Dirichlet realizations, spectral projectors, and local extensions

The central construction in this lesson is a bounded point-evaluation functional on a spectral subspace. Elliptic estimates bound that functional, and its Hilbert-space representative produces a spectral kernel. This construction works for a sharp spectral cutoff even when the spectrum is continuous. Compactness has a separate role: on a bounded smooth Dirichlet domain it supplies an eigenbasis and a finite trace.

Throughout, $D_j=-i\partial_j$, and Hilbert inner products are linear in the first variable. The precise analytic entry results used below are the integer half-space trace and extension construction in Section 2 of Boundary energy, local inverses, and harmonic data, supplemented by the complete real-order trace proof in Section 2 below; the local boundary $H^1$-to-$H^2$ theorem Section 6 of Boundary energy, local inverses, and harmonic data, with Lipschitz principal and bounded lower coefficients; its smooth integer bootstrap Section 7 of Boundary energy, local inverses, and harmonic data; and the interior counterpart Section 8 of Local inverses and distance-weighted elliptic estimates together with the differentiated-equation argument. We check their hypotheses at each use. The other Fourier entry facts are inversion and Plancherel with inverse factor $(2\pi)^{-n}$. For the arbitrary lower-bounded extension in Section 10, the complete spectral construction is proved in Spectral measures with the original operator domain retained, Sections 1–4 of Spectral measures with the original operator domain retained: scalar measures, the bounded positive calculus, all multiplier domains, and the exact inverse correspondence back to the given operator. The bounded-domain calculus is also proved directly in Section 5. With the original inner-product convention and the spectral measure $F_A$, these proofs give

$$\|m(A)f\|^2=\int_{\mathbb R}|m(t)|^2\,d(F_A(t)f,f), \qquad D(A^q)=\left\{f:\int |t|^{2q}\,d(F_A(t)f,f)<\infty\right\}. \tag{DSP1}$$

The analytic entry results are used with their stated hypotheses. The spectral identities just displayed have complete proofs in the companion, including the recursive domains of the original operator. The elliptic regularity-to-kernel implication and the boundary-domain implication are proved here. Nothing below identifies this second-order scalar result with a theorem for general systems or higher-order boundary conditions.

1. Weighted Dirichlet realization

Let $n\geq1$, and let $X\subset\mathbb R^n$ be a nonempty bounded open set with smooth boundary. Let $r\in C^\infty(\overline X)$ be real and strictly positive. Consider the unchanged differential expression

$$Pu=-\partial_j(g^{jk}\partial_k u)+b^j\partial_j u+cu, \qquad (u,v)_r=\int_Xu\overline v\,r\,dx. \tag{DSP2}$$

Repeated indices are summed from $1$ to $n$. The coefficients are smooth on a neighborhood of the closure; $g^{jk}=g^{kj}$ are real and $g^{jk}(x)\xi_j\xi_k\geq\theta|\xi|^2$ there for some $\theta>0$. The coefficients $b^j,c$ may be complex. Assume exactly that $P$ is symmetric on $C_c^\infty(X)$ for $(\cdot,\cdot)_r$.

Exact correspondence for an arbitrary real principal matrix. The symmetric presentation above also applies to a given real matrix $G=(G^{jk})$ whose quadratic form is positive, without changing its original coefficients or differential expression. Retain those original data $(G,B,c,r)$, and define the additional coefficient fields $$S^{jk}=\frac{G^{jk}+G^{kj}}2,\qquad A^{jk}=\frac{G^{jk}-G^{kj}}2,\qquad \beta^k=B^k-\sum_j\partial_j A^{jk}. \tag{DSP2a}$$ For every distribution $u$, commuting weak derivatives and pairing the $(j,k)$ and $(k,j)$ terms give $$\sum_{j,k}A^{jk}\partial_j\partial_k u=0,\qquad -\partial_j(G^{jk}\partial_k u)+B^k\partial_k u+cu =-\partial_j(S^{jk}\partial_k u)+\beta^k\partial_k u+cu. \tag{DSP2b}$$ The product rule with smooth coefficients proves the displayed expansion for distributions as well as for smooth functions. Also $S^{jk}\xi_j\xi_k=G^{jk}\xi_j\xi_k$ for every real $\xi$, with exactly the same ellipticity lower bound. Thus the map $(G,B,c,r)\mapsto(S,B-\operatorname{div}A,c,r)$ is an explicit map of coefficient presentations of the same distributional operator on the same weighted space. The condition defining (DSP3) and its operator action are identical for the two presentations. Their adjoints, symmetry, self-adjoint realizations, recursive powers, and spectral projections therefore agree as operators; this follows from equality of their graphs, not from a change of coordinates or of density. Coefficient estimates can be pulled back to the given fields through (DSP2a): for every multiindex $\alpha$, $$\partial^\alpha S^{jk}=\tfrac12(\partial^\alpha G^{jk} +\partial^\alpha G^{kj}),\qquad \partial^\alpha\beta^k=\partial^\alpha B^k -\tfrac12\sum_j\partial^{\alpha+e_j}(G^{jk}-G^{kj}). \tag{DSP2c}$$ The triangle inequality gives their exact componentwise norm bounds, including the additional derivative of the antisymmetric part. The original fields remain available throughout this correspondence. Every subsequent formula in the symmetric presentation uses $g=S,b=\beta$ when it is applied to these more general input data.

Define

$$D(P_D)=\{u\in H_0^1(X):Pu\in L^2(X,r\,dx)\},\qquad P_Du=Pu. \tag{DSP3}$$

There is a real number $\mu\geq1$ such that $T=P_D+\mu I$ is self-adjoint, $T\geq I$, and

$$(Tu,u)_r\geq a\|u\|_{H^1(X)}^2,\qquad \|u\|_{H^1(X)}\leq C\|Tu\|_{L^2_r},\qquad u\in D(P_D). \tag{DSP4}$$

The original $P_D$ is retained; the symbol $T$ always records the shift explicitly.

Hilbert representation, with both functional conventions. We first give the representation argument used by the form and kernel constructions. Let $H$ be a complex Hilbert space with inner product linear in its first argument and let $\ell:H\to\mathbb C$ be bounded and linear. If $\ell=0$, its representing vector is zero. Otherwise the closed affine set $V=\{v:\ell(v)=1\}$ is nonempty. Its distance $d$ from zero satisfies $d\geq\|\ell\|^{-1}>0$. Choose $v_j\in V$ with $\|v_j\|\to d$. The midpoint belongs to $V$, so the parallelogram identity gives $$\|v_j-v_k\|^2 =2\|v_j\|^2+2\|v_k\|^2-4\|(v_j+v_k)/2\|^2 \leq2\|v_j\|^2+2\|v_k\|^2-4d^2\longrightarrow0.$$ Completeness and closedness give a limit $v\in V$ with $\|v\|=d$. For $z\in\ker\ell$, the inequality $\|v+t z\|^2\geq d^2$, first for all real $t$ and then for all purely imaginary $t$, forces $(v,z)=0$. Since $u-\ell(u)v\in\ker\ell$, it follows that $(u,v)=\ell(u)d^2$. Thus $h=v/d^2$ satisfies $$\ell(u)=(u,h),\qquad\|\ell\|=\|h\|.$$ The norm equality follows from Cauchy–Schwarz and evaluation at $u=h/\|h\|$; uniqueness follows by testing the difference of two representing vectors against itself. For a bounded conjugate-linear functional $F$, apply this result to $\ell(u)=\overline{F(u)}$. Its vector $w$ satisfies $F(u)=(w,u)$ with the same norm. This proves the exact forms of representation used below, including the equivalent inner product $q_\mu$, without a spectral or orthonormal-basis theorem.

Proof of the realization. Integration by parts on compact interior tests gives the form

$$q(u,v)=\int_X rg^{jk}\partial_k u\,\overline{\partial_jv}\,dx +\int_X\bigl(g^{jk}\partial_jr+rb^k\bigr)\partial_ku\,\overline v\,dx +\int_Xrcu\overline v\,dx. \tag{DSP5}$$

Every term is continuous on $H_0^1\times H_0^1$. Density extends symmetry from tests to that space. Positivity of the first term and boundedness of the other coefficients imply

$$q(u,u)\geq r_{\min}\theta\|\nabla u\|_2^2 -C\|u\|_2\|\nabla u\|_2-C\|u\|_2^2.$$

Young’s inequality, $2ab\leq\varepsilon a^2+\varepsilon^{-1}b^2$, gives a sufficiently large $\mu\geq1$ for which $q_\mu=q+\mu(\cdot,\cdot)_r$ controls $\|u\|_{H^1}^2$ and also $\|u\|_{L^2_r}^2$. It is an inner product with an equivalent complete norm. Applying the Hilbert representation theorem in this norm to $v\mapsto(f,v)_r$ produces a unique $u\in H_0^1$ with $q_\mu(u,v)=(f,v)_r$ for every $v\in H_0^1$. Interior testing says $(P+\mu)u=f$ distributionally. Conversely, if $u\in H_0^1$ and $(P+\mu)u=f\in L^2_r$, the distributional identity extends to this form identity by density. Thus the variational operator is exactly $T$ on (DSP3), and $T$ maps its domain onto $L^2_r$.

This domain is dense, since it contains $C_c^\infty(X)$. The form identity makes $T$ symmetric. If $w\in D(T^*)$, solve $Tv=T^*w$. For every $u\in D(T)$, symmetry and adjointness give $(Tu,w-v)_r=0$. Surjectivity of $T$ forces $w=v\in D(T)$. Hence $T=T^*$, in particular it is closed. Coercivity gives the first estimate in (DSP4), and Cauchy–Schwarz gives

$$a\|u\|_{H^1}^2\leq\|Tu\|_{L^2_r}\|u\|_{L^2_r} \leq C_r\|Tu\|_{L^2_r}\|u\|_{H^1},$$

which proves the second, including the case $u=0$. The same form lower bound proves $T\geq I$. ∎

2. Inhomogeneous Sobolev convention

For an integer $k\geq0$, write

$$\overline H^k(X)=H^k(X),\qquad \|u\|_{\overline H^k(X)}^2=\sum_{|\alpha|\leq k}\|D^\alpha u\|_{L^2(X)}^2. \tag{DSP6}$$

The derivatives are weak derivatives. The bar indicates this inhomogeneous norm; it does not indicate closure of $C_c^\infty(X)$. In particular $\overline H^1(X)$ and $H_0^1(X)$ have different boundary conditions. Weighted and unweighted norms obey the exact comparison

$$r_{\min}\|u\|_2^2\leq\|u\|_{L^2_r}^2\leq r_{\max}\|u\|_2^2. \tag{DSP7}$$

On a compact coordinate patch the same comparison uses the minimum and maximum there and the chart Jacobian. All constants involving these comparisons retain that dependence.

For reference, all local and transformed Sobolev symbols used below have the following fixed meaning. If $\Phi:U\to V$ is a smooth chart and $J_\Phi=|\det D\Phi^{-1}|$, then $$\|u\|_{H^s_\Phi(U)}^2 :=\sum_{|\alpha|\le s}\int_V \left|D_y^\alpha(u\circ\Phi^{-1})(y)\right|^2\,dy \quad(s\in\mathbb N),$$ while the weighted transformed norm is $$\|u\|_{L^2_{r,\Phi}(U)}^2 =\int_V|u\circ\Phi^{-1}(y)|^2 (r\circ\Phi^{-1})(y)J_\Phi(y)\,dy.$$ For a compact $K\subset\overline X$, $H^s(K)$ denotes the fixed localized quantity obtained from a finite family of cutoffs $\zeta_a\in C_c^\infty(U_a)$ whose regions of value one cover $K$. The charts, cutoffs, and their larger neighborhoods remain part of this notation in each estimate. Coordinate changes of those same localized functions have the two-sided bounds proved below; arbitrary cutoffs with different outer supports need not give equivalent seminorms on all functions. For real noninteger $s\geq0$, the localized coordinate spaces use the restriction norms (RT1)–(RT2). A finite partition covering the whole domain gives a norm equivalent to that restriction norm by the explicit extension and reconstruction below. On a smooth boundary chart, $\gamma u$ denotes the one-sided trace on $t=0$, taking $H^s$ continuously to $H^{s-1/2}$ for $s>1/2$; $H_0^1$ is the kernel of this trace (equivalently the $H^1$-closure of $C_c^\infty(X)$). For complete precision, put $\Omega_a=\Phi_a(U_a\cap X)$ and $v_a=((\zeta_a u)\circ\Phi_a^{-1})|_{\Omega_a}$. The aggregate just specified is the sum $$\|u\|_{H^s(K;\mathcal A,\zeta)} =\sum_a N_{s,\Omega_a}(v_a),\qquad N_{k,\Omega}(v)= \left(\sum_{|\alpha|\leq k}\int_\Omega|D_y^\alpha v|^2\,dy\right)^{1/2} \quad(k\in\mathbb N_0), \tag{DSP7a}$$ where at a noninteger $s\geq0$ the symbol $N_{s,\Omega}$ instead means the restriction infimum in (RT2), with the whole-space norm (RT1). The abbreviation $\|u\|_{H^s(K)}$ retains the fixed atlas, cutoffs and domains $\Omega_a$. It is a seminorm of functions on those fixed neighborhoods; it is not determined by their values on $K$ alone. The integer formula keeps every coefficient in (DSP6) equal to one.

These definitions apply to every occurrence of transformed $H^s$, trace spaces, and local $H^s(K)$ below.

Real-order trace and boundary density

The following argument proves the real-order trace assertion, its fixed right inverse, the coordinate comparisons, and the zero-trace energy-space identity. Its numbered parts and RT formula labels belong to this argument. The derivative-sum norm (DSP6) retains its original coefficients.

Trace proof 1. Spaces and precise statement

Put $D_j=-i\partial_j$, $$\widehat v(\xi)=\int_{\mathbb R^N}e^{-ix\cdot\xi}v(x)\,dx, \qquad v(x)=(2\pi)^{-N}\int e^{ix\cdot\xi}\widehat v(\xi)\,d\xi,$$ and define the Bessel-potential norm by $$\|v\|_{H^s(\mathbb R^N)}^2 =(2\pi)^{-N}\int_{\mathbb R^N}(1+|\xi|^2)^s|\widehat v(\xi)|^2\,d\xi. \tag{RT1}$$ For $s\geq0$ and an open set $X$, $H^s(X)$ is the restriction space with norm $$\|u\|_{H^s(X),\mathrm{res}} =\inf\{\|U\|_{H^s(\mathbb R^n)}:U|_X=u\}. \tag{RT2}$$ The kernel of restriction is closed: convergence in (RT1) implies convergence in $L^2$, and a limit of functions zero almost everywhere on $X$ is zero there. Thus (RT2) is a Hilbert quotient norm. Restriction is equality almost everywhere, or equivalently equality of distributions on $X$.

For integer $k\geq0$, the actual squared norm in (DSP6) remains $\sum_{|\alpha|\leq k}\|D^\alpha u\|_{L^2(X)}^2$. Already on the whole space its relation to (RT1) is the explicit identity $$\|U\|_{H^k(\mathbb R^n)}^2 =\sum_{j=0}^k\binom{k}{j} \sum_{|\alpha|=j}\frac{j!}{\alpha!}\|D^\alpha U\|_2^2. \tag{RT3}$$ On a smooth domain the restriction and weak-derivative spaces agree, with equivalent norms, by the integer reflection extension in Section 2 of Boundary energy, local inverses, and harmonic data and finite smooth localization. The coordinate estimates proved below justify that localization. Equation (RT3) specifies the coefficients of the whole-space comparison; no equality between the quotient norm and the derivative-sum norm on $X$ is asserted.

Let $X$ be bounded with smooth boundary. For every real $s>1/2$ there is a unique bounded trace $$\gamma_X:H^s(X)\longrightarrow H^{s-1/2}(\partial X) \tag{RT4}$$ that agrees with boundary restriction for functions smooth on a neighborhood of $\overline X$. The surface space and its chart norm are defined in Trace proof 5, with an explicit comparison to surface measure. There is a bounded linear right inverse $L_X$, and one fixed choice of $L_X$ works simultaneously at all these orders, with an order-dependent norm. In every localized flattened chart the trace is the limit of the positive slices in $H^{s-1/2}$, independent of the extension in (RT2). Finally, $$\ker(\gamma_X:H^1(X)\to H^{1/2}(\partial X)) =\overline{C_c^\infty(X)}^{\,H^1(X)}=H_0^1(X). \tag{RT5}$$ We now prove all assertions in (RT4)–(RT5).

Trace proof 2. Whole-space slices and quotient independence

Write $n=d+1$, $x=(z,t)$, $\xi=(\zeta,\tau)$, and retain the inhomogeneous factor $\lambda(\zeta)=(1+|\zeta|^2)^{1/2}$. For $s>1/2$, put $$C_s=\int_{\mathbb R}(1+r^2)^{-s}\,dr =\frac{\sqrt\pi\,\Gamma(s-1/2)}{\Gamma(s)}. \tag{RT6}$$ For completeness, the gamma evaluation follows by inserting $(1+r^2)^{-s}=\Gamma(s)^{-1}\int_0^\infty a^{s-1}e^{-a(1+r^2)}\,da$, applying Tonelli, and using $\int e^{-ar^2}dr=\sqrt{\pi/a}$. In particular $$\int_{\mathbb R}(\lambda^2+\tau^2)^{-s}\,d\tau =C_s\lambda^{1-2s}. \tag{RT7}$$

For $U\in H^s(\mathbb R^{d+1})$, weighted Cauchy–Schwarz and Fubini define, for almost every $\zeta$ and every $t$, $$\widehat{T_tU}(\zeta) =\frac1{2\pi}\int_{\mathbb R}e^{it\tau}\widehat U(\zeta,\tau)\,d\tau. \tag{RT8}$$ The integral is absolutely convergent for those $\zeta$. Equations (RT7)–(RT8) give $$\begin{aligned} \|T_tU\|_{H^{s-1/2}(\mathbb R^d)}^2 &\leq (2\pi)^{-d-2}\int \lambda^{2s-1} C_s\lambda^{1-2s} \int(\lambda^2+\tau^2)^s|\widehat U(\zeta,\tau)|^2\,d\tau\,d\zeta\\ &=\frac{C_s}{2\pi}\|U\|_{H^s(\mathbb R^{d+1})}^2. \end{aligned} \tag{RT9}$$ This is uniform for all real $t$.

Let $\mathcal T_hU(z,t)=U(z,t+h)$. Its Fourier multiplier is $e^{ih\tau}$, so dominated convergence in (RT1) proves $\mathcal T_hU\to U$ in $H^s$. Formula (RT8) gives $T_{t+h}U=T_t\mathcal T_hU$. Consequently (RT9) proves that $t\mapsto T_tU$ is continuous into $H^{s-1/2}(\mathbb R^d)$.

This continuous family is a representative of the actual slices of $U$. Indeed choose Schwartz $U_j\to U$ in $H^s$. Such density follows by approximating $\widehat U$ in the weighted $L^2$ space by smooth compactly supported functions; the inverse Fourier transforms are Schwartz. On any bounded interval $I$, (RT9) gives convergence of $T_tU_j$ to $T_tU$ in $L^2(I;L^2(\mathbb R^d))$, while $U_j\to U$ in $L^2(\mathbb R^{d+1})$. Fourier inversion identifies the slices for each Schwartz approximant, and uniqueness of an $L^2$ limit identifies them almost everywhere for $U$.

Suppose now that $U=0$ almost everywhere on $\{t>0\}$. The preceding identification implies $T_tU=0$ for almost every $t>0$; continuity implies this for every $t>0$, and then for $t=0$. Thus two full-space extensions of the same $u\in H^s(\mathbb R^{d+1}_+)$ have the same $T_0$, and even the same continuous slice family for $t\geq0$. Define $\gamma_+u=T_0U$ using either extension and take the infimum in (RT9): $$\sup_{t\geq0}\|u(\cdot,t)\|_{H^{s-1/2}} \leq\left(\frac{C_s}{2\pi}\right)^{1/2} \|u\|_{H^s(\mathbb R^{d+1}_+),\mathrm{res}}, \qquad u(\cdot,t)\longrightarrow\gamma_+u\text{ in }H^{s-1/2} \quad(t\downarrow0). \tag{RT10}$$ The slice notation in (RT10) means this continuous representative. This argument proves quotient independence without presupposing a trace on the restriction space.

For $d=0$, interpret $\mathbb R^0$ as one point with measure one, $\lambda=1$, and its $H^q$ norm as absolute value for every $q$. Equations (RT6)–(RT10) then are the same one-variable proof, with no tangential integration and with $(2\pi)^0=1$.

Trace proof 3. The same Poisson lift at every real order

Choose once and for all $\chi\in C^\infty(\mathbb R)$ with $\chi(r)=0$ for $r\leq-2$, $\chi(r)=1$ for $r\geq-1$, and $0\leq\chi\leq1$, and put $\vartheta(r)=\chi(r)e^{-r}$. Then $\vartheta$ is Schwartz, $\vartheta(0)=1$, and $\vartheta(r)=e^{-r}$ on $[0,\infty)$. For $f\in H^{s-1/2}(\mathbb R^d)$, define $E f$ by its full Fourier transform $$\widehat{E f}(\zeta,\tau) =\lambda(\zeta)^{-1}\widehat\vartheta\bigl(\tau/\lambda(\zeta)\bigr) \widehat f(\zeta). \tag{RT11}$$ Change variables $\tau=\lambda\rho$ in (RT1). There is the exact identity $$\|Ef\|_{H^s(\mathbb R^{d+1})}^2 =K_{s,\vartheta}\|f\|_{H^{s-1/2}(\mathbb R^d)}^2, \qquad K_{s,\vartheta}=\frac1{2\pi}\int_{\mathbb R}(1+\rho^2)^s|\widehat\vartheta(\rho)|^2\,d\rho<\infty. \tag{RT12}$$ Normal Fourier inversion, first for smooth data and then by (RT12), gives $$\mathcal F_z(Ef)(\zeta,t)=\vartheta(\lambda t)\widehat f(\zeta), \qquad \mathcal F_z(Lf)(\zeta,t)=e^{-\lambda t}\widehat f(\zeta) \quad(t>0),\qquad Lf=(Ef)|_{t>0}. \tag{RT13}$$ In (RT8), $(2\pi)^{-1}\int\widehat\vartheta(\rho)d\rho=\vartheta(0)=1$; hence $\gamma_+Lf=f$. Equivalently, dominated convergence applied to $|e^{-\lambda t}-1|^2\lambda^{2s-1}|\widehat f|^2$ proves the same boundary limit. Equations (RT12)–(RT13) prove $$\|Lf\|_{H^s(\mathbb R^{d+1}_+),\mathrm{res}} \leq K_{s,\vartheta}^{1/2}\|f\|_{H^{s-1/2}},\qquad\gamma_+L=I \quad\text{for every real }s>1/2. \tag{RT14}$$ The formula for $L$ is exactly the positive-half-space lift in Section 2 of Boundary energy, local inverses, and harmonic data, formula (B4), independent of $s$. Its full-space extension (RT11) was chosen smooth across $t=0$; none of the estimates assumes that an even reflection of the Poisson exponential has arbitrarily high regularity.

Trace proof 4. Fractional coordinate estimates with their dependencies

We prove the coordinate facts needed for both the domain and the surface. This part is in dimension $N\geq1$. For $0<\sigma<1$, set $$[v]_{\sigma,N}^2=\iint_{\mathbb R^N\times\mathbb R^N} \frac{|v(x)-v(y)|^2}{|x-y|^{N+2\sigma}}\,dx\,dy.$$ Plancherel for translations and Tonelli give the exact identity $$[v]_{\sigma,N}^2 =c_{N,\sigma}(2\pi)^{-N}\int |\xi|^{2\sigma}|\widehat v(\xi)|^2\,d\xi, \qquad c_{N,\sigma}= \frac{2\pi^{N/2}\Gamma(1-\sigma)}{\sigma\,4^\sigma\Gamma(N/2+\sigma)}. \tag{RT15}$$ Here is the constant calculation. The inner translation integral is $\int |e^{ih\cdot\xi}-1|^2|h|^{-N-2\sigma}dh$. Rotations and scaling reduce it to $|\xi|^{2\sigma}$ times its value at the first coordinate unit vector. Insert $$|h|^{-N-2\sigma} =\Gamma(N/2+\sigma)^{-1} \int_0^\infty a^{N/2+\sigma-1}e^{-a|h|^2}\,da.$$ The Gaussian integral of $2-2\cos h_1$ is $2(\pi/a)^{N/2}(1-e^{-1/(4a)})$. Substituting $r=1/(4a)$ leaves $2\pi^{N/2}4^{-\sigma}\Gamma(N/2+\sigma)^{-1} \int_0^\infty r^{-\sigma-1}(1-e^{-r})dr$. Integration by parts evaluates the final integral as $\Gamma(1-\sigma)/\sigma$; its boundary terms vanish because $0<\sigma<1$. This proves (RT15), including its constants. It holds for every $L^2$ function, with both sides allowed to be infinite, by Plancherel for the $L^2$ translation difference and Tonelli.

Let $m\geq0$ be an integer and define $$\mathcal E_{m,\sigma}(v) =\sum_{|\alpha|\leq m}\|D^\alpha v\|_2^2 +c_{N,\sigma}^{-1}\sum_{|\alpha|=m}[D^\alpha v]_{\sigma,N}^2.$$ Fourier transformation, the multinomial formula, and (RT15) give $$\frac1{2^{m+\sigma}m!}\|v\|_{H^{m+\sigma}}^2 \leq\mathcal E_{m,\sigma}(v) \leq(m+2)\|v\|_{H^{m+\sigma}}^2. \tag{RT16}$$ Indeed the multiplier of $\mathcal E$ is $P_m(\xi)+|\xi|^{2\sigma}Q_m(\xi)$, where $P_m=\sum_{|\alpha|\leq m}\xi^{2\alpha}$ and $Q_m=\sum_{|\alpha|=m}\xi^{2\alpha}$. The inequalities $|\xi|^{2m}/m!\leq Q_m\leq|\xi|^{2m}$ and $P_m\leq(m+1)\max(1,|\xi|^{2m})$, split at $|\xi|=1$, prove exactly (RT16). The case $m=0$ uses $0!=1$.

We next give a local composition bound that does not assume the desired fractional invariance. Let $\Psi:V\to W$ be a smooth diffeomorphism, and let $b\in C_c^\infty(V_0)$, where $V_0\Subset V$. Choose the smaller coordinate set so that for $y,z\in V_0$ $$L^{-1}|y-z|\leq|\Psi(y)-\Psi(z)|\leq L|y-z|, \qquad J=\sup_{\Psi(V_0)}|\det D\Psi^{-1}|<\infty, \tag{RT17}$$ with $L\geq1$. Smooth coordinate maps admit such sets: on a sufficiently small ball about any point, the derivative differs from its invertible value at the center by less than half that value’s smallest singular value. Integrating the derivative on line segments proves (RT17); finitely many such smaller sets cover a prescribed compact support. Thus using several such sets entails only a finite smooth partition.

Write $B=\|b\|_\infty$, $M=\|\nabla b\|_\infty$, $\delta=\operatorname{dist}(\operatorname{supp}b,\mathbb R^N\setminus V_0)>0$, and let $\omega_{N-1}$ be the area of the unit sphere in $\mathbb R^N$. Set $$A_{b,\sigma}=\omega_{N-1} \left(\frac{M^2}{2-2\sigma}+\frac{4B^2}{2\sigma}\right).$$ For $v(y)=b(y)f(\Psi(y))$ on $V_0$, extended by zero outside, change of variables gives $$\|v\|_2^2\leq B^2J\|f\|_2^2,$$ and direct separation of differences gives $$[v]_{\sigma,N}^2 \leq2B^2L^{N+2\sigma}J^2[f]_{\sigma,N}^2 +\left(2A_{b,\sigma}J+ \frac{\omega_{N-1}}{\sigma}\delta^{-2\sigma}B^2J\right)\|f\|_2^2. \tag{RT18}$$ To verify every term, on $V_0\times V_0$ use $$|b(y)f(\Psi(y))-b(z)f(\Psi(z))|^2 \leq2B^2|f(\Psi(y))-f(\Psi(z))|^2 +2|b(y)-b(z)|^2|f(\Psi(z))|^2.$$ Two Jacobians and (RT17) bound the first integral. For the second, extend $b$ smoothly by zero; then $|b(y)-b(z)|\leq\min(M|y-z|,2B)$. Integration on $|y-z|\leq1$ and $|y-z|>1$ yields $A_{b,\sigma}$, followed by one Jacobian. Finally the two cross regions with one point outside $V_0$ contribute at most $2\|v\|_2^2\omega_{N-1}\int_\delta^\infty r^{-1-2\sigma}dr$. This is the last term of (RT18). Thus no boundary term in the zero extension of a localized composition was omitted.

For a cutoff $a\in C_c^\infty(V_0)$, apply the usual chain and product rules to $a(f\circ\Psi)$. In ordinary derivative notation their exact coefficients can be generated recursively by $$b_{0,0}=a,\qquad b_{\alpha+e_j,\beta} =\partial_jb_{\alpha,\beta} +\sum_{\ell=1}^N b_{\alpha,\beta-e_\ell}\partial_j\Psi_\ell, \qquad \partial^\alpha(a(f\circ\Psi)) =\sum_{|\beta|\leq|\alpha|}b_{\alpha,\beta} ((\partial^\beta f)\circ\Psi). \tag{RT19}$$ Absent coefficients are zero; for each multiindex fix any ordered sequence of the indicated differentiations. The product rule proves this recursion by induction. The norms of $D^\alpha v$ and $\partial^\alpha v$ are equal, so these ordinary derivative identities do not change the convention $D=-i\partial$.

For integer order, change of variables applied to each term of (RT19) gives the bounded composition operator in $H^m$. For order $m+\sigma$, use (RT18) on each term of each order-$m$ derivative, then (RT16). For every $|\beta|\leq m$, Plancherel also gives $$\|D^\beta f\|_2^2\leq\|f\|_{H^{m+\sigma}}^2, \qquad [D^\beta f]_{\sigma,N}^2 \leq c_{N,\sigma}\|f\|_{H^{m+\sigma}}^2.$$ Using $|\sum_{j=1}^{r}v_j|^2\leq r\sum|v_j|^2$ for each finite sum completes the estimate $$\|a(f\circ\Psi)\|_{H^r(\mathbb R^N)} \leq C_{r,a,\Psi,V_0}\|f\|_{H^r(\mathbb R^N)},\qquad r\geq0, \tag{RT20}$$ where the left side is the zero extension just used. Its squared constant is obtained by the finite sums in (RT19), (RT18), and (RT16). In particular it depends only on $N,m,\sigma$, the displayed $L,J,\delta$, the derivatives of $a$ through order $m+1$, and those of $\Psi$ through order $m+1$. At an integer order no fractional constant occurs and derivatives through order $m$ suffice. Applying the same proof to inverse coordinate maps gives the reverse local comparisons. General compactly supported coordinate changes follow by the finite subdivision explained after (RT17).

All these identities were initially proved on smooth inputs. Density of Schwartz functions in (RT1), already proved in Trace proof 2, extends (RT20) to $H^r$. The resulting extension is the actual composition almost everywhere, since it converges in $L^2$ on the support by the Jacobian bound. Thus its weak derivatives are also those supplied by (RT19).

Multiplication by a fixed compact smooth cutoff is (RT20) with $\Psi=I$. Restrictions of $C_c^\infty(\mathbb R^N)$ are dense in every quotient (RT2): after Schwartz approximation, multiply a fixed Schwartz function by cutoffs $\theta(x/R)$. The difference tends to zero in every integer $H^k$ by the product rule, the rapid decay of all derivatives, and dominated convergence; choosing $k\geq r$ and using (RT1) gives convergence in $H^r$. Approximate a chosen full-space extension and restrict. This also proves density of functions smooth on a neighborhood of $\overline X$ in $H^r(X)$.

Trace proof 5. Surface norms and atlas independence

For $n\geq2$, put $d=n-1$. Fix a finite smooth boundary atlas $\kappa_a:O_a\subset\mathbb R^d\to\partial X$, and a nonnegative smooth partition $\lambda_a$ on $\partial X$ subordinate to relatively compact parts of these charts, with $\sum_{a=1}^A\lambda_a=1$. For $q\geq0$, define $$\|f\|_{H^q(\partial X),\mathcal A}^2 =\sum_{a=1}^A \|((\lambda_af)\circ\kappa_a)^0\|_{H^q(\mathbb R^d)}^2, \tag{RT21}$$ where superscript zero means extension by zero outside $O_a$. The surface space consists of $L^2(dS)$ functions with finite (RT21).

Let $J_a(z)=\sqrt{\det(D\kappa_a(z)^TD\kappa_a(z))}$. This is positive and smooth. If $j_{\min},j_{\max}$ are the minimum and maximum of all $J_a$ on the compact supports concerned, change of variables gives the exact order-zero formula and bounds $$\begin{aligned} \|f\|_{H^0(\partial X),\mathcal A}^2 &=\int_{\partial X}|f(x)|^2 \sum_{a:x\in\kappa_a(O_a)} \frac{\lambda_a(x)^2}{J_a(\kappa_a^{-1}(x))}\,dS(x),\\ \frac1{A j_{\max}}\|f\|_{L^2(dS)}^2 &\leq\|f\|_{H^0(\partial X),\mathcal A}^2 \leq\frac1{j_{\min}}\|f\|_{L^2(dS)}^2. \end{aligned} \tag{RT22}$$ Terms with $\lambda_a=0$ are zero. The inequalities use $A^{-1}\leq\sum\lambda_a^2\leq1$. If a positive surface weight $w$ is used, its exact measure comparison is $w_{\min}\|f\|_{L^2(dS)}^2\leq\|f\|_{L^2(wdS)}^2\leq w_{\max}\|f\|_{L^2(dS)}^2$; substituting these bounds in (RT22) retains both the weight and the chart Jacobians.

To prove atlas independence, take another chart partition $(\widetilde\kappa_b,\widetilde\lambda_b)$. In a $b$-chart write $$(\widetilde\lambda_b f)\circ\widetilde\kappa_b =\sum_a (\widetilde\lambda_b\circ\widetilde\kappa_b) \bigl[((\lambda_a f)\circ\kappa_a) \circ(\kappa_a^{-1}\circ\widetilde\kappa_b)\bigr]. \tag{RT23}$$ For each overlap insert an extra smooth cutoff equal to one on the compact intersection of the two original supports and supported inside the overlap. The summands are then exactly the localized compositions of (RT20), in dimension $d$, at order $q$. The number of summands is finite. Equation (RT20) therefore bounds the new norm by the old; exchanging the atlases proves the reverse bound. The constants have exactly the dependence described in Trace proof 4, including the overlap cutoffs and their distances to chart edges.

For completeness, (RT21) defines a complete space: a Cauchy sequence converges in $L^2(dS)$ by (RT22); each localized coordinate sequence converges in $H^q(\mathbb R^d)$, whose $L^2$ limit is the localization of that same surface function by change of variables. The limits therefore have finite (RT21), and convergence holds in that norm. Smooth surface functions are dense. To see this, each compactly supported coordinate function $((\lambda_af)\circ\kappa_a)^0$ can be convolved with a smooth approximate identity and then multiplied by a fixed cutoff equal to one near its original support. Approximation in $H^q$ follows from Fourier dominated convergence for convolution and bounded cutoff multiplication (RT20); for sufficiently small convolution radius the support remains in $O_a$. Push these smooth compact functions to the surface and sum over $a$. Formula (RT23) and (RT20) show convergence in the surface norm, and the limiting sum is $\sum_a\lambda_af=f$.

In dimension $n=1$, a compact smooth boundary is a finite zero-dimensional manifold: its single-point coordinate neighborhoods form an open cover, so compactness gives finitely many points. Define every $H^q(\partial X)$ as functions on these points with the sum of squared absolute values. Surface measure is counting measure. There are no fractional coordinate changes to prove in dimension zero.

Trace proof 6. Curved-boundary trace, independence, and lifts

Take the fixed boundary atlas from Trace proof 5 as the restriction of ambient smooth flattenings $\Phi_a:U_a\to V_a\subset\mathbb R^{d+1}$, with $\Phi_a(X\cap U_a)=V_a\cap\{t>0\}$ and $\kappa_a(z)=\Phi_a^{-1}(z,0)$. Extend each $\lambda_a$ to a compactly supported smooth ambient cutoff $\eta_a\in C_c^\infty(U_a)$ with $\eta_a|_{\partial X}=\lambda_a$. This is achieved by multiplying its coordinate function on $t=0$ by a smooth normal cutoff equal to one near zero. All supports remain in the chosen coordinate patches.

For a full-space $H^s$ extension $U$ of $u$, set $$v_a(y)= \begin{cases} (\eta_aU)(\Phi_a^{-1}(y)),&y\in V_a\text{ (the chart range)},\\ 0,&\text{elsewhere}. \end{cases} \tag{RT24}$$ By (RT20), $\|v_a\|_{H^s}\leq C_a\|U\|_{H^s}$. Its flat trace $T_0v_a$ is bounded by (RT9). For smooth $U$, it is precisely $(\lambda_aU|_{\partial X})\circ\kappa_a$, extended by zero.

Approximate a general $U$ in $H^s$ by $C_c^\infty$ functions. The traces of these approximants form a Cauchy sequence in the surface norm (RT21), because $$\|U|_{\partial X}\|_{H^{s-1/2}(\partial X),\mathcal A}^2 \leq\frac{C_s}{2\pi}\sum_a C_a^2\|U\|_{H^s(\mathbb R^n)}^2 \tag{RT25}$$ holds for their differences. Completeness gives a limit. Bounded localization and (RT20) show that its $a$-coordinate component is exactly $T_0v_a$, so it is independent of the approximating sequence.

If $U$ vanishes on $X$, each localized function $v_a$ in (RT24) vanishes on the entire positive half-space: inside its chart this follows from the flattening, and outside the chart it follows from its zero definition. Trace proof 2 gives $T_0v_a=0$. Hence the resulting surface limit is zero by (RT21). It follows that the boundary value depends only on $u=U|_X$. Taking the infimum over $U$ in (RT25) proves (RT4) with $$\|\gamma_Xu\|_{H^{s-1/2}(\partial X),\mathcal A} \leq\left(\frac{C_s}{2\pi}\sum_a C_a^2\right)^{1/2} \|u\|_{H^s(X),\mathrm{res}}. \tag{RT26}$$ Smooth restrictions are dense by Trace proof 4, so this continuous extension of smooth restriction is unique. Consequently changing the flattenings or partition yields exactly the same trace, with the equivalent surface norms of Trace proof 5. The positive-slice limit in each cutoff chart is exactly (RT10) applied to $v_a|_{t>0}$. This proves the asserted local one-sided interpretation at every real $s>1/2$.

Here are also the precise local Sobolev comparisons implicit in the notation of Section 2. Choose a finite partition of unity on a neighborhood of $\overline X$, subordinate to boundary and interior patches, and in each patch take the corresponding localized coordinate restriction norm of order $s$. Such an ambient partition can retain boundary values $\lambda_a$: choose the nonnegative extensions above, divide by their positive sum near the boundary, multiply the resulting functions by a collar cutoff equal to one on a smaller boundary neighborhood, and partition the remaining interior term. Equation (RT20), applied to any extension $U$, bounds the finite localized norm by $C\|u\|_{H^s(X),\mathrm{res}}$ after taking the infimum. Conversely extend each localized coordinate restriction to the whole coordinate space with norm within $\varepsilon$ of its quotient infimum. Multiply by a fixed slightly larger cutoff equal to one on the original localized support and pull back by the inverse chart, extending by zero beyond that chart. Equation (RT20) bounds each resulting global $H^s$ function. The sum restricts to $u$, since the original cutoffs sum to one. The triangle inequality and then $\varepsilon\downarrow0$ bound (RT2) by the finite localized norm. This proves both directions, with cutoff, Jacobian, and chart constants retained. For an estimate near a compact subset, fix the localizing functions and larger patch first and apply this construction to those localized functions; the constants retain both choices. No equivalence between seminorms on unrelated larger neighborhoods is needed. Interior pieces are treated by whole-space coordinate charts and the same proof.

To construct a right inverse, first choose once and for all $\rho_a\in C_c^\infty(V_a)$ whose value on $t=0$ is one on a neighborhood of the fixed compact set $\operatorname{supp}((\lambda_a\circ\kappa_a)^0)$. These cutoffs depend only on the atlas and partition, not on the datum or the Sobolev order. For $f\in H^{s-1/2}(\partial X)$, put $f_a=((\lambda_af)\circ\kappa_a)^0$, whose support lies in that fixed set, and take $Ef_a$ from (RT11). Define the global function $$W_a(x)= \begin{cases} (\rho_aEf_a)(\Phi_a(x)),&x\in U_a,\\ 0,&x\notin U_a, \end{cases} \qquad L_Xf=\left.\sum_aW_a\right|_X. \tag{RT27}$$ Equations (RT12), (RT20), and the finite sum give $$\|L_Xf\|_{H^s(X),\mathrm{res}} \leq K_{s,\vartheta}^{1/2} \left(\sum_a C_{a,s}'{}^2\right)^{1/2} \|f\|_{H^{s-1/2}(\partial X),\mathcal A}, \tag{RT28}$$ where $C_{a,s}'$ is the bounded composition-and-cutoff constant for the $a$-term. For smooth $f$, direct evaluation and $\vartheta(0)=1$ give $\gamma_XW_a=\lambda_af$; hence $\gamma_XL_Xf=f$. Surface density and (RT26), (RT28) extend this identity to all $f$. The choices of $\vartheta$, all cutoffs, and the atlas were fixed independently of $s$. Therefore the operator (RT27) is a single right inverse on all real scales $s>1/2$, with the displayed order-dependent bounds. This construction also works at each boundary point when $n=1$, using the one-variable instance of (RT11).

Trace proof 7. The zero-trace energy space

For clarity we prove (RT5) with the derivative-sum $H^1$ norm of (DSP6), which is equivalent to (RT2). The inclusion of the closure into the kernel follows immediately from the trace bound and the zero trace of every compact interior smooth function.

For the reverse inclusion, first work on the flat half-space with a compactly supported $v\in H^1(\mathbb R^{d+1}_+)$ whose trace is zero. Write $v^0$ for zero extension. Smooth approximation by restrictions of whole-space tests, proved in Trace proof 4, and integration by parts give the distribution formulas $$\partial_t v^0=(\partial_t v)^0+(\gamma_+v)\otimes\delta_0, \qquad \partial_{z_j}v^0=(\partial_{z_j}v)^0. \tag{RT29}$$ Indeed, for a smooth approximant and a compact smooth test $\psi$, integration on $t>0$ yields $-\int v\partial_t\psi=\int(\partial_tv)\psi+\int v(z,0)\psi(z,0)dz$. The volume terms converge by $H^1$ convergence; the boundary term converges by (RT10) at $s=1$, and the test boundary restriction is smooth compactly supported. This proves the normal formula, with its positive sign. Tangential integration has no boundary term and proves the other formulas. Since the trace is zero, every derivative on the right of (RT29) is $L^2$, so $v^0\in H^1(\mathbb R^{d+1})$.

Set $v_\varepsilon(z,t)=v^0(z,t-\varepsilon)$. It vanishes for $t<\varepsilon$, and Fourier dominated convergence gives $v_\varepsilon\to v^0$ in $H^1$. Convolve it with a compact smooth mollifier of radius less than $\varepsilon/2$, choosing its radius sufficiently small that the $H^1$ convolution error is less than $\varepsilon$. The resulting smooth function is supported in $t\geq\varepsilon/2$ and converges to $v^0$ in $H^1$. The original compact support and sufficiently small translations and radii keep it within any fixed slightly larger coordinate neighborhood. A fixed cutoff equal to one near the original support, if needed, preserves convergence by (RT20). Thus the restrictions approximate $v$ by compact interior smooth functions.

For a zero-trace $u\in H^1(X)$, use a finite smooth partition subordinate to boundary and interior patches. Each boundary-localized pullback has zero flat trace by (RT24)–(RT26) and is approximated by the preceding construction. Pull those approximants back by the smooth flattening and extend by zero beyond its compact patch. They are $C_c^\infty(X)$, and (RT20) at order one proves their convergence in $H^1(X)$. The interior-localized pieces are approximated by ordinary mollification with radius smaller than their distance from the boundary. Summing the finite approximations proves that $u$ belongs to the closure in (RT5).