Shifted characteristic primitive on a torus

Let $F(s,y)$ be smooth on $\mathbb R\times\mathbb T^n$ with support in a fixed bounded interval in $s$ . For fixed $M\in\mathbb R$ and $v\in\mathbb R^n$ , define

I^-F(U,y)=\int_{-\infty}^U F(s,y+M(s-U)v)\,ds, \qquad I^+F(U,y)=\int_U^\infty F(s,y+M(s-U)v)\,ds.

These are characteristic primitives for the transport derivative $D_M=\partial_U+M v\cdot\nabla_y$ . They satisfy $D_MI^-F=F$ and $D_MI^+F=-F$ .

Fixed-shift representation

With $s=U+z$ , the two integration intervals become $( -\infty,0)$ and $(0,\infty)$ , and the integrand is $F(U+z,y+Mzv)$ . Differentiating in $U,y$ , or any additional smooth parameter on which only $F$ depends, now differentiates the source without a factor of $M$ . If the source interval has length $L$ , each such derivative is bounded by $L$ times the corresponding source supremum. The parameters $M,v$ are held fixed in these estimates.

The sum $JF=I^-F+I^+F$ solves $D_MJF=0$ . It generally does not have compact support in $U$ , even though $F$ does.