We prove a scale-invariant boundary Harnack inequality and H\"older continuity up to the boundary of quotients of positive fractional $p$-harmonic functions in Reifenberg flat domains with sufficiently small flatness constant, for $0<s<1$ and $2\le p<2/(1-s)$. The solutions are globally nonnegative, vanish on a common...
We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=\Delta_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable \(y_m\). Earlier theory required independenc...
Let $m\geq1$, $k\geq0$, $n=m+k$, and write $x=(v,z)\in\mathbb R^m\times \mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y+\partial_t$. We consider \[ \operatorname{div}_x(A\nabla_xu)-Y(wu)=0, \] where $w=w(x)\in A_2(\mathbb R^n)$ and the measurable matrix $A$ h...
We prove global reverse H\"older estimates in \(\mathrm L^2\) for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable \(X=(\lambda,x)\), while no periodicity in time is assumed. If its boundary trace...
We develop a finite-width geometric framework describing how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. Incompatibility among weight-generated covariance, gates, and backward sensitivities is quantified through three families of commutators: between gates and...
K. Nyström· 1 citation· ⚡1
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