Let \(a, f\) be measurable on a bounded domain \(\Omega\subseteq \mathbb{R}^d\). Furthermore, assume there exist positive constants \(M\) and \(L\), such that
\[ |a(z)| \leq M, \quad |a(x) - a(y)| \leq L \]for every \(x,y,z\in \Omega\).
Then, there exists a constant \(C\) depending on \(d\), such that
\[ [af]_{H^{1/2}(\Omega)}^2 \leq C\Bigl(M^2 [f]_{H^{1/2}(\Omega)}^2 +\diam (\Omega) \, L^2 \int_\Omega |f(x)|^2 dx\Bigr). \]
Proof
The proof is based on [1, Proposition 4.2].
Using the bounds of \(a\), we obtain
\begin{align*} [af]_{H^{1/2}(\Omega)}^2 &= \iint_{\Omega \times \Omega} \frac{|a(x)f(x) - a(y)f(y)|^2}{|x-y|^{d+1}} \, dx \, dy \\ &= \iint_{\Omega \times \Omega} \frac{|a(x)(f(x)-f(y)) + f(y)(a(x)-a(y))|^2}{|x-y|^{d+1}} \, dx \, dy\\ &\leq 2 \iint_{\Omega \times \Omega} \frac{|a(x)|^2 |f(x)-f(y)|^2}{|x-y|^{d+1}} \, dx \, dy + 2 \iint_{\Omega \times \Omega} \frac{|f(y)|^2 |a(x)-a(y)|^2}{|x-y|^{d+1}} \, dx \, dy \\ &\leq 2M^2 [f]_{H^{1/2}(\Omega)}^2 + 2L^2 \iint_{\Omega \times \Omega} \frac{|f(y)|^2}{|x-y|^{d-1}} \, dx \, dy. \end{align*}Furthermore, by radial integration formula ,
\begin{align*} \iint_{\Omega \times \Omega} \frac{|f(y)|^2}{|x-y|^{d-1}} \, dx \, dy &\leq \int_\Omega \int_{B_{\mathrm{diam}\,\Omega}(y)} \frac{|f(y)|^2}{|x-y|^{d-1}} \, dx \, dy\\ &\leq \omega_d \int_\Omega \int_0^{\mathrm{diam}\,\Omega} \frac{|f(y)|^2}{r^{d-1}} \, r^{d-1} dr \, dy\\ &= \omega_d \, \mathrm{diam}(\Omega) \int_\Omega |f(y)|^2 dy. \end{align*}Thus,
\[ [af]_{H^{1/2}(\Omega)}^2 \le 2M^2 [f]_{H^{1/2}(\Omega)}^2 + 2\omega_d \, L^2\, \mathrm{diam}(\Omega) \int_\Omega |f(y)|^2 dy. \]See also Link to heading
References Link to heading
- M. Di Cristo, E. Francini, C. Lin, S. Vessella, and J. Wang,
Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface,
Journal de Mathématiques Pures et Appliquées, vol. 108, no. 2, pp. 163–206, 2017. doi:10.1016/j.matpur.2016.10.015