On Fractional Smoothness of Modulus of Functions

Dong Li

SUSTech International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology,Shenzhen, Guangdong 518055, China

Abstract. We consider the Nemytskii operators u→|u|and u→u±in a bounded domain ? with C2 boundary. We give elementary proofs of the boundedness in Hs(?) with 0≤s<3/2.

Key words: Nemytskii, nonlocal extension, fractional Laplacian.

1 Introduction

Let ? be a nonempty open bounded set in Rd. For 0<γ<1 andf ∈C1(?), define the nonlocal ˙Hγsemi-norm as

Forf ∈C2(?) and 0<γ<1, we define

where (and throughout this note)?f=(?x1f,···,?xnf) denotes the usual gradient.Throughout this note we shall only be concerned with real-valued functions, however with some additional work the results can be generalized to complex-valued functions. Define the Nemytskii operators

The purpose of this note is to give an elementary proof of the following.

where α1>0depends on(s,?,d).

We refer to[1,7]and the references therein for a more detailed survey of composition operators in function spaces with various fractional order of smoothness.

The rest of this note is organized as follows. In Section 2 we give the proof for the one dimensional case. In Section 3 we give the details for general dimensionsd≥2.

1.1 Notation

2 The one dimensional case

Lemma 2.1.Let0<α<1and0<δ<1?α. Consider

Sending?→0 then yields the result.

Lemma 2.3.Let0<α<1. Assume u is bounded on[0,1]. Suppose

More generally, there are constants β1>0, β2>0depending only on α, such that for any finite interval onR, it holds that(below|I|denote the length of the interval))

where β3>0depends only on α.

One can see [3] for an extensive discussion on general fractional order Hardy inequalities and counterexamples.

Proof.The first identity is obvious. For the second inequality,one can apply Lemma 2.2 tou(x)χ(x),whereχis a smooth cut-offfunction supported in[0,2/3]. This then yields

The desired inequality follows easily by using the first identity. To get the extra factor(1?x)?αone can invoke the symmetryx→1?x. The inequality (2.8)follows from rescaling and reducing to the caseI=(0,1).

The following theorem is a special case of Bourdaud-Meyer[6]. We reproduce the proof here to highlight the needed changes for the finite domain case (see Theorem 2.2).

Theorem 2.1.Let0

Proof.Setα=2s. WriteI={x:u(x)>0}as a countable disjoint union of intervalsIjsuch that eachIj=(aj,bj) satisfiesu(aj)=u(bj)=0. Here ifajorbjare infinity the value ofu(aj) oru(bj) are understood in the limit sense. By using Lemma 2.3,we have

Thus, we complete the proof.Corollary 2.1.Let0

Proof.By a partition of unity in the frequency space, we have

Theorem 2.2(Boundedness ofT2on a finite interval).Let0

Proof.Setα=2s. If{u>0}=(0,1) there is nothing to prove. Otherwise write{x∈(0,1):u(x)>0}as a countable union of maximal intervalsIjsuch that for eachIj=(aj,bj),eitheru(aj)=u(bj)=0 with 00,u(b)=0. We discuss several further cases.Case 1:b≥1/4. Clearly then

Thus this case is also proved.

3 The general case d≥2

Lemma 3.1.Let0

where C1>0depends only on(s,K,f).

Figure 1: The set Q+ and K in Lemma 3.1.

Proof.Denote ?gas an extension ofgto R2such that ?ghas compact support and

Thus, we complete the proof.

Lemma 3.2.Let0

Then

where C2>0depends only on(s,K,f).

Proof.Observe that

Figure 2: The set F0 and the curve γ in Lemma 3.3.

Assume u is compactly supported in

Proof.With no loss we consider the two dimensional case. The argument then follows from a standard partition of unity. The extension for the interior piece is quite straightforward. The localized boundary piece follows from(after rotation and relabelling coordinate axes if necessary) Lemma 3.3.Proof.By Theorem 3.1, we extendfto ?fdefined on the whole Rd. The result then follows from the boundedness in the whole space case (see Corollary 2.1).

Finally we remark that Theorem 1.1 follows from Corollary 3.1 since the proofs forT1andT3are similar.

Acknowledgements

The author is indebted to Prof. Xiaoming Wang for raising this intriguing question.