• $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and $u \in \dot{B}_{p,q}^s(\mathbb{R}_+^n)$ (resp. $u \in \dot{B}_{p,q}^s(\mathbb{R}^n)$);

then we have the estimate

$$ \|Eu\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,m,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. \quad (\text{resp. } \|P_0u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,m,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)}.) $$

In particular, E (resp. $P_0$) is a bounded operator from $\dot{B}_{p,q}^s(\mathbb{R}_+^n)$ to $\dot{B}_{p,q}^s(\mathbb{R}^n)$ (resp. from $\dot{B}_{p,q}^s(\mathbb{R}^n)$ to $\dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$) whenever $(C_{s,p,q})$ is satisfied.

*Proof.* — Let $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s > -1 + \frac{1}{p}$, $m \in \mathbb{N}$, such that $s < m + 1 + \frac{1}{p}$. Without loss of generality, it suffices to prove the result for the operator E, since we have the identity $P_0 = I - E^{-}[1_{\mathbb{R}_+^n}]$, as written in the proof of Lemma 3.6.

The boundedness of E on $\dot{B}_{p,q}^s(\mathbb{R}^n)$ for $(p,q) \in (1, +\infty) \times [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ is again a direct consequence of Proposition 2.9.

It remains to prove boundedness for $s \ge \frac{1}{p}$. To do so, we proceed via a manual real interpolation scheme.

Let $u \in B_{p,q}^s(\mathbb{R}_+^n)$, $\theta \in (0,1)$ such that $\theta s_1 = s$, where $s_1 \in (s, m + 1 + \frac{1}{p})$. One has,

$$ u \in (L^p(\mathbb{R}_+^n), H^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \hookrightarrow (L^p(\mathbb{R}_+^n), \dot{H}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \subset L^p(\mathbb{R}_+^n) + \dot{H}^{s_1,p}(\mathbb{R}_+^n). $$

Hence, for $a \in L^p(\mathbb{R}_+^n)$, $b \in \dot{H}^{s_1,p}(\mathbb{R}_+^n)$ such that $f = a + b$, we can deduce that

$$ b = u - a \in B_{p,q}^s(\mathbb{R}_+^n) + L^p(\mathbb{R}_+^n) \subset L^p(\mathbb{R}_+^n), $$

so that $b \in L^p(\mathbb{R}_+^n) \cap \dot{H}^{s_1,p}(\mathbb{R}_+^n) = H^{s_1,p}(\mathbb{R}_+^n)$ thanks to Proposition 3.3. Hence, $Eu = Ea + Eb$, with $Ea \in L^p(\mathbb{R}_+^n)$, $Eb \in H^{s_1,p}(\mathbb{R}_+^n)$, with the homogeneous estimates provided by Proposition 3.1. Then $Eu|_{\mathbb{R}_+^n} = u$, and we have the estimates

$$ K(t, Eu, L^p(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n)) \le \|Ea\|_{L^p(\mathbb{R}^n)} + t\|Eb\|_{\dot{H}^{s_1,p}(\mathbb{R}^n)} \lesssim_{p,m,n} \|a\|_{L^p(\mathbb{R}_+^n)} + t\|b\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}. $$

Hence, taking infimum on all such functions $a$ and $b$, and multiplying by $t^{-\theta}$ leads to

$$ t^{-\theta} K(t, Eu, L^p(\mathbb{R}^n), \dot{H}^{s_1,p}(\mathbb{R}^n)) \lesssim_{p,s,s_1,n} t^{-\theta} K(t, u, L^p(\mathbb{R}_+^n), \dot{H}^{s_1,p}(\mathbb{R}_+^n)), $$

so one may take the $L^q$-norm of the above inequality and use (3.13) from Lemma 3.13 to deduce that

$$ \|Eu\|_{\dot{B}_{p,q}^s(\mathbb{R}^n)} \lesssim_{p,s,q,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

If $q < +\infty$, then $B_{p,q}^s(\mathbb{R}_+^n)$ is dense in $\dot{B}_{p,q}^s(\mathbb{R}_+^n)$, so that the conclusion holds by density whenever $(C_{s,p,q})$ is satisfied.

If $q = +\infty$, and $(C_{s,p,q})$ is satisfied, necessarily $s < \frac{n}{p}$. We introduce $\mathcal{E} := E[1_{\mathbb{R}_+^n}]$ which is bounded, thanks to the above step, seen as an operator

$$ \mathcal{E} : \dot{B}_{p,q_j}^{s_j}(\mathbb{R}^n) \to \dot{B}_{p,q_j}^{s_j}(\mathbb{R}^n), $$

provided $s_0 < s < s_1 < \frac{n}{p}$, and $q_j \in [1, \infty)$, $j \in \{0, 1\}$. Thus, by real interpolation argument, thanks to Theorem 2.6, for all $U \in \dot{B}_{p,\infty}^s(\mathbb{R}^n)$, we have

$$ \|\mathcal{E}U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)} \lesssim_{p,s,q,n} \|U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)}. $$

In particular, for all $u \in \dot{B}_{p,\infty}^s(\mathbb{R}_+^n)$, and all $U \in \dot{B}_{p,\infty}^s(\mathbb{R}^n)$ such that $U|_{\mathbb{R}_+^n} = u$, we have

$$ \|Eu\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)} \lesssim_{p,s,q,n} \|U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)}. $$

Therefore, taking the infimum on all such functions $U$ gives the result when $q = +\infty$ and $(C_{s,p,q})$ is satisfied. ■

**Proposition 3.15** Let $p, q \in [1, +\infty]$, $s \in (0, n)$, such that

$$ \frac{1}{q} = \frac{1}{p} - \frac{s}{n}. $$
$$
(ii) \quad q \in [1, +\infty], s_j > -1 + \frac{1}{p_j}, \text{ and } (C_{s_j, p_j}) \text{ is satisfied, } j \in \{0, 1\};
$$

If $p_0 = p_1 = p$ and $(C_{s,p,q})$ is satisfied, the following equality is true with equivalence of norms

$$
(\dot{\mathfrak{h}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathfrak{h}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} = \dot{\mathfrak{b}}_{p,q}^s(\mathbb{R}_+^n). \quad (3.17)
$$

If $(C_{s_0, p_0})$ and $(C_{s_1, p_1})$ are true then also is $(C_{s, p_\theta})$ and

$$
[\dot{\mathfrak{h}}^{s_0, p_0}(\mathbb{R}_+^n), \dot{\mathfrak{h}}^{s_1, p_1}(\mathbb{R}_+^n)]_{\theta} = \dot{\mathfrak{h}}^{s, p_\theta}(\mathbb{R}_+^n). \quad (3.18)
$$

Proof. — We start noticing that (3.18) only makes sense under assertion (ii).

Step 1: We prove first (3.18) and (3.17) under assertion (ii).

It suffices to assert that ${\dot{\mathfrak{h}}^{s_0,p_0}(\mathbb{R}_+^n), \dot{\mathfrak{h}}^{s_1,p_1}(\mathbb{R}_+^n)}$ is a retraction of ${\dot{\mathfrak{H}}^{s_0,p_0}(\mathbb{R}^n), \dot{\mathfrak{H}}^{s_1,p_1}(\mathbb{R}^n)}$, thanks to [BL76, Theorem 6.4.2]. Indeed, both retractions are given by

$$
E : \dot{\mathbf{H}}^{s_j, p_j}(\mathbb{R}_+^n) \to \dot{\mathbf{H}}^{s_j, p_j}(\mathbb{R}^n) \text{ and } R_{\mathbb{R}_+^n} : \dot{\mathbf{H}}^{s_j, p_j}(\mathbb{R}^n) \to \dot{\mathbf{H}}^{s_j, p_j}(\mathbb{R}_+^n),
$$

$$
\iota : \dot{\mathbf{H}}_{0}^{s_j, p_j}(\mathbb{R}_+^n) \to \dot{\mathbf{H}}^{s_j, p_j}(\mathbb{R}^n) \text{ and } \mathcal{P}_0 : \dot{\mathbf{H}}^{s_j, p_j}(\mathbb{R}^n) \to \dot{\mathbf{H}}_{0}^{s_j, p_j}(\mathbb{R}_+^n).
$$

Here, $R_{\mathbb{R}_+^n}$ and $\iota$ stand respectively for the restriction and the canonical injection operator. Bound-
edness and range of $E$ and $\mathcal{P}_0$ provided by Lemma 3.6 and Corollary 3.14 lead to (3.18) and (3.17)
under assertion (ii).

Step 2: We prove (3.17) under assertion (i).

Step 2.1: $(\mathfrak{h}, \mathfrak{b}) = (\mathbb{H}, \mathbb{B})$.

Thanks to Lemma 3.13, we have continuous embedding,

$$
\dot{\mathbf{B}}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}. \quad (3.19)
$$

Let us prove the reverse embedding,

$$
\dot{\mathbf{B}}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \leftarrow (\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}.
$$

Without loss of generality, we can assume $s_1 \ge \frac{n}{p}$. Let $f \in S_0(\overline{\mathbb{R}}_+^n) \subset \dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)$, if follows that
$f \in (\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \subset \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n) + \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)$. Thus, for all $(a,b) \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n) \times \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)$ such that $f = a+b$, we have,

$$
b = f - a \in (S_0(\overline{\mathbb{R}}_+^n) + \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n).
$$

In particular, we have $a \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)$ and $b \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)$. Hence, we can introduce
$F := Ea + Eb$, where $F_{|\mathbb{R}_+^n} = f$, $Ea \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)$ and $Eb \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$, with the estimates,
given by Corollary 3.4,

$$
\| \mathrm{E} a \|_{\dot{\mathrm{H}}^{s_0,p}(\mathbb{R}^n)} \lesssim_{s_0,m,p,n} \| a \|_{\dot{\mathrm{H}}^{s_0,p}(\mathbb{R}_+^n)} \quad \text{and} \quad \| \mathrm{E} b \|_{\dot{\mathrm{H}}^{s_1,p}(\mathbb{R}^n)} \lesssim_{s_1,m,p,n} \| b \|_{\dot{\mathrm{H}}^{s_1,p}(\mathbb{R}_+^n)}.
$$

Then, one may bound the K-functional of F, for t > 0,

$$
K(t, F, \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)) \le \|\mathrm{E}a\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)} + t\|\mathrm{E}b\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)} \lesssim_{s_j,p,n} \|a\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)} + t\|b\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)}
$$

Taking the infimum over all such functions $a$ and $b$, we obtain

$$
K(t, F, \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)) \lesssim_{s_j,p,n} K(t, f, \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)),
$$

from which we obtain, after multiplying by $t^{-\theta}$, taking the $L_*^q$-norm with respect to $t$, and applying
Theorem 2.6,

$$
\|f\|_{\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)} \le \|F\|_{\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,p,n} \|f\|_{(\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}}.
$$

Finally, thanks to the first embedding (3.19), we have

$$
\|f\|_{\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)} \sim_{p,s,n} \|f\|_{(\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}}, \forall f \in S_0(\overline{\mathbb{R}}_+^n).
$$

Since $q < +\infty$, we can conclude by density of $S_0(\overline{\mathbb{R}}_+^n)$ in both $\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)$ and in the interpolation space $(\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}$. Density argument for the later one is carried over by Lemma 2.5 and [BL76, Theorem 3.4.2].

**Step 2.2:** $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{\mathbf{B}}_{p,q,0}^s$, provided $-1 + \frac{1}{p} < s < \frac{1}{p}$, $p \in (1, +\infty)$, $q \in [1, +\infty)$.
We have the following estimates,

$$
\begin{align*}
\|u\|_{L^q(\mathbb{R}_+^n)} &\lesssim_{n,s,p,q,r} \|u\|_{\dot{B}_{p,r}^s(\mathbb{R}_+^n)}, \quad \forall u \in \dot{B}_{p,r}^s(\mathbb{R}_+^n), \ r \in [1, q] \\
\|u\|_{\dot{B}_{q,r,0}^{-s}(\mathbb{R}_+^n)} &\lesssim_{n,s,p,q,r} \|u\|_{L^p(\mathbb{R}_+^n)}, \quad \forall u \in L^p(\mathbb{R}_+^n), \ r \in [q, +\infty].
\end{align*}
$$

Moreover, we also have $\dot{B}_{p,1}^{\frac{n}{p}}(\mathbb{R}_+^n) \hookrightarrow C_0^0(\overline{\mathbb{R}}_+^n)$, whenever $p$ is finite.

Proof. — Each embedding is a direct consequence of the definition of each space and the correspond-
ing ones on $\mathbb{R}^n$, see point (iv) of Proposition 2.1.

■

**Lemma 3.16** Let $p \in (1, +\infty)$, $q \in [1, +\infty)$ and $s > 0$. The function space $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$ whenever $(C_{s,p,q})$ is satisfied.

Proof. — As in the proof of Proposition 3.9, in the case of non-negative index: by a successive approx-
imations scheme, we use density of $B_{p,q}^s(\mathbb{R}^n)$ in $\dot{B}_{p,q}^s(\mathbb{R}^n)$, to approximate functions in $\dot{B}_{p,q,0}^s(\mathbb{R}^n)$.
Then the boundedness of $\mathcal{P}_0$ on $\dot{B}_{p,q}^s(\mathbb{R}^n)$, and the density of $C_c^\infty(\mathbb{R}^n)$ in $B_{p,q,0}^s(\mathbb{R}^n)$ yields the
result.

■

**Proposition 3.17** Let $(p_0, p_1, p, q) \in (1, +\infty)^3 \times [1, +\infty]$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 < s_1$, let $(\mathfrak{h}, \mathfrak{b}) \in \{(\mathrm{H}, \mathrm{B}), (\mathrm{H}_0, \mathrm{B}_{.,.0})\}$, and set

$$
\left(s, \frac{1}{p_{\theta}}\right) := \left(1 - \theta\right) \left(s_0, \frac{1}{p_0}\right) + \theta \left(s_1, \frac{1}{p_1}\right).
$$

If either one of following assertions is satisfied,

$$
(i) \quad q \in [1, +\infty), s_j > -1 + \frac{1}{p_j}, j \in \{0, 1\};
$$

$$
(ii) \quad q \in [1, +\infty], s_j > -1 + \frac{1}{p_j}, \text{ and } (C_{s_j, p_j}) \text{ is satisfied, } j \in \{0, 1\};
$$

If $p_0 = p_1 = p$ and $(C_{s,p,q})$ is satisfied, the following equality is true with equivalence of norms

$$
(\dot{h}^{s_0,p}(\mathbb{R}_+^n), \dot{h}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} = \dot{b}_{p,q}^s(\mathbb{R}_+^n). \quad (3.17)
$$

If $(C_{s_0, p_0})$ and $(C_{s_1, p_1})$ are true then also is $(C_{s, p_\theta})$ and

$$
[\dot{h}^{s_0, p_0}(\mathbb{R}_+^n), \dot{h}^{s_1, p_1}(\mathbb{R}_+^n)]_{\theta} = \dot{h}^{s, p_{\theta}}(\mathbb{R}_+^n). \quad (3.18)
$$

Proof. — We start noticing that (3.18) only makes sense under assertion (ii).

**Step 1:** We prove first (3.18) and (3.17) under assertion (ii).

It suffices to assert that {$\dot{h}^{s_0,p_0}(\mathbb{R}_+^n), \dot{h}^{s_1,p_1}(\mathbb{R}_+^n)$} is a retraction of {$\dot{H}^{s_0,p_0}(\mathbb{R}^n), \dot{H}^{s_1,p_1}(\mathbb{R}^n)$},
thanks to [BL76, Theorem 6.4.2]. Indeed, both retractions are given by

$$
\begin{align*}
\mathrm{E} : \dot{H}^{s_j, p_j}(\mathbb{R}_+^n) &\to \dot{H}^{s_j, p_j}(\mathbb{R}^n) \quad \text{and} \quad \mathrm{R}_{\mathbb{R}_+} : \dot{H}^{s_j, p_j}(\mathbb{R}^n) &\to \dot{H}^{s_j, p_j}(\mathbb{R}_+^n), \\
\iota : \dot{H}_0^{s_j, p_j}(\mathbb{R}_+^n) &\to \dot{H}^{s_j, p_j}(\mathbb{R}^n) \quad \text{and} \quad \mathcal{P}_0 : \dot{H}^{s_j, p_j}(\mathbb{R}^n) &\to \dot{H}_0^{s_j, p_j}(\mathbb{R}_+^n).
\end{align*}
$$

Here, $\mathrm{R}_{\mathbb{R}_+}^n$ and $\iota$ stand respectively for the restriction and the canonical injection operator. Bound-
edness and range of $\mathrm{E}$ and $\mathcal{P}_0$ provided by Lemma 3.6 and Corollary 3.14 lead to (3.18) and (3.17)
under assertion (ii).

**Step 2:** We prove (3.17) under assertion (i).

**Step 2:** We prove (3.17) under assertion (i).

**Step 2.1:** $(\mathfrak{h}, \mathfrak{b}) = (\mathrm{H}, \mathrm{B})$.

Thanks to Lemma 3.13, we have continuous embedding,

$$
\dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \hookrightarrow (\dot{H}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}. \quad (3.19)
$$

Let us prove the reverse embedding,

$$
\dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) \leftarrow (\dot{H}^{s_0,p}(\mathbb{R}_{+}^{n}), \dot{H}^{s_1,p}(\mathbb{R}_{+}^{n}))_{\theta,q}.
$$

Without loss of generality, we can assume $s_1 \ge \frac{n}{p}$. Let $f \in S_0(\overline{\mathbb{R}}_+^n) \subset \dot{B}_{p,q}^s(\mathbb{R}_+^n)$, it follows that
$f \in (\dot{H}^{s_0,p}(\mathbb{R}_+^n), \dot{H}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \subset \dot{H}^{s_0,p}(\mathbb{R}_+^n) + \dot{H}^{s_1,p}(\mathbb{R}_+^n)$. Thus, for all $(a,b) \in \dot{H}^{s_0,p}(\mathbb{R}_+^n) \times \dot{H}^{s_1,p}(\mathbb{R}_+^n)$
Thanks to **Step 1** one may find, $-1 + \frac{1}{p} < s_0 < s < s_1 < \frac{1}{p}$, $\theta \in (0,1)$, such that, as a consequence of [BL76, Theorem 3.4.2], we have the following dense embedding,

$$ \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow (\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} = \dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n) = \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n). $$

The equality in above line is a direct consequence of Proposition 2.9. In this case, the density of $C_c^\infty(\mathbb{R}_+^n)$ is a straightforward application of Corollary 3.12 by successive approximations.

**Step 2.3:** $(\mathfrak{h}, \mathfrak{b}) = (\mathbb{H}_0, \mathbb{B}_{\cdot, \cdot, 0})$.

Thanks to Lemma 3.13, we have continuous embedding,

$$ (\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \hookrightarrow \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n). $$

We are going to prove the reverse embedding,

$$ (\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \leftarrow \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n). $$

Again, without loss of generality we can assume $s_1 \ge \frac{n}{p}$, otherwise one can go back to **Step 1**. Let us consider $u \in C_c^\infty(\mathbb{R}_+^n)$, then $u$ belongs to $\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) + \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$. In particular for $(a, b) \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) \times \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$, such that $u = a + b$ we have

$$ b = u - a \in (C_c^\infty(\mathbb{R}_+^n) + \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n). $$

in particular we have $a \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)$ and $b \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$. Consequently, we have $u = \mathcal{P}_0u = \mathcal{P}_0a + \mathcal{P}_0b$, with $\mathcal{P}_0a \in \dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n)$ and $\mathcal{P}_0b \in \dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n)$, with the estimates

$$ \|\mathcal{P}_0a\|_{\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n)} \lesssim_{s_0,m,p,n} \|a\|_{\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n)} \text{ and } \|\mathcal{P}_0b\|_{\dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n)} \lesssim_{s_1,m,p,n} \|b\|_{\dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n)}, $$

thanks to Corollary 3.7. Thus, one may follow the lines of **Step 2.1**, to obtain for all $u \in C_c^\infty(\mathbb{R}_+^n)$,

$$ \|u\|_{\dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n)} \sim_{s,p,n} \|u\|_{(\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}}. $$

Again, one can conclude via density arguments since $q < +\infty$, and $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n)$ thanks to **Step 2.2** and Lemma 3.16. ■

The **Step 2.2** in above proof can be turned more formally into,

**Corollary 3.18** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$. Then the following equality holds with equivalence of norms,

$$ \dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n) = \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n). $$

Moreover, the space $C_c^\infty(\mathbb{R}_+^n)$ is dense whenever $q < +\infty$.

From general interpolation theory we are able to deduce the following,

**Corollary 3.19** Let $p \in (1, +\infty)$, $s > -1 + 1/p$, such that $(C_{s,p,\infty})$ is satisfied.

* The space $C_c^\infty(\mathbb{R}_+^n)$ is weak* dense in $\dot{\mathbf{B}}_{p,\infty,0}^s(\mathbb{R}_+^n)$.

* The space $S_0(\mathbb{R}_+^n)$ is weak* dense in $\dot{\mathbf{B}}_{p,\infty}^s(\mathbb{R}_+^n)$.

*Proof.* — The [BL76, Theorem 3.7.1] with the remark at the end of its proof in combination with Lemma 3.12, with the use of [BL76, Theorem 3.4.2], and Proposition 3.17 imply that, for some $-1 + 1/p < s_0 < s < s_1$, with $\theta \in (0,1)$, such that $s = (1 - \theta)s_0 + \theta s_1$, we have the following strongly dense embedding,

$$ C_c^\infty(\mathbb{R}_+^n) \hookrightarrow \dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow (\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta}, $$

and the following weak* dense embedding

$$ (\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta} \hookrightarrow (\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta, \infty} = (\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta, \infty} = \dot{\mathbf{B}}_{p,\infty,0}^s(\mathbb{R}_+^n), $$

so that the result follows. We mention that $(\cdot, \cdot)_\theta$ is the real interpolation functor asking the K-functional to decay at infinity and near the origin, see for instance [Lun18, Definition 1.2].

The same argument apply for the weak* density of $S_0(\mathbb{R}_+^n)$ in $\dot{\mathbf{B}}_{p,\infty}^s(\mathbb{R}_+^n)$. ■
such that $f = a + b$, we have,

$$b = f - a \in (S_0(\overline{\mathbb{R}}_+^n) + \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n).$$

In particular, we have $a \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)$ and $b \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)$. Hence, we can introduce $F := Ea + Eb$, where $F|_{\mathbb{R}_+^n} = f$, $Ea \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)$ and $Eb \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$, with the estimates, given by Corollary 3.4,

$$\|Ea\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)} \lesssim_{s_0,m,p,n} \|a\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)} \text{ and } \|Eb\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)} \lesssim_{s_1,m,p,n} \|b\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)}.$$

Then, one may bound the $K$-functional of $F$, for $t > 0$,

$$K(t, F, \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)) \le \|Ea\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)} + t \|Eb\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)} \lesssim_{s_j,p,n} \|a\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n)} + t \|b\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)}$$

Taking the infimum over all such functions $a$ and $b$, we obtain

$$K(t, F, \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)) \lesssim_{s_j,p,n} K(t, f, \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n)),$$

from which we obtain, after multiplying by $t^{-\theta}$, taking the $L_*^q$-norm with respect to $t$, and applying Theorem 2.6,

$$\|f\|_{\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)} \le \|F\|_{\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}^n)} \lesssim_{s,p,n} \|f\|_{(\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}}.$$

Finally, thanks to the first embedding (3.19), we have

$$\|f\|_{\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)} \sim_{p,s,n} \|f\|_{(\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}}, \forall f \in S_0(\overline{\mathbb{R}}_+^n).$$

Since $q < +\infty$, we can conclude by density of $S_0(\overline{\mathbb{R}}_+^n)$ in both $\dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n)$ and in the interpolation space $(\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}$. The density argument for the later one is carried over by Lemma 2.5 and [BL76, Theorem 3.4.2].

**Step 2.2:** $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{\mathbf{B}}_{p,q,0}^s$, provided $-1 + \frac{1}{p} < s < \frac{1}{p}$, $p \in (1, +\infty)$, $q \in [1, +\infty)$.

Thanks to **Step 1** one may find, $-1 + \frac{1}{p} < s_0 < s < s_1 < \frac{1}{p}$, $\theta \in (0, 1)$, such that, as a consequence of [BL76, Theorem 3.4.2], we have the following dense embedding,

$$\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow (\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} = \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n) = \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n).$$

The equality in the above line is a direct consequence of Proposition 2.9. In this case, the density of $C_c^\infty(\mathbb{R}_+^n)$ is a straightforward application of Corollary 3.12 by successive approximations.

**Step 2.3:** $(\mathfrak{h}, \mathfrak{b}) = (H_0, B_{.,.}^0)$.

Thanks to Lemma 3.13, we have continuous embedding,

$$(\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \hookrightarrow \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n).$$

We are going to prove the reverse embedding,

$$(\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q} \leftarrow \dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n).$$

Again, without loss of generality we can assume $s_1 \ge \frac{n}{p}$, otherwise one can go back to **Step 1**. Let us consider $u \in C_c^\infty(\mathbb{R}_+^n)$, then, $u$ belongs to $\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) + \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$. In particular for $(a, b) \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) \times \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$, such that $u = a + b$ we have

$$b = u - a \in (C_c^\infty(\mathbb{R}_+^n) + \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n).$$

in particular we have $a \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)$ and $b \in \dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n) \cap \dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)$. Consequently, we have $u = P_0u = P_0a + P_0b$, with $P_0a \in \dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n)$ and $P_0b \in \dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n)$, with the estimates

$$\|P_0a\|_{\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n)} \lesssim_{s_0,m,p,n} \|a\|_{\dot{\mathbf{H}}^{s_0,p}(\mathbb{R}^n)} \text{ and } \|P_0b\|_{\dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n)} \lesssim_{s_1,m,p,n} \|b\|_{\dot{\mathbf{H}}^{s_1,p}(\mathbb{R}^n)}.$$

thanks to Corollary 3.7. Thus, one may follow the lines of **Step 2.1**, to obtain for all $u \in C_c^\infty(\mathbb{R}_+^n)$,

$$\|u\|_{\dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n)} \sim_{s,p,n} \|u\|_{(\dot{\mathbf{H}}_0^{s_0,p}(\mathbb{R}_+^n), \dot{\mathbf{H}}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,q}}.$$

Again, one can conclude via density arguments since $q < +\infty$, and $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{\mathbf{B}}_{p,q,0}^s(\mathbb{R}_+^n)$ thanks to **Step 2.2** and Lemma 3.16. ■

The **Step 2.2** in the proof above can be turned more formally into the following corollary.
We state below the Besov analogue of Corollary 3.7, Lemma 2.5 and Proposition 3.3, for which the proofs are similar and left to the reader.

**Proposition 3.20** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty]$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, $m \in \mathbb{N}$, such that $(C_{s_0, p_0, q_0})$ is satisfied and $s_j < m + 1 + \frac{1}{p_j}$, and consider the extension operator E given by Proposition 3.1.

Then for all $u \in \dot{B}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)$, we have $Eu \in \dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}^n)$, $j \in \{0, 1\}$, with the estimate

$$ \|Eu\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}^n)} \lesssim_{s_j, m, p, n} \|u\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n)}. $$

The same result holds replacing $(E, \dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n), \dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}^n))$ by $(\mathcal{P}_0, \dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}^n), \dot{B}_{p_j, q_j, 0}^{s_j}(\mathbb{R}_+^n))$, where $\mathcal{P}_0$ is the projection operator given in Lemma 3.6.

Thus, the following equality of vector spaces holds with equivalence of norms

$$ \dot{B}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n) = [\dot{B}_{p_0, q_0}^{s_0} \cap \dot{B}_{p_1, q_1}^{s_1}](\mathbb{R}_+^n). $$

In particular, $\dot{B}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)$ is a Banach space, and it admits $\mathcal{S}_0(\overline{\mathbb{R}}_+^n)$ as a dense subspace whenever $q_j < +\infty$, $j \in \{0, 1\}$.

Similarly, the following equality with equivalence of norms holds for all $s > 0$, $q \in [1, +\infty]$,

$$ L^p(\mathbb{R}_+^n) \cap \dot{B}_{p, q}^s(\mathbb{R}_+^n) = B_{p, q}^s(\mathbb{R}_+^n). $$

With direct consequence similar to Corollary 3.5:

**Corollary 3.21** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty]$ $m_j \in [1, +\infty]$, $s_j > m_j - 1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0, q_0})$ is satisfied. For all $u \in [\dot{B}_{p_0, q_0}^{s_0} \cap \dot{B}_{p_1, q_1}^{s_1}](\mathbb{R}_+^n)$,

$$ \|\nabla^{m_j} u\|_{\dot{B}_{p_j, q_j}^{s_j-m_j}(\mathbb{R}_+^n)} \sim_{s_j, m_j, p_j, n} \|u\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n)}. $$

Above Proposition 3.20 also implies the expected interpolation result for Besov spaces, for which the proof is similar to the one of Proposition 3.17 and left again to the reader.

**Proposition 3.22** Let $(p_0, p_1, p, q, q_0, q_1) \in (1, +\infty)^3 \times [1, +\infty]^3$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 < s_1$, and let $b \in \{B, B_{\cdot, \cdot, 0}\}$, and set

$$ \left(s, \frac{1}{p_\theta}, \frac{1}{q_\theta}\right) := (1-\theta) \left(s_0, \frac{1}{p_0}, \frac{1}{q_0}\right) + \theta \left(s_1, \frac{1}{p_1}, \frac{1}{q_1}\right). $$

such that the following assertion is satisfied,

* $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, and $(C_{s_0, p_0, q_0})$ is true;

Then if $p_0 = p_1 = p$, and $(C_{s, p, q})$ is satisfied, the following equality holds with equivalence of norms

$$ (\dot{b}_{p, q_0}^{s_0}(\mathbb{R}_+^n), \dot{b}_{p, q_1}^{s_1}(\mathbb{R}_+^n))_{\theta, q} = \dot{b}_{p, q}^s(\mathbb{R}_+^n). \quad (3.20) $$

If $(C_{s_0, p_0, q_0})$ and $(C_{s_1, p_1, q_1})$ are true then also is $(C_{s, p_\theta, q_\theta})$ and with equivalence of norms,

$$ [\dot{b}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), \dot{b}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)]_{\theta} = \dot{b}_{p_\theta, q_\theta}^s(\mathbb{R}_+^n). \quad (3.21) $$

We finish stating a duality result for homogeneous Besov space on the half-space.

**Proposition 3.23** Let $p \in (1, +\infty)$, $q \in (1, +\infty]$, $s > -1 + \frac{1}{p}$, if $(C_{s, p, q})$ is satisfied then the following isomorphisms hold

$$ (\dot{B}_{p', q', 0}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p, q}^s(\mathbb{R}_+^n) \quad \text{and} \quad (\dot{B}_{p', q'}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p, q, 0}^s(\mathbb{R}_+^n). $$

*Proof.* — We only prove $(\dot{B}_{p', q'}^{-s}(\mathbb{R}_+^n))' = \dot{B}_{p, q, 0}^s(\mathbb{R}_+^n)$, the other equality can be shown in a similar way. First let $q < +\infty$, and choose $u \in \dot{B}_{p, q, 0}^s(\mathbb{R}_+^n)$, it follows that $u$ induce a linear form on $\dot{B}_{p', q'}^{-s}(\mathbb{R}_+^n)$,

$$ v \mapsto \langle u, \tilde{v} \rangle_{\mathbb{R}^n} $$
**Corollary 3.18** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$. Then the following equality holds with equivalence of norms,

$$ \dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) = \dot{B}_{p,q,0}^{s}(\mathbb{R}_{+}^{n}). $$

Moreover, the space $C_c^\infty(\mathbb{R}_+^n)$ is dense whenever $q < +\infty$.

From general interpolation theory, we are able to deduce the following,

**Corollary 3.19** Let $p \in (1, +\infty)$, $s > -1 + 1/p$, such that $(C_{s,p,\infty})$ is satisfied.

* The space $C_c^\infty(\mathbb{R}_+^n)$ is weak* dense in $\dot{B}_{p,\infty,0}^s(\mathbb{R}_+^n)$.

* The space $S_0(\overline{\mathbb{R}}_+^n)$ is weak* dense in $\dot{B}_{p,\infty}^s(\mathbb{R}_+^n)$.

*Proof.* — The [BL76, Theorem 3.7.1] with the remark at the end of its proof in combination with Lemma 3.12, with the use of [BL76, Theorem 3.4.2], and Proposition 3.17 imply that, for some $-1+1/p < s_0 < s < s_1$, with $\theta \in (0, 1)$, such that $s = (1-\theta)s_0 + \theta s_1$, we have the following strongly dense embedding,

$$ C_c^\infty(\mathbb{R}_+^n) \hookrightarrow \dot{H}_0^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}_0^{s_1,p}(\mathbb{R}_+^n) \hookrightarrow (\dot{H}_0^{s_0,p}(\mathbb{R}_+^n), \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))_\theta, $$

and the following weak* dense embedding

$$ (\dot{H}_0^{s_0,p}(\mathbb{R}_+^n), \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))_\theta \hookrightarrow (\dot{H}_0^{s_0,p}(\mathbb{R}_+^n), \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))''_\theta = (\dot{H}_0^{s_0,p}(\mathbb{R}_+^n), \dot{H}_0^{s_1,p}(\mathbb{R}_+^n))_{\theta,\infty} = \dot{B}_{p,\infty,0}^s(\mathbb{R}_+^n), $$

so that the result follows. We mention that $(\cdot, \cdot)_\theta$ is the real interpolation functor asking the K-functional to decay at infinity and near the origin, see for instance [Lun18, Definition 1.2].

The same argument applies for the weak* density of $S_0(\overline{\mathbb{R}}_+^n)$ in $\dot{B}_{p,\infty}^s(\mathbb{R}_+^n)$.

<figure><img src="image_8.png" /></figure>

We state below the Besov analogue of Corollary 3.7, Lemma 2.5 and Proposition 3.3, for which the proofs are similar and left to the reader.

**Proposition 3.20** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty]$, $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, $m \in \mathbb{N}$, such that $(C_{s_0,p_0,q_0})$ is satisfied and $s_j < m + 1 + \frac{1}{p_j}$, and consider the extension operator $E$ given by Proposition 3.1.

Then for all $u \in \dot{B}_{p_0,q_0}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p_1,q_1}^{s_1}(\mathbb{R}_+^n)$, we have $Eu \in \dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}^n)$, $j \in \{0, 1\}$, with the estimate

$$ \|Eu\|_{\dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}^n)} \lesssim_{s_j,m,p,n} \|u\|_{\dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}_+^n)}. $$

The same result holds replacing $(E, \dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}_+^n), \dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}^n))$ by $(\mathcal{P}_0, \dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}^n), \dot{B}_{p_j,q_j,0}^{s_j}(\mathbb{R}_+^n))$, where $\mathcal{P}_0$ is the projection operator given in Lemma 3.6.

Thus, the following equality of vector spaces holds with equivalence of norms

$$ \dot{B}_{p_0,q_0}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p_1,q_1}^{s_1}(\mathbb{R}_+^n) = [\dot{B}_{p_0,q_0}^{s_0} \cap \dot{B}_{p_1,q_1}^{s_1}](\mathbb{R}_+^n). $$

In particular, $\dot{B}_{p_0,q_0}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p_1,q_1}^{s_1}(\mathbb{R}_+^n)$ is a Banach space, and it admits $S_0(\overline{\mathbb{R}}_+^n)$ as a dense subspace whenever $q_j < +\infty$, $j \in \{0, 1\}$.

Similarly, the following equality with equivalence of norms holds for all $s > 0$, $q \in [1, +\infty]$,

$$ L^p(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^s(\mathbb{R}_+^n) = B_{p,q}^s(\mathbb{R}_+^n). $$

With direct consequence similar to Corollary 3.5:

**Corollary 3.21** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty]$, $m_j \in [1, +\infty]$, $s_j > m_j - 1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, such that $(C_{s_0,p_0,q_0})$ is satisfied. For all $u \in [\dot{B}_{p_0,q_0}^{s_0} \cap \dot{B}_{p_1,q_1}^{s_1}](\mathbb{R}_+^n)$,

$$ \|\nabla^{m_j} u\|_{\dot{B}_{p_j,q_j}^{s_j-m_j}(\mathbb{R}_+^n)} \sim s_j, m_j, p_j, n \|u\|_{\dot{B}_{p_j,q_j}^{s_j}(\mathbb{R}_+^n)}. $$

Above Proposition 3.20 also implies the expected interpolation result for Besov spaces, for which the proof is similar to the one of Proposition 3.17 and left again to the reader.
where $\tilde{v} \in \dot{B}_{p',q'}^{-s}(\mathbb{R}^n)$ is any extension of $v \in \dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n)$. If one choose $v'$ to be any other extension of $v$, we have that $\tilde{v} - v' \in \dot{B}_{p',q',0}^{-s}(\mathbb{R}_-^n)$. Since $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$, see either Lemma 3.16 or Corollary 3.18, for $(u_k)_{k \in \mathbb{N}} \subset C_c^\infty(\mathbb{R}_+^n)$ converging to $u$, we have

$$ \langle u, \tilde{v} - v' \rangle_{\mathbb{R}^n} = \lim_{k \to +\infty} \langle u_k, \tilde{v} - v' \rangle_{\mathbb{R}^n} = 0 $$

due to the fact that $\mathbb{R}_+^n \cap \mathbb{R}_-^n = \emptyset$. Thus, the map does not depend on the choice of the extension but is entirely and uniquely determined by $u$. We have the continuous canonical embedding

$$ \dot{B}_{p,q,0}^s(\mathbb{R}_+^n) \hookrightarrow (\dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n))' $$

In fact, the same result holds for $q = +\infty$: the space $C_c^\infty(\mathbb{R}_+^n)$ is sequentially weak* dense in $\dot{B}_{p,\infty,0}^s(\mathbb{R}_+^n)$ by Corollary 3.19.

For the reverse embedding, if $U \in (\dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n))'$, it induces a continuous linear functional on $\dot{B}_{p',q'}^{-s}(\mathbb{R}^n)$ by the mean of

$$ v \mapsto \langle U, 1_{\mathbb{R}_+^n} \tilde{v} \rangle, $$

where again $\tilde{v} \in \dot{B}_{p',q'}^{-s}(\mathbb{R}^n)$ is any extension of $v \in \dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n)$. Thus, $1_{\mathbb{R}_+^n} U \in (\dot{B}_{p',q'}^{-s}(\mathbb{R}^n))'$ and by Proposition 2.8 there exists a unique $u \in \dot{B}_{p,q}^s(\mathbb{R}^n)$ such that, for all $\tilde{v} \in \dot{B}_{p',q'}^{-s}(\mathbb{R}^n)$,

$$ \langle U, 1_{\mathbb{R}_+^n} \tilde{v} \rangle = \langle u, \tilde{v} \rangle_{\mathbb{R}^n}. $$

Finally, if we test with $\tilde{v} \in C_c^\infty(\mathbb{R}_-^n)$, it shows that $\text{supp } u \subset \overline{\mathbb{R}_+^n}$, then $u \in \dot{B}_{p,q,0}^s(\mathbb{R}_+^n)$ which close the proof. ■

# 4 On traces of functions

Dealing with function spaces on domains implies that one may need to investigate the meaning of traces on the boundary if those exist, i.e. to see in our setting if the trace operator

$$ \gamma_0 : u \mapsto u|_{\partial \mathbb{R}_+^n} $$

still has the expected behavior on $\dot{H}^{s,p}(\mathbb{R}_+^n)$ and $\dot{B}_{p,q}^s(\mathbb{R}_+^n)$. In fact, in the complete case, it behaves as in the case of inhomogeneous function spaces.

The idea here is to give some appropriate trace theorems for homogeneous Sobolev and Besov spaces. It seems there is no clear trace theorem for homogeneous function spaces in the literature, except maybe [Jaw78], but in this case the corresponding results were obtained in a different framework.

## 4.1 On inhomogeneous function spaces.

We discuss first about the usual well known trace theorem on $\mathbb{R}^n$ with trace on $\mathbb{R}^{n-1} \times \{0\}$ in the inhomogeneous case, the result is a rewritten weaker version adapted to our context.

**Theorem 4.1 ( [BL76, Theorem 6.6.1] )** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (\frac{1}{p}, +\infty)$, and consider the following operator

$$ \gamma_0 : \left\{ \begin{array}{rcl} \mathcal{S}(\mathbb{R}^n) & \to & \mathcal{S}(\mathbb{R}^{n-1}) \\ u & \mapsto & u(\cdot, 0) \end{array} \right. , $$

then following statements are true:

(i) the trace operator $\gamma_0 : \dot{H}^{s,p}(\mathbb{R}^n) \to \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$ is a bounded surjection, in particular for all $u \in \dot{H}^{s,p}(\mathbb{R}^n)$,

$$ \|\gamma_0 u\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}^n)}; $$
**Proposition 3.22** Let $(p_0, p_1, p, q, q_0, q_1) \in (1, +\infty)^3 \times [1, +\infty]^3$, $s_0, s_1 \in \mathbb{R}$, such that $s_0 < s_1$, and
let $b \in \{B, B_{\cdot, \cdot, 0}\}$, and set

$$ \left( s, \frac{1}{p_\theta}, \frac{1}{q_\theta} \right) := (1 - \theta) \left( s_0, \frac{1}{p_0}, \frac{1}{q_0} \right) + \theta \left( s_1, \frac{1}{p_1}, \frac{1}{q_1} \right). $$

such that the following assertion is satisfied,

• $s_j > -1 + \frac{1}{p_j}$, $j \in \{0, 1\}$, and $(C_{s_0, p_0, q_0})$ is true;

Then if $p_0 = p_1 = p$, and $(C_{s,p,q})$ is satisfied, the following equality holds with equivalence of norms

$$ (\dot{\mathbf{b}}_{p,q_0}^{s_0}(\mathbb{R}_+^n), \dot{\mathbf{b}}_{p,q_1}^{s_1}(\mathbb{R}_+^n))_{\theta,q} = \dot{\mathbf{b}}_{p,q}^{s}(\mathbb{R}_+^n). \qquad (3.20) $$

If $(C_{s_0, p_0, q_0})$ and $(C_{s_1, p_1, q_1})$ are true then also is $(C_{s, p_0, q_0})$ and with equivalence of norms,

$$ [\dot{\mathbf{b}}_{p_0, q_0}^{s_0}(\mathbb{R}_+^n), \dot{\mathbf{b}}_{p_1, q_1}^{s_1}(\mathbb{R}_+^n)]_{\theta} = \dot{\mathbf{b}}_{p_\theta, q_\theta}^{s}(\mathbb{R}_+^n), \qquad (3.21) $$

whenever $q_\theta < +\infty$.

We finish stating a duality result for homogeneous Besov space on the half-space.

**Proposition 3.23** Let $p \in (1, +\infty)$, $q \in (1, +\infty]$, $s > -1 + \frac{1}{p}$, if $(C_{s,p,q})$ is satisfied then the following isomorphisms hold

$$ (\dot{\mathbf{B}}_{p', q', 0}^{-s}(\mathbb{R}_+^n))' = \dot{\mathbf{B}}_{p,q}^{s}(\mathbb{R}_+^n) \quad \text{and} \quad (\dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}_+^n))' = \dot{\mathbf{B}}_{p,q,0}^{s}(\mathbb{R}_+^n). $$

*Proof.* — We only prove $(\dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}_+^n))' = \dot{\mathbf{B}}_{p,q,0}^{s}(\mathbb{R}_+^n)$, the other equality can be shown similarly. First let $q < +\infty$, and choose $u \in \dot{\mathbf{B}}_{p,q,0}^{s}(\mathbb{R}_+^n)$, it follows that $u$ induce a linear form on $\dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}_+^n)$,

$$ v \mapsto \langle u, \tilde{v} \rangle_{\mathbb{R}^n} $$

where $\tilde{v} \in \dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}^n)$ is any extension of $v \in \dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}_+^n)$. If one choose $v'$ to be any other extension of $v$, we have that $\tilde{v} - v' \in \dot{\mathbf{B}}_{p', q', 0}^{-s}(\mathbb{R}_-)$. Since the space $C_c^\infty(\mathbb{R}_+^n)$ is dense in $\dot{\mathbf{B}}_{p,q,0}^{s}(\mathbb{R}_+^n)$, see either Lemma 3.16 or Corollary 3.18, for $(u_k)_{k \in \mathbb{N}} \subset C_c^\infty(\mathbb{R}_+^n)$ converging to $u$, we have

$$ \langle u, \tilde{v} - v' \rangle_{\mathbb{R}^n} = \lim_{k \to +\infty} \langle u_k, \tilde{v} - v' \rangle_{\mathbb{R}^n} = 0 $$

due to the fact that $\mathbb{R}_+^n \cap \mathbb{R}_-^n = \emptyset$. Thus, the map does not depend on the choice of the extension, but is entirely and uniquely determined by $u$. We have the continuous canonical embedding

$$ \dot{\mathbf{B}}_{p,q,0}^{s}(\mathbb{R}_+^n) \hookrightarrow (\dot{\mathbf{B}}_{p',q'}^{-s}(\mathbb{R}_+^n))'. $$

In fact, the same result holds for $q = +\infty$: the space $C_c^\infty(\mathbb{R}_+^n)$ is sequentially weak* dense in $\dot{\mathbf{B}}_{p, \infty, 0}^{s}(\mathbb{R}_+^n)$ by Corollary 3.19.

For the reverse embedding, if $U \in (\dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}_+^n))'$, it induces a continuous linear functional on $\dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}^n)$ by the mean of

$$ v \mapsto \langle U, 1_{\mathbb{R}_+^n} \tilde{v} \rangle, $$

where again $\tilde{v} \in \dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}^n)$ is any extension of $v \in \dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}_+^n)$. Thus, $1_{\mathbb{R}_+^n} U \in (\dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}^n))'$ and by Proposition 2.8 there exists a unique $u \in \dot{\mathbf{B}}_{p,q}^{s}(\mathbb{R}^n)$ such that, for all $\tilde{v} \in \dot{\mathbf{B}}_{p', q'}^{-s}(\mathbb{R}^n)$,

$$ \langle U, 1_{\mathbb{R}_+^n} \tilde{v} \rangle = \langle u, \tilde{v} \rangle_{\mathbb{R}^n}. $$

Finally, if we test with $\tilde{v} \in C_c^\infty(\mathbb{R}_-^n)$, it shows that $\text{supp } u \subset \overline{\mathbb{R}_+^n}$, then $u \in \dot{\mathbf{B}}_{p,q,0}^{s}(\mathbb{R}_+^n)$ which closes the proof. ■

# 4 On traces of functions

Dealing with function spaces on domains implies that one may need to investigate the meaning of traces at the boundary if those exist, i.e., to see in our setting if the trace operator

$$ \gamma_0 : u \mapsto u|_{\partial \mathbb{R}_+^n} $$
(ii) the trace operator $\gamma_0 : B_{p,q}^s(\mathbb{R}^n) \to B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$ is a bounded surjection, in particular for all $u \in B_{p,q}^s(\mathbb{R}^n)$,

$$ \|\gamma_0 u\|_{B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s,p,n,q} \|u\|_{B_{p,q}^s(\mathbb{R}^n)}; $$

(iii) the trace operator $\gamma_0 : B_{p,1}^{\frac{1}{p}}(\mathbb{R}^n) \to L^p(\mathbb{R}^{n-1})$ is a bounded surjection, in particular for all $u \in B_{p,1}^{\frac{1}{p}}(\mathbb{R}^n)$,

$$ \|\gamma_0 u\|_{L^p(\mathbb{R}^{n-1})} \lesssim_{p,n} \|u\|_{B_{p,1}^{\frac{1}{p}}(\mathbb{R}^n)}; $$

Moreover the trace operator $\gamma_0$ admits a linear right bounded inverse $\text{Ext}$ in cases (i) and (ii).

**Remark 4.2** One also mention [Sch10, Theorems 2.2 & 2.10], [JW84, Sections V-VII], which give different proofs of the trace theorem. Notice that in [Sch10, Theorems 2.2 & 2.10] and [Saw18, Theorems 4.47, 4.48] the right bounded inverse they give is not linear but covers case (iii).

## 4.2 On homogeneous function spaces.

**Theorem 4.3** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (\frac{1}{p}, +\infty)$, then for $(\mathfrak{h}, \mathfrak{b}) \in \{(\mathbb{H}, \mathbb{B}), (\dot{\mathbb{H}}, \dot{\mathbb{B}})\}$, we consider the trace operator

$$ \gamma_0 : u \mapsto u(\cdot, 0). $$

The following assertions are true.

(i) For all $u \in H^{s,p}(\mathbb{R}_+^n)$, we have $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, \mathfrak{b}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))$, with estimate

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \mathfrak{b}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathfrak{b}_{p,p}^{s,p}(\mathbb{R}_+^n)}; $$

In particular, the trace operator extends uniquely to a bounded linear operator

$$ \gamma_0 : \dot{\mathbb{H}}^{s,p}(\mathbb{R}_+^n) \to \dot{\mathbb{B}}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) $$

whenever $(C_{s,p})$ is satisfied, and the following continuous embedding holds

$$ \dot{\mathbb{H}}^{s,p}(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{\mathbb{B}}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

(ii) For all $u \in B_{p,q}^s(\mathbb{R}_+^n)$, we have $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, \mathfrak{b}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))$, with estimate

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \mathfrak{b}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathfrak{b}_{p,q}^s(\mathbb{R}_+^n)}; $$

In particular, the trace operator extends uniquely to a bounded linear operator

$$ \gamma_0 : \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n) \to \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) $$

whenever $(C_{s,p,q})$ is satisfied, and the following continuous embedding holds

$$ \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

If $q = +\infty$, the result still holds with uniform boundedness and weak* continuity only.

(iii) For all $u \in B_{p,1}^{1/p}(\mathbb{R}_+^n)$, we have $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, L^p(\mathbb{R}^{n-1}))$, with estimate

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, L^p(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathfrak{b}_{p,1}^{1/p}(\mathbb{R}_+^n)}; $$

In particular, the trace operator extends uniquely to a bounded linear operator

$$ \gamma_0 : \dot{\mathbb{B}}_{p,1}^{1/p}(\mathbb{R}_+^n) \to L^p(\mathbb{R}^{n-1}) $$
still has the expected behavior on $\dot{H}^{s,p}(\mathbb{R}_+^n)$ and $\dot{B}_{p,q}^s(\mathbb{R}_+^n)$. In fact, in the complete case, it behaves as in the case of inhomogeneous function spaces.

The idea here is to give some appropriate trace theorems for homogeneous Sobolev and Besov spaces. It seems there is no clear trace theorem for homogeneous function spaces in the literature, except maybe [Jaw78], but in this case the corresponding results were obtained in a different framework.

## 4.1 On inhomogeneous function spaces.

We discuss first about the usual well known trace theorem on $\mathbb{R}^n$ with trace on $\mathbb{R}^{n-1} \times \{0\}$ in the inhomogeneous case, the result is a rewritten weaker version adapted to our context.

**Theorem 4.1** ([BL76, Theorem 6.6.1]) Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (\frac{1}{p}, +\infty)$, and consider the following operator

$$ \gamma_0 : \left\{ \begin{array}{ll} S(\mathbb{R}^n) & \to S(\mathbb{R}^{n-1}) \\ u & \mapsto u(\cdot, 0) \end{array} \right. , $$

then following statements are true:

(i) the trace operator extends uniquely as a bounded surjection $\gamma_0 : H^{s,p}(\mathbb{R}^n) \to B_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, in particular for all $u \in H^{s,p}(\mathbb{R}^n)$,

$$ \|\gamma_0 u\|_{B_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s,p,n} \|u\|_{H^{s,p}(\mathbb{R}^n)}; $$

(ii) the trace operator extends uniquely as a bounded surjection $\gamma_0 : B_{p,q}^s(\mathbb{R}^n) \to B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, in particular for all $u \in B_{p,q}^s(\mathbb{R}^n)$,

$$ \|\gamma_0 u\|_{B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s,p,n,q} \|u\|_{B_{p,q}^s(\mathbb{R}^n)}; $$

(iii) the trace operator extends uniquely as a bounded surjection $\gamma_0 : B_{p,1}^{\frac{1}{p}}(\mathbb{R}^n) \to L^p(\mathbb{R}^{n-1})$, in particular for all $u \in B_{p,1}^{\frac{1}{p}}(\mathbb{R}^n)$,

$$ \|\gamma_0 u\|_{L^p(\mathbb{R}^{n-1})} \lesssim_{p,n} \|u\|_{B_{p,1}^{\frac{1}{p}}(\mathbb{R}^n)}; $$

Moreover the trace operator $\gamma_0$ admits a linear right bounded inverse Ext in cases (i) and (ii).

**Remark 4.2** • About the boundedness for the trace operator $\gamma_0 : B_{p,\infty}^s(\mathbb{R}^n) \to B_{p,\infty}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$. The construction does not follow directly by density argument since $S(\mathbb{R}^n)$ is not strongly but only weakly* dense in $B_{p,\infty}^s(\mathbb{R}^n)$. However, $S(\mathbb{R}^n)$ is a dense subspace of $B_{p,1}^{s_0}(\mathbb{R}^n) \cap B_{p,1}^{s_1}(\mathbb{R}^n)$, so that the trace is consistent on both spaces $B_{p,1}^{s_0}(\mathbb{R}^n)$ and $B_{p,1}^{s_1}(\mathbb{R}^n)$, and we can define the trace operator

$$ \gamma_0 : B_{p,1}^{s_0}(\mathbb{R}^n) + B_{p,1}^{s_1}(\mathbb{R}^n) \to B_{p,1}^{s_0-\frac{1}{p}}(\mathbb{R}^{n-1}) + B_{p,1}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1}). $$

But by real interpolation, if $1/p < s_0 < s < s_1$ we have

$$ B_{p,\infty}^s(\mathbb{R}^n) = (B_{p,1}^{s_0}(\mathbb{R}^n), B_{p,1}^{s_1}(\mathbb{R}^n))_{\frac{s-s_0}{s_1-s_0}, \infty} \hookrightarrow B_{p,1}^{s_0}(\mathbb{R}^n) + B_{p,1}^{s_1}(\mathbb{R}^n). $$

Thus, we have the well-defined (and bounded) trace operator

$$ \gamma_0 : B_{p,\infty}^s(\mathbb{R}^n) \to B_{p,1}^{s_0-\frac{1}{p}}(\mathbb{R}^{n-1}) + B_{p,1}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1}). $$

It remains then to prove the boundedness $\gamma_0 : B_{p,\infty}^s(\mathbb{R}^n) \to B_{p,\infty}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, which is given by real interpolation.

A simpler argument is available (arguing that $B_{p,\infty}^s(\mathbb{R}^n) \hookrightarrow B_{p,1}^{s-\varepsilon}(\mathbb{R}^n)$) but the one presented here will also apply to the trace operator in the case of homogeneous Besov spaces, for the Theorem 4.3.
and the following continuous embedding holds

$$ \dot{B}_{p,1}^{1/p}(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, L^p(\mathbb{R}^{n-1})). $$

Moreover,

(a) If $(\mathfrak{h}, \mathfrak{b}) = (\mathrm{H}, \mathrm{B})$, the trace operator $\gamma_0$ admits a linear right bounded inverse $\mathrm{Ext}_{\mathbb{R}_+^n}$ in cases (i) and (ii).

(b) If $(\mathfrak{h}, \mathfrak{b}) = (\dot{\mathrm{H}}, \dot{\mathrm{B}})$, the trace operator $\gamma_0$ admits a linear right bounded inverse $\underline{\mathrm{Ext}}_{\mathbb{R}_+^n}$ in cases (i) and (ii).

**Proof.** — We cut the proof in several steps.

**Step 1:** The case $(\mathfrak{h}, \mathfrak{b}) = (\mathrm{H}, \mathrm{B})$.

We first check validity of the embedding

$$ \mathrm{H}^{s,p}(\mathbb{R}^n) \hookrightarrow C_{0,x_n}^0(\mathbb{R}, \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

Let $u \in \mathrm{H}^{s,p}(\mathbb{R}^n)$, for $t > 0$, for almost every $x = (x', x_n) \in \mathbb{R}^n$, we introduce $u_t(x', x_n) := u(x', x_n + t)$, we have $u_t \in \mathrm{H}^{s,p}(\mathbb{R}^n)$, and by Theorem 4.1,

$$ \|\gamma_0 u_t\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{\mathrm{H}^{s,p}(\mathbb{R}^n)}, $$

$$ \|\gamma_0(u_t - u)\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u_t - u\|_{\mathrm{H}^{s,p}(\mathbb{R}^n)}. $$

Therefore, by strong continuity of translation in Lebesgue spaces, then in Sobolev spaces, we obtain

$$ \|\gamma_0(u_t - u)\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u_t - u\|_{\mathrm{H}^{s,p}(\mathbb{R}^n)} \xrightarrow{t \to 0} 0. $$

Hence, $u \in C_{b,x_n}^0(\mathbb{R}, \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))$, with estimate,

$$ \|t \mapsto \|u(\cdot, t)\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}\|_{L^{\infty}(\mathbb{R})} \lesssim_{p,s,n} \|u\|_{\mathrm{H}^{s,p}(\mathbb{R}^n)}. $$

Finally, one can approximate $u$ by Schwartz functions to deduce

$$ u \in C_{0,x_n}^0(\mathbb{R}, \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

One may perform a similar proof for all other cases, and one may check [Gui91, Proposition 1.9] for the continuity of translation in Besov spaces, one may also use a density and an interpolation argument.

Now, one can use the definition of function spaces by restriction.

One choose $\mathrm{Ext}_{\mathbb{R}_+^n} = [\mathrm{Ext} \cdot]_{\mathbb{R}_+^n}$ introduced in the proof of Theorem 4.1 which satisfies the desired boundedness properties.

**Step 2.1:** The case $(\mathfrak{h}, \mathfrak{b}) = (\dot{\mathrm{H}}, \dot{\mathrm{B}})$. Boundedness of the trace operator.

We only achieve the case (ii) other ones can be done similarly. From **Step 1**, and for fixed $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s > \frac{1}{p}$, and $u \in \dot{B}_{p,q}^s(\mathbb{R}_+^n)$, we have

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{p,s,n,q} \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

Thus, one may use the fact that $\dot{B}_{p,q}^s(\mathbb{R}_+^n) = L^p(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}_+^n)$, which comes from Proposition 3.20, to obtain

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n,q} \|u\|_{L^p(\mathbb{R}_+^n)} + \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

So that, by a dilation argument, replacing $u$, by $u_\lambda := u(\lambda \cdot)$, for $\lambda \in 2^\mathbb{N}$,

$$ \lambda^{s-\frac{1}{p}} \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n,q} \lambda^{-\frac{1}{p}} \|u\|_{L^p(\mathbb{R}_+^n)} + \lambda^{s-\frac{1}{p}} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$
• One also mentions [Sch10, Theorems 2.2 & 2.10], [JW84, Sections V-VII], which give different proofs of the trace theorem (in a more general geometric setting). Notice that in [Sch10, Theorems 2.2 & 2.10] and [Saw18, Theorems 4.47, 4.48] the right bounded inverse they give is not linear but covers case (iii).

## 4.2 On homogeneous function spaces.

**Theorem 4.3** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (\frac{1}{p}, +\infty)$, then for $(\mathfrak{h}, \mathfrak{b}) \in \{(\mathbb{H}, \mathbb{B}), (\dot{\mathbb{H}}, \dot{\mathbb{B}})\}$, we consider the trace operator

$$ \gamma_0 : u \mapsto u(\cdot, 0). $$

The following assertions are true.

(i) For all $u \in H^{s,p}(\mathbb{R}_+^n)$, we have $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, \mathfrak{b}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))$, with the estimate

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \mathfrak{b}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathfrak{b}^{s,p}(\mathbb{R}_+^n)}; $$

In particular, the trace operator extends uniquely to a bounded linear operator

$$ \gamma_0 : \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) $$

whenever $(C_{s,p})$ is satisfied, and the following continuous embedding holds

$$ \dot{H}^{s,p}(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

(ii) For all $u \in B_{p,q}^s(\mathbb{R}_+^n)$, we have $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, \mathfrak{b}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))$, with the estimate

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \mathfrak{b}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathfrak{b}_{p,q}^s(\mathbb{R}_+^n)}; $$

In particular, the trace operator extends uniquely to a bounded linear operator

$$ \gamma_0 : \dot{B}_{p,q}^s(\mathbb{R}_+^n) \to \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) $$

whenever $(C_{s,p,q})$ is satisfied, and the following continuous embedding holds

$$ \dot{B}_{p,q}^s(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

If $q = +\infty$, the result still holds with uniform boundedness and weak* continuity only.

(iii) For all $u \in B_{p,1}^{1/p}(\mathbb{R}_+^n)$, we have $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, L^p(\mathbb{R}^{n-1}))$, with the estimate

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, L^p(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathfrak{b}_{p,1}^{1/p}(\mathbb{R}_+^n)}; $$

In particular, the trace operator extends uniquely to a bounded linear operator

$$ \gamma_0 : \dot{B}_{p,1}^{1/p}(\mathbb{R}_+^n) \to L^p(\mathbb{R}^{n-1}) $$

and the following continuous embedding holds

$$ \dot{B}_{p,1}^{1/p}(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, L^p(\mathbb{R}^{n-1})). $$

Moreover,

(a) If $(\mathfrak{h}, \mathfrak{b}) = (\mathbb{H}, \mathbb{B})$, the trace operator $\gamma_0$ admits a linear right bounded inverse $\mathrm{Ext}_{\mathbb{R}_+^n}$ in cases (i) and (ii).

(b) If $(\mathfrak{h}, \mathfrak{b}) = (\dot{\mathbb{H}}, \dot{\mathbb{B}})$, the trace operator $\gamma_0$ admits a linear right bounded inverse $\mathrm{Ext}_{\mathbb{R}_+^n}$ in cases (i) and (ii).

*Proof.* — We cut the proof in several steps.
Hence, we can divide by $\lambda^{s-\frac{n}{p}}$ on both sides and pass to the limit $\lambda \to +\infty$,

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n,q} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

Therefore, if $q < +\infty$, and $(C_{s,p,q})$ is satisfied, the embedding

$$ \dot{B}_{p,q}^s(\mathbb{R}_+^n) \hookrightarrow C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})) $$

holds by density. If $q = +\infty$, and $(C_{s,p,q})$ is satisfied, the result follows from real interpolation.

**Step 2.2:** The case $(\mathfrak{h}, \mathfrak{b}) = (\dot{H}, \dot{B})$. Boundedness of the extension operator.

The operator $T$ given by Proposition B.2 is an appropriate extension operator which satisfies the desired boundedness properties. Thus $\underline{\mathrm{Ext}}_{\mathbb{R}_+^n} := T$ behaves as expected. ■

A raised question is about what happens when we want to deal with intersection of homogeneous Sobolev and Besov spaces.

**Proposition 4.4** Let $p \in (1, +\infty)$, $q \in [1, +\infty)$, $-1 + \frac{1}{p} < s_0 < \frac{1}{p} < s_1$, and $\theta \in (0, 1)$ such that

$$ \frac{1}{p} = (1 - \theta)s_0 + \theta s_1. $$

Then the following assertions hold.

(i) for all $u \in \dot{H}^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}^{s_1,p}(\mathbb{R}_+^n)$, we have $\gamma_0 u \in B_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})$, with estimate

$$ \|\gamma_0 u\|_{B_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s_0,s_1,p,n} \|u\|_{\dot{H}^{s_0,p}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}^{\theta} + \|u\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)} $$

We also have,

$$ \|\gamma_0 u\|_{\dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s_0,s_1,p,n} \|u\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}; $$

(ii) for all $u \in \dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)$, we have $\gamma_0 u \in B_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})$, with estimate

$$ \|\gamma_0 u\|_{B_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s_0,s_1,p,n} \|u\|_{\dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}^{\theta} + \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}. $$

We also have,

$$ \|\gamma_0 u\|_{\dot{B}_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s_0,s_1,p,n} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}; $$

(iii) for all $u \in \dot{B}_{p,\infty}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,\infty}^{s_1}(\mathbb{R}_+^n)$, we have $\gamma_0 u \in L^p(\mathbb{R}^{n-1})$, with estimate

$$ \|\gamma_0 u\|_{L^p(\mathbb{R}^{n-1})} \lesssim_{s_0,s_1,p,n} \|u\|_{\dot{B}_{p,\infty}^{s_0}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{B}_{p,\infty}^{s_1}(\mathbb{R}_+^n)}^{\theta}. $$

*Proof.* — We only start proving the point (ii), and claim that point (i) can be achieved in a similar manner. We start noticing, the following continuous embedding,

$$ \dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n) \stackrel{\iota}{\hookrightarrow} (\dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n), \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n))_{\theta,1} = \dot{B}_{p,1}^{\frac{1}{p}}(\mathbb{R}_+^n) \xrightarrow{\gamma_0} L^p(\mathbb{R}^{n-1}). $$

Here, $\iota$ is the canonical embedding obtained via standard interpolation theory, and the last embedding via the trace operator is a direct consequence of Theorem 4.3, and everything can be turned into the following inequality,

$$ \|\gamma_0 u\|_{L^p(\mathbb{R}^{n-1})} \lesssim_{s_0,s_1,p,n} \|u\|_{\dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}, \forall u \in \dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n). $$

Again, from Theorem 4.3 we obtain for all $u \in S_0(\overline{\mathbb{R}}_+^n)$,

$$ \|\gamma_0 u\|_{\dot{B}_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s_0,p,n} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}. $$

Then one may sum both inequality, notice that $L^p(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1}) = B_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})$ and use the density argument provided by Proposition 3.20 so that each estimate holds. ■
**Step 1:** The case $(\mathfrak{h}, \mathfrak{b}) = (\mathbb{H}, \mathbb{B})$.

We first check validity of the embedding

$$ \mathbb{H}^{s,p}(\mathbb{R}^n) \hookrightarrow \mathbb{C}_{0,x_n}^0(\mathbb{R}, \mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

Let $u \in \mathbb{H}^{s,p}(\mathbb{R}^n)$, for $t > 0$, for almost every $x = (x', x_n) \in \mathbb{R}^n$, we introduce $u_t(x', x_n) := u(x', x_n + t)$, we have $u_t \in \mathbb{H}^{s,p}(\mathbb{R}^n)$, and by Theorem 4.1,

$$ \|\gamma_0 u_t\|_{\mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{\mathbb{H}^{s,p}(\mathbb{R}^n)}, $$

$$ \|\gamma_0(u_t - u)\|_{\mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u_t - u\|_{\mathbb{H}^{s,p}(\mathbb{R}^n)}. $$

Therefore, by strong continuity of translation in Lebesgue spaces, then in Sobolev spaces, we obtain

$$ \|\gamma_0(u_t - u)\|_{\mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u_t - u\|_{\mathbb{H}^{s,p}(\mathbb{R}^n)} \xrightarrow{t \to 0} 0. $$

Hence, $u \in \mathbb{C}_{b,x_n}^0(\mathbb{R}, \mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))$, with the estimate,

$$ \|t \mapsto \|u(\cdot, t)\|_{\mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}\|_{L^{\infty}(\mathbb{R})} \lesssim_{p,s,n} \|u\|_{\mathbb{H}^{s,p}(\mathbb{R}^n)}. $$

Finally, one can approximate $u$ by Schwartz functions to deduce

$$ u \in \mathbb{C}_{0,x_n}^0(\mathbb{R}, \mathbb{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

One may perform a similar proof for all other cases, and one may check [Gui91, Proposition 1.9] for the continuity of translation in Besov spaces, one may also use a density and an interpolation argument. Concerning the case of Besov spaces with index $q = +\infty$, the argument is postponed to the end of the next **Step 2.1**.

Now, one can use the definition of function spaces by restriction.

One choose $\mathrm{Ext}_{\mathbb{R}_+^n} = [\mathrm{Ext} \cdot]_{\mathbb{R}_+^n}$ introduced in the proof of Theorem 4.1 which satisfies the desired boundedness properties.

**Step 2.1:** The case $(\mathfrak{h}, \mathfrak{b}) = (\dot{\mathbb{H}}, \dot{\mathbb{B}})$.

We only treat the case (ii) other ones can be done similarly. From **Step 1**, and for fixed $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s > \frac{1}{p}$, and $u \in \mathbb{B}_{p,q}^s(\mathbb{R}_+^n)$, we have

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{p,s,n} \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \mathbb{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|u\|_{\mathbb{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

Thus, one may use the fact that $\mathbb{B}_{p,q}^s(\mathbb{R}_+^n) = L^p(\mathbb{R}_+^n) \cap \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n)$, which comes from Proposition 3.20, to obtain

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n,q} \|u\|_{L^p(\mathbb{R}_+^n)} + \|u\|_{\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n)}. $$

So that, by a dilation argument, replacing $u$, by $u_\lambda := u(\lambda \cdot)$, for $\lambda \in 2^\mathbb{N}$,

$$ \lambda^{s-\frac{n}{p}} \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n,q} \lambda^{-\frac{n}{p}} \|u\|_{L^p(\mathbb{R}_+^n)} + \lambda^{s-\frac{n}{p}} \|u\|_{\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n)}. $$

Hence, we can divide by $\lambda^{s-\frac{n}{p}}$ on both sides and pass to the limit $\lambda \to +\infty$,

$$ \|u\|_{L_{x_n}^{\infty}(\mathbb{R}_+, \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n,q} \|u\|_{\dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n)}. $$

Therefore, if $q < +\infty$, and $(C_{s,p,q})$ is satisfied, the embedding

$$ \dot{\mathbb{B}}_{p,q}^s(\mathbb{R}_+^n) \hookrightarrow \mathbb{C}_{0,x_n}^0(\mathbb{R}_+, \dot{\mathbb{B}}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})). $$

holds by density. If $q = +\infty$, and $(C_{s,p,q})$ is satisfied, the result follows from real interpolation applied to the map $u \mapsto u(\cdot, x_n) = \gamma_0 u_{x_n}$ yielding for all $x_n \in [0, +\infty)$,

$$ \|u(\cdot, x_n)\|_{\dot{\mathbb{B}}_{p,\infty}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{s,p,n} \|u\|_{\dot{\mathbb{B}}_{p,\infty}^s(\mathbb{R}_+^n)} \quad (4.1) $$

for more details about the meaning of traces in this case $q = +\infty$, see the discussion in the first part
**Remark 4.5** As in Theorem 4.3, above Proposition 4.4 could be turned into a $C_{0,x_n}^0$-embedding in the appropriate Besov space.

**Proposition 4.6** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty)$, $s_j > 1/p_j$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ (resp. $(C_{s_0, p_0, q_0})$) is satisfied. Then,

(i) For all $u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n)$, we have $\gamma_0 u \in \dot{B}_{p_j, p_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})$, $j \in \{0, 1\}$, with estimate
$$ \|\gamma_0 u\|_{\dot{B}_{p_j, p_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})} \lesssim_{s_j, p_j, n} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}; $$

(ii) For all $u \in [\dot{B}_{p_0, q_0}^{s_0} \cap \dot{B}_{p_1, q_1}^{s_1}](\mathbb{R}_+^n)$, we have $\gamma_0 u \in \dot{B}_{p_j, q_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})$, $j \in \{0, 1\}$, with estimate
$$ \|\gamma_0 u\|_{\dot{B}_{p_j, q_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})} \lesssim_{s_j, p_j, n} \|u\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n)}; $$

**Remark 4.7** Corollary B.3 yields the ontoness of the trace operator on intersection spaces given by above Proposition 4.6.

**Lemma 4.8** Let $p_j \in (1, +\infty)$, $s \in (1/p_j, 1 + 1/p_j)$, $j \in \{0, 1\}$ such that $(C_{s_0, p_0})$ is satisfied. For all $u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C})$ such that $u|_{\partial \mathbb{R}_+^n} = 0$, the extension $\tilde{u}$ to $\mathbb{R}^n$ by 0, satisfies

$$ \tilde{u} \in [\dot{H}_0^{s_0, p_0} \cap \dot{H}_0^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C}) $$

with estimate

$$ \|\tilde{u}\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)} \lesssim_{p_j, s_j, n} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}, \quad j \in \{0, 1\}. $$

Hence, we have the following canonical isomorphism of Banach spaces

$$ \{ u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C}) \mid u|_{\partial \mathbb{R}_+^n} = 0 \} \simeq [\dot{H}_0^{s_0, p_0} \cap \dot{H}_0^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C}). $$

The result still holds replacing $\dot{H}^{s_j, p_j}$ by $\dot{B}_{p_j, q_j}$, $q_j \in [1, +\infty]$, $j \in \{0, 1\}$ assuming that $(C_{s_0, p_0, q_0})$ is satisfied.

**Proof.** — Let $u \in [\dot{H}_0^{s_0, p_0} \cap \dot{H}_0^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C})$ such that $u|_{\partial \mathbb{R}_+^n} = 0$, then for all $\phi \in [\dot{H}^{1-s_j, p'_j} \cap \mathcal{S}](\mathbb{R}^n, \mathbb{C}^n)$, we have

$$ \int_{\mathbb{R}_+^n} \nabla u \cdot \phi = - \int_{\mathbb{R}_+^n} \mathrm{div}(\phi) u. $$

So that introducing the extensions by 0 to $\mathbb{R}^n$, $\tilde{u}$ and $\tilde{\nabla}u$,

$$ \int_{\mathbb{R}^n} \tilde{\nabla}u \cdot \phi = \int_{\mathbb{R}_+^n} \nabla u \cdot \phi = - \int_{\mathbb{R}_+^n} \mathrm{div}(\phi) u = - \int_{\mathbb{R}^n} \mathrm{div}(\phi) \tilde{u} = \langle \nabla \tilde{u}, \phi \rangle_{\mathbb{R}^n}. $$

Therefore, for all $\phi \in [\dot{H}^{1-s_j, p'_j} \cap \mathcal{S}](\mathbb{R}^n, \mathbb{C}^n)$,

$$ \int_{\mathbb{R}^n} \tilde{\nabla}u \cdot \phi = \langle \nabla \tilde{u}, \phi \rangle_{\mathbb{R}^n}. $$

Hence $\tilde{\nabla}u = \nabla \tilde{u}$ in $S'(\mathbb{R}^n, \mathbb{C}^n)$. Thus, by Propositions 2.7 and 2.9, we deduce that

$$ \begin{aligned} |\langle \nabla \tilde{u}, \phi \rangle_{\mathbb{R}^n}| & \le \|\phi\|_{\dot{H}^{1-s_j, p'_j}(\mathbb{R}^n)} \|\tilde{\nabla}u\|_{\dot{H}^{s_j-1, p_j}(\mathbb{R}^n)} \\ & \lesssim_{p_j, n, s_j} \|\phi\|_{\dot{H}^{1-s_j, p'_j}(\mathbb{R}^n)} \|\nabla u\|_{\dot{H}^{s_j-1, p_j}(\mathbb{R}_+^n)} \\ & \lesssim_{p_j, n, s_j} \|\phi\|_{\dot{H}^{1-s_j, p'_j}(\mathbb{R}^n)} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}. \end{aligned} $$

One may conclude thanks to Proposition 2.7, and Corollary 3.5. The case of Besov spaces follows the same lines. The isomorphism is then a direct consequence. ■
of Remark 4.2. Notice that the bound is uniform with respect to $x_n$.

It remains to show weak* continuity with respect to $x_n$. This follows from Lemma C.1 in Appendix C. Let $\phi \in \dot{B}_{p',1}^{-s+1-1/p'}(\mathbb{R}^{n-1})$. One has by Lemma C.1

$$ \langle u(\cdot, x_n), \phi \rangle_{\mathbb{R}^{n-1}} = \langle \gamma_0 u_{x_n}, \phi \rangle_{\mathbb{R}^{n-1}} = \langle u_{x_n}, \gamma_0^* \phi \rangle_{\mathbb{R}^n} \underset{x_n \to 0}{\longrightarrow} \langle u, \gamma_0^* \phi \rangle_{\mathbb{R}^n} = \langle \gamma_0 u, \phi \rangle_{\mathbb{R}^{n-1}}, $$

where $\gamma_0^* = \mathrm{I} \otimes [\delta_0^*] : \dot{B}_{p',1}^{-s+1-1/p'}(\mathbb{R}^{n-1}) \to \dot{B}_{p',1,0}^{-s}(\mathbb{R}^n)$ is the (pre-)dual map of the trace operator.

This yields the weak* continuity with respect to $x_n$, from which we deduce the (weak*) measurability of $x_n \mapsto u(\cdot, x_n)$, and this allows to take the supremum in (4.1).

**Step 2.2:** The case $(\mathfrak{h}, \mathfrak{b}) = (\dot{H}, \dot{B})$. Boundedness of the extension operator.

The operator $T$ given by Proposition B.2 is an appropriate extension operator which satisfies the desired boundedness properties. Thus $\mathrm{Ext}_{\mathbb{R}^n} := T$ behaves as expected. ■

A raised question is about what happens when we want to deal with intersection of homogeneous Sobolev and Besov spaces.

**Proposition 4.4** Let $p \in (1, +\infty)$, $q \in [1, +\infty)$, $-1 + \frac{1}{p} < s_0 < \frac{1}{p} < s_1$, and $\theta \in (0, 1)$ such that

$$ \frac{1}{p} = (1 - \theta)s_0 + \theta s_1. $$

Then the following assertions hold.

(i) for all $u \in \dot{H}^{s_0,p}(\mathbb{R}_+^n) \cap \dot{H}^{s_1,p}(\mathbb{R}_+^n)$, we have $\gamma_0 u \in \dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})$, with the estimate

$$ \|\gamma_0 u\|_{\dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim s_{0,s_1,p,n} \|u\|_{\dot{H}^{s_0,p}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}^{\theta} + \|u\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}. $$

We also have,

$$ \|\gamma_0 u\|_{\dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim s_{0,s_1,p,n} \|u\|_{\dot{H}^{s_1,p}(\mathbb{R}_+^n)}; $$

(ii) for all $u \in \dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)$, we have $\gamma_0 u \in \dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})$, with the estimate

$$ \|\gamma_0 u\|_{\dot{B}_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim s_{0,s_1,p,n} \|u\|_{\dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}^{\theta} + \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}. $$

We also have,

$$ \|\gamma_0 u\|_{\dot{B}_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim s_{0,s_1,p,n} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}; $$

(iii) for all $u \in \dot{B}_{p,\infty}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,\infty}^{s_1}(\mathbb{R}_+^n)$, we have $\gamma_0 u \in L^p(\mathbb{R}^{n-1})$, with the estimate

$$ \|\gamma_0 u\|_{L^p(\mathbb{R}^{n-1})} \lesssim s_{0,s_1,p,n} \|u\|_{\dot{B}_{p,\infty}^{s_0}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{B}_{p,\infty}^{s_1}(\mathbb{R}_+^n)}^{\theta}. $$

*Proof.* — We only start proving the point (ii), and claim that point (i) can be achieved similarly. We start noticing, the following continuous embedding,

$$ \dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n) \stackrel{\iota}{\hookrightarrow} (\dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n), \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n))_{\theta,1} = \dot{B}_{p,1}^{\frac{1}{p}}(\mathbb{R}_+^n) \stackrel{\gamma_0}{\hookrightarrow} L^p(\mathbb{R}^{n-1}). $$

Here, $\iota$ is the canonical embedding obtained via standard interpolation theory, and the last embedding via the trace operator is a direct consequence of Theorem 4.3, and everything can be turned into the following inequality,

$$ \|\gamma_0 u\|_{L^p(\mathbb{R}^{n-1})} \lesssim s_{0,s_1,p,n} \|u\|_{\dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n)}^{1-\theta} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}^{\theta}, \forall u \in \dot{B}_{p,q}^{s_0}(\mathbb{R}_+^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n). $$

Again, from Theorem 4.3 we obtain for all $u \in S_0(\mathbb{R}_+^n)$,

$$ \|\gamma_0 u\|_{\dot{B}_{p,q}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim s_{1,p,n} \|u\|_{\dot{B}_{p,q}^{s_1}(\mathbb{R}_+^n)}. $$

Then one may sum both inequality, notice that $L^p(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1}) = \dot{B}_{p,p}^{s_1-\frac{1}{p}}(\mathbb{R}^{n-1})$ and use the density argument provided by Proposition 3.20 so that each estimate holds. ■
# 5 Additional notations and some remarks

## 5.1 Non-exhaustivity of the construction

The goal of presenting here a definitive construction of homogeneous Sobolev and Besov spaces on the half-space is certainly not reached:

* The way arguments are done Section 3 always requires a ground function space to intersect with so that it ensures we deal with restriction of elements of $S'_h(\mathbb{R}^n)$, e.g. see the proof of Proposition 3.3. Hence, with this kind of methods, obtaining more general results like an exhaustive description of dual spaces of homogeneous Besov and Sobolev spaces on $\mathbb{R}_+^n$ in the non-complete case seems to be difficult to reach.

* A related problem is that the extension operator we use is not general enough and disallow to recover too much negative index of regularity in case of homogeneous function spaces. It would be of interest to know if one can also recover non-complete positive index independently, without using intersection or density tricks. As mentioned at the beginning of this section, to know if one can construct an operator similar to Rychkov's extension operator, from [Ryc99], $\mathcal{E}$ such that $\mathcal{E}(S'_h(\mathbb{R}_+^n)) \subset S'_h(\mathbb{R}^n)$ with homogeneous estimates would be a sufficiently powerful result to overcome such troubles.

* Other definitions are possible for $S'_h(\mathbb{R}^n)$. We have chosen here the one with the strongest convergence for the sum of low frequencies to continue the work started in [BCD11, Chapter 2] and [DHMT21, Chapter 3]. The choice of possible definitions and their functional analytic consequences on Besov spaces' construction are reviewed by Cobb in [Cob21, Appendix] and [Cob22].

Not to further burden the actual presentation, we just mention that one could also investigate spaces such as

$$ \dot{B}_{p,\infty}^s(\mathbb{R}_+^n) \text{ and } \dot{B}_{p,\infty,0}^s(\mathbb{R}_+^n). $$

For those spaces, the space $S_0(\mathbb{R}_+^n)$ is dense in the first one by construction, and we can show that the space $C_c^\infty(\mathbb{R}_+^n)$ is dense in the second one, and both may be recovered from interpolation of other appropriate homogeneous Sobolev and Besov spaces. We can also prove corresponding duality and traces results. Details are left to the interested reader.

## 5.2 Operators on Sobolev and Besov spaces

We introduce domains for an operator $A$ acting on Sobolev or Besov spaces, denoting

* $D_p^s(A)$ (resp. $\dot{D}_p^s(A)$) its domain on $H^{s,p}$ (resp. $\dot{H}^{s,p}$);

* $D_{p,q}^s(A)$ (resp. $\dot{D}_{p,q}^s(A)$) its domain on $B_{p,q}^s$ (resp. $\dot{B}_{p,q}^s$);

* $D_p(A) = D_p^0(A) = \dot{D}_p^0(A)$ its domain on $L^p$.

Similarly, $N_p^s(A)$, $N_{p,q}^s(A)$ will stand for its nullspace on $H^{s,p}$ and $B_{p,q}^s$, and range spaces will be given respectively by $R_p^s(A)$ and $R_{p,q}^s(A)$. We replace $N$ and $R$ by $\dot{N}$ and $\dot{R}$ for their corresponding corresponding sets on homogeneous function spaces.

If the operator $A$ has different realizations depending on various function spaces and on the considered open set, we may write its domain $D(A, \Omega)$, and similarly for its nullspace $N$ and range space $R$. We omit the open set $\Omega$ if there is no possible confusion.
**Remark 4.5** As in Theorem 4.3, above Proposition 4.4 could be turned into an $C_{0,x_n}^0$-embedding in the appropriate Besov space.

**Proposition 4.6** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty)$, $s_j > 1/p_j$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ (resp. $(C_{s_0, p_0, q_0})$) is satisfied. Then,

(i) For all $u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n)$, we have $\gamma_0 u \in \dot{B}_{p_j, p_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})$, $j \in \{0, 1\}$, with the estimate
$$ \|\gamma_0 u\|_{\dot{B}_{p_j, p_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})} \lesssim_{s_j, p_j, n} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}; $$

(ii) For all $u \in [\dot{B}_{p_0, q_0}^{s_0} \cap \dot{B}_{p_1, q_1}^{s_1}](\mathbb{R}_+^n)$, we have $\gamma_0 u \in \dot{B}_{p_j, q_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})$, $j \in \{0, 1\}$, with the estimate
$$ \|\gamma_0 u\|_{\dot{B}_{p_j, q_j}^{s_j - \frac{1}{p_j}}(\mathbb{R}^{n-1})} \lesssim_{s_j, p_j, n} \|u\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n)}; $$

**Remark 4.7** Corollary B.3 yields the ontoness of the trace operator on intersection spaces given by above Proposition 4.6.

From now on, since we justified that the meaning of traces does not depend on the underlying function spaces (they agree on their intersection, when it is well-defined), and since we have the description $\partial \mathbb{R}_+^n = \mathbb{R}^{n-1} \times \{0\}$, we will refer to the trace of $u$ as $u|_{\partial \mathbb{R}_+^n}$ instead of $\gamma_0 u$.

**Lemma 4.8** Let $p_j \in (1, +\infty)$, $s \in (1/p_j, 1 + 1/p_j)$, $j \in \{0, 1\}$ such that $(C_{s_0, p_0})$ is satisfied. For all $u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C})$ such that $u|_{\partial \mathbb{R}_+^n} = 0$, the extension $\tilde{u}$ to $\mathbb{R}^n$ by 0, satisfies

$$ \tilde{u} \in [\dot{H}_0^{s_0, p_0} \cap \dot{H}_0^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C}) $$

with the estimate

$$ \|\tilde{u}\|_{\dot{H}^{s_j, p_j}(\mathbb{R}^n)} \lesssim_{p_j, s_j, n} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}, \quad j \in \{0, 1\}. $$

Hence, we have the following canonical isomorphism of Banach spaces

$$ \{ u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C}) \mid u|_{\partial \mathbb{R}_+^n} = 0 \} \simeq [\dot{H}_0^{s_0, p_0} \cap \dot{H}_0^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C}). $$

The result still holds replacing $\dot{H}^{s_j, p_j}$ by $\dot{B}_{p_j, q_j}^{s_j}$, $q_j \in [1, +\infty]$, $j \in \{0, 1\}$ assuming that $(C_{s_0, p_0, q_0})$ is satisfied.

**Proof.** — Let $u \in [\dot{H}^{s_0, p_0} \cap \dot{H}^{s_1, p_1}](\mathbb{R}_+^n, \mathbb{C})$ such that $u|_{\partial \mathbb{R}_+^n} = 0$, then for all $\phi \in [\dot{H}^{1-s_j, p'_j} \cap \mathcal{S}](\mathbb{R}^n, \mathbb{C}^n)$, we have

$$ \int_{\mathbb{R}_+^n} \nabla u \cdot \phi = - \int_{\mathbb{R}_+^n} \mathrm{div}(\phi) u. $$

So that introducing the extensions by 0 to $\mathbb{R}^n$, $\tilde{u}$ and $\tilde{\nabla}u$,

$$ \int_{\mathbb{R}^n} \tilde{\nabla} u \cdot \phi = \int_{\mathbb{R}_+^n} \nabla u \cdot \phi = - \int_{\mathbb{R}_+^n} \mathrm{div}(\phi) u = - \int_{\mathbb{R}^n} \mathrm{div}(\phi) \tilde{u} = \langle \nabla \tilde{u}, \phi \rangle_{\mathbb{R}^n}. $$

Therefore, for all $\phi \in [\dot{H}^{1-s_j, p'_j} \cap \mathcal{S}](\mathbb{R}^n, \mathbb{C}^n)$,

$$ \int_{\mathbb{R}^n} \tilde{\nabla} u \cdot \phi = \langle \nabla \tilde{u}, \phi \rangle_{\mathbb{R}^n}. $$

Hence $\tilde{\nabla}u = \nabla \tilde{u}$ in $S'(\mathbb{R}^n, \mathbb{C}^n)$. Thus, by Propositions 2.7 and 2.9, we deduce that

$$ \begin{aligned} |\langle \nabla \tilde{u}, \phi \rangle_{\mathbb{R}^n}| & \le \|\phi\|_{\dot{H}^{1-s_j, p'_j}(\mathbb{R}^n)} \|\tilde{\nabla} u\|_{\dot{H}^{s_j-1, p_j}(\mathbb{R}^n)} \\ & \lesssim_{p_j, n, s_j} \|\phi\|_{\dot{H}^{1-s_j, p'_j}(\mathbb{R}^n)} \|\nabla u\|_{\dot{H}^{s_j-1, p_j}(\mathbb{R}_+^n)} \\ & \lesssim_{p_j, n, s_j} \|\phi\|_{\dot{H}^{1-s_j, p'_j}(\mathbb{R}^n)} \|u\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)}. \end{aligned} $$

One may conclude thanks to Proposition 2.7, and Corollary 3.5. The case of Besov spaces follows the same lines. The isomorphism is then a direct consequence. ■
# 6 Applications: the Dirichlet and Neumann Laplacians on the half-space

We introduce the following subsets of the complex plane

$$ \Sigma_{\mu} := \{ z \in \mathbb{C}^* : |\arg(z)| < \mu \}, \text{ if } \mu \in (0, \pi), $$

we also define $\Sigma_0 := (0, +\infty)$, and later we are going to consider $\bar{\Sigma}_\mu$ its closure.

An operator $(D(A), A)$ on complex valued Banach space $X$ is said to be $\omega$-sectorial, if for a fixed $\omega \in (0, \pi)$, both conditions are satisfied

(i) $\sigma(A) \subset \bar{\Sigma}_\omega$, where $\sigma(A)$ denotes the spectrum of $A$;

(ii) For all $\mu \in (\omega, \pi)$, $\sup_{\lambda \in \mathbb{C} \setminus \bar{\Sigma}_\mu} \|\lambda(\lambda I - A)^{-1}\|_{X \to X} < +\infty$.

Sectorial operators is widely reviewed in several references but we mention here Haase's book [Haa06]. One may also check [Ege15, Chapter 3].

Before starting the analysis of Dirichlet, Neumann on the half-space, we introduce two appropriate extension operators. We denote $E_J$, for $J \in \{\mathcal{D}, \mathcal{N}\}$, the extension operator defined for any measurable function $u$ on $\mathbb{R}_+^n$, for almost every $x = (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_+$:

$$ E_{\mathcal{D}}u(x', x_n) := \begin{cases} u(x', x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_+, \\ -u(x', -x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_-^*; \end{cases} $$

$$ E_{\mathcal{N}}u(x', x_n) := \begin{cases} u(x', x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_+, \\ u(x', -x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_-^*. \end{cases} $$

Obviously, for $J \in \{\mathcal{D}, \mathcal{N}\}$, $s \in (-1 + 1/p, 1/p)$, $p \in (1, +\infty)$, the Proposition 2.9 leads to boundedness of

$$ E_J : \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n). \qquad (6.1) $$

The same result holds replacing $\dot{H}^{s,p}$ by either $H^{s,p}$, $B_{p,q}^s$, or even by $\dot{B}_{p,q}^s$, $q \in [1, +\infty]$.

We are going to use the properties of Laplacian acting on the whole space to build the resolvent estimates for both the the Dirichlet and the Neumann Laplacian. Usual Dirichlet and Neumann Laplacians are the operators $(D(\Delta_J), -\Delta_J)$, for $J \in \{\mathcal{D}, \mathcal{N}\}$, where the subscript $\mathcal{D}$ (resp. $\mathcal{N}$) stands for the Dirichlet (resp. Neumann) Laplacian, with, for $p \in (1, +\infty)$,

$$ D_p(\Delta_{\mathcal{D}}) := \left\{ u \in H^{1,p}(\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in L^p(\mathbb{R}_+^n, \mathbb{C}) \text{ and } u|_{\partial \mathbb{R}_+^n} = 0 \right\}, $$

$$ D_p(\Delta_{\mathcal{N}}) := \left\{ u \in H^{1,p}(\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in L^p(\mathbb{R}_+^n, \mathbb{C}) \text{ and } \partial_{\nu} u|_{\partial \mathbb{R}_+^n} = 0 \right\}. $$

For $J \in \{\mathcal{D}, \mathcal{N}\}$, and all $u \in D_p(\Delta_J)$,

$$ -\Delta_J u := -\Delta u. $$

When $p=2$, one can also realize both Dirichlet and Neumann Laplacians by the mean of densely defined, symmetric, accretive, continuous, closed, sesquilinear forms on $L^2(\mathbb{R}_+^n, \mathbb{C})$, for $J \in \{\mathcal{D}, \mathcal{N}\}$,

$$ \mathfrak{a}_J : D_2(\mathfrak{a}_J)^2 \ni (u, v) \mapsto \int_{\mathbb{R}_+^n} \nabla u \cdot \overline{\nabla v} \qquad (6.2) $$

with $D_2(\mathfrak{a}_\mathcal{D}) = H_0^1(\mathbb{R}_+^n, \mathbb{C})$, $D_2(\mathfrak{a}_\mathcal{N}) = H^1(\mathbb{R}_+^n, \mathbb{C})$, so that it is easy to see, and well-known, that both, the Neumann and Dirichlet Laplacians, are closed, densely defined, non-negative self-adjoint operators on $L^2(\mathbb{R}_+^n, \mathbb{C})$, see [Ouh05, Chapter 1, Section 1.2]. We can be even more precise.

**Proposition 6.1** Provided $J \in \{\mathcal{D}, \mathcal{N}\}$, the operator $(D_2(\Delta_J), -\Delta_J)$ is an injective non-negative self-adjoint and 0-sectorial operator on $L^2(\mathbb{R}_+^n, \mathbb{C})$.

Moreover, the following hold

(i) $D_2(\Delta_J)$ is a closed subspace of $H^2(\mathbb{R}_+^n, \mathbb{C})$;
## 4.3 Several remarks

We finish this section giving several remarks. The goal of presenting here a definitive construction of homogeneous Sobolev and Besov spaces on the half-space is certainly not reached:

* A related problem is that the extension operator we use is not general enough and disallow to recover too much negative index of regularity in case of homogeneous function spaces. It would be of interest to know if one can also recover non-complete positive index independently, without using intersection or density tricks. As mentioned at the beginning of this section, to know if one can construct an operator similar to Rychkov's extension operator, from [Ryc99], $\mathcal{E}$ such that $\mathcal{E}(S'_h(\mathbb{R}_+^n)) \subset S'_h(\mathbb{R}^n)$ with homogeneous estimates would be a sufficiently powerful result to overcome such troubles.

* Other definitions are possible for $S'_h(\mathbb{R}^n)$. We have chosen here the one with the strongest convergence for the sum of low frequencies to continue the work started in [BCD11, Chapter 2] and [DHMT21, Chapter 3]. The choice of possible definitions and their functional analytic consequences on Besov spaces' construction are reviewed by Cobb in [Cob21, Appendix] and [Cob23].

Not to further burden the actual presentation, we just mention that one could also investigate spaces such as

$$ \dot{B}_{p,\infty}^{s}(\mathbb{R}_{+}^{n}) \text{ and } \dot{B}_{p,\infty,0}^{s}(\mathbb{R}_{+}^{n}). $$

For those spaces, the space $S_0(\mathbb{R}_+^n)$ is (strongly) dense in the first one by construction, and we can show that the space $C_c^\infty(\mathbb{R}_+^n)$ is dense in the second one, and both may be recovered from interpolation of other appropriate homogeneous Sobolev and Besov spaces. We can also prove corresponding duality and traces results with values in $\dot{B}_{p,\infty}^{s-1/p}(\mathbb{R}^{n-1})$ as presented in Theorem (4.3). Details are left to the interested reader.

# 5 Applications: the Dirichlet and Neumann Laplacians on the half-space

**Notations and Definitions :** Sectorial operators, Operators on Sobolev and Besov spaces.

We introduce domains for an operator $A$ acting on Sobolev or Besov spaces, denoting

* $D_p^s(A)$ (resp. $\dot{D}_p^s(A)$) its domain on $H^{s,p}$ (resp. $\dot{H}^{s,p}$);

* $D_{p,q}^s(A)$ (resp. $\dot{D}_{p,q}^s(A)$) its domain on $B_{p,q}^s$ (resp. $\dot{B}_{p,q}^s$);

* $D_p(A) = D_p^0(A) = \dot{D}_p^0(A)$ its domain on $L^p$.

Similarly, $N_p^s(A)$, $N_{p,q}^s(A)$ will stand for its nullspace on $H^{s,p}$ and $B_{p,q}^s$, and range spaces will be given respectively by $R_p^s(A)$ and $R_{p,q}^s(A)$. We replace $N$ and $R$ by $\dot{N}$ and $\dot{R}$ for their corresponding sets on homogeneous function spaces.

If the operator $A$ has different realizations depending on various function spaces and on the considered open set, we may write its domain $D(A, \Omega)$, and similarly for its nullspace $N$ and range space $R$. We omit the open set $\Omega$ if there is no possible confusion.

We introduce the following subsets of the complex plane

$$ \Sigma_{\mu} := \{ z \in \mathbb{C}^* : |\arg(z)| < \mu \}, \text{ if } \mu \in (0, \pi), $$

we also define $\Sigma_0 := (0, +\infty)$, and later we are going to consider $\bar{\Sigma}_\mu$ its closure.

An operator $(D(A), A)$ on complex valued Banach space $X$ is said to be $\omega$-sectorial, if for a fixed $\omega \in (0, \pi)$, both conditions are satisfied
(ii) Provided $\mu \in [0, \pi)$, for $\lambda \in \Sigma_\mu$, $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$, then $u := (\lambda I - \Delta_J)^{-1} f$ satisfies

$$|\lambda||u||_{L^2(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{L^2(\mathbb{R}_+^n)} + ||\nabla^2 u||_{L^2(\mathbb{R}_+^n)} \lesssim_{n,\mu} ||f||_{L^2(\mathbb{R}_+^n)};$$

(iii) The following resolvent identity holds for all $\mu \in [0, \pi)$, $\lambda \in \Sigma_\mu$, $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$,

$$E_J(\lambda I - \Delta_J)^{-1} f = (\lambda I - \Delta)^{-1} E_J f.$$

**Remark 6.2** For $u : \mathbb{R}_+^n \to \mathbb{C}$, we set

$$\tilde{u}_{\mathcal{J}} := [E_{\mathcal{J}} u]_{|\mathbb{R}_+^n}$$

for $\mathcal{J} \in \{\mathcal{D}, \mathcal{N}\}$. We notice that in $\mathcal{D}'(\mathbb{R}_-^n, \mathbb{C})$,

$$\partial_{x_n}[\tilde{u}_{\mathcal{N}}] = [\tilde{\partial}_{x_n} u]_{\mathcal{D}} \text{ and } \partial_{x_n}[\tilde{u}_{\mathcal{D}}] = [\tilde{\partial}_{x_n} u]_{\mathcal{N}}.$$

*Proof.* — One may use self-adjointness and (6.2) which gives, by standard hilbertian theory, the following resolvent estimate

$$|\lambda||u||_{L^2(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{L^2(\mathbb{R}_+^n)} + ||\Delta u||_{L^2(\mathbb{R}_+^n)} \lesssim_{\mu} ||f||_{L^2(\mathbb{R}_+^n)},$$

where $u := (\lambda I - \Delta_J)^{-1} f$, $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$, $\lambda \in \Sigma_\mu$, $\mu \in [0, \pi)$.

Now, for fixed $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$, $\lambda \in \Sigma_\mu$, $\mu \in [0, \pi)$, we consider $u := (\lambda I - \Delta_J)^{-1} f$. Assuming $\mathcal{J} = \mathcal{N}$, we have for $\phi \in S(\mathbb{R}^n, \mathbb{C})$,

$$\begin{aligned} \langle E_{\mathcal{N}} u, -\Delta \phi \rangle_{\mathbb{R}^n} &= \langle u, -\Delta \phi \rangle_{\mathbb{R}^n_+} + \langle \tilde{u}_{\mathcal{N}}, -\Delta \phi \rangle_{\mathbb{R}^n_-} \\ &= \langle \nabla u, \nabla \phi \rangle_{\mathbb{R}^n_+} + \langle u, \nabla \phi \cdot \mathfrak{e}_n \rangle_{\partial \mathbb{R}^n_+} - \langle \tilde{u}_{\mathcal{N}}, \nabla \phi \cdot \mathfrak{e}_n \rangle_{\partial \mathbb{R}^n_-} \\ &\quad + \langle [\widetilde{\nabla' u}]_{\mathcal{N}}, \nabla' \phi \rangle_{\mathbb{R}^n_+} + \langle [\widetilde{\partial_{x_n} u}]_{\mathcal{D}}, \partial_{x_n} \phi \rangle_{\mathbb{R}^n_-} \end{aligned}$$

Since $\partial \mathbb{R}_+^n = \partial \mathbb{R}_-^n = \mathbb{R}^{n-1} \times \{0\}$, with traces $\tilde{u}_{\mathcal{N}|_{\partial \mathbb{R}_+^n}} = u_{|_{\partial \mathbb{R}_+^n}}$, we deduce $\langle u_{|_{\partial \mathbb{R}_+^n}}, \nabla \phi \cdot \mathfrak{e}_n \rangle_{\partial \mathbb{R}_+^n} - \langle \tilde{u}_{\mathcal{N}|_{\partial \mathbb{R}_+^n}}, \nabla \phi \cdot \mathfrak{e}_n \rangle_{\partial \mathbb{R}_-^n} = 0$. Then, thanks to Remark 6.2 and the boundary condition on $u$, i.e. $\partial_{x_n} u_{|_{\partial \mathbb{R}_+^n}} = 0$, we have

$$\begin{aligned} \langle E_{\mathcal{N}} u, -\Delta \phi \rangle_{\mathbb{R}^n} &= \langle \nabla u, \nabla \phi \rangle_{\mathbb{R}^n_+} + \langle [\widetilde{\nabla' u}]_{\mathcal{N}}, \nabla' \phi \rangle_{\mathbb{R}^n_-} + \langle [\widetilde{\partial_{x_n} u}]_{\mathcal{D}}, \partial_{x_n} \phi \rangle_{\mathbb{R}^n_-} \\ &= \langle -\Delta u, \phi \rangle_{\mathbb{R}^n_+} + \langle [-\widetilde{\Delta' u}]_{\mathcal{N}}, \phi \rangle_{\mathbb{R}^n_-} + \langle [-\widetilde{\partial_{x_n}^2 u}]_{\mathcal{N}}, \phi \rangle_{\mathbb{R}^n_-} \\ &\quad - \langle \partial_{x_n} u, \phi \rangle_{\partial \mathbb{R}^n_+} - \langle [\widetilde{\partial_{x_n} u}]_{\mathcal{D}}, \phi \rangle_{\partial \mathbb{R}^n_-} \\ &= \langle E_{\mathcal{N}}[-\Delta u], \phi \rangle_{\mathbb{R}^n_-}. \end{aligned}$$

Thus, $-\Delta E_{\mathcal{N}} u = E_{\mathcal{N}}[-\Delta u]$ in $S'(\mathbb{R}^n, \mathbb{C})$. One may reproduce above calculations for $\mathcal{J} = \mathcal{D}$. So for $\mathcal{J} \in \{\mathcal{D}, \mathcal{N}\}$, $E_{\mathcal{J}} u$ is a solution of

$$\lambda U - \Delta U = E_{\mathcal{J}} f.$$

We have $E_{\mathcal{J}} f \in L^2(\mathbb{R}^n, \mathbb{C})$. By uniqueness of the solution provided in $\mathbb{R}^n$, we necessarily have $U = E_{\mathcal{J}} u$, which be can written as

$$E_{\mathcal{J}}(\lambda I - \Delta_{\mathcal{J}})^{-1} f = (\lambda I - \Delta)^{-1} E_{\mathcal{J}} f.$$

Thus one deduces point (iii), from the definition of function spaces by restriction, (ii) follows, and finally setting $\lambda = 1$ in point (ii) yields (i).

We want to show some sharp regularity results on the Dirichlet an Neumann resolvent problems, on the scale of inhomogeneous and homogeneous Sobolev and Besov spaces. To do so, we introduce their corresponding domains on each space. Provided $p \in (1, +\infty)$ $s \in (-1 + 1/p, 1 + 1/p)$, if is satisfied $(C_{s,p})$:
(i) $\sigma(A) \subset \bar{\Sigma}_{\omega}$, where $\sigma(A)$ denotes the spectrum of $A$;

(ii) For all $\mu \in (\omega, \pi)$, $\sup_{\lambda \in \mathbb{C} \setminus \bar{\Sigma}_{\mu}} \|\lambda(\lambda I - A)^{-1}\|_{X \to X} < +\infty$.

Sectorial operators is widely reviewed in several references but we mention here Haase's book [Haa06]. One may also check [Ege15, Chapter 3].

## The Dirichlet and Neumann Laplacians.

Before starting the analysis of the Dirichlet and Neumann Laplacians on the half-space, we introduce two appropriate extension operators. We denote $E_J$, for $J \in \{\mathcal{D}, \mathcal{N}\}$, the extension operator defined for any measurable function $u$ on $\mathbb{R}_+^n$, for almost every $x = (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_+$:

$$ E_{\mathcal{D}}u(x', x_n) := \begin{cases} u(x', x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_+, \\ -u(x', -x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_-^*; \end{cases} $$

$$ E_{\mathcal{N}}u(x', x_n) := \begin{cases} u(x', x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_+, \\ u(x', -x_n) & , \text{ if } (x', x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}_-^*. \end{cases} $$

Obviously, for $J \in \{\mathcal{D}, \mathcal{N}\}$, $s \in (-1 + 1/p, 1/p)$, $p \in (1, +\infty)$, the Proposition 2.9 leads to boundedness of

$$ E_J : \dot{H}^{s,p}(\mathbb{R}_+^n) \to \dot{H}^{s,p}(\mathbb{R}^n). \quad (5.1) $$

The same result holds replacing $\dot{H}^{s,p}$ by either $H^{s,p}$, $B_{p,q}^s$, or even by $\dot{B}_{p,q}^s$, $q \in [1, +\infty]$.

We are going to use the properties of Laplacian acting on the whole space to build the resolvent estimates for both the Dirichlet and the Neumann Laplacian. Usual Dirichlet and Neumann Laplacians are the operators $(D(\Delta_J), -\Delta_J)$, for $J \in \{\mathcal{D}, \mathcal{N}\}$, where the subscript $\mathcal{D}$ (resp. $\mathcal{N}$) stands for the Dirichlet (resp. Neumann) Laplacian, with, for $p \in (1, +\infty)$,

$$ \begin{aligned} D_p(\Delta_{\mathcal{D}}) &:= \left\{ u \in H^{1,p}(\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in L^p(\mathbb{R}_+^n, \mathbb{C}) \text{ and } u|_{\partial \mathbb{R}_+^n} = 0 \right\}, \\ D_p(\Delta_{\mathcal{N}}) &:= \left\{ u \in H^{1,p}(\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in L^p(\mathbb{R}_+^n, \mathbb{C}) \text{ and } \partial_{\nu} u|_{\partial \mathbb{R}_+^n} = 0 \right\}. \end{aligned} $$

For $J \in \{\mathcal{D}, \mathcal{N}\}$, and all $u \in D_p(\Delta_J)$,

$$ -\Delta_J u := -\Delta u. $$

When $p=2$, one can also realize both Dirichlet and Neumann Laplacians by the mean of densely defined, symmetric, accretive, continuous, closed, sesquilinear forms on $L^2(\mathbb{R}_+^n, \mathbb{C})$, for $J \in \{\mathcal{D}, \mathcal{N}\}$,

$$ \mathbf{a}_J : D_2(\mathbf{a}_J)^2 \ni (u, v) \mapsto \int_{\mathbb{R}_+^n} \nabla u \cdot \overline{\nabla v} \quad (5.2) $$

with $D_2(\mathbf{a}_\mathcal{D}) = H_0^1(\mathbb{R}_+^n, \mathbb{C})$, $D_2(\mathbf{a}_\mathcal{N}) = H^1(\mathbb{R}_+^n, \mathbb{C})$, so that it is easy to see, and well-known, that both, the Neumann and Dirichlet Laplacians, are closed, densely defined, non-negative self-adjoint operators on $L^2(\mathbb{R}_+^n, \mathbb{C})$, see [Ouh05, Chapter 1, Section 1.2]. We can be even more precise.

**Proposition 5.1** *Provided $J \in \{\mathcal{D}, \mathcal{N}\}$, the operator $(D_2(\Delta_J), -\Delta_J)$ is an injective non-negative self-adjoint and 0-sectorial operator on $L^2(\mathbb{R}_+^n, \mathbb{C})$.*

Moreover, the following hold

(i) $D_2(\Delta_J)$ is a closed subspace of $H^2(\mathbb{R}_+^n, \mathbb{C})$;

(ii) Provided $\mu \in [0, \pi)$, for $\lambda \in \Sigma_\mu$, $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$, then $u := (\lambda I - \Delta_J)^{-1} f$ satisfies

$$ |\lambda| \|u\|_{L^2(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}} \|\nabla u\|_{L^2(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{L^2(\mathbb{R}_+^n)} \lesssim_{n,\mu} \|f\|_{L^2(\mathbb{R}_+^n)}; $$

(iii) The following resolvent identity holds for all $\mu \in [0, \pi)$, $\lambda \in \Sigma_\mu$, $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$,

$$ E_J(\lambda I - \Delta_J)^{-1} f = (\lambda I - \Delta)^{-1} E_J f. $$
$$
\dot{D}_p^s(\Delta_D) := \left\{ u \in [\dot{H}_0^{s,p} \cap \dot{H}^{s+1,p}](\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in \dot{H}_0^{s,p}(\mathbb{R}_+^n, \mathbb{C}) \text{ and } u|_{\partial \mathbb{R}_+^n} = 0 \right\} \subset \dot{H}_0^{s,p}(\mathbb{R}_+^n, \mathbb{C}),
$$

$$
\dot{D}_p^s(\Delta_N) := \left\{ u \in [\dot{H}^{s,p} \cap \dot{H}^{s+1,p}](\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C}) \text{ and } \partial_{\nu} u|_{\partial \mathbb{R}_+^n} = 0 \right\} \subset \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C}).
$$

We can also consider their domains on inhomogeneous Sobolev and Besov spaces, as well as
homogeneous spaces, replacing ($\dot{D}_p^s, \dot{H}^{s,p}$) by either ($D_p^s, H^{s,p}$), ($D_{p,q}^s, B_{p,q}^s$) and finally ($\dot{D}_{p,q}^s, \dot{B}_{p,q}^s$)
provided $q \in [1, +\infty]$, and ($C_{s,p,q}$) is satisfied.

**Remark 6.3** We allowed us a slight abuse of notation here: we identified $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$ with either

• $\dot{H}^{s,p}(\mathbb{R}_+^n)$ when $s \in (-1 + 1/p, 1/p)$, thanks to Proposition 2.9;

• $\dot{H}^{s,p}(\mathbb{R}_+^n)$ with homogeneous Dirichlet boundary condition when $s \in (1/p, 1+1/p)$, thanks to Lemma 4.8.

The same identification is made for Besov spaces, and inhomogeneous function spaces.

It is then not difficult to see that the Dirichlet and Neumann Laplacians are well defined un-
bounded closed linear operators, densely defined, if $q \in [1, +\infty)$ in the case of inhomogeneous and
homogeneous Besov spaces. If $q = +\infty$, the domain of the Dirichlet (resp. Neumann) Laplacian is
only known to be weak* dense in $B_{p,\infty,0}$ (resp. in $B_{p,\infty}$) and $\dot{B}_{p,\infty,0}^s$ (resp. $\dot{B}_{p,\infty}^s$).

**Proposition 6.4** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1+\frac{1}{p}, 1+\frac{1}{p})$, $s \neq 1/p$, $\alpha \in (-1+\frac{1}{\tilde{p}}, 1+\frac{1}{\tilde{p}})$, $\alpha \neq 1/\tilde{p}$, and $\lambda \in \Sigma_\mu$ provided $\mu \in [0, \pi)$. We assume that $(C_{s,p})$, and we let $f \in \dot{H}_0^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. Let us consider the resolvent Dirichlet problem with homogeneous boundary condition:

$$
\left\{
\begin{array}{ll}
\lambda u - \Delta u = f, & \text{in } \mathbb{R}_+^n, \\
u|_{\partial \mathbb{R}_+^n} = 0, & \text{on } \partial \mathbb{R}_+^n.
\end{array}
\right.
\qquad (\mathcal{DL}_\lambda)
$$

The problem $(\mathcal{DL}_{\lambda})$ admits a unique solution $u \in [\dot{H}_{0}^{s,p} \cap \dot{H}^{s+2,p}](\mathbb{R}_{+}^{n}, \mathbb{C})$ with the estimate

$$
|\lambda||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + ||\nabla^2 u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} ||f||_{\dot{H}^{s,p}(\mathbb{R}_+^n)}.
$$

If moreover $\alpha \neq 1/\tilde{p}$ and $f \in \dot{H}_0^{\alpha, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$, then we also have $u \in [\dot{H}^{\alpha, \tilde{p}} \cap \dot{H}^{\alpha+2, \tilde{p}}](\mathbb{R}_+^n, \mathbb{C})$ with
the corresponding estimate

$$
|\lambda||u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + ||\nabla^2 u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha, \mu} ||f||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)}.
$$

The result still holds replacing $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{H}^{\alpha, \tilde{p}}, \dot{H}^{\alpha+2, \tilde{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+2})$ whenever $(C_{s,p,q})$ is satisfied.

**Remark 6.5** • For this specific Proposition 6.4, we have excluded the cases $s = 1/p$ and $\alpha = 1/\tilde{p}$.
Both require to introduce, e.g. in case of Sobolev spaces, the homogeneous counterpart of the Lions-
Magenes Sobolev space $\dot{H}_{00}^{1/q,q}(\mathbb{R}_+^n)$, $q \in \{p, \tilde{p}\}$. See for instance [LM72, Chapter 1, Theorem 11.7]
for the inhomogeneous space in the case $q = 2$.

• We bring to the attention of the reader that ($C_{\alpha, \tilde{p}}$) is **NEVER** assumed, only ($C_{s,p}$) is. This is in order to echo the principle of decoupled estimates in intersection spaces when one wants to deal with higher regularities involving some non-complete spaces. All the other results presented below follow the same principle.

Proof. — Provided $p \in (1, +\infty)$, and firstly that $s \in (-1+1/p, 1/p)$, for $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, it follows
from Proposition 2.9 that for $U := (\lambda I - \Delta)^{-1} E_D f$

$$
|\lambda||U||_{\dot{H}^{s,p}(\mathbb{R}^n)} + |\lambda|^{\frac{1}{2}}||\nabla U||_{\dot{H}^{s,p}(\mathbb{R}^n)} + ||\nabla^2 U||_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p,n,s,\mu} ||f||_{\dot{H}^{s,p}(\mathbb{R}^n)}.
$$

Thus, by definition of restriction restriction space, we set $u := U|_{\mathbb{R}_+^n}$ which satisfies

$$
|\lambda||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + ||\nabla^2 u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} ||f||_{\dot{H}^{s,p}(\mathbb{R}_+^n)},
$$
**Remark 5.2** For $u : \mathbb{R}_+^n \to \mathbb{C}$, we set

$$ \tilde{u}_{\mathcal{J}} := [\mathrm{E}_{\mathcal{J}} u]_{|\mathbb{R}_+^n} $$

for $\mathcal{J} \in \{\mathcal{D}, \mathcal{N}\}$. We notice that in $\mathcal{D}'(\mathbb{R}_-^n, \mathbb{C})$,

$$ \partial_{x_n}[\tilde{u}_{\mathcal{N}}] = [\widetilde{\partial_{x_n} u}]_{\mathcal{D}} \text{ and } \partial_{x_n}[\tilde{u}_{\mathcal{D}}] = [\widetilde{\partial_{x_n} u}]_{\mathcal{N}}. $$

*Proof.* — One may use self-adjointness and (5.2) which gives, by standard Hilbertian theory, the following resolvent estimate

$$ |\lambda|||u||_{L^2(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{L^2(\mathbb{R}_+^n)} + ||\Delta u||_{L^2(\mathbb{R}_+^n)} \lesssim_{\mu} ||f||_{L^2(\mathbb{R}_+^n)}, $$

where $u := (\lambda I - \Delta_{\mathcal{J}})^{-1}f$, $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$, $\lambda \in \Sigma_{\mu}$, $\mu \in [0, \pi)$.

Now, for fixed $f \in L^2(\mathbb{R}_+^n, \mathbb{C})$, $\lambda \in \Sigma_{\mu}$, $\mu \in [0, \pi)$, we consider $u := (\lambda I - \Delta_{\mathcal{J}})^{-1}f$. Assuming $\mathcal{J} = \mathcal{N}$, we have for $\phi \in S(\mathbb{R}^n, \mathbb{C})$,

$$ \begin{aligned} \langle \mathrm{E}_{\mathcal{N}} u, -\Delta \phi \rangle_{\mathbb{R}^n} &= \langle u, -\Delta \phi \rangle_{\mathbb{R}_+^n} + \langle \tilde{u}_{\mathcal{N}}, -\Delta \phi \rangle_{\mathbb{R}_-^n} \\ &= \langle \nabla u, \nabla \phi \rangle_{\mathbb{R}_+^n} + \langle u, \nabla \phi \cdot \mathbf{e}_n \rangle_{\partial \mathbb{R}_+^n} - \langle \tilde{u}_{\mathcal{N}}, \nabla \phi \cdot \mathbf{e}_n \rangle_{\partial \mathbb{R}_-^n} \\ &\quad + \langle [\widetilde{\nabla' u}]_{\mathcal{N}}, \nabla' \phi \rangle_{\mathbb{R}_+^n} + \langle [\widetilde{\partial_{x_n} u}]_{\mathcal{D}}, \partial_{x_n} \phi \rangle_{\mathbb{R}_-^n} \end{aligned} $$

Since $\partial \mathbb{R}_+^n = \partial \mathbb{R}_-^n = \mathbb{R}^{n-1} \times \{0\}$, with traces $\tilde{u}_{\mathcal{N}|_{\partial \mathbb{R}_+^n}} = u|_{\partial \mathbb{R}_+^n}$, we deduce $\langle u|_{\partial \mathbb{R}_+^n}, \nabla \phi \cdot \mathbf{e}_n \rangle_{\partial \mathbb{R}_+^n} - \langle \tilde{u}_{\mathcal{N}|_{\partial \mathbb{R}_-^n}}, \nabla \phi \cdot \mathbf{e}_n \rangle_{\partial \mathbb{R}_-^n} = 0$. Then, thanks to Remark 5.2 and the boundary condition on $u$, i.e. $\partial_{x_n} u|_{\partial \mathbb{R}_+^n} = 0$, we have

$$ \begin{aligned} \langle \mathrm{E}_{\mathcal{N}} u, -\Delta \phi \rangle_{\mathbb{R}^n} &= \langle \nabla u, \nabla \phi \rangle_{\mathbb{R}_+^n} + \langle [\widetilde{\nabla' u}]_{\mathcal{N}}, \nabla' \phi \rangle_{\mathbb{R}_-^n} + \langle [\widetilde{\partial_{x_n} u}]_{\mathcal{D}}, \partial_{x_n} \phi \rangle_{\mathbb{R}_-^n} \\ &= \langle -\Delta u, \phi \rangle_{\mathbb{R}_+^n} + \langle [-\Delta' u]_{\mathcal{N}}, \phi \rangle_{\mathbb{R}_-^n} + \langle [-\partial_{x_n}^2 u]_{\mathcal{N}}, \phi \rangle_{\mathbb{R}_-^n} \\ &\quad - \langle \partial_{x_n} u, \phi \rangle_{\partial \mathbb{R}_+^n} - \langle [\widetilde{\partial_{x_n} u}]_{\mathcal{D}}, \phi \rangle_{\partial \mathbb{R}_-^n} \\ &= \langle \mathrm{E}_{\mathcal{N}}[-\Delta u], \phi \rangle_{\mathbb{R}^n}. \end{aligned} $$

Thus, $-\Delta \mathrm{E}_{\mathcal{N}} u = \mathrm{E}_{\mathcal{N}}[-\Delta u]$ in $S'(\mathbb{R}^n, \mathbb{C})$. One may reproduce the above calculations for $\mathcal{J} = \mathcal{D}$. So for $\mathcal{J} \in \{\mathcal{D}, \mathcal{N}\}$, $\mathrm{E}_{\mathcal{J}} u$ is a solution of

$$ \lambda U - \Delta U = \mathrm{E}_{\mathcal{J}} f. $$

We have $\mathrm{E}_{\mathcal{J}} f \in L^2(\mathbb{R}^n, \mathbb{C})$. By uniqueness of the solution provided in $\mathbb{R}^n$, we necessarily have $U = \mathrm{E}_{\mathcal{J}} u$, which can be written as

$$ \mathrm{E}_{\mathcal{J}}(\lambda I - \Delta_{\mathcal{J}})^{-1} f = (\lambda I - \Delta)^{-1} \mathrm{E}_{\mathcal{J}} f. $$

Thus one deduces point (iii), from the definition of function spaces by restriction, (ii) follows, and finally setting $\lambda = 1$ in point (ii) yields (i).

We want to show some sharp regularity results on the Dirichlet and Neumann resolvent problems, on the scale of inhomogeneous and homogeneous Sobolev and Besov spaces. To do so, we introduce their corresponding domains on each space. Provided $p \in (1, +\infty)$, $s \in (-1 + 1/p, 1 + 1/p)$, if is satisfied $(C_{s,p})$:

$$ \dot{D}_p^s(\Delta_D) := \left\{ u \in [\dot{H}_0^{s,p} \cap \dot{H}^{s+1,p}](\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in \dot{H}_0^{s,p}(\mathbb{R}_+^n, \mathbb{C}) \text{ and } u|_{\partial \mathbb{R}_+^n} = 0 \right\} \subset \dot{H}_0^{s,p}(\mathbb{R}_+^n, \mathbb{C}), $$

$$ \dot{D}_p^s(\Delta_N) := \left\{ u \in [\dot{H}^{s,p} \cap \dot{H}^{s+1,p}](\mathbb{R}_+^n, \mathbb{C}) \mid \Delta u \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C}) \text{ and } \partial_{\nu} u|_{\partial \mathbb{R}_+^n} = 0 \right\} \subset \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C}). $$

We can also consider their domains on inhomogeneous Sobolev and Besov spaces, as well as homogeneous spaces, replacing $(\dot{D}_p^s, \dot{H}^{s,p})$ by either $(D_p^s, H^{s,p})$, $(D_{p,q}^s, B_{p,q}^s)$ and finally $(\dot{D}_{p,q}^s, \dot{B}_{p,q}^s)$ provided $q \in [1, +\infty]$, and $(C_{s,p,q})$ is satisfied. Certainly, the conditions $(C_{s,p})$ and $(C_{s,p,q})$ are no longer necessary when one only deals with inhomogeneous function spaces.
then the map $f \mapsto [(\lambda I - \Delta)^{-1} E_D f]_{\mathbb{R}_+^n}$ is a bounded map on $\dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. Everything goes similarly for $H^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. One may check, as in the proof of Proposition 6.1, and by a limiting argument, given the density of $[L^2 \cap \dot{H}^{s,p}](\mathbb{R}_+^n, \mathbb{C})$ in $\dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, that $u|_{\partial \mathbb{R}_+^n} = 0$, and

$$ \lambda u - \Delta u = f \quad \text{in } \mathbb{R}_+^n. $$

Again, as in the proof of Proposition 6.1, one may check that any solution $u$ to above resolvent Dirichlet problem necessarily satisfies $E_D u = (\lambda I - \Delta)^{-1} E_D f$.

Now if $s \in (1/p, 1 + 1/p)$, $f \in [\dot{H}_0^{s-1,p} \cap \dot{H}_0^{s,p}](\mathbb{R}_+^n, \mathbb{C})$ then we have, thanks to previous considerations, $U := (\lambda I - \Delta)^{-1} E_D f \in \dot{H}^{s-1,p}(\mathbb{R}^n, \mathbb{C})$. It suffices to show that $U \in \dot{H}^{s,p}(\mathbb{R}^n, \mathbb{C})$, which is true. Indeed, we have

$$ \begin{aligned} |\lambda|||U||_{\dot{H}^{s,p}(\mathbb{R}^n)} &\lesssim_{s,p,n,\mu} |\lambda||\nabla U||_{\dot{H}^{s-1,p}(\mathbb{R}^n)} \\ &\lesssim_{s,p,n,\mu} \|\nabla E_D f\|_{\dot{H}^{s-1,p}(\mathbb{R}^n)} \\ &\lesssim_{s,p,n,\mu} \sum_{k=1}^n \|\partial_{x_k} E_D f\|_{\dot{H}^{s-1,p}(\mathbb{R}^n)}. \end{aligned} $$

Since equalities $\partial_{x_k} E_D f = E_D \partial_{x_k} f$, $k \in [1, n-1]$ and $\partial_{x_n} E_D f = E_N \partial_{x_n} f$ occur in $S'(\mathbb{R}^n, \mathbb{C})$, we deduce

$$ |\lambda|||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \le |\lambda|||U||_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n,\mu} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

One may proceed similarly as before to obtain the full estimate

$$ |\lambda|||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}\|\nabla u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

Thus the estimates still hold by density for all $f \in \dot{H}_0^{s,p}(\mathbb{R}_+^n)$, $s \in (-1 + 1/p, 1 + 1/p)$, $s \ne 1/p$, whenever $(C_{s,p})$ is satisfied.

The $\dot{H}^{\alpha, \tilde{p}}$-estimate for $f \in [\dot{H}_0^{s,p} \cap \dot{H}_0^{\alpha, \tilde{p}}](\mathbb{R}_+^n)$ can be obtained the same way, whenever $(C_{s,p})$ is satisfied.

The case of Besov spaces $\dot{B}_{p,q,0}^s$ can be achieved via similar argument for $q < +\infty$, the case $q = +\infty$ is obtained via real interpolation. The case of the $\dot{B}_{\tilde{p}, \tilde{q}, 0}^{\alpha}$-estimate for $f \in \dot{B}_{p,q,0}^s \cap \dot{B}_{\tilde{p}, \tilde{q}, 0}^{\alpha}$ can be done as above. ■

The proof for the Neumann resolvent problem in the proposition below is fairly similar to the proof of Proposition 6.4, a complex interpolation argument allows values $s = 1/p$ and $\alpha = 1/\tilde{p}$.

**Proposition 6.6** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1 + 1/p, 1 + 1/p)$, $\alpha \in (-1 + 1/p, 1 + 1/p)$ and $\lambda \in \Sigma_\mu$ provided $\mu \in [0, \pi)$. We assume that $(C_{s,p})$, and we let $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. Let us consider the resolvent Neumann problem with homogeneous boundary condition:

$$ \left\{ \begin{array}{ll} \lambda u - \Delta u = f, & \text{in } \mathbb{R}_+^n, \\ \\ \partial_\nu u|_{\partial \mathbb{R}_+^n} = 0, & \text{on } \partial \mathbb{R}_+^n. \end{array} \right. \qquad (\mathcal{N}\mathcal{L}_\lambda) $$

The problem $(\mathcal{N}\mathcal{L}_\lambda)$ admits a unique solution $u \in [\dot{H}^{s,p} \cap \dot{H}^{s+2,p}](\mathbb{R}_+^n, \mathbb{C})$ with the estimate

$$ |\lambda|||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}\|\nabla u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

If moreover $f \in \dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$, then we also have $u \in [\dot{H}^{\alpha, \tilde{p}} \cap \dot{H}^{\alpha+2, \tilde{p}}](\mathbb{R}_+^n, \mathbb{C})$ with corresponding the estimate

$$ |\lambda|||u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}\|\nabla u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha, \mu} \|f\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)}. $$

The result still holds replacing $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{H}^{\alpha, \tilde{p}}, \dot{H}^{\alpha+2, \tilde{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+2})$ whenever $(C_{s,p,q})$ is satisfied.

**Proposition 6.7** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1 + 1/p, +\infty)$, $\alpha \in (-1 + 1/p, +\infty)$ such that $(C_{s+2,p})$ is satisfied. For $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, $g \in \dot{B}_{p,q}^{s+2-1/p}(\mathbb{R}^{n-1}, \mathbb{C})$, let us consider the Dirichlet
**Remark 5.3** We allowed us a slight abuse of notation here: we identified $\dot{H}_0^{s,p}(\mathbb{R}_+^n)$ with either

• $\dot{H}^{s,p}(\mathbb{R}_+^n)$ when $s \in (-1 + 1/p, 1/p)$, thanks to Proposition 2.9;

• $\dot{H}^{s,p}(\mathbb{R}_+^n)$ with homogeneous Dirichlet boundary condition when $s \in (1/p, 1 + 1/p)$, thanks to Lemma 4.8.

The same identification is made for Besov spaces, and inhomogeneous function spaces.

It is then not difficult to see that the Dirichlet and Neumann Laplacians are well-defined unbounded closed linear operators, densely defined, if $q \in [1, +\infty)$ in the case of inhomogeneous and homogeneous Besov spaces. If $q = +\infty$, the domain of the Dirichlet (resp. Neumann) Laplacian is only known to be weak* dense in $B_{p,\infty,0}^s$ (resp. in $B_{p,\infty}^s$) and $\dot{B}_{p,\infty,0}^s$ (resp. $\dot{B}_{p,\infty}^s$).

**Proposition 5.4** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1+\frac{1}{\tilde{p}}, 1+\frac{1}{\tilde{p}})$, $s \neq 1/p$, $\alpha \in (-1+\frac{1}{\tilde{p}}, 1+\frac{1}{\tilde{p}})$, $\alpha \neq 1/\tilde{p}$, and $\lambda \in \Sigma_\mu$ provided $\mu \in [0, \pi)$. We assume that $(C_{s,p})$, and we let $f \in \dot{H}_0^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. Let us consider the resolvent Dirichlet problem with homogeneous boundary condition:

$$ \left\{ \begin{array}{ll} \lambda u - \Delta u = f, & \text{in } \mathbb{R}_+^n, \\ u|_{\partial \mathbb{R}_+^n} = 0, & \text{on } \partial \mathbb{R}_+^n. \end{array} \right. \qquad (\mathcal{DL}_\lambda) $$

The problem $(\mathcal{DL}_\lambda)$ admits a unique solution $u \in [\dot{H}_0^{s,p} \cap \dot{H}^{s+2,p}](\mathbb{R}_+^n, \mathbb{C})$ with the estimate

$$ |\lambda||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + ||\nabla^2 u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} ||f||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} $$

If moreover $\alpha \neq 1/\tilde{p}$ and $f \in \dot{H}_0^{\alpha, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$, then we also have $u \in [\dot{H}^{\alpha, \tilde{p}} \cap \dot{H}^{\alpha+2, \tilde{p}}](\mathbb{R}_+^n, \mathbb{C})$ with the corresponding estimate

$$ |\lambda||u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + ||\nabla^2 u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha, \mu} ||f||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} $$

The result still holds replacing $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{H}^{\alpha, \tilde{p}}, \dot{H}^{\alpha+2, \tilde{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{\tilde{p},\tilde{q}}^{\alpha}, \dot{B}_{\tilde{p},\tilde{q}}^{\alpha+2})$ whenever $(C_{s,p,q})$ is satisfied.

The whole result still holds for inhomogeneous function spaces, replacing $(\dot{H}, \dot{B})$ by $(H, B)$. However, in this case, the consideration of intersection spaces and the conditions $(C_{s,p})$ and $(C_{s,p,q})$ are no longer necessary.

**Remark 5.5** • For this specific Proposition 5.4, we have excluded the cases $s = 1/p$ and $\alpha = 1/\tilde{p}$. Both require to introduce, e.g. in case of Sobolev spaces, the homogeneous counterpart of the Lions-Magenes Sobolev space $\dot{H}_{00}^{1/q,q}(\mathbb{R}_+^n)$, $q \in \{p, \tilde{p}\}$. See for instance [LM72, Chapter 1, Theorem 11.7] for the inhomogeneous space in the case $q = 2$.

• We bring to the attention of the reader that $(C_{\alpha, \tilde{p}})$ is **NEVER** assumed, only $(C_{s,p})$ is. This is in order to echo the principle of decoupled estimates in intersection spaces when one wants to deal with higher regularities involving some non-complete spaces. All the other results presented below follow the same principle.

*Proof.* — Provided $p \in (1, +\infty)$, and firstly that $s \in (-1 + 1/p, 1/p)$, for $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, it follows from Proposition 2.9 that for $U := (\lambda I - \Delta)^{-1}E_D f$

$$ |\lambda||U||_{\dot{H}^{s,p}(\mathbb{R}^n)} + |\lambda|^{\frac{1}{2}}||\nabla U||_{\dot{H}^{s,p}(\mathbb{R}^n)} + ||\nabla^2 U||_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{p,n,s,\mu} ||f||_{\dot{H}^{s,p}(\mathbb{R}^n)} $$

Thus, by the definition of function spaces by restriction, we set $u := U|_{\mathbb{R}_+^n}$ which satisfies

$$ |\lambda||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}||\nabla u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + ||\nabla^2 u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} ||f||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} $$

then the map $f \mapsto [(\lambda I - \Delta)^{-1}E_D f]|_{\mathbb{R}_+^n}$ is a bounded map on $\dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. Everything goes similarly for $H^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. One may check, as in the proof of Proposition 5.1, and by a limiting argument, given the density of $[L^2 \cap \dot{H}^{s,p}](\mathbb{R}_+^n, \mathbb{C})$ in $\dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, that $u|_{\partial \mathbb{R}_+^n} = 0$, and

$$ \lambda u - \Delta u = f \quad \text{in } \mathbb{R}_+^n. $$
problem with inhomogeneous boundary condition:

$$ \left\{ \begin{array}{ll} -\Delta u = f, & \text{in } \mathbb{R}_+^n, \\ u|_{\partial \mathbb{R}_+^n} = g, & \text{on } \partial \mathbb{R}_+^n. \end{array} \right. \qquad (\mathcal{D}\mathcal{L}_0) $$

The problem $(\mathcal{D}\mathcal{L}_0)$ admits a unique solution $u$ such that

$$ u \in \dot{H}^{s+2,p}(\mathbb{R}_+^n, \mathbb{C}) \subset C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})) $$

with the estimate

$$ \|u\|_{L^{\infty}(\mathbb{R}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

If moreover $f \in \dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n, \mathbb{C})$ and $g \in \dot{B}_{\bar{p}, \bar{p}}^{\alpha+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ then the solution $u$ also satisfies $u \in \dot{H}^{\alpha+2, \bar{p}}(\mathbb{R}_+^n, \mathbb{C})$ with the corresponding estimate

$$ \|\nabla^2 u\|_{\dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n)} \lesssim_{\bar{p}, n, \alpha} \|f\|_{\dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{\bar{p}, \bar{p}}^{\alpha+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

The result still holds if we replace both $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{B}_{p,p}^{s+2-\frac{1}{p}})$ and $(\dot{H}^{\alpha, \bar{p}}, \dot{H}^{\alpha+2, \bar{p}}, \dot{B}_{\bar{p}, \bar{p}}^{\alpha+2-\frac{1}{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{p,q}^{s+2-\frac{1}{p}})$ and $(\dot{B}_{\bar{p}, \bar{q}}^{\alpha}, \dot{B}_{\bar{p}, \bar{q}}^{\alpha+2}, \dot{B}_{\bar{p}, \bar{q}}^{\alpha+2-\frac{1}{p}})$ whenever $(C_{s+2,p,q})$ is satisfied, $q < +\infty$.

If $q = +\infty$, everything still holds except $x_n \mapsto u(\cdot, x_n)$ is no more strongly continuous but only weak* continuous with values in $\dot{B}_{p,q}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$.

Proof. — Let $p \in (1, +\infty)$, $s > -1 + 1/p$, such that $(C_{s+2,p})$ is satisfied. Then for $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, $g \in \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ we can write the problem $(\mathcal{D}\mathcal{L}_0)$ as an evolution problem in the $x_n$ variable,

$$ \left\{ \begin{array}{ll} -\partial_{x_n}^2 u - \Delta' u = f, & \text{in } \mathbb{R}^{n-1} \times (0, +\infty), \\ u(\cdot, 0) = g, & \text{on } \mathbb{R}^{n-1}. \end{array} \right. \qquad (6.3) $$

Thanks to [ABHN11, Theorem 3.8.3], considering the semigroup $(e^{-x_n(-\Delta')^{1/2}})_{x_n \ge 0}$ and its mapping properties given by Proposition B.2 and Theorem 4.3, if $f = 0$, above problem admits unique solution $u \in C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C}))$. Thus, by linearity, we also have uniqueness of the solution $u$ in $C_{0,x_n}^0(\overline{\mathbb{R}}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C}))$ for non-identically zero function $f$. Therefore, it suffices to construct a solution.

Since $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, by definition, there exists $F \in \dot{H}^{s,p}(\mathbb{R}^n, \mathbb{C})$ such that

$$ F|_{\mathbb{R}_+^n} = f, \quad \text{and} \quad \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \sim \|F\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Let $v := (-\Delta)^{-1} F \in \dot{H}^{s+2,p}(\mathbb{R}^n, \mathbb{C})$, we also have

$$ \|v\|_{\dot{H}^{s+2,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|F\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

So it suffices to prove the result for $w \in \dot{H}^{s+2,p}(\mathbb{R}_+^n, \mathbb{C})$, such that

$$ \left\{ \begin{array}{ll} -\Delta w = 0, & \text{in } \mathbb{R}^{n-1} \times (0, +\infty), \\ w|_{\partial \mathbb{R}_+^n} = \tilde{g}, & \text{on } \mathbb{R}^{n-1}, \end{array} \right. $$

where $\tilde{g} \in \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ can be seen as $g - v(\cdot, 0)$. But such a $w$ exists and is unique thanks to Proposition B.2 and [ABHN11, Theorem 3.8.3], and satisfies

$$ \|w\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|\tilde{g}\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$
Again, as in the proof of Proposition 5.1, one may check that any solution $u$ to the above resolvent Dirichlet problem necessarily satisfies $E_D u = (\lambda I - \Delta)^{-1} E_D f$.

Now if $s \in (1/p, 1+1/p)$, $f \in [\dot{H}_0^{s-1,p} \cap \dot{H}_0^{s,p}](\mathbb{R}_+^n, \mathbb{C})$ then we have, thanks to previous considerations, $U := (\lambda I - \Delta)^{-1} E_D f \in \dot{H}^{s-1,p}(\mathbb{R}^n, \mathbb{C})$. It suffices to show that $U \in \dot{H}^{s,p}(\mathbb{R}^n, \mathbb{C})$, which is true. Indeed, we have

$$ 
\begin{aligned}
|\lambda|||U||_{\dot{H}^{s,p}(\mathbb{R}^n)} & \lesssim_{s,p,n,\mu} |\lambda|||\nabla U||_{\dot{H}^{s-1,p}(\mathbb{R}^n)} \\
& \lesssim_{s,p,n,\mu} \|\nabla E_D f\|_{\dot{H}^{s-1,p}(\mathbb{R}^n)} \\
& \lesssim_{s,p,n,\mu} \sum_{k=1}^n \|\partial_{x_k} E_D f\|_{\dot{H}^{s-1,p}(\mathbb{R}^n)}.
\end{aligned}
$$

Since equalities $\partial_{x_k} E_D f = E_D \partial_{x_k} f$, $k \in [1, n-1]$ and $\partial_{x_n} E_D f = E_N \partial_{x_n} f$ occur in $S'(\mathbb{R}^n, \mathbb{C})$, we deduce

$$ |\lambda|||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \le |\lambda|||U||_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n,\mu} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} $$

One may proceed similarly as before to obtain the full estimate

$$ |\lambda|||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}\|\nabla u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} $$

Thus the estimates still hold by density for all $f \in \dot{H}_0^{s,p}(\mathbb{R}_+^n)$, $s \in (-1 + 1/p, 1 + 1/p)$, $s \ne 1/p$, whenever $(C_{s,p})$ is satisfied.

The $\dot{H}^{\alpha, \tilde{p}}$-estimate for $f \in [\dot{H}_0^{s,p} \cap \dot{H}_0^{\alpha, \tilde{p}}](\mathbb{R}_+^n)$ can be obtained the same way, whenever $(C_{s,p})$ is satisfied.

The case of Besov spaces $\dot{B}_{p,q,0}^s$ can be achieved via similar argument for $q < +\infty$, the case $q = +\infty$ is obtained via real interpolation. The case of the $\dot{B}_{\tilde{p}, \tilde{q}, 0}^{\alpha}$-estimate for $f \in \dot{B}_{p,q,0}^s \cap \dot{B}_{\tilde{p}, \tilde{q}, 0}^{\alpha}$ can be done as above. ■

The proof for the Neumann resolvent problem in the proposition below is fairly similar to the proof of Proposition 5.4, a complex interpolation argument allows values $s = 1/p$ and $\alpha = 1/\tilde{p}$.

**Proposition 5.6** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1 + 1/p, 1 + 1/p)$, $\alpha \in (-1 + 1/\tilde{p}, 1 + 1/\tilde{p})$ and $\lambda \in \Sigma_\mu$ provided $\mu \in [0, \pi)$. We assume that $(C_{s,p})$, and we let $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$. Let us consider the resolvent Neumann problem with homogeneous boundary condition:

$$ 
\left\{ 
\begin{array}{ll} 
\lambda u - \Delta u = f, & \text{in } \mathbb{R}_+^n, \\ 
\partial_\nu u|_{\partial \mathbb{R}_+^n} = 0, & \text{on } \partial \mathbb{R}_+^n. 
\end{array} 
\right. 
\qquad (\mathcal{N}\mathcal{L}_\lambda) $$

The problem $(\mathcal{N}\mathcal{L}_\lambda)$ admits a unique solution $u \in [\dot{H}^{s,p} \cap \dot{H}^{s+2,p}](\mathbb{R}_+^n, \mathbb{C})$ with the estimate

$$ |\lambda|||u||_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}\|\nabla u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s,\mu} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} $$

If moreover $f \in \dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$, then we also have $u \in [\dot{H}^{\alpha, \tilde{p}} \cap \dot{H}^{\alpha+2, \tilde{p}}](\mathbb{R}_+^n, \mathbb{C})$ with corresponding the estimate

$$ |\lambda|||u||_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + |\lambda|^{\frac{1}{2}}\|\nabla u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + \|\nabla^2 u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha, \mu} \|f\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} $$

The result still holds replacing $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{H}^{\alpha, \tilde{p}}, \dot{H}^{\alpha+2, \tilde{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+2})$ whenever $(C_{s,p,q})$ is satisfied.

The whole result still holds for inhomogeneous function spaces, replacing $(\dot{H}, \dot{B})$ by $(H, B)$. However, in this case, the consideration of intersection spaces and the conditions $(C_{s,p})$ and $(C_{s,p,q})$ are no longer necessary.

**Proposition 5.7** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1 + 1/p, +\infty)$, $\alpha \in (-1 + 1/\tilde{p}, +\infty)$ such that $(C_{s+2,p})$ is satisfied. For $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, $g \in \dot{B}_{p,p}^{s+2-1/p}(\mathbb{R}^{n-1}, \mathbb{C})$, let us consider the Dirichlet
Now, we can set $u := v + w$ which is a solution of $(\mathcal{D}\mathcal{L}_0)$, and triangle inequality leads to

$$ \begin{aligned} \|u\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} &\le \|v\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} + \|w\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} \\ &\lesssim_{p,n,s} \|v\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|v(\cdot, 0)\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})} \\ &\lesssim_{p,n,s} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})} \end{aligned} $$

which was the desired bound.

The Besov spaces case for $(f,g) \in \dot{B}_{p,q}^s(\mathbb{R}_+^n, \mathbb{C}) \times \dot{B}_{p,q}^{s+2-1/p}(\mathbb{R}^{n-1}, \mathbb{C})$, whenever $(C_{s+2,p,q})$ is satisfied, follows the same lines as before, except when $q = +\infty$ where the uniqueness argument can only be checked in a weak sense since $(e^{-x_n(-\Delta')^{1/2}})_{x_n \ge 0}$ is only weak* continuous in $\dot{B}_{p,\infty}^{s+2-1/p}(\mathbb{R}^{n-1}, \mathbb{C})$.

Now, if we assume that $f \in [\dot{H}^{s,p} \cap \dot{H}^{\alpha, \bar{p}}](\mathbb{R}_+^n, \mathbb{C})$ and $g \in [\dot{B}_{p,p}^{s+2-1/p} \cap \dot{B}_{\bar{p},\bar{p}}^{\alpha+2-1/\bar{p}}](\mathbb{R}^{n-1}, \mathbb{C})$, then with the same notations as above, by Proposition 4.6, we have

$$ v = (-\Delta)^{-1} F \in [\dot{H}^{s+2,p} \cap \dot{H}^{\alpha+2, \bar{p}}](\mathbb{R}^n, \mathbb{C}) \text{ and } v(\cdot, 0) \in \dot{B}_{p,p}^{s+2-1/p} \cap \dot{B}_{\bar{p},\bar{p}}^{\alpha+2-1/\bar{p}}](\mathbb{R}^{n-1}, \mathbb{C}). $$

From this, one may reproduce the estimates as above to obtain

$$ \|\nabla^2 u\|_{\dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n)} \lesssim_{\bar{p}, n, \alpha} \|f\|_{\dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{\bar{p}, \bar{p}}^{\alpha+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

The case of intersection of Besov spaces follows the same lines.

<figure><img src="image_7.png" /></figure>

We state the same result for the corresponding Neumann problem, for which the proof is very close.

**Proposition 6.8** Let $p, \bar{p} \in (1, +\infty)$, $q, \bar{q} \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, +\infty)$, $\alpha \in (-1 + \frac{1}{\bar{p}}, +\infty)$, such that $(C_{s+2,p})$ is satisfied. For $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, $g \in \dot{B}_{p,p}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$, let us consider the Neumann problem with inhomogeneous boundary condition:

$$ \left\{ \begin{array}{ll} -\Delta u = f, & \text{in } \mathbb{R}_+^n, \\ \partial_\nu u|_{\partial \mathbb{R}_+^n} = g, & \text{on } \partial \mathbb{R}_+^n. \end{array} \right. \qquad (\mathcal{N}\mathcal{L}_0) $$

The problem $(\mathcal{N}\mathcal{L}_0)$ admits a unique solution $u$ such that

$$ u \in \dot{H}^{s+2,p}(\mathbb{R}_+^n, \mathbb{C}) \subset C_{0,x_n}^0 (\overline{\mathbb{R}_+}, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})) $$

with the estimate

$$ \|u\|_{L^\infty(\mathbb{R}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

If moreover $f \in \dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n, \mathbb{C})$ and $g \in \dot{B}_{\bar{p}, \bar{p}}^{\alpha+1-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ then the solution $u$ also satisfies $u \in \dot{H}^{\alpha+2, \bar{p}}(\mathbb{R}_+^n, \mathbb{C})$ with the corresponding estimate

$$ \|\nabla^2 u\|_{\dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n)} \lesssim_{\bar{p}, n, \alpha} \|f\|_{\dot{H}^{\alpha, \bar{p}}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{\bar{p}, \bar{p}}^{\alpha+1-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

The result still holds replacing $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{B}_{p,p}^{s+1-\frac{1}{p}}, \dot{B}_{p,p}^{s+2-\frac{1}{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{p,q}^{s+1-\frac{1}{p}}, \dot{B}_{p,q}^{s+2-\frac{1}{p}})$ and $(\dot{H}^{\alpha, \bar{p}}, \dot{H}^{\alpha+2, \bar{p}}, \dot{B}_{\bar{p}, \bar{p}}^{\alpha+1-\frac{1}{p}})$ by $(\dot{B}_{\bar{p}, \bar{q}}^{\alpha}, \dot{B}_{\bar{p}, \bar{q}}^{\alpha+2}, \dot{B}_{\bar{p}, \bar{q}}^{\alpha+1-\frac{1}{p}})$ whenever $(C_{s+2,p,q})$ is satisfied and $q < +\infty$.

If $q = +\infty$, everything still holds except $x_n \mapsto u(\cdot, x_n)$ is no more strongly continuous but only weak* continuous with values in $\dot{B}_{p,\infty}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$.

We notice that similar, but a little bit different, results of well-posedness and regularity are also available in [DM15, Chapter 3] with arguments of a different nature, and the case of Sobolev spaces and the resolvent problems are not treated.
problem with inhomogeneous boundary condition:

$$ \left\{ \begin{array}{ll} -\Delta u = f, & \text{in } \mathbb{R}_+^n, \\ u|_{\partial \mathbb{R}_+^n} = g, & \text{on } \partial \mathbb{R}_+^n. \end{array} \right. \qquad (\mathcal{D}\mathcal{L}_0) $$

The problem $(\mathcal{D}\mathcal{L}_0)$ admits a unique solution $u$ such that

$$ u \in \dot{H}^{s+2,p}(\mathbb{R}_+^n, \mathbb{C}) \subset C_{0,x_n}^0(\overline{\mathbb{R}_+}, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})) $$

with the estimate

$$ \|u\|_{L^{\infty}(\mathbb{R}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

If moreover $f \in \dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$ and $g \in \dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ then the solution $u$ also satisfies $u \in \dot{H}^{\alpha+2, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$ with the corresponding estimate

$$ \|\nabla^2 u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha} \|f\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

The result still holds if we replace both families of spaces $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{B}_{p,p}^{s+2-\frac{1}{p}})$ and $(\dot{H}^{\alpha, \tilde{p}}, \dot{H}^{\alpha+2, \tilde{p}}, \dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+2-\frac{1}{p}})$ by respectively $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{p,q}^{s+2-\frac{1}{p}})$ and $(\dot{B}_{\tilde{p}, \tilde{q}}^{\alpha}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+2}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+2-\frac{1}{p}})$ whenever $(C_{s+2,p,q})$ is satisfied, $q < +\infty$.

If $q = +\infty$, everything still holds except $x_n \mapsto u(\cdot, x_n)$ is no more strongly continuous but only weak* continuous with values in $\dot{B}_{p,q}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$.

Proof. — Let $p \in (1, +\infty)$, $s > -1 + 1/p$, such that $(C_{s+2,p})$ is satisfied. Then for $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, $g \in \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ we can write the problem $(\mathcal{D}\mathcal{L}_0)$ as an evolution problem in the $x_n$ variable,

$$ \left\{ \begin{array}{ll} -\partial_{x_n}^2 u - \Delta' u = f, & \text{in } \mathbb{R}^{n-1} \times (0, +\infty), \\ u(\cdot, 0) = g, & \text{on } \mathbb{R}^{n-1}. \end{array} \right. \qquad (5.3) $$

Thanks to [ABHN11, Theorem 3.8.3], considering the semigroup $(e^{-x_n(-\Delta')^{1/2}})_{x_n \ge 0}$ and its mapping properties given by Proposition B.2 and Theorem 4.3, if $f = 0$, the problem (5.3) admits a unique solution $u \in C_{0,x_n}^0(\overline{\mathbb{R}_+}, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C}))$. Thus, by linearity, we also have uniqueness of the solution $u$ in $C_{0,x_n}^0(\overline{\mathbb{R}_+}, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C}))$ for non-identically zero function $f$. Therefore, it suffices to construct a solution.

Since $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, by definition, there exists $F \in \dot{H}^{s,p}(\mathbb{R}^n, \mathbb{C})$ such that

$$ F|_{\mathbb{R}_+^n} = f, \quad \text{and} \quad \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \sim \|F\|_{\dot{H}^{s,p}(\mathbb{R}^n)}. $$

Let $v := (-\Delta)^{-1} F \in \dot{H}^{s+2,p}(\mathbb{R}^n, \mathbb{C})$, this follows from the isomorphism property, which is itself due to the point (ii) of Proposition 2.2, $-\Delta : \dot{H}^{s+2,p}(\mathbb{R}^n, \mathbb{C}) \to \dot{H}^{s,p}(\mathbb{R}^n, \mathbb{C})$ since $(C_{s+2,p})$ is assumed (one may argue by density). One obtains consequently the estimates

$$ \|v\|_{\dot{H}^{s+2,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|F\|_{\dot{H}^{s,p}(\mathbb{R}^n)} \lesssim_{s,p,n} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

So it suffices to prove the result for $w \in \dot{H}^{s+2,p}(\mathbb{R}_+^n, \mathbb{C})$, such that

$$ \left\{ \begin{array}{ll} -\Delta w = 0, & \text{in } \mathbb{R}^{n-1} \times (0, +\infty), \\ w|_{\partial \mathbb{R}_+^n} = \tilde{g}, & \text{on } \mathbb{R}^{n-1}, \end{array} \right. $$

where $\tilde{g} \in \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$ can be seen as $g - v(\cdot, 0)$. But such a $w$ exists and is unique thanks to Proposition B.2 and [ABHN11, Theorem 3.8.3], and satisfies

$$ \|w\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|\tilde{g}\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$
# A Complex interpolation for intersection of homogeneous Besov spaces

The next result is direct. Thanks to the fact that for all $a, b > 0$, $\theta \in [0, 1]$,

$$a + a^{1-\theta}b^{\theta} \le (a+b)^{\theta}a^{1-\theta} \le 2^{\theta}(a+a^{1-\theta}b^{\theta}),$$

and since for $q \in [1, +\infty)$, $s_0, s_1 \in \mathbb{R}$, and for $\theta \in (0, 1)$, if $s = (1 - \theta)s_0 + \theta s_1$, we have with equivalence of norms

$$\ell_{s_0}^q(\mathbb{Z}) \cap \ell_s^q(\mathbb{Z}) = \ell^q(\mathbb{Z}, (2^{ks_0q} + 2^{ksq})dk) = \ell^q(\mathbb{Z}, (2^{ks_0q} + 2^{ks_1q})^{\theta}2^{ks_0q(1-\theta)}dk),$$

therefore, by complex interpolation of weighted $\ell^q$ spaces, see [Tri78, Section 1.18.5], we obtain

**Proposition A.1** Let $q \in [1, +\infty)$, $s_0, s_1 \in \mathbb{R}$, consider a complex Banach space $X$, and for $\theta \in (0, 1)$ let's introduce $s := (1 - \theta)s_0 + \theta s_1$. The following equality holds with equivalence of norms

$$[\ell_{s_0}^q(\mathbb{Z}, X), \ell_{s_0}^q(\mathbb{Z}, X) \cap \ell_{s_1}^q(\mathbb{Z}, X)]_\theta = \ell_{s_0}^q(\mathbb{Z}, X) \cap \ell_s^q(\mathbb{Z}, X).$$

The result still holds with $\mathbb{N}$ instead of $\mathbb{Z}$.

The next corollary will have its importance in the proof of the next Proposition B.2.

**Corollary A.2** Let $p \in [1, +\infty]$, $q \in [1, +\infty)$, $s_j \in \mathbb{R}$, $j \in \{0, 1\}$ such that $(C_{s_0,p,q})$ is satisfied. Then for $\theta \in (0, 1)$, let's introduce $s := (1 - \theta)s_0 + \theta s_1$. Then the following equality holds with equivalence for norms

$$[\dot{B}_{p,q}^{s_0}(\mathbb{R}^n), \dot{B}_{p,q}^{s_0}(\mathbb{R}^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}^n)]_\theta = \dot{B}_{p,q}^{s_0}(\mathbb{R}^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}^n).$$

*Proof.* — Both function spaces $\dot{B}_{p,q}^{s_0}(\mathbb{R}^n)$, and $\dot{B}_{p,q}^{s_0}(\mathbb{R}^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}^n)$ are complete normed vector spaces, see [BCD11, Theorem 2.25].

Now, we apply [BL76, Theorem 6.4.2] and Proposition A.1, claiming that, for all $s \in \mathbb{R}$, $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is a retraction of $\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))$ through the homogeneous Littlewood-Paley decomposition $(\Delta_j)_{j \in \mathbb{Z}}$. ■

# B Estimates for the Poisson semigroup

**Lemma B.1** Let $s > 0$, $\alpha \ge 0$ and $p, q \in [1, +\infty]^2$. For all $u \in S'_h(\mathbb{R}^n)$,

$$\|u\|_{\dot{B}_{p,q}^{\alpha-s}(\mathbb{R}^n)} \sim_{p,s,\alpha,n,q} \|t \mapsto \|t^s(-\Delta)^{\frac{\alpha}{2}} e^{-t(-\Delta)^{\frac{1}{2}}} u\|_{L^p(\mathbb{R}^n)} \|u\|_{L_*^q(\mathbb{R}_+)}.$$

*Proof.* — It suffices to show the result for $\alpha = 0$. But in this case, the proof is straightforward the same as the one of [BCD11, Theorem 2.34] for the heat semigroup. ■

The following result was already proven in the case of homogeneous Besov spaces only, see [DM09, Lemma 2]. It is extended here to the case of homogeneous Sobolev spaces with a new proof that also cover the case of Besov spaces.

**Proposition B.2** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$. The map

$$T : f \mapsto [(x', x_n) \mapsto e^{-x_n(-\Delta')^{\frac{1}{2}}} f(x')]$$

is such that

(i) Given $s \ge 0$, for all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, we have

$$\|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{s,p,n} \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}.$$

In particular, $T$ extends uniquely as a bounded linear operator $T : \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) \to \dot{H}^{s,p}(\mathbb{R}_+^n)$ whenever $(C_{s,p})$ is satisfied.
Now, we can set $u := v + w$ which is a solution of $(\mathcal{D}\mathcal{L}_0)$, and the triangle inequality leads to

$$
\begin{aligned}
\|u\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} &\le \|v\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} + \|w\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} \\
&\lesssim_{p,n,s} \|v\|_{\dot{H}^{s+2,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|v(\cdot, 0)\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})} \\
&\lesssim_{p,n,s} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1})}
\end{aligned}
$$

which was the desired bound.

The Besov spaces case for $(f, g) \in \dot{B}_{p,q}^s(\mathbb{R}_+^n, \mathbb{C}) \times \dot{B}_{p,q}^{s+2-1/p}(\mathbb{R}^{n-1}, \mathbb{C})$, whenever $(C_{s+2,p,q})$ is satisfied, follows the same lines as before, except when $q = +\infty$ where the uniqueness argument can only be checked in a weak sense since $(e^{-x_n(-\Delta')^{1/2}})_{x_n \ge 0}$ is only weak* continuous in $\dot{B}_{p,\infty}^{s+2-1/p}(\mathbb{R}^{n-1}, \mathbb{C})$.

Now, if we assume that $f \in [\dot{H}^{s,p} \cap \dot{H}^{\alpha, \tilde{p}}](\mathbb{R}_+^n, \mathbb{C})$ and $g \in [\dot{B}_{p,p}^{s+2-1/p} \cap \dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+2-1/\tilde{p}}](\mathbb{R}^{n-1}, \mathbb{C})$, then with the same notations as above, by Proposition 4.6, we have

$$v = (-\Delta)^{-1} F \in [\dot{H}^{s+2,p} \cap \dot{H}^{\alpha+2, \tilde{p}}](\mathbb{R}^n, \mathbb{C}) \text{ and } v(\cdot, 0) \in \dot{B}_{p,p}^{s+2-1/p} \cap \dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+2-1/\tilde{p}}](\mathbb{R}^{n-1}, \mathbb{C}).$$

As before, the fact that $v \in [\dot{H}^{s+2,p} \cap \dot{H}^{\alpha+2, \tilde{p}}](\mathbb{R}^n, \mathbb{C})$ can be obtained arguing by density, by mean of point (ii) of Proposition 2.2 and Lemma 2.5. From this, one may reproduce the estimates as above to obtain

$$\|\nabla^2 u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha} \|f\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+2-\frac{1}{p}}(\mathbb{R}^{n-1})}.$$

The case of intersection of Besov spaces follows the same lines.

<figure><img src="image_9.png" /></figure>

We state the same result for the corresponding Neumann problem, for which the proof is very close.

**Proposition 5.8** Let $p, \tilde{p} \in (1, +\infty)$, $q, \tilde{q} \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, +\infty)$, $\alpha \in (-1 + \frac{1}{\tilde{p}}, +\infty)$, such that $(C_{s+2,p})$ is satisfied. For $f \in \dot{H}^{s,p}(\mathbb{R}_+^n, \mathbb{C})$, $g \in \dot{B}_{p,p}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$, let us consider the Neumann problem with inhomogeneous boundary condition:

$$\left\{
\begin{array}{ll}
-\Delta u = f, & \text{in } \mathbb{R}_+^n, \\
\partial_\nu u|_{\partial \mathbb{R}_+^n} = g, & \text{on } \partial \mathbb{R}_+^n.
\end{array}
\right.
\qquad (\mathcal{N}\mathcal{L}_0)$$

The problem $(\mathcal{N}\mathcal{L}_0)$ admits a unique solution $u$ such that

$$u \in \dot{H}^{s+2,p}(\mathbb{R}_+^n, \mathbb{C}) \subset C_{0,x_n}^0 (\overline{\mathbb{R}_+}, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C}))$$

with the estimate

$$\|u\|_{L^\infty(\mathbb{R}_+, \dot{B}_{p,p}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}))} \lesssim_{s,p,n} \|\nabla^2 u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{p,p}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1})}.$$

If moreover $f \in \dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$ and $g \in \dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+1-\frac{1}{\tilde{p}}}(\mathbb{R}^{n-1}, \mathbb{C})$ then the solution $u$ also satisfies $u \in \dot{H}^{\alpha+2, \tilde{p}}(\mathbb{R}_+^n, \mathbb{C})$ with the corresponding estimate

$$\|\nabla^2 u\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} \lesssim_{\tilde{p}, n, \alpha} \|f\|_{\dot{H}^{\alpha, \tilde{p}}(\mathbb{R}_+^n)} + \|g\|_{\dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+1-\frac{1}{p}}(\mathbb{R}^{n-1})}.$$

The result still holds, replacing $(\dot{H}^{s,p}, \dot{H}^{s+2,p}, \dot{B}_{p,p}^{s+1-\frac{1}{p}}, \dot{B}_{p,p}^{s+2-\frac{1}{p}})$ by $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+2}, \dot{B}_{p,q}^{s+1-\frac{1}{p}}, \dot{B}_{p,q}^{s+2-\frac{1}{p}})$ and $(\dot{H}^{\alpha, \tilde{p}}, \dot{H}^{\alpha+2, \tilde{p}}, \dot{B}_{\tilde{p}, \tilde{p}}^{\alpha+1-\frac{1}{\tilde{p}}})$ by $(\dot{B}_{\tilde{p}, \tilde{q}}^{\alpha}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+2}, \dot{B}_{\tilde{p}, \tilde{q}}^{\alpha+1-\frac{1}{\tilde{p}}})$ whenever $(C_{s+2,p,q})$ is satisfied and $q < +\infty$.

If $q = +\infty$, everything still holds except $x_n \mapsto u(\cdot, x_n)$ is no more strongly continuous but only weak* continuous with values in $\dot{B}_{p, \infty}^{s+2-\frac{1}{p}}(\mathbb{R}^{n-1}, \mathbb{C})$.

We notice that similar, but a little bit different, results of well-posedness and regularity are also available in [DM15, Chapter 3] with arguments of a different nature, moreover the case of Sobolev spaces and the resolvent problems does not seem to be treated.
(ii) Given $s > 0$, for all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, we have
$$ \|Tf\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \lesssim_{s,p,n} \|f\|_{\dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} $$

In particular, $T$ extends uniquely as a bounded linear operator $T: \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) \to \dot{B}_{p,q}^s(\mathbb{R}_+^n)$ whenever $(C_{s,p,q})$ is satisfied.

**Proof.** — **Point (i):** For $p \in (1, +\infty)$, let's consider $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})$. We apply Lemma B.1 to obtain,

$$ \begin{aligned} \|Tf\|_{L^p(\mathbb{R}_+^n)} &= \left( \int_0^{+\infty} \|e^{-x_n(-\Delta')^{\frac{1}{2}}} f\|_{L^p(\mathbb{R}^{n-1})}^p dx_n \right)^{\frac{1}{p}} \\ &= \left( \int_0^{+\infty} \left( t^{\frac{1}{p}} \|e^{-t(-\Delta')^{\frac{1}{2}}} f\|_{L^p(\mathbb{R}^{n-1})} \right)^p \frac{dt}{t} \right)^{\frac{1}{p}} \\ &\lesssim_{p,n} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot \end{aligned} $$

We continue noticing that for all $f \in S'_h(\mathbb{R}^{n-1})$, $m \in \mathbb{N}$, $\partial_{x_n}^m Tf = (-\Delta')^{\frac{m}{2}} Tf = T(-\Delta')^{\frac{m}{2}} f$ and $Tf \in S'_h(\mathbb{R}^{n-1})$, thus if $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})$ we may apply previous inequality to obtain,

$$ \begin{aligned} \|Tf\|_{\dot{H}^{m,p}(\mathbb{R}_+^n)} &\lesssim_{p,n,m} \|\partial_{x_n}^m Tf\|_{L^p(\mathbb{R}_+^n)} + \|(-\Delta')^{\frac{m}{2}} Tf\|_{L^p(\mathbb{R}_+^n)} \\ &\sim_{p,n,m} \|T(-\Delta')^{\frac{m}{2}} f\|_{L^p(\mathbb{R}_+^n)} \\ &\lesssim_{p,n,m} \|f\|_{\dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot \end{aligned} $$

So that for all $m \in \mathbb{N}$, all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{H^{m,p}(\mathbb{R}_+^n)} \lesssim_{p,n,m} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$

Thus, by complex interpolation and Corollary A.2, for all $s \ge 0$, all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{H^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$

Hence, thanks to Proposition 3.3, $H^{s,p}(\mathbb{R}_+^n) = L^p(\mathbb{R}_+^n) \cap \dot{H}^{s,p}(\mathbb{R}_+^n)$,

$$ \|Tf\|_{L^p(\mathbb{R}_+^n)} + \|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$

Therefore, if $\lambda \in 2^\mathbb{N}$, we can consider $f_\lambda$ is the dilation by factor $\lambda$ of $f$, so that plugging $f_\lambda$ instead of $f$ in above inequality, and checking the fact that $Tf_\lambda = (Tf)_\lambda$, we obtain

$$ \lambda^{-\frac{n}{p}} \|Tf\|_{L^p(\mathbb{R}_+^n)} + \lambda^{s-\frac{n}{p}} \|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \lambda^{-\frac{n}{p}} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \lambda^{s-\frac{n}{p}} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$

One may divide above inequality by $\lambda^{s-\frac{n}{p}}$, so as $\lambda$ tends to infinity, it yields

$$ \|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$

So that the result holds by density whenever $(C_{s,p})$ is satisfied.

**Point (ii):** Now let $q \in [1, +\infty]$, since for all $s \ge 0$, all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{H^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$

Hence, by real interpolation, using [Haa06, Proposition B.2.7] instead of Corollary A.2, we obtain that for all $s > 0$ all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{B_{p,q}^s(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \cdot $$
# A Complex interpolation for intersection of homogeneous Besov spaces

The next result is direct. Thanks to the fact that for all $a, b > 0$, $\theta \in [0, 1]$,

$$a + a^{1-\theta}b^{\theta} \le 2(a+b)^{\theta}a^{1-\theta} \le 2(a+a^{1-\theta}b^{\theta}),$$

and since for $q \in [1, +\infty)$, $s_0, s_1 \in \mathbb{R}$, and for $\theta \in (0, 1)$, if $s = (1 - \theta)s_0 + \theta s_1$, we have with equivalence of norms

$$\ell_{s_0}^q(\mathbb{Z}) \cap \ell_s^q(\mathbb{Z}) = \ell^q(\mathbb{Z}, (2^{ks_0q} + 2^{ksq})dk) = \ell^q(\mathbb{Z}, (2^{ks_0q} + 2^{ks_1q})^{\theta}2^{ks_0q(1-\theta)}dk),$$

therefore, by complex interpolation of weighted $\ell^q$ spaces, see [Tri78, Section 1.18.5], we obtain

**Proposition A.1** Let $q \in [1, +\infty)$, $s_0, s_1 \in \mathbb{R}$, consider a complex Banach space $X$, and for $\theta \in (0, 1)$ let's introduce $s := (1 - \theta)s_0 + \theta s_1$. The following equality holds with equivalence of norms

$$[\ell_{s_0}^q(\mathbb{Z}, X), \ell_{s_0}^q(\mathbb{Z}, X) \cap \ell_{s_1}^q(\mathbb{Z}, X)]_{\theta} = \ell_{s_0}^q(\mathbb{Z}, X) \cap \ell_s^q(\mathbb{Z}, X).$$

The result still holds with $\mathbb{N}$ instead of $\mathbb{Z}$.

The next corollary will have its importance in the proof of the next Proposition B.2.

**Corollary A.2** Let $p \in [1, +\infty)$, $q \in [1, +\infty)$, $s_j \in \mathbb{R}$, $j \in \{0, 1\}$ such that $(C_{s_0,p,q})$ is satisfied. Then for $\theta \in (0, 1)$, let's introduce $s := (1 - \theta)s_0 + \theta s_1$. Then the following equality holds with equivalence of norms

$$[\dot{B}_{p,q}^{s_0}(\mathbb{R}^n), \dot{B}_{p,q}^{s_0}(\mathbb{R}^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}^n)]_{\theta} = \dot{B}_{p,q}^{s_0}(\mathbb{R}^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}^n).$$

*Proof.* — Both function spaces $\dot{B}_{p,q}^{s_0}(\mathbb{R}^n)$, and $\dot{B}_{p,q}^{s_0}(\mathbb{R}^n) \cap \dot{B}_{p,q}^{s_1}(\mathbb{R}^n)$ are complete normed vector spaces, see [BCD11, Theorem 2.25].

Now, we apply [BL76, Theorem 6.4.2] and Proposition A.1, claiming that, for all $s \in \mathbb{R}$, $\dot{B}_{p,q}^s(\mathbb{R}^n)$ is a retraction of $\ell_s^q(\mathbb{Z}, L^p(\mathbb{R}^n))$ through the homogeneous Littlewood-Paley decomposition $(\dot{\Delta}_j)_{j \in \mathbb{Z}}$. ■

# B Estimates for the Poisson semigroup

**Lemma B.1** Let $s > 0$, $\alpha \ge 0$ and $p, q \in [1, +\infty]^2$. For all $u \in S'_h(\mathbb{R}^n)$,

$$\|u\|_{\dot{B}_{p,q}^{\alpha-s}(\mathbb{R}^n)} \sim_{p,s,\alpha,n,q} \|t \mapsto \|t^s(-\Delta)^{\frac{\alpha}{2}} e^{-t(-\Delta)^{\frac{1}{2}}} u\|_{L^p(\mathbb{R}^n)} \|f\|_{L_s^q(\mathbb{R}_+)}.$$

*Proof.* — It suffices to show the result for $\alpha = 0$. But in this case, the proof is straightforward the same as the one of [BCD11, Theorem 2.34] for the heat semigroup. ■

The following result was already proven in the case of homogeneous Besov spaces only, see [DM09, Lemma 2]. It is extended here to the case of homogeneous Sobolev spaces, with a new proof that also cover the case of Besov spaces.

**Proposition B.2** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$. The map

$$T: f \mapsto [(x', x_n) \mapsto e^{-x_n(-\Delta')^{\frac{1}{2}}} f(x')]$$

is such that

(i) Given $s \ge 0$, for all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, we have

$$\|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{s,p,n} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})}.$$

In particular, $T$ extends uniquely as a bounded linear operator $T: \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) \to \dot{H}^{s,p}(\mathbb{R}_+^n)$ whenever $(C_{s,p})$ is satisfied.
Then the same dilation procedure as before, yields

$$ \|Tf\|_{\dot{B}_{p,q}^{s}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}, $$

which again allows to conclude via density argument if $q < +\infty$ and $(C_{s,p,q})$ is satisfied. The case $q = +\infty$, when $(C_{s,p,q})$ is satisfied, follows from real interpolation with the last estimate. ■

Proposition B.2 can be self-improved as

**Corollary B.3** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty)$, $j \in \{0, 1\}$. The map

$$ T : f \mapsto [(x', x_n) \mapsto e^{-x_n(-\Delta')}^{\frac{1}{2}} f(x')] $$

is such that

(i) Let $s_j \ge 0$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ is satisfied. For all $f \in [\dot{B}_{p_0, p_0}^{s_0-\frac{1}{p_0}} \cap \dot{B}_{p_1, p_1}^{s_1-\frac{1}{p_1}}](\mathbb{R}^{n-1})$, we have

$$ \|Tf\|_{\dot{H}^{s_j, p_j}(\mathbb{R}_+^n)} \lesssim_{s_j, p_j, n} \|f\|_{\dot{B}_{p_j, p_j}^{s_j-\frac{1}{p_j}}(\mathbb{R}^{n-1})}, \quad j \in \{0, 1\}. $$

(ii) Let $s_j > 0$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0, q_0})$ is satisfied. For all $f \in [\dot{B}_{p_0, q_0}^{s_0-\frac{1}{p_0}} \cap \dot{B}_{p_1, q_1}^{s_1-\frac{1}{p_1}}](\mathbb{R}^{n-1})$, we have

$$ \|Tf\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n)} \lesssim_{s_j, p_j, n} \|f\|_{\dot{B}_{p_j, q_j}^{s-\frac{1}{p_j}}(\mathbb{R}^{n-1})}, \quad j \in \{0, 1\}. $$

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[DHMT21] R. Danchin, M. Hieber, P.B. Mucha, and P. Tolksdorf. Free Boundary Problems via Da Prato-Grisvard Theory. *arXiv e-prints*, November 2021. arXiv:2011.07918v2.

[DM09] R. Danchin and P. B. Mucha. A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space. *J. Funct. Anal.*, 256(3):881-927, 2009.

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(ii) Given $s > 0$, for all $f \in \dot{B}_{p,q}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$, we have

$$ \|Tf\|_{\dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n})} \lesssim_{s,p,n} \|f\|_{\dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

In particular, $T$ extends uniquely as a bounded linear operator $T: \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}) \to \dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n})$ whenever $(C_{s,p,q})$ is satisfied.

**Proof.** — **Point (i):** For $p \in (1, +\infty)$, let's consider $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})$. We apply Lemma B.1 to obtain,

$$ \begin{aligned} \|Tf\|_{L^p(\mathbb{R}_+^n)} &= \left( \int_0^{+\infty} \|e^{-x_n(-\Delta')^{\frac{1}{2}}} f\|_{L^p(\mathbb{R}^{n-1})}^p dx_n \right)^{\frac{1}{p}} \\ &= \left( \int_0^{+\infty} \left( t^{\frac{1}{p}} \|e^{-t(-\Delta')^{\frac{1}{2}}} f\|_{L^p(\mathbb{R}^{n-1})} \right)^p \frac{dt}{t} \right)^{\frac{1}{p}} \\ &\lesssim_{p,n} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})}. \end{aligned} $$

We continue noticing that for all $f \in S'_h(\mathbb{R}^{n-1})$, $m \in \mathbb{N}$, $\partial_{x_n}^m Tf = (-\Delta')^{\frac{m}{2}} Tf = T(-\Delta')^{\frac{m}{2}} f$ and $Tf \in S'_h(\mathbb{R}^{n-1})$, thus if $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})$ we may apply previous inequality to obtain,

$$ \begin{aligned} \|Tf\|_{\dot{H}^{m,p}(\mathbb{R}_+^n)} &\lesssim_{p,n,m} \|\partial_{x_n}^m Tf\|_{L^p(\mathbb{R}_+^n)} + \|(-\Delta')^{\frac{m}{2}} Tf\|_{L^p(\mathbb{R}_+^n)} \\ &\sim_{p,n,m} \|T(-\Delta')^{\frac{m}{2}} f\|_{L^p(\mathbb{R}_+^n)} \\ &\lesssim_{p,n,m} \|f\|_{\dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})}. \end{aligned} $$

So that for all $m \in \mathbb{N}$, all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{H^{m,p}(\mathbb{R}_+^n)} \lesssim_{p,n,m} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{m-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

Thus, by complex interpolation and Corollary A.2, for all $s \ge 0$, all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{H^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

Hence, thanks to Proposition 3.3, $H^{s,p}(\mathbb{R}_+^n) = L^p(\mathbb{R}_+^n) \cap \dot{H}^{s,p}(\mathbb{R}_+^n)$,

$$ \|Tf\|_{L^p(\mathbb{R}_+^n)} + \|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

Therefore, if $\lambda \in 2^{\mathbb{N}}$, we can consider $f_{\lambda}$ the dilation by factor $\lambda$ of $f$, so that plugging $f_{\lambda}$ instead of $f$ in above inequality, and checking the fact that $Tf_{\lambda} = (Tf)_{\lambda}$, we obtain

$$ \lambda^{-\frac{n}{p}} \|Tf\|_{L^p(\mathbb{R}_+^n)} + \lambda^{s-\frac{n}{p}} \|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \lambda^{-\frac{n}{p}} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \lambda^{s-\frac{n}{p}} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

One may divide above inequality by $\lambda^{s-\frac{n}{p}}$, so as $\lambda$ tends to infinity, it yields

$$ \|Tf\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

So that the result holds by density whenever $(C_{s,p})$ is satisfied.

**Point (ii):** Now let $q \in [1, +\infty]$, since for all $s \ge 0$, all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{H^{s,p}(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$

Hence, by real interpolation, using [Haa06, Proposition B.2.7] instead of Corollary A.2, we obtain that for all $s > 0$ all $f \in \dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1}) \cap \dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})$,

$$ \|Tf\|_{B_{p,q}^{s}(\mathbb{R}_{+}^{n})} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,p}^{-\frac{1}{p}}(\mathbb{R}^{n-1})} + \|f\|_{\dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}. $$
[Gra14b] L. Grafakos. *Modern Fourier analysis*, volume 250 of *Graduate Texts in Mathematics*. Springer, New York, 3rd edition, 2014.

[Gui91] D. Guidetti. On interpolation with boundary conditions. *Math. Z.*, 207(3):439–460, 1991.

[Haa06] M. Haase. *The functional calculus for sectorial operators*, volume 169 of *Operator Theory: Advances and Applications*. Birkhäuser Verlag, Basel, 2006.

[IMT19] T. Iwabuchi, T. Matsuyama, and K. Taniguchi. Besov spaces on open sets. *Bull. Sci. Math.*, 152:93–149, 2019.

[Jaw78] Björn Jawerth. The trace of Sobolev and Besov spaces if $0 < p < 1$. *Studia Math.*, 62(1):65–71, 1978.

[JK95] D. Jerison and C. E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains. *J. Funct. Anal.*, 130(1):161–219, 1995.

[JW84] A. Jonsson and H. Wallin. *Function Spaces on Subsets of $\mathbb{R}^n$*, volume 2. Harwood Academic Publishers, 1984.

[KMM07] N. Kalton, S. Mayboroda, and M. Mitrea. Interpolation of Hardy-Sobolev-Besov-Triebel Lizorkin spaces and applications to problems in partial differential equations. In *Interpolation Theory and Applications (Contemporary Mathematics 445)*, pages 121–177, 2007.

[LM72] J. L. Lions and E. Magenes. *Non-homogeneous boundary value problems and applications. Vol. I.*, volume 181 of *Grundlehren Math. Wiss*. Springer, Cham, 1972. Translated from the French by P. Kenneth.

[Lun18] A. Lunardi. *Interpolation theory. Appunti*. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 3rd edition, 2018.

[Ouh05] E. M. Ouhabaz. *Analysis of heat equations on domains*, volume 31 of *London Mathematical Society Monographs Series*. Princeton University Press, Princeton, NJ, 2005.

[Ryc99] V.S. Rychkov. On Restrictions and Extensions of the Besov and Triebel-Lizorkin spaces with Respect to Lipschitz Domains. *Journal of The London Mathematical Society*, 60:237–257, 08 1999.

[Saw18] Y. Sawano. *Theory of Besov spaces*, volume 56 of *Developments in Mathematics*. Springer, Singapore, 2018.

[Sch10] C. Schneider. Trace Operators in Besov and Triebel-Lizorkin Spaces. *Z. Anal. Anwend.*, 29(1):275–302, 2010.

[Ste70] E.M. Stein. *Singular Integrals and Differentiability Properties of Functions*. Monographs in Harmonic Analysis. Princeton University Press, 1970.

[Str67] R.S. Strichartz. Multipliers on Fractional Sobolev Spaces. *Journal of Mathematics and Mechanics*, 16(9):1031–1060, 1967.

[Tri78] H. Triebel. *Interpolation Theory, Function Spaces, Differential Operators*. North-Holland Mathematical Library. Elsevier, 1978.

[Tri83] H. Triebel. *Theory of Function Spaces*. Modern Birkhäuser Classics. Birkhäuser Basel, 1983.

[Tri92] H. Triebel. *Theory of Function Spaces II*. Modern Birkhäuser Classics. Birkhäuser Basel, 1992.

[Tri15] H. Triebel. *Tempered homogeneous function spaces*. EMS Ser. Lect. Math. Zürich: European Mathematical Society (EMS), 2015.
Then the same dilation procedure as before, yields

$$ \|Tf\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \lesssim_{p,n,s} \|f\|_{\dot{B}_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})}, $$

which again allows concluding via a density argument if $q < +\infty$ and $(C_{s,p,q})$ is satisfied. The case $q = +\infty$, when $(C_{s,p,q})$ is satisfied, follows from real interpolation with the last estimate. ■

Proposition B.2 can be self-improved as

**Corollary B.3** Let $p_j \in (1, +\infty)$, $q_j \in [1, +\infty]$, $j \in \{0, 1\}$. The map

$$ T : f \mapsto \left[ (x', x_n) \mapsto e^{-x_n(-\Delta')}^{\frac{1}{2}} f(x') \right] $$

is such that

(i) If $s_j \ge 0$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0})$ is satisfied. For all $f \in [\dot{B}_{p_0, p_0}^{s_0-\frac{1}{p_0}} \cap \dot{B}_{p_1, p_1}^{s_1-\frac{1}{p_1}}](\mathbb{R}^{n-1})$, we have

$$ \|Tf\|_{\dot{B}^{s_j, p_j}(\mathbb{R}_+^n)} \lesssim_{s_j, p_j, n} \|f\|_{\dot{B}_{p_j, p_j}^{s_j-\frac{1}{p_j}}(\mathbb{R}^{n-1})}, \quad j \in \{0, 1\}. $$

(ii) If $s_j > 0$, $j \in \{0, 1\}$, such that $(C_{s_0, p_0, q_0})$ is satisfied. For all $f \in [\dot{B}_{p_0, q_0}^{s_0-\frac{1}{p_0}} \cap \dot{B}_{p_1, q_1}^{s_1-\frac{1}{p_1}}](\mathbb{R}^{n-1})$, we have

$$ \|Tf\|_{\dot{B}_{p_j, q_j}^{s_j}(\mathbb{R}_+^n)} \lesssim_{s_j, p_j, n} \|f\|_{\dot{B}_{p_j, q_j}^{s_j-\frac{1}{p_j}}(\mathbb{R}^{n-1})}, \quad j \in \{0, 1\}. $$

## C Weak* continuity of translations for endpoint Besov spaces

**Lemma C.1** Let $p \in (1, +\infty)$, $s \in \mathbb{R}$. Then for all $t \in \mathbb{R}$, the map

$$ \tau_t : u \mapsto [(x', x_n) \mapsto u(x', x_n + t)] $$

is uniformly bounded on $B_{p, \infty}^s(\mathbb{R}^n)$, that is for all $u \in B_{p, \infty}^s(\mathbb{R}^n)$

$$ \|\tau_t u\|_{B_{p, \infty}^s(\mathbb{R}^n)} \le \|u\|_{B_{p, \infty}^s(\mathbb{R}^n)}, \quad \forall t \in \mathbb{R}, $$

and has a limit

$$ \tau_t u \xrightarrow{t \to 0} u, \text{ weakly* in } B_{p, \infty}^s(\mathbb{R}^n). $$

* Assuming $s < n/p$, the results still hold with $\dot{B}$ instead of $B$.

* The whole result still holds with $\mathbb{R}_+^n$ instead of $\mathbb{R}^n$, with $t \ge 0$, assuming $s \in (-1 + 1/p, n/p)$ in the case of homogeneous Besov spaces.

*Proof.* — We only treat the case of homogeneous function spaces, the case of inhomogeneous function spaces can be treated with the same arguments.

**Step 1:** The case of the whole space $\mathbb{R}^n$. The proof relies on the fact that, by Proposition 2.8, $\dot{B}_{p, \infty}^s(\mathbb{R}^n) = (\dot{B}_{p', 1}^{-s}(\mathbb{R}^n))' = (\dot{B}_{p, \infty}^s(\mathbb{R}^n))''$.

Let $t \in \mathbb{R}$, $\phi \in \dot{B}_{p', 1}^{-s}(\mathbb{R}^n)$, for $u \in \dot{B}_{p, \infty}^s(\mathbb{R}^n)$, one obtains by strong continuity of translation on $\dot{B}_{p', 1}^{-s}(\mathbb{R}^n)$, see [Gui91, Proposition 1.9] (one may use instead an interpolation and density argument),

$$ \langle \tau_t u, \phi \rangle_{\mathbb{R}^n} = \langle u, \tau_{-t} \phi \rangle_{\mathbb{R}^n} \xrightarrow{t \to 0} \langle u, \phi \rangle_{\mathbb{R}^n}. $$

The norm estimate follows from the fact that for all $t \in \mathbb{R}$, $j \in \mathbb{Z}$, $\dot{\Delta}_j \tau_t u = \tau_t \dot{\Delta}_j u$ in $S'(\mathbb{R}^n)$, so that by translation invariance of Lebesgue norms,

$$ \|\tau_t u\|_{\dot{B}_{p, \infty}^s(\mathbb{R}^n)} = \sup_{j \in \mathbb{Z}} 2^{js} \|\tau_t \dot{\Delta}_j u\|_{L^p(\mathbb{R}^n)} = \sup_{j \in \mathbb{Z}} 2^{js} \|\dot{\Delta}_j u\|_{L^p(\mathbb{R}^n)} = \|u\|_{\dot{B}_{p, \infty}^s(\mathbb{R}^n)}. $$

**Step 2:** Now, the case of the half-space $\mathbb{R}_+^n$. We show the estimate first. Let $t \ge 0$ and $u \in \dot{B}_{p, \infty}^s(\mathbb{R}_+^n)$, we choose an arbitrary extension $U \in \dot{B}_{p, \infty}^s(\mathbb{R}^n)$ of $u$, i.e. such that $U|_{\mathbb{R}_+^n} = u$, then
$\tau_t U$ is an extension of $\tau_t u$. From this point, we deduce the following estimate by the definition of function spaces by restriction

$$ \|\tau_t u\|_{\dot{B}_{p,\infty}^s(\mathbb{R}_+^n)} \le \|\tau_t U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)} = \|U\|_{\dot{B}_{p,\infty}^s(\mathbb{R}^n)}. $$

One takes the infimum over all such $U$ to obtain

$$ \|\tau_t u\|_{\dot{B}_{p,\infty}^s(\mathbb{R}_+^n)} \le \|u\|_{\dot{B}_{p,\infty}^s(\mathbb{R}_+^n)}. $$

Now, we proceed as in **Step 1** using the fact that $\dot{B}_{p,\infty}^s(\mathbb{R}_+^n) = (\dot{B}_{p',1,0}^{-s}(\mathbb{R}_+^n))'$, see Proposition 3.23. Let $\phi \in \dot{B}_{p',1,0}^{-s}(\mathbb{R}_+^n)$, in particular, we have $\phi \in \dot{B}_{p',1}^{-s}(\mathbb{R}^n)$ with supp $\phi \subset \mathbb{R}_+^n$. Then for an arbitrary extension $U \in \dot{B}_{p,\infty}^s(\mathbb{R}^n)$ of $u$, we have $(\tau_t U)|_{\mathbb{R}_+^n} = \tau_t u$ for all $t \ge 0$. Therefore, by the previous **Step 1**, using the same arguments one can find in the proof of Proposition 3.23, it follows that

$$ \langle \tau_t u, \phi \rangle_{\mathbb{R}_+^n} = \langle \tau_t U, \phi \rangle_{\mathbb{R}^n} \xrightarrow{t \to 0} \langle U, \phi \rangle_{\mathbb{R}^n} = \langle u, \phi \rangle_{\mathbb{R}_+^n}, $$

which is the desired result.

<figure><img src="image_10.png" /></figure>

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[Haa06] M. Haase. *The functional calculus for sectorial operators*, volume 169 of *Operator Theory: Advances and Applications*. Birkhäuser Verlag, Basel, 2006.

[IMT19] T. Iwabuchi, T. Matsuyama, and K. Taniguchi. Besov spaces on open sets. *Bull. Sci. Math.*, 152:93–149, 2019.

[Jaw78] Björn Jawerth. The trace of Sobolev and Besov spaces if $0 < p < 1$. *Studia Math.*, 62(1):65–71, 1978.
[JK95] D. Jerison and C. E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains. *J. Funct. Anal.*, 130(1):161–219, 1995.

[JW84] A. Jonsson and H. Wallin. *Function Spaces on Subsets of $\mathbb{R}^n$, volume 2*. Harwood Academic Publishers, 1984.

[KMM07] N. Kalton, S. Mayboroda, and M. Mitrea. Interpolation of Hardy-Sobolev-Besov-Triebel Lizorkin spaces and applications to problems in partial differential equations. In *Interpolation Theory and Applications (Contemporary Mathematics 445)*, pages 121–177, 2007.

[LM72] J. L. Lions and E. Magenes. *Non-homogeneous boundary value problems and applications. Vol. I.*, volume 181 of *Grundlehren Math. Wiss*. Springer, Cham, 1972. Translated from the French by P. Kenneth.

[Lun18] A. Lunardi. *Interpolation theory*. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 3rd edition, 2018.

[Ouh05] E. M. Ouhabaz. *Analysis of heat equations on domains*, volume 31 of *London Mathematical Society Monographs Series*. Princeton University Press, Princeton, NJ, 2005.

[Ryc99] V.S. Rychkov. On Restrictions and Extensions of the Besov and Triebel-Lizorkin spaces with Respect to Lipschitz Domains. *Journal of The London Mathematical Society*, 60:237–257, 08 1999.

[Saw18] Y. Sawano. *Theory of Besov spaces*, volume 56 of *Developments in Mathematics*. Springer, Singapore, 2018.

[Sch10] C. Schneider. Trace Operators in Besov and Triebel-Lizorkin Spaces. *Z. Anal. Anwend.*, 29(1):275–302, 2010.

[Ste70] E.M. Stein. *Singular Integrals and Differentiability Properties of Functions*. Monographs in Harmonic Analysis. Princeton University Press, 1970.

[Str67] R.S. Strichartz. Multipliers on Fractional Sobolev Spaces. *Journal of Mathematics and Mechanics*, 16(9):1031–1060, 1967.

[Tri78] H. Triebel. *Interpolation Theory, Function Spaces, Differential Operators*. North-Holland Mathematical Library. Elsevier, 1978.

[Tri83] H. Triebel. *Theory of Function Spaces*. Modern Birkhäuser Classics. Birkhäuser Basel, 1983.

[Tri92] H. Triebel. *Theory of Function Spaces II*. Modern Birkhäuser Classics. Birkhäuser Basel, 1992.

[Tri15] H. Triebel. *Tempered homogeneous function spaces*. EMS Ser. Lect. Math. Zürich: European Mathematical Society (EMS), 2015.