Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the Hormander condition for the existence of product of distributions. Study of nonlinear pdes and the renormalization problem in Quantum Field Theory are two major applications of product of distributions.

An important tool in this context is the wavefront set of a distribution. Roughly speaking, the wavefront set locates the singularities of a distribution in space and also helps in the analysis of the 'direction' of singularities. For $u\in \mathcal{D}'(\mathbb{R}^n)$ the wavefront set $WF(u)\in T^{*}(\mathbb{R}^n)$. If $(x,\xi)\in WF(u)$ then $\xi\in \mathbb{R}^{n}/\{0\}$ is the direction of 'propagation of singularity' that is located at $x\in\mathbb{R}^{n}$.

Hormonder condition for existence of product: Let $u,v\in \mathcal{D}'({\mathbb{R}^n})$. If for every $(x,\xi)\in WF(u)$ we have $(x,-\xi)\notin WF(v)$ then the product $uv$ exists and the Leibniz rule $\partial(uv)=\partial(u)v+u\partial(v)$ holds. Moreover, $WF(uv)\subseteq WF(u)\cup WF(v)\cup \{(x,\xi+\eta):(x,\xi)\in WF(u), (x,\eta)\in WF(v)\}$.

We can note the following from the above condition:

1. If the point of singularity is same for both distributions then for the product to exist the 'directions' of the singularity should not cancel each other.
2. We can also estimate the wavefront set of the product.
3. The condition is only a sufficiency condition. The condition only provides information on the product of distributions that are relevant for pdes - those that satisfy the Lebiniz rule. For example, the Heaviside function can be multiplied with itself but the Lebniz rule does not hold for the square of Heaviside function. So, the Hormander condition rules out square of Heaviside function.

Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the Hormander condition for the existence of product of distributions. Study of nonlinear pdes and the renormalization problem in Quantum Field Theory are two major applications of product of distributions.

An important tool in this context is the wavefront set of a distribution. Roughly speaking, the wavefront set locates the singularities of a distribution in space and also helps in the analysis of the 'direction' of singularities. For $u\in \mathcal{D}'(\mathbb{R}^n)$ the wavefront set $WF(u)\subset T^{*}(\mathbb{R}^n)$. If $(x,\xi)\in WF(u)$ then $\xi\in \mathbb{R}^{n}-\{0\}$ is the direction of 'propagation of singularity' that is located at $x\in\mathbb{R}^{n}$.

Hormander condition for existence of product: Let $u,v\in \mathcal{D}'({\mathbb{R}^n})$. If for every $(x,\xi)\in WF(u)$ we have $(x,-\xi)\notin WF(v)$ then the product $uv$ exists and the Leibniz rule $\partial(uv)=\partial(u)v+u\partial(v)$ holds. Moreover, $WF(uv)\subseteq WF(u)\cup WF(v)\cup \{(x,\xi+\eta):(x,\xi)\in WF(u), (x,\eta)\in WF(v)\}$.

We can note the following from the above condition:

1. If the point of singularity is same for both distributions then for the product to exist the 'directions' of the singularity should not cancel each other.
2. We can also estimate the wavefront set of the product.
3. The condition is only a sufficiency condition. The condition only provides information on the product of distributions that are relevant for pdes - those that satisfy the Lebiniz rule. For example, the Heaviside function can be multiplied with itself but the Lebniz rule does not hold for the square of Heaviside function. So, the Hormander condition rules out square of Heaviside function.

Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the Hormander condition for the existence of product of distributions. Study of nonlinear pdes and the renormalization problem in Quantum Field Theory are two major applications of product of distributions.

An important tool in this context is the wavefront set of a distribution. Roughly speaking, the wavefront set locates the singularities of a distribution in space and also helps in the analysis of the 'direction' of singularities. For $u\in \mathcal{D}'(\mathbb{R}^n)$ the wavefront set $WF(u)\in T^{*}(\mathbb{R}^n)$. If $(x,\xi)\in WF(u)$ then $\xi\in \mathbb{R}^{n}/\{0\}$ is the direction of 'propagation of singularity' that is located at $x\in\mathbb{R}^{n}$.

Hormonder condition for existence of product: Let $u,v\in \mathcal{D}'({\mathbb{R}^n})$. If for every $(x,\xi)\in WF(u)$ we have $(x,-\xi)\notin WF(v)$ then the product $uv$ exists and the Leibniz rule $D^{\alpha}(uv)=D^{\alpha}(u)v+uD^{\alpha}(v)$ holds for all multi-indices $\alpha$. Moreover, $WF(uv)\subseteq WF(u)\cup WF(v)\cup \{(x,\xi+\eta):(x,\xi)\in WF(u), (x,\eta)\in WF(v)\}$.

We can note the following from the above condition:

1. If the point of singularity is same for both distributions then for the product to exist the 'directions' of the singularity should not cancel each other.
2. We can also estimate the wavefront set of the product.
3. The condition is only a sufficiency condition. The condition only provides information on the product of distributions that are relevant for pdes - those that satisfy the Lebiniz rule. For example, the Heaviside function can be multiplied with itself but the Lebniz rule does not hold for the square of Heaviside function. So, the Hormander condition rules out square of Heaviside function.

Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the Hormander condition for the existence of product of distributions. Study of nonlinear pdes and the renormalization problem in Quantum Field Theory are two major applications of product of distributions.

An important tool in this context is the wavefront set of a distribution. Roughly speaking, the wavefront set locates the singularities of a distribution in space and also helps in the analysis of the 'direction' of singularities. For $u\in \mathcal{D}'(\mathbb{R}^n)$ the wavefront set $WF(u)\in T^{*}(\mathbb{R}^n)$. If $(x,\xi)\in WF(u)$ then $\xi\in \mathbb{R}^{n}/\{0\}$ is the direction of 'propagation of singularity' that is located at $x\in\mathbb{R}^{n}$.

Hormonder condition for existence of product: Let $u,v\in \mathcal{D}'({\mathbb{R}^n})$. If for every $(x,\xi)\in WF(u)$ we have $(x,-\xi)\notin WF(v)$ then the product $uv$ exists and the Leibniz rule $\partial(uv)=\partial(u)v+u\partial(v)$ holds. Moreover, $WF(uv)\subseteq WF(u)\cup WF(v)\cup \{(x,\xi+\eta):(x,\xi)\in WF(u), (x,\eta)\in WF(v)\}$.

We can note the following from the above condition:

1. If the point of singularity is same for both distributions then for the product to exist the 'directions' of the singularity should not cancel each other.
2. We can also estimate the wavefront set of the product.
3. The condition is only a sufficiency condition. The condition only provides information on the product of distributions that are relevant for pdes - those that satisfy the Lebiniz rule. For example, the Heaviside function can be multiplied with itself but the Lebniz rule does not hold for the square of Heaviside function. So, the Hormander condition rules out square of Heaviside function.