Critical trajectories in kinetic geometry are constructed as an almost exponential map, yielding a fundamental-solution-free proof of the kinetic Sobolev inequality and an optimal weak Harnack inequality for the Kolmogorov equation with rough coefficients.
The matched projections of idempotents on Hilbert $C^*$-modules
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abstract
The aim of this paper is to give new characterizations of some fundamental issues about idempotents. In the general setting of adjointable operators on Hilbert $C^*$-modules, a new term of quasi-projection pair is introduced. For each idempotent $Q$, a projection $m(Q)$, called the matched projection of $Q$, is constructed. It is shown that $Q$ and $m(Q)$ as idempotents are homotopic, and $\big(m(Q),Q\big)$ is a quasi-projection pair. Some formulas for $m(Q)$ are derived. Based on these formulas, representations and norm estimations associated with $m(Q)$ are dealt with.
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Critical trajectories in kinetic geometry
Critical trajectories in kinetic geometry are constructed as an almost exponential map, yielding a fundamental-solution-free proof of the kinetic Sobolev inequality and an optimal weak Harnack inequality for the Kolmogorov equation with rough coefficients.