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We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(\Omega) \] with possibly complex coefficients. We study three problems: 1) Boundedness on $C^\nu$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) H\"older and $L_p$-bounds for the harmonic lifting associated with ${\cal A}$. 3) Poisson bounds for the heat kernel of ${\cal N}$. We solve these problems in the case where the coefficients are H\"older continuous and the underlying domain is bounded and of class $C^{1+\kappa}$ for some $\kappa > 0$. For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function $G$ of the elliptic operator with Dirichlet boundary conditions., Comment: This is the final version, to appear in Calculus of Variations and PDE

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$ with real symmetric and H\''older continuous coefficients. We prove $L^p(\Gamma)\to L^q(\Gamma)$ off-diagonal bounds of the form$$ \| 1_F e^{-t N_0} 1_E f \|_q \lesssim (t \wedge 1)^{\frac{n}{q}-\frac{n}{p}} \left( 1 + \frac{dist(E,F)}{t} \right)^{-1} \| 1_E f \|_p$$for all measurable subsets $E$ and $F$ of $\Gamma$. If $\Gamma$ is $C^{1+ \kappa}$ for some $\kappa > 0$ and $m=1$, we obtain a sharp estimate in the sense that $ \left( 1 + \frac{dist(E,F)}{t} \right)^{-1}$ can be replaced by$ \left( 1 + \frac{dist(E,F)}{t} \right)^{-(1 + \frac{n}{p} - \frac{n}{q})}$. Such bounds are also valid for complex time. For $n=1$, we apply our off-diagonal bounds to prove that the Dirichlet-to-Neumann operator associated with a system generates an analytic semigroup on $L^p(\Gamma)$ for all $p \in (1, \infty)$. In addition, the corresponding evolution problem has $L^q(L^p)$-maximal regularity., Comment: final version, to appear in J. Math. Anal. Appl

Let $\Omega$ be a bounded open subset with $C^{1+\kappa}$-boundary for some $\kappa > 0$. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator $- \sum \partial_l ( c_{kl} \, \partial_k ) + V$, where the $c_{kl} = c_{lk}$ are H\"older continuous and $V \in L_\infty(\Omega)$ are real valued. We prove that the Dirichlet-to-Neumann operator generates a $C_0$-semigroup on the space $C(\partial \Omega)$ which is in addition holomorphic with angle $\frac{\pi}{2}$. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence we obtain an optimal holomorphic functional calculus and maximal regularity on $L_p(\Gamma)$ for all $p \in (1,\infty)$.

We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert space $H$ . We assume that these forms all have the same domain and satisfy some regularity assumption with respect to t (e.g., piecewise $\alpha$-H{\"o}lder continuous for some $\alpha\textgreater{} 1/2$). We prove maximal Lp-regularity for all initial values in the real-interpolation space $(H, D(A(0)))\_{1/p,p}$ . The particular case where $p = 2$ improves previously known results and gives a positive answer to a question of J.L. Lions [11] on the set of allowed initial data $u\_0$ ., Comment: 19 pages. To appear in Math. Ann

We prove Poisson upper bounds for the kernel $K$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a $C^\infty$-boundary. We also prove Poisson bounds for $K_z$ for all $z$ in the right half-plane and for all its derivatives.

Let $A = - \sum \partial_k \, c_{kl} \, \partial_l$ be a degenerate sectorial differential operator with complex bounded mesaurable coefficients. Let $\Omega \subset \mathds{R}^d$ be open and suppose that $A$ is strongly elliptic on $\Omega$. Further, let $\chi \in C_{\rm b}^\infty(\mathds{R}^d)$ be such that an $\varepsilon$-neighbourhood of $\supp \chi$ is contained in $\Omega$. Let $\nu \in (0,1]$ and suppose that the ${c_{kl}}_{|\Omega} \in C^{0,\nu}(\Omega)$. Then we prove (H\"older) Gaussian kernel bounds for the kernel of the operator $u \mapsto \chi \, S_t (\chi \, u)$, where $S$ is the semigroup generated by $-A$. Moreover, if $\nu = 1$ and the coefficients are real, then we prove Gaussian bounds for the kernel of the operator $u \mapsto \chi \, S_t u$ and for the derivatives in the first variable. Finally we show boundedness on $L_p(\mathds{R}^d)$ of various Riesz transforms.

We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $\chi \in C_b^\infty(\mathbb{R}^d)$ (respectively, $\Omega$ a bounded Lipschitz domain) and suppose that $(a_{kj}) \ge \mu > 0$ a.e.\ on $ \supp \chi$ (resp., a.e.\ on $\Omega$). We prove a spectral multiplier type result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \| \varphi(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $\varphi \in C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_\chi F(I+A) M_\chi$ is weak type $(1,1)$ (resp.\ $P_\Omega F(I+A) P_\Omega$ is weak type $(1,1)$). We also prove boundedness on $L^p$ for all $p \in (1,2]$ of the partial Riesz transforms $M_\chi \nabla (I + A)^{-1/2}M_ \chi$. The proofs are based on a criterion for a singular integral operator to be weak type $(1,1)$., Comment: To appear in Revista Matem\'atica Iberoamericana

For general non-symmetric operators $A$, we prove that the moment of order $\gamma \ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $\gamma$ of negative eigenvalues of its symmetric part $H = {1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schr\"odinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schr\"odinger self-adjoint operators.