf\Vert_{X_{\sim} <\Vert f\Vert_{X}.
Then\{$\psi$^{j}*f\}_{j=1}^{\infty} forms bounded set in X since f\in HX(\mathbb{R}^{n})\{$\psi$^{j}*f\}_{j=1}^{\infty} converges to g\in X(\mathbb{R}^{n}) in thea.

Thus, if we pass to a subsequence, thenweak‐ * topology of X(\mathbb{R}^{n}) thanks to the Banach‐Alaoglu theorem.

\{$\psi$^{j}*f\}_{j=1}^{\infty}

converges to f in S'(\mathbb{R}^{n})f=g Thus, f\in X(\mathbb{R}^{n}).

.

Meanwhile,

Since HX(\mathbb{R}^{n}) is embedded into S'(\mathbb{R}^{n}) ,\square.

Putting together this proposition and our main result, we can obtain decom‐
position results for many function spaces; see [1, 9] for some recent works.

Main idea of the proof of the main result4
4.1

Some problems

One of the attractive ways to prove the main result is to reexamine the book[16]. We need to show HX(\mathbb{R}^{n})\cap L_{1\mathrm{o}\mathrm{c} ^{1}(\mathbb{R}^{n}) is dense in HX(\mathbb{R}^{n}) However, as theexample of the Morrey space \mathcal{M}_{q}^{p}(\mathbb{R}^{n}) with 0<q<1\leq p<\infty shows, this does
not seem to be true. One sufficient condition that makes this argument possible
is the notion of absolute continuity of the norms. A quasi‐Banach function space
X is said to have an absolutely continuous quasi‐norm, if | $\chi$_{E_{j} \Vert_{X} \downarrow 0 whenever\{E_{j}\}_{j=1}^{\infty} is a sequence decreasing to the empty set. In this case, we can also show.

that HX(\mathbb{R}^{n})\cap L_{\mathrm{c} ^{\infty}(\mathbb{R}^{n}) is dense in HX(\mathbb{R}^{n})4.2.

What does assumption (3.6)

implies?

Assume that X is a ball quasi‐Banach function space satisfying (3.6). To consider

the meaning of (3.6), we consider non‐homogeneous Herz spaces. Let Q_{0} and C_{j}asbefore. Then as the inequality

M^{( $\eta$)}[f$\chi$_{C_{j} ](x)_{\sim}> (\displaystyle \frac{1}{|C_{j}| \int_{C_{j} |f(y)|^{ $\eta$}dy)^{\frac{1}{ $\eta$} (x\in Q_{0})implies, X \rightarrow K_{ $\eta$ q}^{-n/ $\eta$}(\mathbb{R}^{n})Note that

1\not\in K_{ $\eta$ q}^{-n/ $\eta$}(\mathbb{R}^{n}).

.

We use this fact to prove the main theorem in

[11].

8

Some problems

5
5.1

Improvement of our key assumption

We postulated assumption (3.6) because it appeared many times in the proof.
In general, it is demanding that we verify assumption (3.6). Here and below
we

write

M^{( $\eta$)} by

\{a\}

=

\sqrt{1+|a|^{2}} fora\in \mathbb{R}^{n}

So we propose here replace the operator.

$\varphi$\displaystyle \mapsto\sup_{z\in \mathb {R}^{n} \langle z\}^{-\frac{n}{ $\eta$} | $\varphi$(\cdot-z)|

motivated by the well‐known Plancherel‐Polya‐Nikolskii inequality:\displaystyle \sup_{z\in \mathb {R}^{n} \{z\rangle^{-\frac{n}{ $\eta$} | $\varphi$(x-z)|\sim<M^{( $\eta$)} $\varphi$(x)(5.1)

when \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathcal{F} $\varphi$) is contained in a fixed compact set. The operator

f\displaystyle \mapsto\sup_{z\in \mathb {R}^{n} \{z\}^{-\frac{n}{ $\eta$} |f(\cdot-z)|
is called the Peetre maximal

operator.

For Besov spaces and Triebel‐Lizorkinsucceeded in this attempt; see [12].
definition of the Peetre maximal operator is thatspaces,we

A direct consequence of the\displaystyle \sup_{z\in \mathb {R}^{n} \{z\}^{-\frac{n}{ $\eta$} |f(\cdot+a-z)| \leq 2^{\frac{a}{2} \langle a\}^{\frac{n}{ $\eta$} \sup_{z\in \mathb {R}^{n} \langle z\rangle^{-\frac{n}{ $\eta$} |f(\cdot-z)|
for all a\in \mathbb{R}^{n}.
We can control
the factor of\langlea\rangle^{\frac{n}{$\eta$} .

\displaystyle \sup_{z\in \mathb {R}^{n} \langle z\rangle^{-\frac{n}{ $\eta$} |f(\cdot+a-z)| by \displaystyle \sup_{z\in \mathb {R}^{n} \langle z\rangle^{-\frac{n}{ $\eta$} |f(\cdot-z)|

at the cost of

Although this attempt does not work, we are also interested in the following\displaystyle \Vert\sup_{z\in \mathb {R}^{n} \langle z\rangle^{-\frac{n}{ $\eta$} |f(\cdot-z)|\Vert_{X}parametrized space: \Vert f\Vert_{HX}\equiv.

For Besov spaces, thefollowing condition is sufficient: \Vert f(\cdot+x)\Vert_{X_{\sim}}< \langle x\}^{N}\Vert f\Vert_{X} for some N\in \mathbb{N} See.

[14] for more details.

5.2

The boundedness of the fractional integral operators

Let I_{ $\alpha$} be the fractional integral operator of order $\alpha$ given by

I_{ $\alpha$}f(x)\displaystyle \equiv\int_{\mathb {R}^{n} \frac{f(y)}{|x-y|^{n- $\alpha$} dy.

Here we igore the problem of the convergence of the integral which will bejustified

later. In fact, in many cases

f\in X

converge for almost all x\in \mathbb{R}^{n}.



is a sufficient condition of the integral to9Problem 5.1. Let X, Y be ball Banach function spaces.
bounded5.3

When is I_{ $\alpha$}:

X \rightarrow Y

2

The condition on X for (3.6) to holdIt may be interesting to look for the condition for (3.6) to hold. If X is a Banachfunction space, then a beautiful result is known; see [6].

5.4

The characterization of HX(\mathbb{R}^{n}) in terms of the Riesz

transform
It is not known whether HX(\mathbb{R}^{n})transform.

canbe characterized in terms of the Riesz

Maybe, the method in [18] can be used.

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Department of Mathematics and Information Sciences,
Tokyo Metropolitan University
1‐1 Minami‐Ohsawa

Hachioji, Tokyo, 192‐0397
Japan \mathrm{E} ‐mail address: yoshihiro‐sawano@celery.ocn.ne.jp
Yoshihiro Sawano