Research based on Metric Measure Spaces in Machine Learning part4

Assume that (X,d,μ) is a metric space endowed with a non-negative Borel measure μ satisfying the doubling condition and the additional condition that μ(B(x,r))≳rn for any x∈X,r>0 and some n≥1. Let L be a non-negative self-adjoint operator on L2(X,μ). We assume that e−tL satisfies a Gaussian upper bound and the Schrödinger operator eitL satisfies an L1→L∞ decay estimate of the form ∥eitL∥L1→L∞≲|t|−n2. Then for a general class of dispersive semigroup eitφ(L), where φ:R+→R is smooth, we establish a similar L1→L∞ decay estimate by a suitable subordination formula connecting it with the Schrödinger operator eitL. As applications, we derive new Strichartz estimates for several dispersive equations related to Hermite operators, twisted Laplacians and Laguerre operators. 2.Weak porosity on metric measure spaces (arXiv) Author : <a href="https://arxiv.org/search/?searchtype=author&query=Mudarra%2C+C" rel="noopener ugc nofollow" target="_blank">Carlos Mudarra</a> Abstract : We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power dist(⋅,E)−α belongs to the Muckenhoupt A1 class of weights in X. To this end, we introduce the weakly porous and dyadic weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest E-free holes, these sets provide the mentioned A1-characterization. We exhibit examples showing the optimality of these conditions, and also see how they can be simplified in the particular case where the underlying measure possesses certain qualitative version of the annular decay property. Moreover, for these classes of sets E and every 1≤p<∞, we determine the range of exponents α for which dist(⋅,E)−α∈Ap(X) <a href="https://medium.com/@monocosmo77/research-based-on-metric-measure-spaces-in-machine-learning-part4-c7d0d8dc9fa1">Click Here</a>