A note on maximal singular integrals with rough kernels

In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

Continuum Deformation of Multi-Agent Systems Under Directed Communication Topologies

A leader follower model has recently been proposed for homogeneous deformation of a multi-agent system (MAS) in ℝn. Researchers have shown how a desired homogeneous transformation can be designed by choosing proper trajectories for n + 1 leader agents and can be learned by every follower through local communication. However, existing work requires every follower to communicate with n + 1 adjacent agents, where communication between every two adjacent followers is constrained to be bidirectional. These requirements limit the total allowable number of agents, so an arbitrary number of agents may not be able to acquire a desired homogeneous mapping by local interaction. Additionally, if followers are not allowed to communicate with more than n + 1 neighboring agents, the convergence rate of actual positions to the desired positions (defined by a homogeneous transformation) may not be sufficiently high. The system may then considerably deviate from the desired configuration during transition. The main contribution of this article is to address these two issues, where each follower is considered to be a general linear system. It will be proven that followers can acquire desired positions prescribed by a homogeneous mapping in the presence of disturbance and measurement noise by applying a new finite-time reachability model under either fixed or switching topologies, if: (i) communication among followers is defined by a directed and strongly connected subgraph, (ii) each follower applies a consensus protocol with communication weights that are consistent with the positions of the agents in the initial configuration, and (iii) every follower i is allowed to communicate with mi≥n+1 local agents. With this strategy, an MAS with an arbitrary number of agents with linear dynamics can acquire a desired homogeneous mapping in the presence of disturbance and measurement noise, where convergence rate can be enhanced by increasing the number of communication links.

Singular integrals related to homogeneous mapping with rough kernels on product spaces

In this paper, we study the $ L^{p} $ mapping properties of singular integral operators related to homogeneous mappings on product spaces with kernels which belong to block spaces. Our results extend as well as improve some known results on singular integrals.