The asymptotic of elements belonging to $\mathcal D_{L^p}$ and $\mathcal D'_{L^p}$


Bogoljub Stanković




We know that a distribution $T\in\mathcal D_{L^p}$, $1\leq p<\infty$ has the $S$-asymptotic related to $c(h)=1$ with the limit just zero; namely \[ im_{\|h\|oıfty}angle T(x+h),hi(x)\rangle>0,\quad hiı\mathcal D. \] We examine how fast $\langle T(x+h),\phi(x)\rangle$ can tend to zero when $\phi\in\mathcal D$ or $\phi\in\mathcal D_{L^q}$, $\frac1p+\frac1q\geq1$.