# Almost adjointness propertie for distributions

Suppose $$h\in S(\mathbb{R}^d)$$ (Schwartz space) and a family $$\mathcal{F}=\{f(\,\cdot\,;s)\}_{s\in\mathbb{R}^d}\subseteq S^\prime(\mathbb{R}^d)$$ of tempered distributions. Then, for each fixed $$s$$ we have $$f(h;s)\in\mathbb{R}$$.

Now, assume that the map $$s\mapsto f(h;s)$$ is again in $$S(\mathbb{R}^d)$$ and take $$g\in S^\prime(\mathbb{R}^d)$$. Denote the action of $$g$$ over $$s\mapsto f(h;s)$$ by $$\langle g_\cdot,f(h;\,\cdot\,)\rangle$$.

Is there some theorem/propertie that guarantees that there is a continuous function $$M\{g,\mathcal{F}\}:\mathbb{R}^d\to\mathbb{R}$$ such that for all $$h\in S(\mathbb{R}^d)$$ we have $$\langle g_\cdot,f(h;\,\cdot\,)\rangle=\int_{\mathbb{R}^d}h(s)M\{g,\mathcal{F}\}(s)\,\mathrm{d}s\,?$$

EDIT Boris'answer is correct. I'll change the question:

Is there some tempered distribution $$M\{g,\mathcal{F}\}$$ such that $$\langle g_\cdot,f(h;\,\cdot\,)\rangle=M\{g,\mathbb{F}\}(h)$$?

In Boris answer, the tempered distribution $$M\{g,\mathbb{F}\}$$ is $$g$$.

• Can't give a detailed answer now but essentially the answer is yes modulo suitable continuity hypotheses on the family, e.g., the map $h\mapsto (s\mapsto f(h;s))$ being continuous from Schwartz space to itself. The relevant theory has to do with the nuclear theorem and the sequels to Schwartz books: numdam.org/item/AIF_1957__7__1_0 and numdam.org/item/AIF_1958__8__1_0 – Abdelmalek Abdesselam May 14 at 23:43

Let $$f(\cdot;s)=\delta(x-s)$$ and $$g=\delta(x)$$, then $$f(h;\cdot)=h(\cdot)$$ and $$\langle g_\cdot,f(h;\cdot)\rangle=h(0)$$, so there is no such function, that $$h(0)=\int_{\mathbb{R}^d}h(s)M\{g,\mathcal{F}\}(s)\mathrm{d}s$$