Weak and strong type estimates for multilinear Calderón–Zygmund operators on differential forms

In this paper, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of the operators. Based on nonhomogeneous A-harmonic tensor, the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms are obtained.

On Some Integral Inequalities in Quantum Calculus

The objective of this paper is to establish q -analogue of some well-known inequalities in analysis, namely, Poincaré-type inequalities, Sobolev-type inequalities, and Lyapunov-type inequalities. Our obtained results may serve as a useful source of inspiration for future works in quantum calculus.

Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.