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If by 'finitely supported' you mean supported on a set made by a finite number of points, the answer is no. As a simple example, let $D=\partial_t-\partial_{xx}$ be the heat operator, $E=\{(t,x):t>0,x\in R\}$ and $u(t,x)=t^{-1/2}e^{-|x-y|^2/4t}$ the fundamental solution. Then $D(u\chi_E)$ is (a multiple of) the delta at $(0,0)$.

EDIT: I did not notice the set $E$ was assumed to be bounded in the question. In this case let $E$ be the square $(0,1)^2$ in $R^2$, $D=\partial_x\partial_y$ and $u=1$. The support of $D(\chi_E u)$ is the set of corners of $E$.

Let $E$ be the square $(0,1)^2$ in $R^2$, $D=\partial_x\partial_y$ and $u=1$. The support of $D(\chi_E u)$ is the set of corners of $E$.

If by 'finitely supported' you mean supported on a set made by a finite number of points, the answer is no. As a simple example, let $D=\partial_t-\partial_{xx}$ be the heat operator, $E=\{(t,x):t>0,x\in R\}$ and $u(t,x)=t^{-1/2}e^{-|x-y|^2/4t}$ the fundamental solution. Then $D(u\chi_E)$ is (a multiple of) the delta at $(0,0)$.

If by 'finitely supported' you mean supported on a set made by a finite number of points, the answer is no. As a simple example, let $D=\partial_t-\partial_{xx}$ be the heat operator, $E=\{(t,x):t>0,x\in R\}$ and $u(t,x)=t^{-1/2}e^{-|x-y|^2/4t}$ the fundamental solution. Then $D(u\chi_E)$ is (a multiple of) the delta at $(0,0)$.

EDIT: I did not notice the set $E$ was assumed to be bounded in the question. In this case let $E$ be the square $(0,1)^2$ in $R^2$, $D=\partial_x\partial_y$ and $u=1$. The support of $D(\chi_E u)$ is the set of corners of $E$.

If by 'finitely supported' you mean supported on a set made by a finite number of points, the answer is no. As a simple example, let $D=\partial_t-\partial_{xx}$ be the heat operator, $E=\{(t,x):t>0,x\in R\}$ and $u(t,x)=t^{-1/2}e^{-|x-y|^2/4t}$ the fundamental solution. Then $D(u\chi_E)$ is the delta at $(0,0)$.

If by 'finitely supported' you mean supported on a set made by a finite number of points, the answer is no. As a simple example, let $D=\partial_t-\partial_{xx}$ be the heat operator, $E=\{(t,x):t>0,x\in R\}$ and $u(t,x)=t^{-1/2}e^{-|x-y|^2/4t}$ the fundamental solution. Then $D(u\chi_E)$ is (a multiple of) the delta at $(0,0)$.