Bramble–Hilbert approximation on a reference simplex

The monomials $x^{\alpha }$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le r-1$ form a basis of ${\mathbb{P}}_{r-1}$. Thus if $N$ is the number of these multi-indices, then $N=\dim {\mathbb{P}}_{r-1}$. Define the linear moment map $M:{\mathbb{P}}_{r-1}\to {\mathbb{R}}^N$ by $M(q{)}_{\alpha }=\mathrm{\mid }{\widehat{S}}_d{\mathrm{\mid }}^{-1}{\int }_{{\widehat{S}}_d}{D}^{\alpha }q$. If every component is zero, take the largest degree $m$ of a nonzero homogeneous part of $q$. For each $\mathrm{\mid }\alpha \mathrm{\mid }=m$, $D^{\alpha }q$ is the constant $\alpha !$ times the coefficient of $x^{\alpha }$ in that highest homogeneous part; its zero mean forces that coefficient to vanish. This contradicts the choice of $m$. Repeating downward shows $q=0$, so $M$ is injective and, since its domain and codomain have the same finite dimension, bijective.

The vector of means of $D^{\alpha }v$, $\mathrm{\mid }\alpha \mathrm{\mid }\le r-1$, is defined because each derivative is in $L^p$ on the finite-measure simplex and therefore in $L^1$ by [F4]. By bijectivity in step 1.1 there is a unique $q\in {\mathbb{P}}_{r-1}$ with those moments. For $w=v-q$, each derivative $D^{\alpha }w$, $\mathrm{\mid }\alpha \mathrm{\mid }\le r-1$, has zero mean; and every derivative of $q$ of order at least $r$ is zero.

Apply [F1] to every mean-zero derivative of order $r-1$ in step 2.1; its $L^p$ norm is bounded by the $L^p$ norms of derivatives of order $r$ of $v$, since $D^{r}q=0$. Apply [F1] successively to derivatives of orders $r-2,r-3,\dots ,0$, each of which also has zero mean; at each stage its norm is bounded by the next derivative order already controlled. Thus all derivatives of $w$ of order at most $r$ are bounded by $C\mathrm{\mid }v{\mathrm{\mid }}_{W^{r,p}}$. If $s\le r$, retaining only orders through $s$ proves the first clause.

If $s>r$, the same argument controls all derivative orders at most $r-1$ by $C\mathrm{\mid }v{\mathrm{\mid }}_{W^{r,p}}$. For each $r\le \mathrm{\mid }\beta \mathrm{\mid }\le s$, $D^{\beta }w=D^{\beta }v$ because $D^{\beta }q=0$. Adding the finitely many derivative bounds through order $s$ gives $\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{s,p}}\le C{\sum }_{j=r}^s\mathrm{\mid }v{\mathrm{\mid }}_{W^{j,p}}$, proving the second clause.