Abstract
Let \(\Omega \subseteq {\mathbb {R}}^{d}\) be open and A a complex uniformly strictly accretive \(d\times d\) matrix-valued function on \(\Omega \) with \(L^{\infty }\) coefficients. Consider the divergence-form operator \({{\mathscr {L}}}^{A}=-\mathrm{div}\,(A\nabla )\) with mixed boundary conditions on \(\Omega \). We extend the bilinear inequality that we proved in Carbonaro and Dragičević (J Eur Math Soc, to appear) in the special case when \(\Omega ={\mathbb {R}}^{d}\). As a consequence, we obtain that the solution to the parabolic problem \(u^{\prime }(t)+{{\mathscr {L}}}^{A}u(t)=f(t)\), \(u(0)=0\), has maximal regularity in \(L^{p}(\Omega )\), for all \(p>1\) such that A satisfies the p-ellipticity condition that we introduced in Carbonaro and Dragičević (to appear). This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on \(\Omega \), in particular, we do not assume any regularity of \(\partial \Omega \), nor the existence of a Sobolev embedding. The methods of Carbonaro and Dragičević (to appear) do not apply directly to the present case and a new argument is needed.
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Notes
In the definition of \({{\mathcal {E}}}\) we implicitly identify \(T^{A,{{\mathscr {V}}}}_{t}f\) and \(T^{B,{{\mathscr {V}}}^{\prime }}_{t}g\) with their real counterparts; see Sect. 2.4 for a more accurate statement.
In [38] the author also considered systems. The results are stated under geometric assumptions on \(\Omega \) which are stronger than (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\) . These stronger assumptions are used, for example, for proving results on Riesz transforms. However, for the specific result stated here (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\) suffices; see [39].
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Acknowledgements
The first author was partially supported by the “National Group for Mathematical Analysis, Probability and their Applications” (GNAMPA-INdAM). The second author was partially supported by the Ministry of Higher Education, Science and Technology of Slovenia (research program Analysis and Geometry, Contract No. P1-0291) and Slovenian Research Agency (Grant no. J1-1690). The authors would like to thank the referee for his/her valuable comments which improved the presentation of this paper.
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Appendices
Appendix A: Comparison with known results
Under stronger assumptions on A and/or \(\Omega \) than those of Theorem 3 it is known that \((T^{A,{{\mathscr {V}}}}_{t})_{t>0}\), where \({{\mathscr {V}}}\) is one of the subspaces of Sect. 1.1, extrapolates to a bounded strongly continuous semigroup on \(L^{p}(\Omega )\) in a range of p’s larger than the range given by p-ellipticity, and the negative generator \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity in this larger range of exponents.
1.1 A.1 Semigroup extrapolation
Let \({{\mathscr {V}}}\) denote one of the subspaces of Sect. 1.1.
(i) For every \(\Omega \subseteq {\mathbb {R}}^{d}\) and every real-valued\(A\in {{\mathcal {A}}}(\Omega )\) the semigroup \((T^{A,{{\mathscr {V}}}}_{t})_{t>0}\) is sub-Markovian (see [70, 71] and [72, Corollary 4.3 and Corollary 4.10]), so \(\omega _{H^{\infty }}({{\mathscr {L}}}^{A}_{p})<\pi /2\) for all \(p>1\); see [18, 22, 57] for symmetric real-valued A and [51, Corollary 5.2] for nonsymmetric real-valued A. It follows from the Dore-Venni theorem [33, 76] that, in this case, \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity for all \(p>1\).
(ii) As for complex-valued\(A\in {{\mathcal {A}}}(\Omega )\), define the upper and lower Sobolev exponent by the rule \(2^{*}=2d/(d-2)\) if \(d\geqslant 3\) and \(2^{*}=+\infty \) if \(d=1,2\); \(2_{*}=(2^{*})^{\prime }\). For \(A\in {{\mathcal {A}}}(\Omega )\), \(\delta \geqslant 0\) and \(\vartheta \in [0,\pi /2)\) denote by \(J(A,{{\mathscr {V}}},\delta ,\vartheta )\) the maximal open interval in \((1,+\infty )\) such that \((e^{-\delta z}T^{A,{{\mathscr {V}}}}_{z})_{z\in \mathbf{S}_{\vartheta }}\) is uniformly bounded in \(L^{p}(\Omega )\), for all \(p\in J(A,{{\mathscr {V}}},\delta ,\vartheta )\). Denote by \(\omega ({{\mathscr {L}}}^{A}_{2})\) the sectoriality angle of \({{\mathscr {L}}}^{A}_{2}\).
-
(a)
When \(\Omega ={\mathbb {R}}^{d}\), \(d=1,2\) and \(A\in {{\mathcal {A}}}({\mathbb {R}}^{d})\), we have \(J(A,W^{1,2}({\mathbb {R}}^{d}),0,\vartheta )=(1,+\infty )\), for all \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\) [6]. When \(\Omega ={\mathbb {R}}^{d}\), \(d\geqslant 3\) and \(A\in {{\mathcal {A}}}({\mathbb {R}}^{d})\), Auscher proved [5] that there exists \(\varepsilon >0\) depending only on dimension and the ellipticity constants of A such that \((2_{*}-\varepsilon ,2^{*}+\varepsilon )\subset J(A,W^{1,2}({\mathbb {R}}^{d}),0,\vartheta )\), for all \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\).
-
(b)
The results in (a) are sharp [48]: for all \(d\geqslant 3\) and all \(p\in [1,\infty ]{\setminus }[2_{*},2^{*}]\) there exists \(A\in {{\mathcal {A}}}({\mathbb {R}}^{d})\) such that \((T^{A}_{t})_{t>0}\) is not bounded on \(L^{p}({\mathbb {R}}^{d})\).
-
(c)
In the case when \({{\mathscr {V}}}\) has the embedding property (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\), Egert implicitlyFootnote 2 proved [38, Theorem 1.6] that \((2_{*},2^{*})\subset J(A,{{\mathscr {V}}},\delta ,\vartheta )\), for every \(\delta >0\) and \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\). Also, Egert extended the result in (a) (\(d\geqslant 3\)) by proving that if \(\Omega \) is bounded and connected, the boundary is Lipschitz regular around the Neumann part \(\overline{\partial \Omega {\setminus } D}\) and D satisfies further geometric assumptions (see [38, 46, 79]) one always has that \((2_{*}-\varepsilon ,2^{*}+\varepsilon )\subset J(A,{{\mathscr {V}}},\delta ,\vartheta )\), for all \(\delta >0\) and \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\) and some \(\varepsilon >0\) depending only on dimension, the ellipticity constants of A and the geometry of \(\Omega \). Note that under the above mentioned geometric assumptions there exists a Sobolev extension operator \(E: {{\mathscr {V}}}\rightarrow W^{1,2}({\mathbb {R}}^{d})\) and so (\(\mathrm{SE}_{q}\)) holds true with \(q=2^{*}\). A result similar to (c), but without any further geometric assumption on D, has been previously obtained by Tolksdorf [80], who also proved maximal regularity in the range \((2_{*}-\varepsilon ,2^{*}+\varepsilon )\).
-
(d)
Recently, Egert [39] improved the result in (c) by combining our notion of p-ellipticity and its properties with the technology developed in [5, 38].
A similar result has been proved by ter Elst et al. [79, Theorem 3.1] by means of a different technique, but still using p-ellipticity.
For \(p\geqslant 2\) set
Proposition A.1
([39, Theorem 1]) Let \(d\geqslant 3\). Let \(\Omega \subset {\mathbb {R}}^{d}\) be open. Let \({{\mathscr {V}}}\) denote one of the subspaces of Sect. 1.1. Assume the Sobolev embedding (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\). Let \(p^{}_{0}>2\) be such that \(\Delta _{p_{0}}(A)>0\). Then
for all \(\delta >0\) and \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\).
1.2 A.2. Functional calculus and maximal regularity
The next result follows by combining Proposition A.1 with a result of Blunck and Kunstmann [10] that was simplified by Auscher in [5, Theorem 1.1] and extended to domains \(\Omega \subset {\mathbb {R}}^{d}\) by Egert in [38, Proposition 5.2].
Corollary A.2
([38, 39]) Under the assumptions of Proposition A.1 we have
for every \(p\in ((p^{\circ }_{0})^{\prime },p^{\circ }_{0})\) and \(\delta >0\).
As a consequence [33, 76], \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity, for every \(p\in [(p^{\circ }_{0})^{\prime },p^{\circ }_{0}]\).
1.3 A.3. Absence of Sobolev embeddings
It is well-known that (\(\mathrm{SE}_{q}\)) for \(q>2\) requires geometric assumptions on \(\Omega \) and does not hold in general, see [1, Theorem 4.46, Theorem 4.48 and Example 4.55] and [11, Proposition 3 and Example 6]. For simplicity, we only discuss the case when \({{\mathscr {V}}}=W^{1,2}(\Omega )\).
When \(\Omega \subset {\mathbb {R}}^{d}\) has finite measure, by the Rellich–Kondrachov theorem [45, Theorem 5 and Corollary 1], the Sobolev embedding (\(\mathrm{SE}_{q}\)) for some \(q>2\) implies the compactness of the resolvent operator \((I+{{\mathscr {L}}}^{A}_{2})^{-1}\) for all \(A\in {{\mathcal {A}}}(\Omega )\) (see also [11, Theorem 7]). So, in this case, the spectrum of \({{\mathscr {L}}}^{I}_{2}\) is discrete.
-
(a)
A classical example of Courant and Hilbert [21, p. 531] shows that there exists a “rooms and passages” connected bounded domain \(\Omega \subset {\mathbb {R}}^{2}\) for which \(0\in \sigma _{\mathrm{ess}}({{\mathscr {L}}}^{I}_{2})\). Actually, for \(d\geqslant 2\), given any closed subset S of \([0,+\infty )\) there exists an open connected subset \(\Omega \) of the unit ball in \({\mathbb {R}}^{d}\) such that \(\sigma _{\mathrm{ess}}({{\mathscr {L}}}^{I}_{2})=S\), see [47].
-
(b)
Let \(d\geqslant 2\). By using a criterion of Evans and Harris [41] (see also [26, p. 106]), one can easily construct unbounded “horn-shaped” \(\Omega \subset {\mathbb {R}}^{d}\) of finite measure for which the Neumann Laplacian \({{\mathscr {L}}}^{I}_{2}\) does not have compact resolvent operators. A simple example of this phenomenon is given by the regions
$$\begin{aligned} \Omega _{\alpha }=\{(x,x^{\prime })\in {\mathbb {R}}\times {\mathbb {R}}^{d-1}: x>0,\ |x^{\prime }|<ce^{-\alpha x}\},\quad \alpha>0,\ c>0. \end{aligned}$$(A.1)
1.4 A.4. Sharpness of Theorem 3
For open sets like those described above in (b), the conclusions of Proposition A.1 and Corollary A.2 are false, because the analyticity angle of the semigroup and the functional calculus angle of the generator may depend on p, even for smooth A and pure Neumann boundary conditions.
Kunstmann [56] further developed a result of Davies and Simon [26] for \(p=2\) and proved that for \(d=2\) and \(p>1\) the \(L^{p}\) spectrum of the Neumann Laplacian in the region \(\Omega _{\alpha }\) defined in (A.1) satisfies the inclusions
where
Moreover, \(\sigma _{\mathrm{ess}}({{\mathscr {L}}}^{I}_{p})=\mathbf{P}_{p,\alpha }\).
For \(p\in (1,+\infty )\) and \(\alpha >0\) the parabolic region \(\mathbf{P}_{p,\alpha }\) is tangent to the critical sector \(\mathbf{S}_{\phi ^{*}_{p}}\), where \(\phi ^{*}_{p}=\arcsin |2/p-1|\). Recall that \(\phi _{p}=\pi /2-\phi ^{*}_{p}\) and \(\phi ^{*}_{p}\) are, respectively, the optimal analyticity angle in \(L^{p}\) [53, 60] and the optimal functional calculus angle in \(L^{p}\) [18], for all generators of symmetric contraction semigroups.
By attaching to, say, a ball in \({\mathbb {R}}^{2}\) countably many disjoint horns \({\widetilde{\Omega }}_{\alpha _{n}}\), each one congruent to some \(\Omega _{\alpha _{n}}\), \(\alpha _{n}>0\), Kunstmann [56] was able to construct a domain \(\Omega _{\max }\) of finite measure such that the associated Neumann Laplacian has maximal \(L^{p}\) spectrum:
Consider the region \(\Omega _{\mathrm{max}}\) described above. Recall from [17, Lemma 5.22] that for \(\phi \in (-\pi /2,\pi /2)\) and \(p>1\) we have
Fix \(\phi \in (0,\pi /2)\). Since \((T^{I}_{t})_{t>0}\) is symmetric and sub-Markovian, we obtain from [18], (A.4) and (A.3) that
Let \(p>1\) be such that \(\Delta _{p}(e^{i\phi }I)\leqslant 0\), that is, such that \(\phi \geqslant \phi _{p}\). Then \({{\mathscr {L}}}^{e^{i\phi } I}_{p}\) does not have parabolic maximal regularity in \(L^{p}(\Omega _{\mathrm{max}})\), since otherwise, by [32, Theorem 2.2], there would exist \(\varepsilon ,\delta >0\) such that \( \sigma ({{\mathscr {L}}}^{I}_{p})\subseteq {\overline{\mathbf{S }}}_{\pi /2-\phi -\varepsilon }-\delta \), contradicting (A.3).
More generally, by combining [18] with Proposition 20 and [55, Example 2.4 and Remark 3.2], we deduce that for all \(d\geqslant 2\) there exist \(\Omega \subset {\mathbb {R}}^{d}\) and \(A\in {{\mathcal {A}}}(\Omega )\) such that the Neumann operator \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity if and only if \(\Delta _{p}(A)>0\).
In this sense Theorem 3 is sharp.
1.5 A.5. Extrapolation in \(L^{\infty }\) for smooth coefficients: counterexamples
Consider the open set \(\Omega _{\mathrm{max}}\) of [56] described above. The equality (A.3) implies that if \(\phi >0\) then \((T^{e^{i\phi }I}_{t})_{t>0}\)=\((T^{I}_{e^{i\phi }t})_{t>0}\) is not exponentially bounded in \(L^{\infty }(\Omega _{\mathrm{max}})\). Indeed, assuming the contrary, by interpolation with \(L^{2}\) and the relation \(\overline{T^{I}_{z}f}=T^{I}_{{\bar{z}}}{\bar{f}}\), \(\mathrm{Re}\,z>0\), (because \((T^{I}_{t})_{t>0}\) is positive and analytic in \(L^{2}\) in \(\{\mathrm{Re}\,z>0\}\)), there would exist \(r_{0}>0\) and \(\phi _{0}\in (0,\pi /2)\) such that
This implies that \(\sigma ({{\mathscr {L}}}^{I}_{p})\subseteq \overline{\mathbf{S}}_{\pi /2-\phi _{0}}-r_{0}\) for all \(p>2\), contradicting (A.3) when p is such that \(\phi ^{*}_{p}>\pi /2-\phi _{0}\).
Example A.3
We would expect that if \(\phi \ne 0\) then one has \(T^{I}_{e^{i\phi }t}(L^{\infty }(\Omega _{\mathrm{max}}))\not \subset L^{\infty }(\Omega _{\mathrm{max}})\) for some \(t>0\), but we could not extract this result from the existing literature. Therefore, we now construct a (disconnected) open set \(\Omega \subset {\mathbb {R}}^2\) such that there exists \(\phi >0\) for which \(T^{e^{i\phi }I}_{1}(L^{\infty }(\Omega ))\not \subset L^{\infty }(\Omega )\).
We first consider the Neumann Laplacian \(\Delta ^{\Omega _{\alpha }} := {{\mathscr {L}}}^{I}\) in the region \(\Omega _{\alpha }\) given by (A.1) with \(d=2\), \(\alpha >0\) and \(c=\alpha ^{-1}\). Note that \(\Omega _{\alpha }=\alpha ^{-1}\cdot \Omega _{1}\). Set \(T^{\Omega _{\alpha }}_{t}=\exp (-t\Delta ^{\Omega _{\alpha }})\).
By arguing much as in the case of \(\Omega _{\mathrm{max}}\) discussed above and using (A.2), we see that
for every \(\phi \in (0,\pi /2)\). Fix \(\phi \in (0,\pi /2)\). By the uniform boundeness principle, there exists a nonzero \(f_{1}\in L^{\infty }(\Omega _{1})\) and a sequence \(t_{n}>0\) such that
We now use a rescaling argument. For \(\alpha >0\), consider the operator
It is not hard to see that
-
(i)
\(J_{\alpha }:L^{2}(\Omega _{1})\rightarrow L^{2}(\Omega _{\alpha })\) is a surjective isometry with \(J^{-1}_{\alpha }=J_{1/\alpha }\);
-
(ii)
\(J_{\alpha }\mathrm{D}(\Delta ^{\Omega _{1}}_{2})=\mathrm{D}(\Delta ^{\Omega _{\alpha }}_{2})\);
-
(iii)
\(J^{-1}_{\alpha }\Delta ^{\Omega _{\alpha }}_{2}J_{\alpha }=\alpha ^{2}\Delta ^{\Omega _{1}}_{2}\).
It follows that
Hence, for \(\alpha =t_{n}\) and \(z=e^{i\phi }\),
Set \(f_{n}=f_{1}(t_{n}\cdot )\). Then \({\left\| {f_{n}}\right\| _{L^{\infty }(\Omega _{t_{n}})}}={\left\| {f_{1}}\right\| _{L^{\infty }(\Omega _{1})}}\) and
For each \(n\in {{\mathbb {N}}}_{+}\) select a rigid motion \(R_{n}\) of the plane such that the congruent copies
are pairwise disjoint; for example \(R_{1}=\mathrm{Id}\) and, for \(n\geqslant 2\), \(R_{n}(x,y)=(x,y+v_{n})\), where \(v_{n}=t^{-1}_{n}+2\sum ^{n-1}_{k=1}t^{-1}_{k}\). Define
Then, \(\Vert f\Vert _{L^{\infty }(\Omega )}={\left\| {f_{1}}\right\| _{L^{\infty }(\Omega _{1})}}\) and
It follows from (A.5) that \(T^{\Omega }_{e^{i\phi }}f\notin L^{\infty }(\Omega )\).
Appendix B: Rigidity of generalized convexity
Proposition B.1
Let \(A\in {{\mathbb {C}}}^{d\times d}\) be an elliptic matrix, \(t_{0}>0\) and let \(\gamma \in C^{1}([0,+\infty );{\mathbb {R}})\cap C^{2}((0,+\infty ){\setminus }\{t_{0}\};{\mathbb {R}})\). Set \(\Gamma (\zeta )=\gamma (|\zeta |)\), \(\zeta \in {\mathbb {R}}^{2}\). Suppose that \(\Gamma \) is A-convex in \({\mathbb {R}}^{2}{\setminus }\{|\zeta |=t_{0}\}\). Then,
-
(i)
The profile function \(\gamma \) is nondecreasing and convex, and \(\Gamma \) is convex.
-
(ii)
If \(\mathrm{Im}\,A\ne 0\), then either \(\gamma =\mathrm{const}\) or \(\gamma ^{\prime }(t)>0\) for all \(t>0\).
Proof
A rapid calculation shows that
for all \(\zeta \in {\mathbb {R}}^{2}{\setminus }(\{|\zeta |=t_{0}\}\cup \{0\})\).
We first prove that \(\gamma ^{\prime }(t)\geqslant 0\) for all \(t>0\). By continuity it suffices to consider \(t\ne t_{0}\). For \(Z=(Z^{1},Z^{2})\in {\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) write
Fix \(\zeta \in {\mathbb {R}}^{2}{\setminus }(\{|\zeta |=t_{0}\}\cup \{0\})\). Take \(X^{1}=(x^{1}_{1},0,\dots ,0)\), \(X^{2}=(x^{2}_{1},0,\dots ,0)\). Define \(X=(X^{1},X^{2})\), \(Y=(Y^{1},Y^{2})={{\mathcal {M}}}(A)X\), \(x=(x^{1}_{1},x^{2}_{1})\) and \(y=(y^{1}_{1},y^{2}_{1})\). Then, by using the notation of Sect. 2.3, we obtain
Now take \(X\ne 0\) of the form above and such that x is orthogonal to \(\zeta \). Then, by the assumption of A-convexity of \(\Gamma \),
It follows that \(\gamma ^{\prime }(|\zeta |)\geqslant 0\), because \(\left\langle x , y\right\rangle = \left\langle X , {{\mathcal {M}}}(A)X\right\rangle _{{\mathbb {R}}^{2d}}>0 \) by (12) and the ellipticity of A.
We now prove that\(\gamma \)is convex. It is well-known (and easy to see by means of a convolution argument for regularising \(\gamma \)) that this is equivalent to proving that \(\gamma ^{\prime \prime }(t)\geqslant 0\) for all \(t\in (0,+\infty ){\setminus }\{t_{0}\}\). Fix \(t\notin \{t_{0},0\}\) and \(\zeta \in {\mathbb {R}}^{2}\) such that \(|\zeta |=t\). We rewrite (B.1) as
where \(F_{1}(\zeta )=|\zeta |\).
Therefore, for all \(X\in {\mathbb {R}}^{2d}\) we have
It follows from (4) that \(\Delta _{1}(A)\leqslant 0\). Hence, by Lemma 4 (v) we have
From this, (B.2), the fact that \(\gamma ^{\prime }\geqslant 0\), and the inequality \(\left\langle X , {{\mathcal {M}}}(A)X\right\rangle _{{\mathbb {R}}^{2d}}\geqslant \lambda |X|^{2}\) we deduce that \(\gamma ^{\prime \prime }(t)\geqslant 0\).
Convexity of\(\Gamma \) is now clear and easily follows from the already proved properties of \(\gamma \). Indeed, since \(\gamma \) is nondecreasing and convex, we have
for every \(\zeta _{1},\zeta _{2}\in {\mathbb {R}}^{2}\) and every \(t\in [0,1]\).
We now prove (ii). Suppose that there exists \(s>0\) such that \(\gamma ^{\prime }(s)=0\), then by item (i) \(\gamma ^{\prime }(t)=0\) for all \(t\in [0,s]\). Assume that \(\gamma \) is nonconstant. Then,
\(\gamma ^{\prime }(t)>0\), for all \(t>t_{1}\) and \(\gamma ^{\prime }(t_{1})=0\). For simplifying the proof, we first assume that \(t_{1}=t_{0}\). For \(t>t_{0}\), define the function \(v(t)=\log (\gamma ^{\prime }(t))\). Then \(\lim _{t\rightarrow t^{+}_{0} }v(t)=-\infty \) and \(v^{\prime }(t)=\frac{\gamma ^{\prime \prime }(t)}{\gamma ^{\prime }(t)}\geqslant 0\), for all \(t>t_{0}\). It follows that
We now show that (B.3) implies \(\mathrm{Im}\,A=0\). Define the function
Then, by (B.1),
The formula above expresses the Hessian of the radial function \(\Gamma \) in terms of Hessians of power functions. More specifically, by (17) we have
where
Since \(\Gamma \) is A-convex, it follows from the identity above that
or, equivalently [see Lemma 4 (v)], \(\Delta _{p(|\zeta |)}(A)\geqslant 0\), for all \(|\zeta |>t_{0}\). Now by (B.3) we have \(\sup _{|\zeta |>t_{0}}p(|\zeta |)=+\infty \), so by Lemma 4 (i) and (iii) we have \(\Delta _{p}(A)\geqslant 0\), for all \(p>1\). Hence, Lemma 4 (vi) implies that \(\mathrm{Im}\,A=0\).
When \(t_{1}\ne t_{0}\) the proof is similar. Just consider the function \(v(t)=\log (\gamma ^{\prime }(t))\) for \(t\in (t_{1},t_{0})\) if \(t_{1}<t_{0}\), or for \(t>t_{1}\) if \(t_{1}>t_{0}\) (and so \(\gamma \) is in \(C^{2}({\mathbb {R}})\)). \(\square \)
Appendix C: Flow regularity
Let \(({{\mathcal {X}}},\mu )\) be a \(\sigma \)-finite measure space. Fix \(p\geqslant 2\) and set \(q=p/(p-1)\). Suppose that \((\exp (-t{{\mathcal {A}}}))_{t>0}\) and \((\exp (-t{{\mathcal {B}}}))_{t>0}\) are analytic and uniformly bounded both in \(L^{p}({{\mathcal {X}}},\mu )\) and \(L^{q}({{\mathcal {X}}},\mu )\). Let \({{\mathcal {Q}}}={{\mathcal {Q}}}_{p,\delta }\) be the Nazarov–Treil Bellman function defined in (10). Fix \(f,g\in (L^{p}\cap L^{q})({{\mathcal {X}}},\mu )\). Consider the flow
where we omit the subscript \({{\mathcal {W}}}\) (see Sect. 2.4).
Proposition C.1
Under the above assumptions, we have:
-
(a)
\({{\mathcal {E}}}\in C[0,+\infty )\);
-
(b)
\({{\mathcal {E}}}\in C^{1}(0,+\infty )\) and
$$\begin{aligned} -{{\mathcal {E}}}^{\prime }(t)=2\mathrm{Re}\,\int _{{{\mathcal {X}}}}\left( (\partial _{\zeta }{{\mathcal {Q}}})(e^{-t{{\mathcal {A}}}}f,e^{-t{{\mathcal {B}}}}g){{\mathcal {A}}}e^{-t{{\mathcal {A}}}}+(\partial _{\eta }{{\mathcal {Q}}})(e^{-t{{\mathcal {A}}}}f,e^{-t{{\mathcal {B}}}}g){{\mathcal {B}}}e^{-t{{\mathcal {B}}}}g\right) . \end{aligned}$$
Proof
We start with (a) We prove only the continuity at 0 since the continuity at other points can be proved exactly in the same way, or it follows from item (b). Set
By the mean value theorem applied to \({{\mathcal {Q}}}\),
Estimates (11) immediately give
where C does not depend on x and t. Now item (a) follows from Hölder’s inequality and the strong continuity of the two semigroups in \(L^{p}({{\mathcal {X}}},\mu )\) and \(L^{q}({{\mathcal {X}}},\mu )\).
We now prove item (b) Analyticity implies that there exists \(C\geqslant 1\) such that, for \(r\in \{p,q\}\),
see, for example, [40, Chapter II, p.104]. Fix \(t_{0}>0\). Then for \(\delta <t_{0}/C\) we have
where the series converges in \((L^{p}\cap L^{q})({{\mathcal {X}}},\mu )\). Moreover,
and similarly,
Possibly taking a smaller \(\delta \), we also get
By using the powers series expansion of \(e^{-t{{\mathcal {A}}}}f\) and \(e^{-t{{\mathcal {B}}}}g\) one can also prove that each \(e^{-t{{\mathcal {A}}}}f\) and each \(e^{-t{{\mathcal {B}}}}g\) can be redefined in a set of measure zero, in such a manner that for almost every \(x\in \Omega \) the functions \(t\mapsto e^{-t{{\mathcal {A}}}}f(x)\) is real-analytic on \((0,\infty )\), for a.e. \(x\in \Omega \); see [78, p. 72]. Now item (b) follows from estimates (11) and the usual theorems of derivation and passage of the limit under the integral sign that are consequence of the Lebesgue’s dominated convergence theorem. \(\square \)
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Carbonaro, A., Dragičević, O. Bilinear embedding for divergence-form operators with complex coefficients on irregular domains. Calc. Var. 59, 104 (2020). https://doi.org/10.1007/s00526-020-01751-3
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DOI: https://doi.org/10.1007/s00526-020-01751-3