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

on ,

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

and

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

for

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 ):

;

; and

.

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)