Regular tempered distribution

A regular tempered distribution on $\mathbb R^n$ is a tempered distribution $T_f$ for which there is a locally integrable function $f$ satisfying

\langle T_f,\varphi\rangle =\int_{\mathbb R^n}f(x)\varphi(x)\,dx \qquad \text{for every }\varphi\in\mathcal S(\mathbb R^n).

Here $\varphi$ ranges over the Schwartz space, and local integrability means that $f$ is Lebesgue integrable on every compact set. In addition, the displayed integral must be absolutely convergent for every Schwartz function and must depend continuously on $\varphi$ in the Schwartz topology. Functions equal almost everywhere determine the same regular tempered distribution.

Sufficient growth conditions

If $f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and

|f(x)|\leq C(1+|x|)^N

almost everywhere for some $C,N$ , then $f$ defines a regular tempered distribution. More generally, it suffices that the integral of $|f|$ over balls grow at most polynomially. Pointwise polynomial growth is therefore a convenient sufficient condition, not the definition.

Examples

Every function in $L^p(\mathbb R^n)$ for $1\leq p\leq\infty$ defines a regular tempered distribution by Hölder's inequality, since Schwartz functions belong to the conjugate $L^q$ space. Polynomials and bounded measurable functions give further examples. A function such as $e^{|x|}$ generally fails because multiplication by a Schwartz function need not be integrable.

Regular versus singular

The adjective regular refers to representation by a function, not to smoothness of that function. The Dirac delta and its derivatives are tempered but not regular, since no locally integrable function represents point evaluation on all test functions. Regular distributions are therefore a proper subclass of tempered distributions.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: §§2.1 and 7.1 on regular and tempered distributions.
Robert S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, CRC Press, 1994. Publisher record. Relevant: introductory examples of distributions and tempered distributions.