Convolution of temperate distributions. (English) Zbl 0627.46046

Kučera, Jan

Publ. Math. Debr. 33, 323-327 (1986).

Let \(L_ n\), \(n\in \mathbb N\), be the Hilbert space of all functions \(\phi:\mathbb R^ r\to \mathbb C\) for which each \(x^{\alpha}D^{\beta}\phi\), \(\alpha,\beta \in \mathbb N^ r\), \(| \alpha +\beta | \leq n\), is square integrable on \(\mathbb R^ r\) and \(L_{-n}\) be the dual space of \(L_ n\). Here Sobolev derivatives are used to ensure the completeness of \(L_ n\). The space \({\mathcal S}'\) of temperate distributions equals \(\mathrm{indlim} \{L_{-n};n\to \infty \}\) and \({\mathcal S}=\mathrm{projlim} \{L_ n;n\to \infty \}\). Denote by \({\mathcal O}^*_ n\), \(n\in \mathbb N\), the space of all continuous operators \(f:\phi \mapsto f^*\phi:L_{-n}\to {\mathcal S}'\). Then the following statements are equivalent:
1. \(f\in {\mathcal O}^*_ n,\)
2. \(f=\Sigma \{D^{\alpha}f_{\alpha};\alpha \in A\}\), where \(f_{\alpha}\in L_ n\) and \(A\subset \mathbb N^ r\) is finite,
3. \(f\in U\{(1-\Delta)^ kL_ n;k\in \mathbb N\}\), where \(\Delta\) is the Laplace operator,
4. \(\phi \mapsto (1+| x|^ 2)^{n}(f^*\phi):{\mathcal S}\to L^ 2(\mathbb R^ r)\) is continuous.
If \({\mathcal O}^*_ n\) carries the topology of \(\mathrm{indlim}\{(1-\Delta)^ kL_ n;k\to \infty \}\) then the classical space \({\mathcal O}_ C'\) of convolution operators on \({\mathcal S}'\) equals \(\mathrm{projlim}\{{\mathcal O}^*_ n;n\to \infty \}\). The convolution \((f,g)\mapsto f^*g:{\mathcal O}^*_ n\times L_{-n}\to {\mathcal S}'\) is continuous while it is not continuous on \({\mathcal O}_ C'\times {\mathcal S}'\). Any temperate distribution \(g\in {\mathcal S}'\) belongs to some \(L_{-n}\). Hence it can be convolved by distributions from \({\mathcal O}^*_ n\), a space strictly larger than \({\mathcal O}_ C'\).

MSC:

46F05	Topological linear spaces of test functions, distributions and ultradistributions
46F10	Operations with distributions and generalized functions

Keywords:

space of convolution operators; inductive and projective limits; Sobolev derivatives; temperate distributions