arXiv subjects
Publications and source records attributed to Yufeng Lu.
The embedding relations between Besov-Triebel-Sobolev spaces and modulation spaces are determined explicitly. We extend the results of Sugimoto[2007]; Wang[2007] and Kobayashi[2011] to the most general cases. And we give the sharp embedding between Fourier $L^{p}$ spaces and modulation spaces.
We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz-Finsler version of the Bonnet-Myers theorem based on a generalized Bishop inequality.
In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,μ)$. The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if $\mathscr{X}$ supports a weak Poincaré inequality and the measure $μ$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Hajłasz gradient, the authors also introduce the Hajłasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Hajłasz-Morrey-Sobolev space when $μ$ is doubling and $\mathscr{X}$ supports a weak Poincaré inequality. In particular, on the Euclidean space ${\mathbb R}^n$, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Hajłasz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when $(\mathscr{X},d)$ is geometrically doubling and $μ$ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when $μ$ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Hajłasz-Morrey-Sobolev spaces.