A degree-uniform polynomial edge lift with the strong edge norm

Since $q(0)=0$, polynomial division gives a polynomial $R$ of degree at most $p-1$ when $p\ge 1$; for $p=0$ or $p=1$ the endpoint conditions force $q=0$. Thus $Lq$ has total degree at most $p$. At $y=0$, $R(1)=q(1)=0$, so $Lq=q(x)$; at $x=0$ it is zero; and on $x+y=1$ one has $1-y=x$, so $xR(1-y)=xR(x)=q(x)$.

Put $A={\int }_0^1\mathrm{\mid }{q}^{\mathrm{\prime }}(t){\mathrm{\mid }}^{2}dt$. Polynomial integration term by term gives $q(t)={\int }_0^t{q}^{\mathrm{\prime }}(s)ds$, so Cauchy–Schwarz [F3] yields $t\mathrm{\mid }R(t){\mathrm{\mid }}^2\le {\int }_0^t\mathrm{\mid }{q}^{\mathrm{\prime }}(s){\mathrm{\mid }}^{2}ds\le A$ and hence ${\int }_0^{1}t\mathrm{\mid }R(t){\mathrm{\mid }}^{2}dt\le A$. The identity $t{R}^{\mathrm{\prime }}(t)=q^{\mathrm{\prime }}(t)-R(t)$ gives ${\int }_0^1{t}^3\mathrm{\mid }{R}^{\mathrm{\prime }}(t){\mathrm{\mid }}^{2}dt\le 2{\int }_0^{1}t\mathrm{\mid }{q}^{\mathrm{\prime }}{\mathrm{\mid }}^2+2{\int }_0^{1}t\mathrm{\mid }R{\mathrm{\mid }}^2\le 4A$. Also $\mathrm{\mid }q(t){\mathrm{\mid }}^2\le tA$, so ${\int }_0^1\mathrm{\mid }q{\mathrm{\mid }}^2\le A$. Writing $t=1-y$, the slices of $T$ are $0<x<t$. Therefore $\mathrm{\parallel }{\mathrm{\partial }}_{x}Lq{\mathrm{\parallel }}_2^2\le 2{\int }_0^1(1-x)\mathrm{\mid }{q}^{\mathrm{\prime }}(x){\mathrm{\mid }}^{2}dx+2{\int }_0^{1}t\mathrm{\mid }R(t){\mathrm{\mid }}^{2}dt\le 4A,$ $\mathrm{\parallel }{\mathrm{\partial }}_{y}Lq{\mathrm{\parallel }}_2^2=\frac{1}{3}{\int }_0^1{t}^3\mathrm{\mid }{R}^{\mathrm{\prime }}(t){\mathrm{\mid }}^{2}dt\le \frac{4}{3}A,$ and $\mathrm{\parallel }Lq{\mathrm{\parallel }}_2^2\le 2{\int }_0^1(1-x)\mathrm{\mid }q(x){\mathrm{\mid }}^{2}dx+\frac{2}{3}{\int }_0^1{t}^3\mathrm{\mid }R(t){\mathrm{\mid }}^{2}dt\le \frac{8}{3}A\mathrm{.}$ Fubini [F3] justifies the slice integrals. Using the Sobolev norm definition [F2], adding the three inequalities gives $\mathrm{\parallel }Lq{\mathrm{\parallel }}_{H^1(T)}^2\le 8A\le 16A$, proving the safe constant four.

Permute the reference vertices to obtain a lift on any selected edge; there are only six such affine maps, so the maximum of their fixed norm factors is independent of $p$. Transport the reference estimate by the shape-regular affine map. The gradient change of variables is bounded by [F1], and the reference edge derivative norm is exactly scaled by ${\mathrm{\ell }}_E^{1\mathrm{/}2}\mathrm{\parallel }{\mathrm{\partial }}_{\tau }q{\mathrm{\parallel }}_{L^2(E)}$. The physical $L^2$ norm has one additional factor bounded by $C_{\gamma }{h}_K$ by [F1]. This gives both stated estimates.

Let $g$ be continuous polynomial data on the three edges, with edge degrees at most $p$. For $p=0$, continuity makes $g$ one constant and that constant extends it. For $p\ge 1$, let $a$ be the affine interpolant of the three vertex values. Each edge residual $g{\mathrm{\mid }}_E-a{\mathrm{\mid }}_E$ vanishes at both endpoints and has degree at most $p$. Lift the three residuals separately by step 1.1; each lift vanishes on the other two edges. Their sum plus $a$ therefore has trace exactly $g$ and degree at most $p$. This works when edge degrees differ and requires no equal-degree assumption.