Research proof index

Swap the second spatial derivative $\partial_{xx}$ with the integral $\int$. -/ lemma swap_xx_heatKernel {α : ℝ} {x t : ℝ} (g : ℝ → ℝ) (ht : 0 < t) (hα : 0 < α) (hg : Integrable g) : HasDerivAt (fun x' => ∫ y, HKx α (x' - y) t * g y ∂volume) (∫ y, HKxx α (x - y) t * g y ∂volume) x := by have mem_nhds : Metric.ball x 1 ∈ 𝓝 x := Metric.ball_mem_nhds x (by...

Swap the time derivative $\partial_t$ with the integral $\int$.

Swap the spatial derivative $\partial_x$ with the integral $\int$.

Swap the second spatial derivative $\partial_{xx}$ with the integral $\int$.