Suppose $u$ is a sign changing classical solution of the fractional laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega; u=g \text{ in } \mathbb R^N -\Omega .$$ (a)Is it true that $\|u\|_{L^{\infty}(\mathbb R^N)}\leq \|g\|_{L^{\infty}(\mathbb R^N -\Omega)}$ when $s\in (0, 1).$

(b) Is this also true for the operator $(-\Delta) ^{s}+I$ where $I$ is the identity.

Suppose $u$ is a sign changing classical solution of the fractional Laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega; u=g \text{ in } \mathbb R^N -\Omega .$$ (a)Is it true that $\|u\|_{L^{\infty}(\mathbb R^N)}\leq \|g\|_{L^{\infty}(\mathbb R^N -\Omega)}$ when $s\in (0, 1).$

(b) Is this also true for the operator $(-\Delta) ^{s}+I$ where $I$ is the identity.

Suppose $u$ is a sign changing classical solution of the fractional laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega; u=g \text{ in } \mathbb R^N -\Omega .$$ Is it true that $\|u\|_{L^{\infty}(\mathbb R^N)}\leq \|g\|_{L^{\infty}(\mathbb R^N -\Omega)}$ when $s\in (0, 1).$

Suppose $u$ is a sign changing classical solution of the fractional laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega; u=g \text{ in } \mathbb R^N -\Omega .$$ (a)Is it true that $\|u\|_{L^{\infty}(\mathbb R^N)}\leq \|g\|_{L^{\infty}(\mathbb R^N -\Omega)}$ when $s\in (0, 1).$

(b) Is this also true for the operator $(-\Delta) ^{s}+I$ where $I$ is the identity.