Hodge theory on compact hermitian manifolds - 4

Now we move on to the proof of Theorem B. The standard Sobolev lemma will be applied without a proof here.

Sobolev lemma and local Sobolev spaces

Recall the statement of Theorem B.

Theorem B. If satisfies that , then .

We see that this is concerned with the regularity of . It is often not easy to consider the continuity and differentiability of such forms directly. To deal with this, the Sobolev lemma is usually applied to turn the proof of regularities to the estimation of -norms.

Theorem (Sobolev lemma). Suppose is an open subset. Then for any nonnegative integers and such that , we have .

Note that here the inclusion means that each agrees with some almost everywhere. As the differentiablity is a local property, we can actually weaken the condition of . Consider the local Sobolev space consisting of measurable functions such that for any . Clearly . By taking to be a bumping function, we see from the Sobolev lemma that whenever . As a consequence, we have

From now on fix an open cover of such that we have holomorphic trivialization of both and on each . Then each can be written locally on as where . By the equvalence of the norm and , we see that if and only if each , and meanwhile clearly if and only if each . Considering a partition of unity subordinate to the open cover , we see that it suffices to prove the assertion of Theorem B for such that vanishes outside some open subset with compact.

Identifying with an open subset with through the coordinate map. Then becomes a bounded open set in . Denote the set of such that vanishes outside by and similarly for and . As there is a correspondence between and , we may analogously consider consisting of those given by . The operator extends to in a natural way. The Sobolev lemma now implies that

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The proof of the following lemma will be given in the next part of this post.

Lemma D. Suppose satisfies that and the support is contained in . Then .

Using Lemma D and the alternate form of the Sobolev lemma, we can provide the proof of Theorem B.

Proof of Theorem B: It suffices to show that if and , then . By the Sobolev lemma, this is equivalent to for any , which will be shown by induction on .

We have got the assumption that . Now assume that and we need to show . Consider an arbitrary . As , we have , i.e., for any . By direct computation, we can see from the smoothness of that As clearly we can apply Lemma D to obtain that . It follows that and hence , which completes the induction.

Smoothing process

The proof of Lemma D involves an important trick call the smoothing process. We regard as an open subset of such that is compact. Fix a smooth function satisfying that , , and For each , define by We see that has the property that For each , define the convolution of with by where we let for . There are some basic properties of convolution which can be verified directly:

• ;
• with the derivatives given by for any multi-index ;
• with ;
• as , converges to in , i.e., .

Lemma. Suppose , for and , and Then as , we have

Proof: Let by the Hilbert space of -tuples of functions in with the norm given by We can see that is a dense subset of .

Let be a linear subspace of containing . For each , there is a linear functional given by We claim that is a bounded operator with bounded by a constant independent of .

Consider any and let Viewing as an element in and as an element in , we see that for each we have It follows that Meanwhile, we have implying that Let which is finite as are and is compact. Since is , is continuous function with compact support. Thus is finite. Therefore for we have and hence

This shows that whenever , and we can see that and are both independent of .

For any , we have where we use the fact that if with respect to , and are all compactly supported, and is then with respect to . Now consider and take any . Since is dense in , there exists such that . Meanwhile, implies that there is some such that for any , . It follows that

We conclude that as .

Corollary. Suppose and are differential operators of order one with coefficiets on . If , then with respect to as .

The proof of Lemma D can be given now.

Proof of Lemma D: For each , let be given by for each , , and . Since is a compact subset of the open subset , we have for sufficiently small . Since , we see that with respect to . Note that there are differential operators of order one with coefficients on such that Hence with respect to , implying that with respect to . Thus and are both Cauchy sequences in .

By Gårding’s inequality, there exists such that

Then for sufficiently small such that , is a Cauchy sequence with respect to . Suppose with respect to . Then clearly converges to with respect to . By the uniqueness of the limit in , we obtain .