For $0<\alpha<1$, let $W_{\alpha}$ and $R_{\alpha}$ denote Weyl fractional integral operator and Riemann-Liouville fractional integral operator, respectively. We establish sharp versions of Muckenhoupt-Wheeden conjecture for these operators. Specifically, we prove that for any weight $w$ on $[0,\infty)$, we have \begin{equation*} \|{W}_{α} f\|_{L^{1/(1-α),∞}(w)}≤ α^{-1}\|{f}\|_{L^{1}((M_{-}w)^{1-α})} \end{equation*} and \begin{equation*} \|{R}_{α} f\|_{L^{1/(1-α),∞}(w)}≤ α^{-1}\|{f}\|_{L^{1}((M_{+}w)^{1-α})}. \end{equation*} Here $M_{-}$, $M_{+}$ denote the one-sided Hardy-Littlewood maximal operators on $[0,\infty)$. In each of the estimates, the constant $\alpha^{-1}$ is the best possible.

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_{g}\mid g\in G)$ by $k$-automorphisms $h(x_{g})=x_{hg}$ for any $g,h\in G$. Noether’s problem asks whether the invariant field $k(G)=k(x_{g}\mid g\in G)^{G}$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_{p}$ of prime order $p$, it is unknown whether there exist infinitely many primes $p$ such that $\mathbf{Q}(C_{p})$ is rational over $\mathbf{Q}$. Only known 17 primes $p$ for which $\mathbf{Q}(C_{p})$ is rational over $\mathbf{Q}$ are $p\leq 43$ and $p=61,67,71$. We show that for primes $p< 20000$, $\mathbf{Q}(C_{p})$ is not (stably) rational over $\mathbf{Q}$ except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbf{Q}(C_{p})$ is not (stably) rational over $\mathbf{Q}$ for undetermined 28 primes $p$ out of 46.