  In this paper, we will introduce and study several types of Kakeya
inequalities by the maximal functions in Hardy spaces in $\RR^n$,\,$(n\geq2)$,
and we could obtain several inequalities associated with the Kakeya
inequalities. We will show that $\big\|M^t_{\delta S_{\alpha,\beta}} f
\big\|_p\lesssim_{p,n,\varphi,\varepsilon}\left(\frac{1}{\delta}\right)^{5\left(\frac{n}{r}+2\right)\varepsilon}\big\|f\big\|_p$,
when $f(x)\in L^p(\RR^n)$ and $supp\,\hat{f}(\xi)\subseteq B(0,1)$.

Classical Analysis and ODEs