Kakeya inequalities by maximal functions in Hardy spaces

Abstract

In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in $\RR^n$,\,$(n\ge 2)$, and we could obtain several inequalities associated with the Kakeya inequalities. We will show that $\Vert M_{\delta S_{\alpha ,\beta }}^{t}f{\Vert }_p{\lesssim }_{p,n,\phi ,\epsilon }{\left(\frac{1}{\delta }\right)}^{5\left(\frac{n}{r}+2\right)\epsilon }\Vert f{\Vert }_p$, when $f(x)\in L^p(\RR^n)$ and $supp\hat{f}(\xi )\subseteq B(0,1)$.

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References in corpus (1)

• An improved bound on the Minkowski dimension of Besicovitch sets in R^3