Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $G\subset{\mathbb R}^m$ be a nonempty compact and Jordan measurable set that coincides with the closure of its interior. Let $K:G\times G\to{\mathbb C} $ be a continuous function. Then we can define an linear operator $A:C(G)\to C(G)$ as $$ (A\phi)(x):=\int_G K(x,y)\phi(y)dy,\quad x\in G. $$ If the kernel $K$ is defined and continuous for all $x,y\in G\subset{\mathbb R}^m, x\neq y$, and there exist positive constants $M$ and $\alpha\in(0,m]$ such that $$ |K(x,y)|\leq M|x-y|^{\alpha-m}, \quad x,y\in G, x\neq y, $$ the kernel $K$ is called weakly singular.

The definitions above are from Linear Integral Equations by Rainer Kress.

Here are my questions:

If the kernel is a rank-2 tensor, for example, the Oseen tensor in this wiki-article about Stokes flow, can I define the concept that "the kernel is weakly singular"? If the answer is YES, how should I do it? One possible way I think is using some norm of matrices.

share|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.