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Sean Lawton
Sean Lawton
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Up onBy the Schwartz kernel theoremKernel Theorem, we know that the kernel of an integral operator belongs to distribution space S'(R^n)$\mathscr{S}'(\mathbb{R}^n)$. Moreover, we know that the kernel K$K$ is $C^{\infty}$ off the diagonal in $\mathbb{R}^n \times \mathbb{R}^n$. 

Now my questions are that

1- Does the Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?:

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?

  1. Does the Schwartz kernel of a pseudo-differential operator of arbitrary order belong to the Schwartz space?

  2. Can we have the Schwartz Kernel Theorem be true with the kernel belonging to $\mathscr{S}(\mathbb{R}^n)$ as opposed to $\mathscr{S}'(\mathbb{R}^n)$?

Up on Schwartz kernel theorem we know that the kernel of an integral operator belongs to distribution space S'(R^n). Moreover, we know that the kernel K is $C^{\infty}$ off diagonal in $\mathbb{R}^n \times \mathbb{R}^n$. Now my questions are that

1- Does the Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?

By the Schwartz Kernel Theorem, we know that the kernel of an integral operator belongs to distribution space $\mathscr{S}'(\mathbb{R}^n)$. Moreover, we know that the kernel $K$ is $C^{\infty}$ off the diagonal in $\mathbb{R}^n \times \mathbb{R}^n$. 

Now my questions are:

  1. Does the Schwartz kernel of a pseudo-differential operator of arbitrary order belong to the Schwartz space?

  2. Can we have the Schwartz Kernel Theorem be true with the kernel belonging to $\mathscr{S}(\mathbb{R}^n)$ as opposed to $\mathscr{S}'(\mathbb{R}^n)$?

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Liviu Nicolaescu
Liviu Nicolaescu
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Up on Schwartz kernel theorem we know that the kernel of an integral operator belongs to distribution space S'(R^n). Moreover, we know that the kernel K is C^{\infty}$C^{\infty}$ off diagonal in R^n \times R^n$\mathbb{R}^n \times \mathbb{R}^n$. Now my questions are that

1-Dose Does the Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?

Up on Schwartz kernel theorem we know that the kernel of an integral operator belongs to distribution space S'(R^n). Moreover, we know that the kernel K is C^{\infty} off diagonal in R^n \times R^n. Now my questions are that

1-Dose Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?

Up on Schwartz kernel theorem we know that the kernel of an integral operator belongs to distribution space S'(R^n). Moreover, we know that the kernel K is $C^{\infty}$ off diagonal in $\mathbb{R}^n \times \mathbb{R}^n$. Now my questions are that

1- Does the Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?

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user80489
user80489
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Schwartz kernel

Up on Schwartz kernel theorem we know that the kernel of an integral operator belongs to distribution space S'(R^n). Moreover, we know that the kernel K is C^{\infty} off diagonal in R^n \times R^n. Now my questions are that

1-Dose Schwartz kernel of pseudo-differential operator of arbitrary order belong to Schwartz space?

2- Can we have the Schwartz kernel Theorem with kernel belong to S(R^n) Not S'(R^n)?