Let ##k:\mathcal{O}\times\mathbb{R}^n\to\mathbb{R}##, with ##\mathcal{O}\subset\mathbb{R}^m## open, be such that ##\forall x\in\mathcal{O}\quad k(x,\cdot)\in L^1(\mathbb{R}^n) ##, i.e. the function ##y\mapsto k(x,y)## is Lebesgue summable on ##\mathbb{R}^n##, according to the usual ##n##-dimensional Lebesgue measure.(adsbygoogle = window.adsbygoogle || []).push({});

I read (theorem 1.d here, p. 2) that, if ##\partial_{x_j}\partial_{x_i} k(\cdot,y)\in C(\mathcal{O})## for almost all ##y\in\mathbb{R}^n## and there exists ## g\in L^1(\mathbb{R}^n)## such that $$|\partial_{x_j}\partial_{x_i} k(x,y)|\le g(y)\quad\forall x\in\mathcal{O}\quad\text{for almost all }y\in\mathbb{R}^n,$$then##^1## $$\partial_{x_j}\partial_{x_i}\int_{\mathbb{R}^n}k(x,y)d\mu_y=\int_{\mathbb{R}^n}\partial_{x_j}\partial_{x_i}k(x,y)d\mu_y.$$

I see, by using a known corollary to the dominated convergence theorem (quoted here), that,providedthat ##\forall x\in\mathcal{O}\quad \partial_{x_i}k(x,\cdot)\in L^1(\mathbb{R}^n) ## (which I do not know how to prove##^2##), then$$\partial_{x_j}\int_{\mathbb{R}^n}\partial_{x_i}k(x,y)d\mu_y=\int_{\mathbb{R}^n}\partial_{x_j}\partial_{x_i}k(x,y)d\mu_y$$but then I do not know how we can "move" ##\partial_{x_i}## outside the integral, although I suspect it may have to do with the rather strong assumption that ##\partial_{x_j}\partial_{x_i} k(\cdot,y)\in C(\mathcal{O})##...

How can we prove that ##\partial_{x_j}\partial_{x_i}\int_{\mathbb{R}^n}k(x,y)d\mu_y=\int_{\mathbb{R}^n}\partial_{x_j}\partial_{x_i}k(x,y)d\mu_y##?

I ##\infty##-ly thank any answerer!

##^1##When I read a text I always assume that it is prefectly worded. Nevertheless, after many fruitless trials, I am beginning to suspect that this result is intended to hold provided that the condition (b) here holds.

##^2##By using Fubini's theorem I only see that, once chosen an arbitrary interval ##[a,t]##, the function ##\partial_{x_i}k(x_t,\cdot)-\partial_{x_i}k(x_a,\cdot)##, where ##x_t## has ##t## as its ##j##-th component, is summable and its integral is ##\int_{[a,t]}\int_{\mathbb{R}^n}\partial_{x_j}\partial_{x_i}k(x_t,y) d\mu_y d\mu_{x_j}##.

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# I Differentiating twice under the integral (with assumptions)

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