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It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

I wonder: Does there exist a function $\varphi_O$ such that $Gf(x):=e^{\varphi_O(x)}f(Ox)$ satisfies for all $f$ but fixed orthogonal $O$

$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

Please let me know if you have any questions.

It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

My question is: Does there exist a function $\varphi_O$ such that $Gf(x):=e^{\varphi_O(x)}f(Ox)$ satisfies for all $f$ but fixed orthogonal $O$

$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

I wonder: Does there exist a function $\varphi_O$ such that $Gf(x):=e^{\varphi_O(x)}f(Ox)$ satisfies for all $f$ but fixed $O$

$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

Please let me know if you have any questions.

It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

I wonder: Does there exist a function $\varphi_O$ such that $Gf(x):=e^{\varphi_O(x)}f(Ox)$ satisfies for all $f$ but fixed orthogonal $O$

$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

Please let me know if you have any questions.

It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

I wonder: Does there exist a function $\varphi$ such that $Gf(x):=e^{\varphi(x)} f(Ox)$ satisfies

$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

Please let me know if you have any questions.

It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.

Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian

$$\Delta_A = \langle \nabla, A \nabla \rangle.$$

I wonder: Does there exist a function $\varphi_O$ such that $Gf(x):=e^{\varphi_O(x)}f(Ox)$ satisfies for all $f$ but fixed $O$

$$\Delta_A Gf(x) = G(\Delta_A f)(x)?$$

Please let me know if you have any questions.