Narrow Results

In this paper, the authors characterize the Sobolev spaces $W^{\alpha ,p}({ℝ}^n)$ with $\alpha \in (0,2]$ and $p\in (max\{1,\frac{2n}{2\alpha +n}\},\infty )$ via a generalized Lusin area function and its corresponding Littlewood–Paley $g_{\lambda }^{\ast }$-function. The range $p\in (max\{1,\frac{2n}{2\alpha +n}\},\infty )$ is also proved to be nearly sharp in the sense that these new characterizations are not true when $\frac{2n}{2\alpha +n}>1$ and $p\in (1,\frac{2n}{2\alpha +n})$. Moreover, in the endpoint case $p=\frac{2n}{2\alpha +n}$, the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

In this paper, we prove the boundedness for the maximal and fractional maximal operators and Riesz potential-type operator associated with the Kontorovich–Lebedev transform (KL transform)in the $L^p({ℝ}_{+},x^{-\beta }dx)$ spaces.