Pointwise convergence approximate identities of dilated radially decreasing kernels

Abstract:

Let $\phi$ be integrable on ${R^n}$ with $\int _{{R^n}} {\phi (y)dy = 1}$. It is shown that \[ {\lim _{\varepsilon \to 0 + }}(\phi _F^*f)(x) = {\lim _{\varepsilon \to 0 + }}{\varepsilon ^{ - n}}\int _{{R^n}} {(\frac {{x - y}} {\varepsilon })f(y)dy = f(x)} \] a.e. on ${R^n}$, whenever the least radially decreasing majorant of $\phi$, defined by $\psi (x) = {\sup _{|y| \geqslant |x|}}|\phi (y)|$, is such that $|x{|^n}\psi (x) = |x{|^n}\psi (|x|)$ is nonincreasing in $|x|$ when $|x|$ is large and $(\psi _{{\varepsilon _0}}^*|f|)({x_0}) < \infty$ for some ${x_0} \in {R^n}$ and ${\varepsilon _0} > 0$.