The bounded of function associated with divergence form operator

Keywords: Littlewood-Paley-Stein, divergence form operator, Riemannian manifold.

Abstract. The -estimates of the Littlewood-Paley-Stein function associated with divergence form operator on complete Riemannian manifolds for .

Introduction

Let be the Euclidean space or a domain of . Let be the sesquilinear form on space , where is a dense subspace of the Sobolev space , given

,

And are bounded, measurable, complex-valued coefficients which satisfy,

,

for some constant , and for all ,where are positive constant.

Let be the divergence operator associated with the form in the sense that is the operator in with largest domain which satisfies

for and all test functions . Let be a complete non-compact Riemannian manifold, be the distance on , and be the Riemannian measure. Denote by the ball of center and radius and by its Riemannian Volume . One says that has the doubling volume property if there exists such that

, , .

Let be the divergence semigroup generated by operator has kernel on and

, , , be the Poisson semigroup on .

In this paper we consider the vertical Littlewood-Paley-Stein function associated to defined by

.

Let be a sublinear operator, defined on ,with values in measurable functions on .We shall say that is bounded on ,for some ,if that exist such that

, ,

and that is of weak type (1,1) if there exists such that

For every and .

The aim of this paper is to prove the following theorem.

Theorem 1.1 Let be a divergence form operator. Then Littlewood-Paley-Stein function is bounded on for , and , is of weak type (1,1).

SOME BASIC FACTS

For , we define the vertical Little-wood-Paley-Stein function associated with the heat kernel semigroup of operator A by

.

The function is pointwise dominated by , that is, there exists a positive constant such that

By using the following equation

.

We can write

.

Since

. (1.1)

Thus, if we prove the -boundedness of , then the same is true for .

Proposition 2.1 There exists a constant such that

.

Proof It has been shown in [1] for operator acting on , the generalized Riesz transform associated with operator is bounded on , and

.

Then, for , we have

.

Therefore

On the other hand

.

The proof of proposition is completed.

THE PROOF OF MAIN RESULT

We can only prove the weak type (1, 1) for , and it is also holds for . The following statement is the main technical tool in [2], it will be instrumental in the proof of Theorem 1.1

Proposition 3.1 Let be a complete Riemannian manifold, with Riemannian measure satisfying the doubling volume property and the heat kernel upper bound

For some and all ,

Let be a bounded operator.

Assume that there exists a kernel such that,

, , for a. e. .

If

Then is weak type (1, 1).

Weighted estimates of derivatives of the heat kernel .In the next sections, we shall work under the assumptions of theorem 1.1.Our first lemma is standard , so we omit the detail (see [2], Lemma 21).

Lemma 3.2 For

.

Recall that assumption (3.1), together with the doubling volume property, the corresponding off-diagonal estimate automatically follows ([4], Theorem 1.1),

.

Lemma 3.3 For small enough, then

.

Proof ， Set

.

.

By the result of [3],

for

We have

And then Lemma 3.3 follows.

Proof of Theorem 1.1 Now we can only give the proof of weak type (1, 1) of and is also true for . The identity operator will be written as .

.

According to Proposition 3.1 , it is enough to prove there exists such that

. (3.4)

For all , set

For

.

For , we write

.

According to Lemma 3.2, we have

Therefore

.

Since

.

According to Lemma 3.3, for small enough,

.

.

Therefore, by (3.5), we have .

Let us now turn to the case .write

Since

,

We obtain .

Similar to , we obtain the following ,

Following the above argument and lemma 3.3 , one obtains

.

Following the above argument there is,

.

Using doubling, we can easily see that the quantity is uniformly bounded form above. Thus and .

Finally, Since and therefore the weak type (1, 1) of , is proved. According to section (1.1),and also has weak type (1,1).

By using the Marcinkiewicz interpolation theorem, we obtain that is bounded on for .

This ends the proof.

References

[1] P. Auscher, S. Hofmann, M. Lacey, A.Mcintosh and Ph. Tchamitchian, The solution of the Kato

square root problem for second order elliptic operators on . Ann. Math. 156 (2002), 633-654.

[2] T. Coulhon and X.T. Duong, Riesz transforms for , Trans. Amer. Math. Soc. 351(1999),

1151-1169.M.A. Green: High Efficiency Silicon Solar Cells (Trans Tech Publications, Switzerland 1987).

[3] X. T. Duong and D. Robinson, Upper bounds of derivatives of the heat kernel on arbitrary

complete mainfold, J. Funct. Anal. 127(1995), 363-389.

[4] A. Grigor’yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds,

J. Diff. Geom. 45(1997), 33-52.

第一作者：姓名：朱素婷 1976.6 籍贯：山东郓城 女 调和分析 学生

第二作者：赵凯 1960.1 籍贯：山东潍坊 男 调和分析 教授