$L^p\to L^q$ estimates for Stein's spherical maximal operators

In this article we consider a modification of the Stein’s spherical maximal operator of complex order $\alpha$ on ${\mathbb{R}}^n$: $${\mathfrak{M}}_{[1,2]}^{\alpha }f(x)=\underset{t\in [1,2]}{\sup }\mid \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_{\mathrm{\mid }y\mathrm{\mid }\le 1}{(1-\mathrm{\mid }y{\mathrm{\mid }}^2)}^{\alpha -1}f(x-ty)dy\mid \mathrm{.}$$ We show that when $n\ge 2$, suppose $\mathrm{\parallel }{\mathfrak{M}}_{[1,2]}^{\alpha }f{\mathrm{\parallel }}_{L^q({\mathbb{R}}^n)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$ holds for some $\alpha \in \mathbb{C}$, $p,q\ge 1$, then we must have that $q\ge p$ and $$\mathrm{R}\mathrm{e}\alpha \ge {\sigma }_n(p,q):=\max \{\frac{1}{p}-\frac{n}{q},\frac{n+1}{2p}-\frac{n-1}{2}(\frac{1}{q}+1),\frac{n}{p}-n+1\}\mathrm{.}$$ Conversely, we show that ${\mathfrak{M}}_{[1,2]}^{\alpha }$ is bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ provided that $q\ge p$ and $\mathrm{R}\mathrm{e}\alpha >{\sigma }_2(p,q)$ for $n=2$; and $\mathrm{R}\mathrm{e}\alpha >\max \{{\sigma }_n(p,q),1\mathrm{/}(2p)-(n-2)\mathrm{/}(2q)-(n-1)\mathrm{/}4\}$ for $n>2$. The range of $\alpha ,p$ and $q$ is almost optimal in the case either $n=2$, or $\alpha =0$, or $(p,q)$ lies in some regions for $n>2$.