This week we come to the study of singular integral operators, that is operators of the form
defined initially for `nice’ functions . Here we typically want to include the case where has a singularity close to the diagonal
which is not locally integrable. Typical examples are
and in one dimension
and so on. Observe that these kernels have a non integrable singularity both at infinity as well as on the diagonal . It is however the local singularity close to the diagonal that is important and will lead us to characterize a kernel as a singular kernel. For example, the kernel
is not a singular kernel since its singularity is locally integrable. Observe that for Schwartz functions it makes perfect sense to define
and in fact the previous integral operator was already considered in the Hardy-Littlewood-Sobolev inequality of Exercise 12 in Notes 5 and can be treated via the standard tools we have seen so far.
Thus, if one insists on writing the representation formula (1) throughout then will not be a function in general. Indeed, the discussion in Notes 4 reveals that if the operator is translation invariant then the kernel must necessarily be of the form for an appropriate tempered distribution :
Bearing in mind that there are tempered distributions which do not arise from functions or measures we see that (1) does not make sense in general and it should be understood in a different way. To give a more concrete example, think of the principal value distribution and write
Here we would like to rewrite this in the form
but this does not make sense even for since the function is not locally integrable on the diagonal .
In fact, the representation (1) of the operator will not be true in general but we will satisfy ourselves with its validity for functions , of compact support, and whenever does not lie in the support of . Indeed, if has compact support and then in (1) and thus we are away from the diagonal. Indeed, returning to the principal value example, observe that the integral
makes perfect sense when has compact support and . Continue reading