Canonical transport of fractional regularity
A change of base coordinates moves covectors by a cotangent map. A homogeneous canonical transformation can move them more generally, mixing base and frequency directions. Fractional microlocal regularity survives this larger change because an elliptic order-zero graph operator and its inverse preserve every Sobolev order. The lower terms created by conjugation are harmless when the loss is below one.
We use Detecting a fractional gain in a cone, especially its weak one-index test and principal-symbol invariance. The graph Sobolev theorem, proper two-sided microlocal inverse and ordered Egorov formula are the proved results in Sections 5–7 of Graph operators, continuity and Egorov. Conic cutoffs and separated smoothing come from Detecting regularity without choosing coordinates. We use these operator results without repeating their proofs.
Basic references are Lerner [L] and Hörmander [H]; Section 25.3 of [H] treats operators whose canonical relation is the graph of a canonical transformation. Scalar operators act on half densities; local positive density trivializations identify their Sobolev spaces with the usual scalar ones. The convention is \(D=-i\partial\).
1. Sobolev order follows the graph
Let \(\chi:U_Y\to U_X\) be a homogeneous canonical diffeomorphism of open punctured cotangent cones, and let
\[ \chi(\eta_0)=\rho_0. \]Choose a proper scalar graph operator \(T\) of order zero, elliptic at the graph point, and a proper inverse-graph operator \(S\) of order zero. The graph prerequisite constructs them so that, on smaller cones,
\[ ST=I,\qquad TS=I \quad\text{modulo microlocally smoothing operators}. \tag{1.1} \]The equalities concern the chosen cones. They impose no global inverse condition.
Lemma 1.1. For every real \(r\) and every distribution \(v\), localized to a compact input neighborhood,
\[ v\in H^r\text{ at }\eta_0 \quad\Longleftrightarrow\quad Tv\in H^r\text{ at }\rho_0. \tag{1.2} \]Proof. Choose an input cutoff \(C\) equal to one on a smaller cone about \(\eta_0\), supported where the assumed regularity holds. Then \(Cv\) is globally \(H^r\) after compact localization. The graph Sobolev theorem puts \(TCv\) in \(H^r_{\mathrm{loc}}\). Choose an output cutoff \(A\) supported in a still smaller cone about \(\rho_0\). The kernel relation of \(AT(I-C)\) is empty on the working input and output cones: the unique graph preimage of that output cone is inside the region where \(C=1\). The separated graph calculus therefore makes \(AT(I-C)v\) smooth. Thus \(Tv\) is \(H^r\) at \(\rho_0\).
Conversely apply the same argument to \(S\), with the inverse graph. Equation (1.1) identifies \(STv\) with \(v\) up to a smooth term at \(\eta_0\). The cutoff choices can all be made proper, with fixed compact base supports. Their smooth errors act on a compact distribution of some finite negative Sobolev order, so every asserted smooth output is meaningful for arbitrary distributions. ∎
In particular, ellipticity and the inverse identities supply the reverse implication. Sobolev continuity alone would supply only the forward one.
2. Transport the weak test first
Let \(P\in\Psi^m_{\mathrm{cl}}(X)\) be proper and scalar, with principal symbol \(p\), and put
\[ Q=SPT. \tag{2.1} \]The ordered Egorov theorem makes \(Q\) a proper pseudodifferential operator of order \(m\), microlocally on \(U_Y\). The normalized inverse identities give
\[ \sigma_m(Q)=p\circ\chi. \tag{2.2} \]Extensions away from the working cone may be chosen arbitrarily and properly; they change neither the stated symbol nor the regularity implication there.
Theorem 2.1. Suppose \(0<\delta<1\). If \(P\) has microlocal loss \(\delta\) at \(\rho_0\), then every proper classical scalar operator of order \(m\), with principal symbol \(p\circ\chi\) near \(\eta_0\), has the same loss at \(\eta_0\). The converse holds after transport by \(\chi^{-1}\).
Proof. First prove the weak implication for (2.1). Fix one real \(s\), and suppose that
\[ v\in H^{s+m-1}\text{ near }\eta_0,\qquad Qv\in H^s\text{ near }\eta_0. \tag{2.3} \]The hypotheses may be conic. The graph relation and compact cutoffs used in Lemma 1.1 transfer the lower input regularity to \(u=Tv\) near \(\rho_0\). On that cone,
\[ Pu-TQv=(I-TS)PTv \quad\text{is smooth}. \tag{2.4} \]To justify the composition in (2.4), insert an input cutoff supported inside the inverse cone and an output cutoff inside its graph image. On those supports \(I-TS\) is smoothing, while the remaining terms have separated graph supports. The proper calculus makes each localized error smooth.
By Lemma 1.1, \(TQv\) is \(H^s\) near \(\rho_0\). Hence \(Pu\) is \(H^s\) there. The loss property for \(P\) gives
\[ u\in H^{s+m-\delta}\text{ at }\rho_0, \]and Lemma 1.1 gives the same index for \(v\) at \(\eta_0\). This proves the weak test. The complete weak-test equivalence in the fractional-gain prerequisite removes its lower input assumption and reaches every real datum index and every distribution.
For any other operator with symbol (2.2), the difference has order \(m-1\) on the working cone. Principal-symbol invariance in that same prerequisite proves the identical full loss property. Apply this argument to \(S\) and the inverse transformation for the converse. ∎
The proof uses the full graph inverse and all real Sobolev orders. It is therefore unaffected by caustics in a particular phase parametrization.
3. Keep the noninverse coefficient
Suppose instead that \(T\) and \(S\) merely have inverse graph relations. They need not be normalized inverses. For scalar \(P\), Egorov gives
\[ \sigma_m(SPT)=c\,(p\circ\chi), \qquad c=\sigma_0(ST). \tag{3.1} \]If both graph symbols are elliptic, then \(c\ne0\) at the point. The homogeneous scalar multiplier theorem in the fractional-gain prerequisite therefore removes \(c\), preserving the loss. If \(c=0\), that theorem does not apply.
For an elliptic \(T\) of real order \(a\), choose its inverse \(S\) of order \(-a\). Its graph mapping shifts \(H^r\) to \(H^{r-a}\). A datum in \(H^s\) for the transported order-\(m\) operator leads to a datum in \(H^{s-a}\) for \(P\), a solution in \(H^{s-a+m-\delta}\), and then, under \(S\), to
\[ H^{s+m-\delta}. \tag{3.2} \]Thus changing the graph quantization order does not change the loss. Order zero gives the most direct identification (1.2).
4. An exact change of variables
Take \(Y=(0,\infty)\), \(X=\mathbb R\), and
\[ (Tv)(x)=e^{x/2}v(e^x),\qquad (Su)(y)=y^{-1/2}u(\log y). \tag{4.1} \]These are inverse half-density pullbacks. Change of variables proves that they are unitary on \(L^2\). Their canonical map is
\[ \chi(y,\eta)=(\log y,y\eta). \tag{4.2} \]Direct differentiation yields
\[ SD_xT=yD_y-\frac i2. \tag{4.3} \]The principal symbol is \(y\eta\). The order-zero coefficient records the half-density Jacobian; it does not change a loss below one. Every assertion is local over compact subsets of \(y>0\), so no uniform bound as \(y\to0\) is being asserted.
5. Exercises with complete solutions
Exercise 1 — a negative graph order, 8 points. Let \(P\) have order \(3/2\) and loss \(2/3\), and choose a graph quantization \(T\) of order \(-1/4\). If the transported datum is in \(H^{-1}\), follow every index through \(T\), the loss property and \(S\).
Solution. The transformed datum for \(P\) is in \(H^{-1+1/4}=H^{-3/4}\). The gain is
\[ 3/2-2/3=5/6. \]It puts the transformed solution in \(H^{1/12}\). The inverse graph operator has order \(1/4\), so it puts the original solution in \(H^{1/12-1/4}=H^{-1/6}\). This equals the order-\(3/2\) target from a datum in \(H^{-1}\).
Exercise 2 — inverse graphs without inverse symbols, 6 points. In (4.1), replace \(S\) by \(3S\). Find the full transported operator and explain why its fractional loss still agrees with that of \(D_x\).
Solution. The operator is \(3yD_y-3i/2\). Its principal symbol is \(3y\eta\), and \(c=3\) in (3.1). This is a nonzero homogeneous order-zero multiplier of \(y\eta\). Multiplier and lower-term invariance preserve every loss in the strict range \(0<\delta<1\).
Exercise 3 — separate the graph supports, 8 points. In Lemma 1.1, why must the output cone be chosen after the input cutoff?
Solution. The complement \(I-C\) is harmless only if the graph preimage of the output cone lies where \(C=1\). Since \(\chi^{-1}\) is continuous, a sufficiently small output cone has that property. For a larger cone, its preimage may meet the uncontrolled part of the input, and the separated-support smoothing argument would fail.
Exercise 4 — preserve the symplectic form, 6 points. Verify directly that (4.2) is canonical and homogeneous.
Solution. The transformed one-form is \((y\eta)d(\log y)=\eta\,dy\), so its exterior derivative is the original symplectic form. Positive cotangent dilation multiplies \(\eta\) and hence \(y\eta\) by the same factor, while the base coordinate \(\log y\) is unchanged.
Exercise 5 — the endpoint requires a new argument, 8 points. Which two steps in Theorem 2.1 use \(\delta<1\)?
Solution. The weak-test equivalence improves the lower input index by \(1-\delta\); that is positive only below one. Principal-symbol invariance also uses that equivalence to remove lower-order terms from the weak test and return to the full implication. At \(\delta=1\), a graph identity may still transport a particular exact operator, but replacing it by an arbitrary operator with the same principal symbol needs additional information. The zero-step example in the fractional-gain lesson shows why.
References
- [L] Nicolas Lerner, Semi-classical estimates for non-selfadjoint operators, Asian Journal of Mathematics 11 (2007), 217–250. Open author's version. Microlocal estimates and principal-symbol geometry.
- [H] Lars Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators, Springer, 2009 reprint. Publisher's record. Graph calculus and canonical invariance of subellipticity.
Written by GPT-6.1 Sol (OpenAI), at Ultra reasoning effort, September 2026. Self-checked by the writing AI. Public domain (CC0).