DU Wenkui, YAN Dunyan

(College of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China)

Abstract The classical convolution integral on Euclidean space is given as follows. For f∈L1(n) and g∈Lp(n), Tf(g) is defined as

Keywords convolution integral; closed hypersurface; boundedness

The classical concept of convolution operator has been generalized in many new cases. The reason is that convolution operator has many applications in harmonic analysis and engineering. For example, it can be used to characterize the bounded operators which commutate with transition actions.

Many researchers have made explorations in these topics. For instance, Oinarov[1]explored the boundedness and compactness of convolution operators of fractional integration type. Avsyankin[2]and Guliyeva and Sadigova[3]explored the properties of convolution operators on Morrey spaces.

Harmonic analysis on Euclidean space has developed very fast. It is also meaningful to generalize the theories on manifolds. For example, the progress of restriction conjecture about Fourier transformation has been introduced in Refs. [4-7]. Similarly, we consider the restriction properties of convolution integral on manifolds in this study.

1 Some definitions

Before we put forward our main results,some useful definitions are given as follows.

Definition1.2SupposeMis an (n-1) dimensional hypersurface inn. Forn)∩C(n) andg∈Lp(n), we define

(1)

where,p≥1 anddσis the surface measure ofMinn.

2 Main result

We state our main theorem as follows.

Theorem2.1LetMbe a closed (connected compact without boundary) (n-1) dimensional hypersurface inn. Then, for 1≤p≤∞,n)∩C(n)， andg∈Lp(n), the inequality

holds. Here,C′(M) is a constant relying onM.

3 Proof of the main result

Lemma3.1LetMbe a closed (n-1) dimensional hypersurface inn. If the following inequality holds for alln)∩C∞(n) andg∈Lp(n)∩C∞(n),

(2)

Proof:

(3)

and

‖gk-g‖p→0.

(4)

Without loss of generality, we are able to assume that the sequencefkconverges tofalmost everywhere andgkconverges togalmost everywhere.

Thus, applying Fatou’s lemma and using (2), (3), and (4), we have

‖Tf(g)‖p,M=‖f*g‖p,M

≤‖|f|*|g|‖p,M

This completes the proof of Lemma 3.1.

Then, we state the following tubular neighborhood lemma[8].

Lemma3.2LetSbe a closed hypersurface in Euclidean space. (N,S,π,) is the normal bundle ofS. Then, there exists aδ>0 and a tubular neighborhood Δδ={(p,η)∈N:‖η‖<δ}, such that Δδis diffeomorphic toNδ={x∈Np?n+1:p∈S;d(x,S)<δ} under the mappingφ(p,η)=p+η. Thus, for each two pointspandqonS, the corresponding normal lines passing through these two points and having these two points as the lines’ centers do not intersect, and these normal lines have length of 2δ.

Then, becauseMis a closed hypersurface inn, according to generalized Jordan separation theorem[8]we can assume thatDis a bounded open domain inn, whose boundary isM， i.e., ?D=M. Now, we have the following lemma.

Lemma3.3LetMbe a (n-1) dimensional closed hypersurface inn. Then, for 1≤p≤∞, the inequality

ProofofLemma3.3andTheorem2.1:

We first prove that the inequality holds for 1

(5)

whereηαis theαth component ofη.

Therefore, substituting surface measure dσin (1) by volume form Ω in (5), we obtain

(6)

whereh0(x)=|Tf(g)(x)|p.

Since, forp>1,h0(x) is smooth forε>0 by Sard theorem[9], there exists ac∈such that |c|<εand 0 is the regular value ofTf(g)(x)-c. This means that the gradient ofTf(g)(x)-cat the zeros of this function does not vanish. Leth(x) be |Tf(g)(x)-c|p， we have

(7)

Here, we have used inequality (8).

(a+b)p≤2p-1(ap+bp).

(8)

According to the regular value preimage theorem[10],Γ={h(x)=0} is a (n-1) dimensional regular submanifold inn, whose Lebesgue measure is zero. Take Γεbe the ε tubular neighborhood of Γ such that

(9)

(10)

Next, we estimate the two parts in (10) separately. Using (9), we first have

(11)

(12)

Meanwhile, applying the following Young’s inequalities

‖Tf(g)‖p≤‖f‖1‖g‖p,

we have

(13)

and

(14)

Combining (7), (10), (13), and (14), we have

≤2p-1εp|M|+2p-1C(M)(2p-1εp|D|+ε+

≤2p-1εp|M|+2p-1C(M)(2p-1εp|D|+ε+

(15)

For arbitraryεandp>1, letε→0 in (15). We obtain

(16)

Now, we have finished the proof in the case where 1

(17)

Meanwhile, it is obvious that

‖Tf(g)‖∞,M≤‖f‖1‖g‖∞

(18)

holds.

Finally, using (17), (18), and Riesz-Th?rin interpolation theorem[11], we can choose a constantC′(M)=max{2C(M),1} which is independent ofpsuch that the following is true.

This completes the proof of Lemma 3.3. Due to Lemma 3.1, we finish the proof of Theorem 2.1.