where as before, $\beta$ includes the lower order terms, $\langle \cdot, \cdot \rangle$ is now the inner product on $L_x^2 = L^2(\Omega)$ and $\langle\langle \cdot, \cdot \rangle\rangle$ the sesquilinear duality extending the $L_t^2 L_x^2$ inner product. As $\mathcal{D}(\mathbb{R}; V)$ is a dense subspace of $\mathcal{V}$, we have

$$ \langle\langle \mathcal{H}u, v \rangle\rangle = -\langle\langle u, \partial_t v \rangle\rangle + \langle\langle \mathcal{L}u, v \rangle\rangle $$

for all $u \in \mathcal{V}$ and $v \in \mathcal{D}(\mathbb{R}; V)$, where $\mathcal{L}$ is defined by the integral above. Hölder's inequality, which is dimensionless in terms of exponents, yields

$$ |\langle\langle \mathcal{L}u, v \rangle\rangle| \le \|A\|_{\infty} \|\nabla u\|_{L_t^2 L_x^2} \|\nabla v\|_{L_t^2 L_x^2} + P_{\bar{r}_1, \bar{q}_1} \|u\|_{\Delta^{r_1, q_1}} \|v\|_{\Delta^{r_1, q_1}} $$

with $P_{\bar{r}_1, \bar{q}_1}$ as in (2.6), so that

$$ |\langle\langle \mathcal{H}u, v \rangle\rangle| \le C \|u\|_{\mathcal{V}} \|v\|_{\mathcal{V}}. $$

Hence, using $\mathcal{H}$ gives access to weak solutions in $L_t^2 V$ of $\partial_t u + \mathcal{L}u = w$ with lateral boundary conditions prescribed by $V$.

**4.5. Main regularity estimates.** Modifying the proofs of Section 2.4 on replacing $-\Delta$ systematically by $(1 - \Delta_{\mathcal{V}})$, the main regularity lemma becomes the following statement.

**Lemma 4.4.** Let $u \in \mathcal{D}'(\mathbb{R}; V')$ with $u \in L_t^2 V$ and $\partial_t u \in \Sigma^{r', q'}$ for $(r, q)$ an admissible pair or $(r, q) = (\infty, 2)$ under Assumption (V). Then $u \in C_0(L_x^2)$ and for some constant $C < \infty$ independent of $u$,

$$ \sup_{t \in \mathbb{R}} \|u(t)\|_{L_x^2} \le C (\|u\|_{L_t^2 V} + \|\partial_t u\|_{\Sigma^{r', q'}}). $$

Moreover, if $(r, q)$ is admissible, then $u \in \mathcal{V}$ with the same estimate on $\|u\|_{\mathcal{V}}$.

*Proof.* We indicate the main changes.

*Modification of the uniqueness Lemma 2.14.* This is now stated for $u \in \mathcal{D}'(\mathbb{R}; V')$ such that $\partial_t u + (1 - \Delta_{\mathcal{V}})u = 0$ in $\mathcal{D}'(\mathbb{R}; V')$: if $u \in L_t^2 V$, then $u = 0$. Indeed, we see that $\partial_t u \in L_t^2 V'$. By Lions' embedding theorem we have $u \in C_0(L_x^2)$, and testing the equation against $u$ yields that $\langle\langle (1 - \Delta_{\mathcal{V}})u, u \rangle\rangle = 0$, which implies $u = 0$.

*Modification of the embedding in Lemma 2.16.* Here, we have to show that with $\theta = 1 - \frac{2}{r}$ we have the continuous inclusion

$$ (L_t^r L_x^q)' \hookrightarrow \dot{H}_t^{-\theta/2} V^{\theta-1}, $$

where

$$ \dot{H}_t^{-\theta/2} V^{\theta-1} := \{D_t^{\theta/2} S^{1-\theta} g : g \in L_t^2 L_x^2\} \quad \text{with} \quad \|w\|_{\dot{H}_t^{-\theta/2} V^{\theta-1}} := \|g\|_{L_t^2 L_x^2}. $$

In the definition of this space, $S^{1-\theta}$ is extended dy duality to a map from $L^2$ into the dual of $V$ with respect to the $L_x^2$ duality. This uses that $V$ is the domain of $S$ and that $0 \le 1 - \theta \le 1$. Hence, we are working with a subspace of $\mathcal{D}'(\mathbb{R}; V')$. The embedding itself is a repetition of the proof of Lemma 2.16 except that now we take $G = S(\mathbb{R}; V)$ as dense subset. This is where we use assumption (V).
Modification of the stronger regularity statement in Lemma 2.17. We need a new dense subspace $G_0$, which we can take as $G_0 := S_{00}(\mathbb{R}; \text{dom}(\Delta_V^2))$ here.

Step 1 then goes through mutadis mutandis if we understand Fourier in the $x$-variable as a special case of the spectral theorem for the Laplacian, compare with the proof of Lemma 4.3.

Step 2 remains unchanged.

For Step 3, we obtain $v'(t) + (1 - \Delta_V)v(t) = w(t)$ in $L_x^2$ for all $t \in \mathbb{R}$, whenever $g \in G_0$. The equation can also be interpreted in $\mathcal{D}'(\mathbb{R}; V')$ for the test functions $\phi \in \mathcal{D}(\mathbb{R}; V)$: this interpretation passes to the limit for $g \in L_t^2 L_x^2$, thanks to Step 1.

Lastly, Step 4, again for $g \in G_0$, has been a Fourier transform argument and now its use in the $x$-variable should be replaced by the spectral theorem. From this perspective, the proof is the same as before.

*End of proof.* Modifications of Proposition 2.18 and of the corollaries that follow are proved similarly with constant $c = 0$. □

**4.6. The resulting theory.** From this point on, the theory can be developed similar to the *inhomogeneous setting* of Section 2.10, assuming $(V)$. The two exceptional topics are pointwise Gaussian upper bounds (Section 2.13) and BMO-coefficients in the principal part (Section 2.16), the extension of which will require finer geometrical properties of the underlying domain $\Omega$ and should be considered open at this point.

The rest works out smoothly, as long as we assume uniform ellipticity in the sense of Gårding: There should exist $\lambda > 0$ and $c_0 \in \mathbb{R}$ such that for almost every $t$ and every $w \in V$ we have

$$ (4.4) \qquad \mathrm{Re}\langle A(t)\nabla w, \nabla w \rangle \geq \lambda \|\nabla w\|_{L_x^2}^2 - c_0 \|w\|_{L_x^2}^2. $$

Then, we can work with lower order coefficients in Lebesgue-Lebesgue mixed spaces with the assumption $(D_\varepsilon)$. This gives access to representation by Green operators for the inverse of $\mathcal{H} + \kappa$ for appropriate $\kappa \ge 0$, causality and fundamental solution operators for the Cauchy problem. The proof of $L^2$ off-diagonal estimates can be adapted if $V$ is invariant under multiplication with bounded Lipschitz functions. For example, the variational spaces for mixed Dirichlet-Neumann boundary conditions have this property [18, Lem. 4].

If (4.4) comes with $c_0 = 0$ and the leading part of $\mathcal{H}$ is a pure second order operator, then one can also develop the theory in the class $L_t^2 V$ similar to Section 2.14.

Finally, for the extension of the definition of $\mathcal{H}$ when coefficients belong to mixed Lorentz spaces, we can use the following self-improvement property to treat all compatible pairs $(\tilde{r}_1, \tilde{q}_1)$ with $\tilde{r}_1 < \infty$.

**Lemma 4.5.** If Assumption (V) holds, then $L^q(\Omega)$ can be replaced with $L^{q,2}(\Omega)$ in (4.1) when $\theta > 0$.

*Proof.* Let us fix $\theta \in (0, 1]$ with corresponding Lebesgue exponent $q$. By open-endedness, we pick $\vartheta \in (0, \theta)$ with corresponding larger exponent $r$. By (4.1) we have a continuous inclusion

$$ [L_x^2, V]_{1-\vartheta} \hookrightarrow L_x^r. $$
For Hilbert spaces the complex method agrees with the $(\cdot, \cdot)_{\theta,2}$-real method [8, Sec. 6].
With $\sigma := \frac{1-\theta}{1-\vartheta} \in (0, 1)$ we obtain the required continuous inclusion

$$[L_x^2, V]_{1-\theta} = (L_x^2, V)_{1-\theta,2} = (L_x^2, (L_x^2, V)_{1-\vartheta,2})_{\sigma,2} \hookrightarrow (L_x^2, L_x^r)_{\sigma,2} = L_x^{q,2}.$$

The second equality is the reiteration theorem [35, Sec. 1.10.2] and the final equality follows from the real interpolation property for Lebesgue spaces [35, Sec. 1.18.6] and the relation $\frac{1-\sigma}{2} + \frac{\sigma}{r} = \frac{1}{q}$. $\square$

With the previous lemma at hand, the extension of the definition of $\mathcal{H}$ with coefficients in mixed Lorentz spaces can be carried out as before for compatible pairs for the coefficients with $\tilde{r}_1 < \infty$. The case $r_1 = 2$ for the conjugate admissible pair $(r_1, q_1)$ is not covered by this statement. Invertibility can be shown under Lorentz-Lorentz mixed norms for the lower order coefficients and causality follows under Lebesgue-Lorentz mixed spaces.

In order to include Lorentz spaces for compatible pairs with $\tilde{r}_1 = \infty$, which is probably the most interesting case in applications, the improvement in Lemma 4.5 for $\theta = 0$ is needed, that is, $d \ge 3$ and the embedding $V \hookrightarrow L_x^{2d/(d-2),2}$ holds. One simple way to guarantee this embedding for $d = n \ge 3$ is to assume that there is a bounded Sobolev extension operator $V \to W^{1,2}(\mathbb{R}^n)$ since then one can use the O'Neil's Sobolev embedding on $\mathbb{R}^n$ and restrict back to $\Omega$. Hence, this always works for pure lateral Dirichlet conditions ($V = W_0^{1,2}(\Omega)$), using the extension by zero. For the existence of an extension operator in the case of mixed lateral boundary conditions, the most general geometric assumptions as far we are aware can be found in [11].

## 5. Gagliardo–Nirenberg inequalities

We prove here a version of Gagliardo–Nirenberg inequalities including Lorentz norms. We work on $\mathbb{R}^n$.

**Proposition 5.1.** Let $m \ge 1$ be an integer, I be an interval and $u \in L^\infty(I; L_x^2)$ with $\nabla^m u \in L^2(I; L_x^2)$. Let $\alpha$ be a multi-index such that $0 \le |\alpha| \le m$ and

$$(r, q) \in [2, \infty)^2, \quad \frac{1}{r} + \frac{n}{2mq} = \frac{n + 2|\alpha|}{4m}.$$

Then

$$\|\partial^\alpha u\|_{L^r(I; L_x^{q,2})} \le C(n, m, |\alpha|, r) \|\nabla^m u\|_{L^2(I; L_x^2)}^{2/r} \|u\|_{L^\infty(I; L_x^2)}^{1-2/r}.$$

*Proof.* Let $a = |\alpha|$. Define $b \ge 0$ by $b - \frac{n}{2} = a - \frac{n}{q}$. Then, for (almost) every $t \in I$, we invoke the boundedness of $\partial^\alpha(-\Delta)^{-a/2}$ on Lorentz spaces and the O'Neil's Sobolev embedding theorem [30],

$$\|\partial^\alpha u(t)\|_{L_x^{q,2}} \lesssim \|(-\Delta)^{a/2} u(t)\|_{L_x^{q,2}} \lesssim \|(-\Delta)^{b/2} u(t)\|_{L_x^2}.$$

We next use the classical interpolation inequality

$$\|(-\Delta)^{b/2} u(t)\|_{L_x^2} \lesssim \|\nabla^m u(t)\|_{L_x^2}^{b/m} \|u(t)\|_{L_x^2}^{(m-b)/m}.$$

Using $u \in L^\infty(I; L_x^2)$, we can conclude by integrating the r-th power and working out the exponents. The interpolation inequality itself is easily seen by using the Fourier transform in $L_x^2$. Indeed, writing $|\xi|^{2b}|\hat{u}(\xi)|^2 = (|\xi|^{2m}|\hat{u}(\xi)|^2)^{b/m}(|\hat{u}(\xi)|^2)^{(m-b)/m}$, it
boils down to Hölder's inequality with exponent $\frac{m}{b} \in [1, \infty)$. This finishes the proof. □

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UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY, FRANCE

Email address: pascal.auscher@universite-paris-saclay.fr

TU DARMSTADT, FACHBEREICH MATHEMATIK, SCHLOSSGARTENSTR. 7, 64289 DARMSTADT, GERMANY

Email address: egert@mathematik.tu-darmstadt.de

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