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Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with inne …

For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form: $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left( …

Let $\nu \propto e^{-f}$ be a probability density on $\mathbb{R}^d$ with full support. We say $\nu$ satisfies the log-Sobolev inequality (LSI) with constant $\alpha$ if for every smooth function $g:\m …

Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that there is a reproducing kernel Hilbert Space $\mathc …

Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ respec …

Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true : $$ A^* f(AA^*) = f(A^* A) A^*$$