Smooth cut-off in homogeneous Besov space

Question

Given a Littlewood-Paley decomposition

$$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$

where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported on an annulus, let's ​consider the homogeneous Besov space

\begin{align*} \dot{B}^{s}_{p,r}(\mathbb R^n) = \{u \in S'(\mathbb R^n) : &\lim \limits_{n \to +\infty}||\chi(2^nD)u||_{L^{\infty}} = 0, \\ &||u||_{\dot{B}^{s}_{p,r}} = ||(2^{js}||\varphi(2^{-j}D)u||_{L^p})_{l \in \mathbb Z}||_{\ell^r} < +\infty\} \end{align*}

For $s > 0$ and $p, r < \infty$, it is shown in G. Leoni "A First Course in Sobolev Spaces" (Ch. 17) that convolution with an approximate identity converges in $\dot{B}^{s}_{p,r}$ to the original function.

Hence, $C^{\infty} \cap \dot{B}^{s}_{p,r}$ is a dense subset of $\dot{B}^{s}_{p,r}$.

Is it possible to obtain density of $C_c^{\infty} \cap \dot{B}^{s}_{p,r}$ by considering a smooth cut-off of an element in $C^{\infty} \cap \dot{B}^{s}_{p,r}$ ? In the inhomogeneous case, it is definitely possible.

I'm interested in approximating elements (by functions which are Schwartz or compactly supported) in intersections such as $\dot{B}^{s}_{p,r} \cap L^q$. If the method of convolution + cut-off works in $\dot{B}^{s}_{p,r}$, then it will work as well in $\dot{B}^{s}_{p,r} \cap L^q$.

I have many tools (Bernstein inequality, Embeddings, Bony Decomposition), but I cannot prove it. We may need to assume $s < n/p$.

Answer 1

I think I've come up with a proof.

Assume that $u \in \dot{B}^{s}_{p,r}(\mathbb R^d)$ with $s < d/p$ and $p,r \in [1,+\infty]$. Write $\dot{\Delta}_j = \varphi(2^{-j}D)$, $j \in \mathbb Z$.

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Approximate identity convergence

Let $(k_{\lambda})_{\lambda > 0} \subset S(\mathbb R^d)$ be any approximate identity that lies in Schwartz space, i.e.

1. $\int_{\mathbb R^d} k_{\lambda} = 1$
2. $\sup_{\lambda > 0} ||k_{\lambda}||_{L^1} < +\infty$
3. $\forall \delta > 0$ : $\lim \limits_{\lambda \to \infty} \int_{|x| \geq \delta}||k_{\lambda}||_{L^1} = 0$

Trivially,

$$||\chi(2^nD)(u * k_{\lambda})||_{L^{\infty}} = ||(\chi(2^nD)u) * k_{\lambda}||_{L^{\infty}} \leq ||\chi(2^nD)u||_{L^{\infty}} \cdot ||k_{\lambda}||_{L^1}$$

Using Young-Convolution inequality, $$2^{js}||\dot{\Delta}_j(u * k_{\lambda})||_{L^p} = 2^{js}||(\dot{\Delta}_ju) * k_{\lambda}||_{L^p} \leq 2^{js}||\dot{\Delta}_ju||_{L^p} \cdot ||k_{\lambda}||_{L^1}$$

Hence, $u * k_{\lambda} \in \dot{B}^{s}_{p,r}(\mathbb R^d)$ and it is uniformly bounded in $\lambda$. By the Fatou property, the limit (along some subsequence) as $\lambda \to +\infty$ exists in $\dot{B}^{s}_{p,r}(\mathbb R^d)$.

Since $\dot{B}^{s}_{p,r}(\mathbb R^d)$ is embedded into $S'(\mathbb R^d)$ (which is Hausdorff) and $u * k_{\lambda}$ converges to $u * \delta = u$ in $S'(\mathbb R^d)$, the limit in $\dot{B}^{s}_{p,r}(\mathbb R^d)$ must be $u$.

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Smooth cut-off convergence

If $f$ is Schwartz, then $$||u \cdot f||_{\dot{B}^{s}_{p,r}} \leq C [ ||f||_{L^{\infty}} + ||f||_{\dot{B}^{d/p}_{p,\infty}} + ||f||_{\dot{B}^{0}_{\infty,\infty}} ] \cdot ||u||_{\dot{B}^{s}_{p,r}}$$ thanks to paraproduct continuity and the embedding $\dot{B}^{s}_{p,r} \subset \dot{B}^{s-d/p}_{\infty,r}$.

Moreover, $$||f||_{\dot{B}^{0}_{\infty,\infty}} \leq C ||f||_{L^{\infty}}$$

Now, we need to study $$||f||_{\dot{B}^{d/p}_{p,\infty}} = \sup_{j \in \mathbb Z} (2^j)^{d/p} ||\dot{\Delta}_j f||_{L^p}$$ We must use something which looks like Bernstein inequality, but with a fractional derivative $\partial^{d/p}$ (defined by means of the Fourier transform). I think we can prove $$(2^j)^{d/p} ||\dot{\Delta}_j f||_{L^p} \leq C ||\partial^{d/p} \dot{\Delta}_j f||_{L^p} \leq C ||\partial^{d/p} f||_{L^p}$$

If $f$ is a smooth cut-off, $f(x) = \chi(x/R)$ with $0 \leq \chi \leq 1$, $\chi \in C^{\infty}_c(\mathbb R^d)$, then $$||u \cdot \chi(x/R)||_{\dot{B}^{s}_{p,r}} \leq C [ ||\chi||_{L^{\infty}} + ||\partial^{d/p} \chi||_{L^p} ] \cdot ||u||_{\dot{B}^{s}_{p,r}}$$ so it is uniformly bounded for $R > 0$. Thanks to Fatou property, the limit (along some subsequence) as $R \to +\infty$ exists in $\dot{B}^{s}_{p,r}(\mathbb R^d)$.

Finally, the limit, denoted by $v$, must satisfy $$\langle v, \varphi \rangle = \langle u, \varphi \rangle$$ for all test functions $\varphi \in C^{\infty}_c(\mathbb R^d)$. Hence, $u$ and $v$ must differ by a constant which is exactly zero.

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For references on theorems used here, look at Bahouri "Fourier Analysis and Nonlinear Partial Differential Equations"