We characterize the weighted weak local Hardy spaces

The theory of classical local Hardy spaces, originally introduced by Goldberg [

On the other hand, the weak

The purpose of this paper is twofold. The first goal is to characterize weighted weak local Hardy spaces by atomic decomposition. The second goal is to show that localized Riesz transforms are bounded on weighted weak local Hardy spaces.

The paper is organized as follows. In Section

Throughout this paper, we let

In this section, we review some notions and notations concerning the weight classes

Let

With regard to the Schrödinger operator

There exist

A ball of the form

There exists a sequence of points

for every

In this paper, we write

A weight always refers to a positive function which is locally integrable. As in [

Clearly, the classes

Since

In what follows, given a Lebesgue measurable set

We remark that balls can be replaced by cubes in definition of

Next, we give some properties of weights class

Let

if

if

let

let

for

if

if

(i)–(viii) have been proved in [

For any

The symbols

Let

If

We now introduce some local maximal functions. For

Let

For convenience’s sake, when

Let

The local vertical maximal function

For

Let

Let

there exists a positive constant

if

The proof of (i) is trivial. For (ii), since

We first prove (

Let

Let

Similarly, the weighted weak local Hardy spaces

In this section, we establish a decomposition theorem of weighted weak local Hardy spaces

We first recall the Calderón-Zygmund decomposition of

Let

Now we take a function

Let

As in [

To obtain the main theorem, we need the following lemmas (Lemmas

There exists a constant

Suppose

Let

Let

If

If

Let

Each

each

Conversely, if a distribution

Moreover, one has

We first suppose

Then by Lemma

Hence,

We set

For the converse, take

Therefore, we get

In this section, we will show the boundedness of localized Riesz transforms on

As in [

Let

Now let us state the main result of this section.

Let

By the definition of

Combining the above two cases with Theorem

The author declares that there is no conflict of interests regarding the publication of this paper.

The research is supported by National Natural Science Foundation of China (Tianyuan Foundation of Mathematics) no. 11426038. The author would like to thank the referees for their very valuable suggestions which made this paper more readable.

^{1}

^{p}estimates for Schrödinger operators with certain potentials