= Mass normalization in mollified singular integrals
{title2=$T(\phi_\varepsilon)*f=\phi_\varepsilon*T(f)\longrightarrow(\int\phi)T(f)$}
For a bounded <Fourier multiplier operator> $T$ on $L^2$ and a <bump function> $\phi$, the displayed convolution identity follows by multiplying Fourier transforms. The convergence holds in $L^2$ and <almost everywhere> by the <approximate identity> theorem, after accounting for the integral of the kernel. The limit is $T(f)$ when $\int\phi=1$. Smoothness and compact support alone do not imply this normalization.
Back to article page
