Research proof index

The Lp norm of the n-th weak derivative of f is bounded by the W^{k,p} norm.

Convergence in W^{k,p}(U) is equivalent to convergence of each weak-derivative eLpNorm component to zero: fₙ → f in W^{k,p} if and only if ‖D^s(fₙ - f)‖_{Lp} → 0 for all multi-indices s with |s| ≤ k.

The W^{k,p}(U) norm vanishes iff the function is zero a.e..

Monotonicity of weighted L^p norms in the exponent, for probability weights.

Any weak multi-derivative Df on U agrees a.e. with the canonical choice. -/ theorem WeakmultiderivU_unique {d : ℕ+} {n : ℕ} {U : Set (Fin d → ℝ)} (hU : IsOpen U) (s : Fin n → Fin d) (f : Lp_locU d 1 U) (h : HasWeakMultiDerivU U f s) (Df : Lp_locU d 1 U) (hDf : IsWeakMultiDerivU U s f Df) : (WeakmultiderivU U f s h : (Fin d → ℝ) →ₘ[μU d U] ℝ) =ᵐ[volume...

For a continuously differentiable smoothing function supported in $[1/2,2]$ and normalized by $\int_0^\infty \nu(x)\,dx/x=1$, the Mellin factor of the smoothed cutoff at $s=1$ is $1+O(\varepsilon)$. Precisely, there is a constant $C$ such that, for every sufficiently small $\varepsilon>0$ and every $X\ge0$, $\left|\mathcal M(\widetilde{1_\varepsilon})(1)X-X\right|\le C\varepsilon X.$