## Structural theorems for quasiasymptotics of distributions at the origin.(English)Zbl 1189.46032

The quasiasymptotics of distributions at the origin is defined with respect to the function $$t^aL(t)$$, where $$a\in\mathbb{R}$$ and $$L$$ is a function slowly varying at zero. It is easy to prove that if a tempered distribution $$T$$ has quasiasymptotics at zero with respect to $$t^aL(t)$$ in $$S'$$, then $$T$$ has it in $$D'$$, as well. In the meantime, it was shown by some authors that, with some limitation on $$L$$ and $$a$$, the converse is also true.
In this paper, it is simply proved (Theorem 6.1): Let $$f\in S'$$. If $$f$$ has quasiasymptotics at $$0$$ in $$D'$$, then $$f$$ has quasiasymptotics at $$0$$ in $$S'$$. The main tool for the proof is a new class of functions, i.e., the class of asymptotically homogeneous functions with respect to a slowly varying function which is elaborated in detail. With this result, an open problem is completely solved.

### MSC:

 46F05 Topological linear spaces of test functions, distributions and ultradistributions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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### References:

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