Pome9767EC7AA70A
Lemma 3.3.2.77E9E0297059 Let $U\subset \mathbb R^ m$ be open, let $C$ be compact, and let $\mu $ be a finite signed Borel measure on $C$. Suppose $u$ and all the partial derivatives $\partial _{x_ j}u$ are jointly continuous on $U\times C$. Then $F(x)=\int _ Cu(x,y)\, d\mu (y)$ is of class $C^1$ on $U$, and
Proof. Choose a closed ball $K\subset U$ whose interior contains $x$. All the functions in the hypotheses are bounded and uniformly continuous on $K\times C$. For sufficiently small $h\neq 0$, the fundamental theorem of calculus gives
This converges to $\partial _{x_ j}u(x,y)$ uniformly in $y\in C$. Integrating the uniform error against the finite measure $|\mu |$ proves the differentiation formula. The same uniform continuity, now applied to $\partial _{x_ j}u$, proves continuity of the derivative integral as $x$ varies inside $K$. Joint continuity of $u$ also makes $F$ finite and continuous. The ball $K$ can be chosen about every point of $U$, proving the assertion. $\square$
Continuity of the $x$ derivatives alone is insufficient. For example, an $x$-independent measurable function $u(x,y)=g(y)$ can fail to be integrable on $C$ even though all those derivatives are zero. The compact-parameter Poisson kernels used below satisfy the corrected hypotheses.
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