subelliptic Sobolev compact embedding theorem

Question

Consider the smooth vector fields $X=(X_1,X_2,...,X_m)$ defined in a open bounded set $\Omega\in R^n$. And the non-isotropic dimension is $Q$ (see https://arxiv.org/pdf/1502.06332.pdf page 398) In the paper above (page 398) the author gave a embedding that if $f\in H_{X,0}^1$, $$\|f\|_{L^{p^{*}}\left(\Omega^{\prime}\right)} \leqslant C\left\|X f\right\|_{L^{p}\left(\Omega^{\prime}\right)}$$ $$\frac{1}{p^{*}}=\frac{1}{p}-\frac{1}{Q}, \quad 1 \leqslant p<Q$$ Now what we concern about is that whether this embedding can be improved to be a compact embedding? We are in trouble in doing the proof. Could anybody give some reference or information about this topic? All advices are greatful. Thank you!

Answer 1

The compactness for exponents below the embedding exponent is standard and true in a great generality. In your case compactness follows from results in Section 8 of [1]. Sobolev embeddings for vector fields are discussed in Section 11 of [1]. See also Theorem 4 in [2] for a very general compactness criteria. Both papers are available on my website.

In many cases one can prove the compactness using the following general result (Theorem 4 in [2]).

Theorem. Let $X$ be a set equipped with a finite measure $\mu$. Assume that a linear normed space $W$ of measurable functions on $X$ has the following two peoperties.

1. There is $q>1$ such that the embedding $W\subset L^q(X,\mu)$ is bounded.
2. Every bounded sequence in $W$ contains a subsequence that converges almost everywhere.

Then the embedding $W\subset L^s(X,\mu)$ is compact for every $1\leq s<q$.

In the case of Sobolev spaces associated with vector fields in order to prove compactness below the embedding exponent it suffices to verify a Poincare inequality on every ball $B\subset\Omega$ such that $2B\subset\Omega$, see Theorem 8.1 in [1].

[1] P. Hajlasz, P. Koskela, Sobolev met Poincare, Memoirs Amer. Math. Soc. 688 (2000), 1--101.

[2] P. Hajlasz, P. Koskela, Isoperimetric inequalities and imbedding theorems in irregular domains, J. London Math. Soc. 58 (1998), 425-450.