Detecting a fractional gain in a cone

A regularity implication refers to every distribution. An estimate refers to one cutoff and one constant. Passing between them is essential: concentrating test functions can test an estimate, while a regularity theorem must also apply to rough solutions. This lesson proves that passage for a scalar operator whose loss is strictly less than one derivative. It also explains why only the principal symbol matters in this range.

We use real Sobolev mapping, proper elliptic weights, conic inverses and separated-support smoothing from Detecting regularity without choosing coordinates, and the product and commutator orders in Symbols, operators and Sobolev scales. The complete-metric Baire theorem is the one stated in Section 6 of Banach estimates, quotient spaces and compact parameter arguments, with Garrett's open proof [G]. We apply it to a complete graph space and a countable family of shrinking cones.

Basic references are Garrett [G], Melrose's microlocalization chapter [M], Lerner's survey [L] and Hörmander [H]. All norms below are Sobolev norms in a coordinate patch, after fixed compact localization. We write $D=-i\partial$.

1. The gain and the lower regularity are different

Let $P\in\Psi^m_{\mathrm{cl}}(X)$ be properly supported and scalar, with $m\in\mathbb R$. Fix $0<\delta<1$ and a nonzero covector $\rho$. We say that $P$ has microlocal loss $\delta$ at $\rho$ if, for every $s\in\mathbb R$ and every distribution, $Pu\in H^s\text{ at }\rho \quad\Longrightarrow\quad u\in H^{s+m-\delta}\text{ at }\rho. \tag{1.1}$ The gain over the datum is $m-\delta$. The gain over the lower regularity $H^{s+m-1}$ is instead $\varepsilon=1-\delta>0. \tag{1.2}$ That second gain drives the rough-solution argument.

Theorem 1.1. Property (1.1) is equivalent to the following weaker assertion at one fixed real index $s_0$: $u\in H^{s_0+m-1}_{\mathrm{loc}}(X),\quad Pu\in H^{s_0}_{\mathrm{loc}}(X) \quad\Longrightarrow\quad u\in H^{s_0+m-\delta}\text{ at }\rho. \tag{1.3}$ The lower regularity in (1.3) is an assumption on the input. We remove it after constructing the quantitative estimate.

2. A fixed cutoff emerges from Baire's theorem

Theorem 2.1. Let $K$ be a compact coordinate neighborhood of the base point of $\rho$. If (1.1) holds, there are a proper classical $A\in\Psi^0$, elliptic at $\rho$, and $C<\infty$ such that $\|Au\|_{m-\delta} \leq C\bigl(\|Pu\|_0+\|u\|_{m-1}\bigr), \qquad u\in C_c^\infty(K). \tag{2.1}$ Here $C_c^\infty(K)$ means smooth functions with support contained in $K$. Conversely, (2.1) implies (1.1) at every covector over $K^\circ$ where $A$ is elliptic.

Proof of necessity. Choose $A_j\in\Psi^0$, $j\geq1$, elliptic at $\rho$, with compact base support and microsupports shrinking to its ray. Spatial and angular cutoffs, extended above a fixed frequency and properly quantized, give such a family. For every distribution that is $H^{m-\delta}$ at $\rho$, some $A_j$ sends it into $H^{m-\delta}$, by the conic Sobolev test in the geometric prerequisite.

Consider $\mathcal H_K= \{u\in H^{m-1}:\operatorname{supp}u\subset K,\ Pu\in L^2\}, \qquad \|u\|_{\mathcal H_K}^2 =\|u\|_{m-1}^2+\|Pu\|_0^2. \tag{2.2}$ Properness puts all outputs in a common compact set. This space is complete. A graph-Cauchy sequence converges to $u$ in $H^{m-1}$ and to $f$ in $L^2$ for its $P$-images. Support is preserved by distributional convergence. Continuity of $P:H^{m-1}_{\mathrm{comp}}\to H^{-1}_{\mathrm{loc}}$ identifies $Pu=f$. Thus the limits lie in (2.2), with convergence in its norm.

For integers $j,N\geq1$, put $E_{j,N}=\{u\in\mathcal H_K:\|A_ju\|_{m-\delta}\leq N\}. \tag{2.3}$ These sets are closed, convex and balanced. For closedness use the Sobolev dual norm formula $\|v\|_q= \sup_{\substack{\phi\in C_c^\infty\\\|\phi\|_{-q}\leq1}} |\langle v,\phi\rangle|. \tag{2.4}$ It follows from the weighted Fourier $L^2$ pairing and density of test functions. A bounded pairing on $H^{-q}$ represents an element of $H^q$ by Hilbert duality. Consequently a distributional limit of elements with $H^q$ norm at most $N$ has the same bound. Graph convergence gives distributional convergence of $A_ju$, proving closedness of (2.3).

For every $u\in\mathcal H_K$, (1.1) at $s=0$ gives the target regularity at $\rho$. Hence $\mathcal H_K=\bigcup_{j,N\geq1}E_{j,N}. \tag{2.5}$ Baire applies to the complete norm metric just verified. Some $E_{j,N}$ contains $B(u_*,r)$. If $\|h\|_{\mathcal H_K}<r$, both $u_*+h$ and $u_*$ belong to that set. Thus $\|A_jh\|_{m-\delta}\leq2N$. Rescaling gives $\|A_ju\|_{m-\delta}\leq C\|u\|_{\mathcal H_K}. \tag{2.6}$ Take $A=A_j$. This proves (2.1). The cutoff is now fixed for every graph input; equality of the individual cutoffs was never assumed. ∎

3. The estimate applies to its full graph domain

Assume (2.1). Let $u\in H^{m-1}_{\mathrm{loc}}$ and $Pu\in L^2_{\mathrm{loc}}$. Choose $\chi\in C_c^\infty(K^\circ)$, equal to one near a base point under consideration, and set $w=\chi u$. Then $Pw=\chi Pu+[P,\chi]u\in L^2, \tag{3.1}$ because the commutator has order $m-1$. Also $w\in H^{m-1}$.

Let $J_h=\vartheta(hD)$, with $\vartheta$ Schwartz and $\vartheta(0)=1$. Choose $\chi_1\in C_c^\infty(K^\circ)$ equal to one near $\operatorname{supp}w$, and put $w_h=\chi_1J_hw$. Each $w_h$ is smooth with support inside $K$, and $w_h\to w$ in $H^{m-1}$. The $J_h$ form a bounded $S^0_{1,0}$ family, so $[P,J_h]\text{ is bounded in }\Psi^{m-1}, \qquad 0<h\leq1. \tag{3.2}$ Indeed derivatives of $\vartheta(h\xi)$ satisfy the uniform symbol inequalities. The first composition difference has one frequency derivative, and the full ordinary remainder preserves this order uniformly. Thus $Pw_h=\chi_1J_hPw+ \chi_1[P,J_h]w+[P,\chi_1]J_hw \tag{3.3}$ is uniformly bounded in $L^2$. Fixed proper localizations handle terms outside the coordinate set.

Equation (2.1) bounds $Aw_h$ in $H^{m-\delta}$. Their distributional limit is $Aw$, so (2.4) gives the same regularity to that limit. At an elliptic covector of $A$ where $\chi=1$, a conic inverse proves $u\in H^{m-\delta}$. Thus the weak implication (1.3) holds at $s_0=0$. We now remove its input assumption and reach every index.

4. One index reaches every distribution

First extend (1.3) from $s_0$ to any $t\in\mathbb R$. Suppose $u\in H^{t+m-1}_{\mathrm{loc}}$ and $Pu\in H^t_{\mathrm{loc}}$. Choose a proper elliptic Sobolev weight $B$ of order $t-s_0$. Then $v=Bu$ is $H^{s_0+m-1}_{\mathrm{loc}}$, and $Pv=BPu+[P,B]u\in H^{s_0}_{\mathrm{loc}}. \tag{4.1}$ The commutator has order $m+t-s_0-1$, giving exactly this output index. Apply (1.3) to $v$, then invert $B$ microlocally. This proves $u\in H^{t+m-\delta}$ at $\rho$. Proper weights exist on $X$; compact partitions make all these mapping statements local.

The weak conclusion also applies when its hypotheses hold only in a larger cone about $\rho$. Choose a compactly based $C\in\Psi^0$, elliptic at $\rho$, with microsupport inside that cone. A larger cutoff $C_1=1$ near that microsupport gives $Cu\in H^{t+m-1}_{\mathrm{loc}}$ and $P(Cu)=CPu+[P,C]C_1u+[P,C](I-C_1)u \in H^t_{\mathrm{loc}}. \tag{4.2}$ The last operator is smoothing by separated microsupports. A compact distribution has some finite negative Sobolev order, which controls this term at every fixed output index. The other terms use the cone regularity and the order $m-1$ of $[P,C]$. Apply the weak conclusion to $Cu$ and invert $C$ at $\rho$.

Now assume only $Pu\in H^s$ at $\rho$. Locally every distribution has some negative Sobolev regularity, since its localized Fourier transform is polynomially bounded. Choose $t_0\leq s$ low enough that $u\in H^{t_0+m-1}$ in the working neighborhood. The preceding step gives $u\in H^{t_0+m-\delta}\text{ in a smaller cone}. \tag{4.3}$ This is the lower input regularity for the next data index $t_0+\varepsilon$, because $t_0+m-\delta=(t_0+\varepsilon)+m-1. \tag{4.4}$ Repeat with finitely many nested cones, increasing $t$ by $\varepsilon>0$ and using a smaller last increment to end at $s$. The datum stays $H^t$ throughout. The last application gives (1.1), proving Theorem 1.1 and the converse of Theorem 2.1. ∎

The same argument shows that the microlocal property at every nonzero covector is equivalent to $Pu\in H^s_{\mathrm{loc}}(X) \quad\Longrightarrow\quad u\in H^{s+m-\delta}_{\mathrm{loc}}(X) \tag{4.5}$ for every index and every distribution. The forward direction uses finitely many conic Sobolev tests over each compact cosphere. Conversely (4.5) supplies (1.3) at each covector, which Theorem 1.1 upgrades to the full microlocal implication.

5. Lower terms and elliptic multipliers

Theorem 5.1. Loss $0<\delta<1$ is unchanged by replacing $P$ with an operator of order $m$ whose principal symbol agrees near $\rho$. More generally, let $a$ be a smooth homogeneous scalar of degree $r$, with $a(\rho)\ne0$. Every proper classical operator of order $m+r$ with principal symbol $ap$ near $\rho$ has the same loss there.

Proof. If $Q-P$ has order $m-1$ near $\rho$, test the weak implication for $Q$. Its input is $H^{s+m-1}$, so $(Q-P)u\in H^s$ there. Thus $Qu\in H^s$ implies $Pu\in H^s$, and then $u\in H^{s+m-\delta}$ by (1.1) for $P$. Theorem 1.1 proves the full property for $Q$. Interchange the operators for the converse. Localization to the working cone adds only separated smoothing terms.

For the multiplier, choose $E\in\Psi^r$ with principal symbol $a$, elliptic at $\rho$. Near that point write $Q=EP+R,\qquad R\in\Psi^{m+r-1}. \tag{5.1}$ In the weak test for $Q$, the input is $H^{s+m+r-1}$. Hence $Ru\in H^s$, and $Qu\in H^s$ gives $EPu\in H^s$. Inverting $E$ puts $Pu$ in $H^{s+r}$. Apply (1.1) for $P$ to obtain $u\in H^{s+m+r-\delta}$. Theorem 1.1 removes the weak input assumption. A conic inverse of $E$ proves the reverse implication by the same argument. All orders are real. ∎

The multiplier changes the operator order and the datum index together. It preserves the loss, not the numerical gain before changing the order.

6. The good covectors form an open cone

Corollary 6.1. For fixed $\delta$, the set where $P$ has loss $\delta$ is open and conic in $T^*X\setminus0$.

Proof. At a good covector choose (2.1). The elliptic set of its classical cutoff $A$, over $K^\circ$, is an open conic neighborhood. The converse of Theorem 2.1 proves the same loss at every point in it. Positive rescaling also preserves the conic Sobolev test defining (1.1). ∎

For one fixed input, its Sobolev regularity already holds on an open cone. Here one neighborhood works for the implication for every distribution. The Baire argument supplies that shared neighborhood.

7. Exercises with complete solutions

Exercise 1 — two different gains, 6 points. Suppose $m=3/2$, $\delta=2/3$, and $Pu$ is $H^{-1}$ in a cone. Find the target regularity and the improvement made by each weak bootstrap application.

Solution. The target is $H^{-1+3/2-2/3}=H^{-1/6}.$ Each weak step improves its lower input index by $1-\delta=1/3$. The datum-to-solution gain $5/6$ differs from that $1/3$ step.

Exercise 2 — a ball away from zero, 8 points. Suppose $B(u_*,r)\subset E_{j,N}$. Derive an explicit bound for $A_j$ without moving the ball's center.

Solution. If $\|h\|_{\mathcal H_K}<r$, the two points $u_*+h,u_*$ have images of norm at most $N$, so the image of their difference has norm at most $2N$. For $u\ne0$, set $h=ru/(2\|u\|_{\mathcal H_K})$. Homogeneity gives $\|A_ju\|_{m-\delta}\leq4N\|u\|_{\mathcal H_K}/r.$ The zero input satisfies this too.

Exercise 3 — a negative-order multiplier, 8 points. Let $P$ have order two and loss $3/4$. Multiply its principal symbol by an elliptic symbol of order $-1/2$. Find the new target for a datum in $H^s$ and verify the error order in (5.1).

Solution. The new order is $3/2$ and the target is $H^{s+3/4}$. The error has order $1/2$, so the weak input $H^{s+1/2}$ puts it in $H^s$. Inverting the multiplier gives $Pu\in H^{s-1/2}$; the old gain $5/4$ then gives $H^{s+3/4}$.

Exercise 4 — a zero bootstrap step, 8 points. Why can the proof not simply set $\delta=1$? Give two members of $\Psi^1$ with the same degree-one principal symbol for which the zero-gain implication differs.

Solution. Then $\varepsilon=0$ in (4.4), so repetition cannot improve a rough input. Take $P=0$ and $Q=I$, regarded as operators of order at most one. Both have zero degree-one principal symbol and differ by an order-zero operator. For $Q$, $Qu\in H^s\Rightarrow u\in H^s$ is immediate. For $P$, the output is smooth for every distribution, including ones outside $H^s$. Thus lower terms can determine the zero-gain property. Neither operator in this example is asserted to have a nonzero degree-one symbol.

Exercise 5 — the graph limit, 8 points. Why must the $L^2$ output limit in (2.2) equal $Pu$? Why is an assumed $H^m$ input unnecessary?

Solution. Input convergence in $H^{m-1}$ gives convergence of the images to $Pu$ in $H^{-1}_{\mathrm{loc}}$, hence in distributions. Their independently obtained $L^2$ limit is the same distributional limit. The graph condition supplies the extra output information directly, one derivative below an $H^m$ input. Replacing it by an assumed $H^m$ input would discard the rough graph elements needed by the argument.

References

[G] Paul Garrett, Review of metric spaces, 2 February 2014. Open author's notes. Complete-metric Baire theorem; reference only, with its own license.
[M] Richard Melrose, Introduction to Microlocal Analysis, Chapter 5, Microlocalization. Open author's chapter. Conic regularity and elliptic testing.
[L] Nicolas Lerner, Semi-classical estimates for non-selfadjoint operators, Asian Journal of Mathematics 11 (2007), 217–250. Author's version. Principal-symbol geometry and estimates.
[H] Lars Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators, Springer, 2009 reprint. Publisher's record. Microlocal subellipticity and its invariances.

Written by GPT-6.1 Sol (OpenAI), at Ultra reasoning effort, September 2026. Self-checked by the writing AI. Public domain (CC0).