Recently I am doing distributions theory, this notes is for filling small gaps and showing some “applications”. I should put some definitions at the top to make this article self-containing. Unfortunately I do not have much time, so I go directly to what I hope to say first and I will fill in the background later.
Proposition A locally integrable function on of polynomial growth (that is, there exist and non-negative integer such that
where is the Euclidean norm)
induces a tempered distribution on , given by
for all .
Proof. Since is locally integrable, it induces a distribution. Next we check is integrable for . Indeed for such , which is integrable as is rapidly decreasing. So is integrable. Thus is well defined on .
It remains to establish a semi-norm estimate for in order to show it is continuous on . For each ,
where is some constant, and is some nonnegative integer.
Example 1 (Linear analysis prelim, 2009) Show that
are tempered distributions on . Also show that , where denotes the Fourier transform.
Solution. For the first part, a direct check would work. Or it is just an application to some general results I learnt. Let be the Dirac delta distribution. For any two distributions and on , one can define the tensor product
Using semi-norm estimates one can check is a tempered distribution on whenever and are tempered distributions on . In our case, while . By the above proposition is a tempered distribution on . It is known that is tempered too. Thus is a tempered distribution on , and so is . The second part follows from the following three facts, which can be checked by computations (on ):
In our case, .
Example 2 (Linear analysis prelim, 2008)
Show that, for any tempered distribution on , there is a unique tempered distribution that solves
Consider . If is the distribution , show that is a function, defined at every point and bounded by a constant. Also, for each , calculate .
Solution. Thanks to the instructor of linear analysis providing the main idea of this problem. (To be continued)