Weighted analytic polygon functions have corner decay and scaled analytic bounds

On the unit square, for smooth $v$ the one-dimensional identity $\mathrm{\mid }g(t)\mathrm{\mid }\le {\int }_0^1\mathrm{\mid }g(s)\mathrm{\mid }ds+{\int }_0^1\mathrm{\mid }{g}^{\mathrm{\prime }}(s)\mathrm{\mid }ds$ applied first in $x$ and then in $y$ gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{L^1}+\mathrm{\parallel }{v}_x{\mathrm{\parallel }}_{L^1}+\mathrm{\parallel }{v}_y{\mathrm{\parallel }}_{L^1}+\mathrm{\parallel }{v}_{xy}{\mathrm{\parallel }}_{L^1}\le {\sum }_{i,j=0}^1\mathrm{\parallel }{\mathrm{\partial }}_x^i{\mathrm{\partial }}_y^{j}v{\mathrm{\parallel }}_{L^2}$ by [F2]. For $v\in H^2((0,1{)}^2)$ extend across $x=0$ by $Ev(x,y)=3v(-x,y)-2v(-2x,y)$ for small $x<0$, and across $x=1$ by $Ev(x,y)=3v(2-x,y)-2v(3-2x,y)$ for small $x>1$; use the same two formulas in $y$. The coefficients satisfy $3-2=1$ for values and $-3+4=1$ for first normal derivatives. For a smooth test $\varphi$ across $x=0$, changing variables in the zeroth-order pairing gives ${\psi }_0(t)=\varphi (t)-3\varphi (-t)+2\varphi (-t\mathrm{/}2)$, so ${\psi }_0(0)=0$. For the first normal derivative, the corresponding test combination is ${\psi }_1(t)=\varphi (t)+3\varphi (-t)-4\varphi (-t\mathrm{/}2)$, also with ${\psi }_1(0)=0$. These formulas follow by rewriting the reflected integrals over $t>0$ and applying the chain rule. Multiply each combined test by a cutoff that is zero on $(0,\epsilon )$ and one on $(2\epsilon ,\mathrm{\infty })$ before using the weak derivative identity on the original open square. If $\mathrm{\mid }{\chi }_{\epsilon }\mathrm{\prime }\mathrm{\mid }\le C\mathrm{/}\epsilon$, then $\mathrm{\mid }{\psi }_j{\chi }_{\epsilon }\mathrm{\prime }\mathrm{\mid }\le C$ on the strip because ${\psi }_j(t)=O(t)$; for any relevant $g\in L^2$, Cauchy--Schwarz bounds the cutoff error by $C\sqrt{2\epsilon }\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(0,2\epsilon )}\le C\sqrt{2\epsilon }\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(0,1)}\to 0$. Thus the reflected piecewise formulas are the weak derivatives through order two, including mixed derivatives. Changes of variables bound each derivative norm by a fixed multiple of $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}$. Reflect successively across the four sides, cut off in the enlarged square, and mollify. For every weak derivative through order two, inserting a translated compactly supported kernel into its weak derivative identity gives $D^{\alpha }(v\ast {\phi }_{\epsilon })=(D^{\alpha }v)\ast {\phi }_{\epsilon }$ on the original square: the sign from integration by parts cancels the sign from differentiating the kernel in its integration variable. Thus [F3] gives smooth approximants converging in $H^2$. The smooth inequality makes them uniformly Cauchy, yielding a continuous representative and the same bound for every $H^2$ class. Scaling gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(Q)}\le C(r^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}_2+\mathrm{\parallel }Dv{\mathrm{\parallel }}_2+r\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_2)$ on a rectangle of side comparable to $r$; a parallelogram with fixed aspect ratio has the same estimate after an affine change of variables.

Choose $R>0$ so distinct corner balls of radius $4R$ are disjoint and each meets only the two sides at its corner. For a point $x$ with $r=\mathrm{\mid }x-c_i\mathrm{\mid }<R$, choose an interior rectangle, or near one straight side an inward rectangle, of diameter comparable to $r$ and lying in $r\mathrm{/}2<\mathrm{\mid }z-c_i\mathrm{\mid }<2r$. A finite angular cover gives aspect ratios and widths bounded in terms of the fixed corner angles. For each individual weak derivative $g_{\alpha }=D^{\alpha }u$ with $\mathrm{\mid }\alpha \mathrm{\mid }=j\ge 2$, the weighted hypotheses imply $g_{\alpha }\in H^2$ on this rectangle. Apply the fixed second-order rectangle extension and estimate of step 1.1 to $g_{\alpha }$ itself. In particular, the coefficients $3,-2$ are not used to extend $u$ in arbitrary Sobolev orders and then differentiated across a reflecting side. The extension of each $g_{\alpha }$ is needed only through order two, so its bound is fixed, independently of $j$. On the annulus the other corner distances are bounded above and below, so at derivative order $s\ge 2$, ${\mathrm{\Phi }}_{\beta +s-2}\ge C^{-s}{r}^{{\beta }_i+s-2}$. The weighted hypotheses at orders $j,j+1,j+2$ therefore bound $\mathrm{\parallel }{g}_{\alpha }{\mathrm{\parallel }}_2$, $\mathrm{\parallel }D{g}_{\alpha }{\mathrm{\parallel }}_2$, and $\mathrm{\parallel }{D}^2{g}_{\alpha }{\mathrm{\parallel }}_2$ respectively by $C{A}^{j}j!r^{2-{\beta }_i-j}$, $C{A}^{j}j!r^{1-{\beta }_i-j}$, and $C{A}^{j}j!r^{-{\beta }_i-j}$. Fixed affine changes of coordinates contribute at most another fixed base to the power $j$; the boundedly many extra derivatives and their factorial factors are absorbed into $A^j$. Multiplying these three norms by $r^{-1},1,r$ as in step 1.1 gives $\mathrm{\mid }{D}^{\alpha }u(x)\mathrm{\mid }\le C{A}^{j}j!r^{1-{\beta }_i-j}\le C{A}^{j}j!r^{a-j}$, uniformly up to an open side. On compact patches away from corners the same argument gives ordinary factorial derivative bounds. There apply step 1.1 also to $g=u$ and to each first derivative: their lower-order $L^2$ terms come from $u\in H^1$, while derivative orders two and three are bounded by the weighted hypotheses since the weights are bounded below on those patches. Thus $u$ and its gradient are bounded there, in particular on the fixed radius-$R$ arcs. These separately constructed continuous representatives of the weak derivatives agree on overlaps by [F4]. They are compatible as derivatives: on a smaller interior patch mollify $u$ itself, so each derivative converges locally in $H^2$ and therefore uniformly by step 1.1 to the representative of that weak derivative; the fundamental theorem along interior line segments passes to these uniform limits. All derivative identities then extend to an open side by continuity. Thus $u$ is smooth up to sides away from corners without an arbitrary-order reflected extension.

On the fixed arc of radius $R$ about $c_i$, step 2.1 bounds $\mathrm{\nabla }u$. Along every ray, the Hessian bound in step 2.1 gives $\mathrm{\mid }\mathrm{\nabla }u(r,\theta )\mathrm{\mid }\le C+C{\int }_r^R{t}^{-1-{\beta }_i}dt$. If ${\beta }_i>0$ this is at most $C{r}^{-{\beta }_i}$; if ${\beta }_i=0$ it is at most $C(1+\mathrm{\mid }\log r\mathrm{\mid })$. Since $a<1-{\beta }_i$, both are bounded by $C{r}^{a-1}$ for $r<R$ after decreasing $R$ if needed. Integrating radially shows that $u(r,\theta )$ has a limit as $r\downarrow 0$ and differs from it by at most $C{r}^a$. Variation along the circular arc of radius $r$ is at most $Cr\cdot r^{a-1}=C{r}^a$, so the radial limits agree for all angles; call the common value $u(c_i)$. This also proves continuity at the corner. The first-derivative estimate gives the claimed bound for $\mathrm{\mid }\alpha \mathrm{\mid }=1$, while step 2.1 gives it for higher orders because $r^{1-{\beta }_i-j}\le r^{a-j}$.

For a regular triangle $K$ in the stated family, when its corner distance is sufficiently small compared with $R$ and the fixed upper bound for $h_K\mathrm{/}\mathrm{dist}(K,\mathcal{C})$, all its points lie in a single corner neighborhood; choose the constant $u(c_i)$ there; the pointwise derivative estimate and $h_K\mathrm{/}\mathrm{dist}(K,\mathcal{C})\asymp 1$ give $h_K^{\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{A}_1^{\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\mid }\alpha \mathrm{\mid }!r_K^a$ after affine pullback and the chain rule. For the remaining triangles, their distance from all corners is bounded below by a fixed positive constant; choose the value at one point of $K$ as the constant; the compact-patch derivative bounds and bounded diameter give the same estimate after increasing $C,A_1$. The affine geometry factors are controlled by [F2].

Let $K$ have vertex $c_i$ and diameter $h<R$. On dyadic annuli $2^{-j-1}h\le \mathrm{\mid }x-c_i\mathrm{\mid }\le 2^{-j}h$, step 3.1 bounds $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$ by $C(2^{-j-1}h{)}^{2a-2}$, while the area of each annulus is at most $C(2^{-j}h{)}^2$. Summing the resulting geometric series gives $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(K)}^2\le C{h}^{2a}$, hence $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2\le C{h}^a$. Every vertex value differs from $u(c_i)$ by at most $C{h}^a$; the barycentric gradients are bounded by $C\mathrm{/}h$ and $\mathrm{\mid }K\mathrm{\mid }\le C{h}^2$ by shape regularity, so the vertex affine interpolant has gradient norm at most $C{h}^a$. The pointwise value estimate gives both $L^2$ errors at most $C{h}^{a+1}$. The triangle inequality proves the stated seminorm and full $L^2$ bounds.