^{1}

^{2}

^{1}

^{2}

We introduce the generalized Carleson measure spaces CM

In 1972, Fefferman and Stein [

We say that a cube

Choose a fixed function

We now introduce a new space

Let

By definition, we immediately have

For a dyadic cube

It is clear that

The definition of

Let

Let

Let

The classical Plancherel-Pôlya inequality [

Using the Calderón reproducing formula (either continuous or discrete version), several authors obtain the variant Plancherel-Pôlya inequalities [

Define a linear map

We now state our first main result as follows.

Suppose that

For

For

Conversely, every continuous linear functional

For

For

Conversely, every continuous linear functional

For

For

In Remark

For

As applications, we first recall the Haar multipliers introduced in [

Using Meyer’s wavelets, we may generalize the above Haar multiplier to

Suppose that

for

for

We consider another application. Let

Suppose that

For

If

When

The paper is organized as follows. In Section

In this section, we introduce sequence spaces

To study the duals of

For

It is obvious that

Suppose that

For

For

Conversely, every continuous linear functional

For

For

Conversely, every continuous linear functional

For

For

We first consider the case

For

On the other hand, suppose that

Let us recall the

Define a linear map

Suppose that

Figures

Diagram for spaces and maps for

Diagram for spaces and maps for

One recalls the almost diagonality given by Frazier and Jawerth [

For

We postpone the proof of Lemma

Let

We summarize that

For

Theorem

First let us consider the case for

Conversely, let

A similar argument gives the desired result for

As pointed out by one of the referees, Yang and Yuan [

In this section we demonstrate the Plancherel-Pôlya inequalities.

Without loss of generality, we may assume that

Next we decompose the set of dyadic cubes

To estimate

By modifying the proof above, we may easily show Theorem

We now return to show Lemma

For

We may assume that

To estimate

When

When

Note that

We define another wavelet multiplier on

Suppose that

for

for

We show the case

In order to prove Theorem

Suppose that

for

for

We still assume that

Conversely, suppose that

For

The “if’’ part follows from Theorem

In order to study the boundedness of the paraproduct operators acting on Triebel-Lizorkin spaces, we need more results described as follows.

For

Define a matrix by

For

We now can prove Theorem

To simplify notations, let

Next suppose that

To prove part (ii), assume that

The authors are grateful to the referees for many invaluable suggestions. Research by both authors was supported by NSC of Taiwan under Grant nos. NSC 100–2115-M-008-002-MY3 and NSC 100–2115-M-259-001, respectively.