We classify the entrywise functions that preserve positive semidefiniteness on discrete Green matrices \(G(p,q)=(p_{\min(i,j)}q_{\max(i,j)})\) with positive parameters, without requiring the resulting matrix to retain Green structure. For matrices of all orders, the preservers are the zero function and the functions \(f(t)=\int_{[0,\infty)}t^\alpha\,d\mu(\alpha)\), where \(\mu\) is a nonzero finite positive measure and the integral is finite for every \(t>0\). Requiring the resulting matrix to be totally nonnegative reduces the nonzero preservers to \(f(t)=ct^\alpha\), where \(c>0\) and \(\alpha\ge0\). These power functions also preserve positive semidefinite Green structure, while strict Green structure is preserved precisely when \(\alpha>0\). No regularity assumption is needed for these classifications. We also characterize continuously differentiable functions that are entrywise Loewner monotone on every fixed-\(q\) Green family: this holds precisely when \(f'\) is a positive mixture of nonnegative real powers, with the zero measure allowed.
Let \(n\ge1\), \(00\), and let \(A\) be a positive definite symmetric matrix. We prove that the unique decaying viscosity solution of \[ \min\{u,\,(-\Delta)^s u-(c-\langle Ax,x\rangle)\}=0 \qquad\text{in }\mathbb R^n \] has the form \[ u(x)=K\max\{1-\langle Bx,x\rangle,0\}^{1+s} \] for some \(K>0\) and some positive de...
Let $U,V$ be matrix $\mathcal A_p$ weights, $1<p<\infty$, and let $B$ be a matrix-valued function. For Calder\'on--Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\otimes I_m]:L^p(U)\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse...
Ji Li, Chong-Wei Liang, Chun-Yen Shen et al.· 2 citations· ⚡1
Let \(q=p^h\) be an odd prime power, and let the affine group \(\mathbb F_q\rtimes\mathbb F_q^\times\) act through its canonical \((q-1)\)-dimensional irreducible representation. Qualitative matrix recovery for these rank-one orbits is known. We determine sharp lower singular-value bounds for explicit real generating w...
We characterize locally finite sets \(T\subset[0,\infty)\) that preserve exact observability from nonnegative integer times for every bounded positive operator \(A\) and bounded observation operator \(C\). The Stein identity reduces universality to uniform upper and lower bounds on radial Stein--Pick cones over all fin...
Let $f:(0,\infty)\to(0,\infty)$ be a nontrivial normalized operator monotone function and set $s=f'(1)$. We introduce the alternative-mean trace functional $$ \altPhi_f(A,B) :=\Tr(A\nabla_s B) -\Tr\!\left( f(A^{-1}\sharp B)\,A\,f(A^{-1}\sharp B) \right). $$ on the positive definite cone. We prove that $\altPhi_f$ is a...
Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=\lambda_0\leq\lambda_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{\phi_j\}_{j\geq0}$. Let $\{\omega_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured...
Talant Talipov· 0 citations
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