We completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball. We show that, for

For any integer

For Toeplitz operators, this problem has been studied for a long time. In the case of the classical Hardy space, Brown and Halmos [

Initiated by Brown and Halmos’s pioneering work, the problem of characterizing when two Toeplitz operators commute has been one of the topics of constant interest in the study of Toeplitz operators on classical function spaces over various domains. On the Bergman space of the unit disk, Axler and Čučković [

The fact that the product of two harmonic functions is no longer harmonic adds some mystery to the study of operators on harmonic Bergman space. Many methods which work for the operators on analytic Bergman space lose their effectiveness on harmonic Bergman space. On the harmonic Bergman space of the unit disk, Ohno [

However, the study on the problem for dual Toeplitz operators started recently. Stroethoff and Zheng [

In recent years the Dirichlet space has received a lot of attention from mathematicians in the areas of modern analysis, probability, and statistical analysis. Many mathematicians are interested in function theory and operator theory on the Dirichlet space. Yu and Wu [

In this paper, we want to characterize commuting dual Toeplitz operators with pluriharmonic symbols on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball.

We state our main result now. We postpone the proofs of these theorems until Section

Suppose that

Both

Both

Either

Suppose that

Both

Both

There are constants

For

A pluriharmonic function in the unit ball is the sum of a holomorphic function and the conjugate of a holomorphic function. It is clear that all pluriharmonic functions on

The difficult part of the proof of Theorem

This question is very subtle. If

The following Lemma has been known to be true for

The set of all polynomials in

We will discuss it in the case of real variables. For

Let

For two multi-indexes

Let

Set

Since polynomials are dense in

The following lemma will be useful for the proof of the main theorem.

Suppose that

Since

The last case is similar; we omit the proof. Hence we get that if

In the following proposition, we give an answer to the question that when a dual Toeplitz operator equals zero.

Suppose that

Assume that

If

Suppose that

To prove the sufficient part, suppose that

Conversely, assume that

In this section, we will present the proofs of the main results.

If

To prove the necessity, suppose that

Both

Both

Either

There is a nonzero constant

Suppose that

From the equation

Assume that

Both

Both

Both

Both

There is a nonzero constant

Both

There is a nonzero constant

In the following proof, assume that none of

Both

Both

There is a nonzero constant

Both

There is a nonzero constant

Note that

Then there exists a multi-index

By Lemma

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by NSFC (nos. 11271059, 11271332, 11431011, and 11301047) and NSF of Zhejiang Province (nos. LY14A010013 and LY14A010021).