From a constant sign to a conic gain

A global estimate for a nonnegative coefficient becomes useful near one covector only after two constructions. The local coefficient must be extended without losing its finite time derivative. The transverse estimate must then control the full frequency on the cone of interest. Both constructions retain the sharp fractional gain.

The analytic prerequisite is Theorem 1.1 of From scalar time estimates to an operator bound. We use the bracket invariants and both Taylor alternatives in Sign-constrained weighted Taylor geometry, Section 4 and Theorem 5.1 of Rotated brackets and finite-jet flow coordinates, and the exact homogeneous normalization in Smooth complex preparation and conic flow coordinates. Positive balanced dilations and their sign limits are proved in Full sign conditions for mixed weighted polynomials.

For operators, the estimate, distributional implication and multiplier theorem are in Detecting a fractional gain in a cone. Homogeneous graph transport is in Canonical transport of fractional regularity. Ordinary symbol and Sobolev bounds come from Symbols, operators and Sobolev scales; proper kernel cutoffs and conic tests come from Detecting regularity without choosing coordinates. The optimality assertion uses the necessary bound in Crossing direction and the necessary finite-type bound.

Basic references are Lerner [L], Evans and Zworski [E], and Hörmander [H]. Our convention is \(D=-i\partial\) and \(H_f g=\{f,g\}\). Scalar operators act on half densities; a smooth positive local trivialization gives the usual Sobolev spaces.

1. A positive extension keeps a finite jet

Write \(\eta=r\omega\), where \(r=|\eta|>0\) and \(\omega\in S^{d-1}\), with \(d\geq1\). Time and position are denoted by \(t,x\). Suppose a real homogeneous degree-one coefficient \(q(t,x,\eta)\) is defined on a product neighborhood in \((t,x,\omega)\) and

throughout that neighborhood, for some \(k\geq1\). A smaller product neighborhood has closure in the original one. Choose a smooth cutoff \(\theta(t,x,\omega)\), compactly supported in the original neighborhood, with

Extend the product \(\theta q\) by zero and put

Lemma 1.1. The function \(q_e\) is globally nonnegative, smooth for \(\eta\ne0\), homogeneous of degree one and equal to \(q\) on the smaller neighborhood. It has uniform ordinary transverse order-one symbol bounds, including every time derivative. For some \(c>0\),

Proof. The support of \(\theta\) stays inside the domain of \(q\), so its zero extension is smooth. Normalize by \(r\) and write \(f=q_e/r\). Outside a compact set in \((t,x,\omega)\), \(f=1\). On that compact set every derivative of \(f\) is bounded. Homogeneity gives the ordinary frequency estimates

It remains to prove that the jets in (1.4) never vanish together. If \(f\ne0\), its zeroth jet suffices. At a zero of \(f\), the two nonnegative summands in (1.3) vanish separately. Hence \(\theta=1\) and \(q=0\) there.

On the time line through such a point, let \(j\leq k\) be the first nonzero derivative of \(q/r\). It exists by (1.1). A nonnegative smooth function has a positive first nonzero Taylor coefficient, and that coefficient has even degree. Multiplication by \(\theta\), whose value is one at the point, retains that coefficient and every earlier zero coefficient.

The other summand \(1-\theta\) is also nonnegative. If it has an earlier first nonzero coefficient, that coefficient is positive and already gives a nonzero jet of \(f\). If its first nonzero coefficient has degree \(j\), the two positive coefficients add. If all its coefficients through degree \(j\) vanish, the coefficient of \(\theta q/r\) survives. Thus some jet of \(f\) through order \(k\) is nonzero in every case.

The continuous function

is positive on the compact set and has a positive minimum. Outside that set it equals one. Multiply its uniform lower bound by \(r\) to obtain (1.4). ∎

The argument allows a cutoff with a nonflat contact at \(\theta=1\). It also allows the first nonzero time derivative at a zero to have order smaller than \(k\). Nonnegativity prevents cancellation in both cases.

2. No sign change forces an even time type

Let \(p\) be a smooth complex function on a symplectic neighborhood. We say that it satisfies condition \((P)\) there if, for every smooth nonzero complex multiplier \(a\), the imaginary part of \(ap\) has no change of sign in either direction on any nonconstant bicharacteristic of \(\operatorname{Re}(ap)\) lying in \(\operatorname{Re}(ap)=0\). Thus it satisfies both orientations defined in the mixed-polynomial prerequisite.

This condition is preserved by nonzero smooth multiplication: a multiplier for the new symbol is a multiplier for the old one after taking their product. A symplectic change preserves it because it carries the Hamilton field and its forward flow parameter to the corresponding field and parameter.

At a characteristic point \(c\), define the bracket depth \(\kappa(c)\) as in the weighted geometry prerequisite: all Poisson-bracket words with at most \(\kappa(c)\) leaves vanish there, while some word with one more leaf does not. Infinite depth means that every finite word vanishes.

Lemma 2.1. Suppose \(p\) satisfies \((P)\) and \(p(c)=0\). Then

Proof. If \(H_{\operatorname{Re}p}\) is zero, the bracket is zero. Otherwise its curve through a characteristic point is nonconstant and stays in \(\operatorname{Re}p=0\). A nonzero derivative of \(\operatorname{Im}p\) at its zero would give opposite strict signs at small positive and negative times. Condition \((P)\) excludes this. ∎

In particular the sign hypothesis of the rotated-bracket equivalence holds.

Lemma 2.2 (a radial field has no finite endpoint). Let \(p\) be homogeneous of degree one, satisfy \((P)\), and have finite bracket depth at a nonzero characteristic covector \(c\). Its complex Hamilton field \(H_p(c)\) is not a complex multiple of the cotangent radial field \(R_c\).

Proof. Suppose \(H_p(c)\in\mathbb C R_c\). For any constant \(z\in\mathbb C\), put

Then \(H_F(c)=\alpha R_c\), with \(\alpha\) real. Degree-one homogeneity makes \(H_F\) commute with \(R\): cotangent dilation carries \(H_F\) to itself. Its restriction to the ray through \(c\) is consequently \(\alpha R\). Uniqueness for the smooth Hamilton equation identifies the curve through \(c\) with

For \(\alpha=0\) this is constant. Homogeneity and \(G(c)=0\) give \(G(\gamma(s))=0\) in either case. Hence

The constant-rotation finite-type test in Corollary 5.2 of the rotated-bracket prerequisite contradicts (2.4) for every \(z\). ∎

Exact homogeneous complex preparation therefore gives homogeneous symplectic coordinates and a nonzero degree-zero multiplier with normalized symbol

The marked point has \(t=x=\tau=0\), \(\eta=\eta_0\ne0\). This construction requires \(n=d+1\geq2\); Lemma 2.2 and the preparation theorem supply that restriction. Both bracket depth and \((P)\) are unchanged.

Proposition 2.3 (the pure jet). If \(\kappa(c)=k<\infty\), then \(k\) is even, \(k\geq2\), and

Proof. Each time derivative of order \(j\) is a bracket word with \(j+1\) leaves. This proves the zero values for \(j<k\). Equation (2.1) also gives \(k\geq2\).

Apply the two Taylor alternatives in Section 4 of the rotated-bracket prerequisite to (2.5). In its one-direction alternative, the pure coefficient is exactly \(\partial_t^k q(c)\). The one-direction theorem of the weighted geometry prerequisite says that a zero value of this coefficient makes every word through \(k+1\) leaves vanish. That contradicts depth \(k\).

In the remaining alternative, the first early transverse differential is straightened on the symplectic slice \(t=\tau=0\). The change is independent of \(t,\tau\). The leading balanced Taylor polynomial is

The polynomial, its weights, and its slope polynomial

are precisely those of the mixed theorem in the weighted geometry prerequisite. For \(s=0\), (2.8) means \(b_t\).

The full sign condition passes to (2.7). Indeed positive balanced dilation has one positive common Hamilton time factor. The compact \(C^2\) limit preserves each of the two orientation conditions by the characteristic-curve limit lemma in the mixed-polynomial prerequisite. Their intersection is \((P)\).

For every \(t\ne0,z\), choose

This makes \(Q=0\). Along the nonconstant characteristic time line of \(H_\tau\), its first derivative must be zero by \((P)\), and that derivative is (2.8). Thus \(B=0\) for \(t\ne0\), and polynomial continuity gives \(B=0\) everywhere. The mixed theorem then makes every bracket through \(k+1\) leaves vanish, again a contradiction.

The mixed alternative is impossible. The pure coefficient in (2.6) is therefore nonzero. Taylor's theorem on the central time line gives

An odd \(k\) would give opposite signs on the two sides of zero, contrary to \((P)\). Hence \(k\) is even. ∎

The slice changes and balanced dilations in this proof test Taylor coefficients. The final coordinates remain the exact homogeneous coordinates in (2.5).

If the last derivative in (2.6) is negative, make the canonical change \((t,\tau)\mapsto(-t,-\tau)\) and multiply the resulting symbol by \(-1\). Its new imaginary coefficient is \(-q(-t,x,\eta)\), whose \(k\)-th derivative is positive because \(k\) is even.

Corollary 2.4. After this choice of sign, there is a product conic neighborhood on which

for some \(c>0\).

Proof. Normalize \(\eta\) to the sphere. Continuity keeps the \(k\)-th derivative positive on a small product neighborhood. Choose a small \(T>0\). Equation (2.10) gives positive values at both \(t=-T\) and \(t=T\) for the marked transverse parameter. These signs persist for all transverse parameters in a smaller neighborhood.

A negative value between these endpoints would give a strict change of sign on that same characteristic time line. Thus \(q\geq0\) throughout the product. Homogeneity turns the uniform positive bound on the normalized \(k\)-th derivative into (2.11). ∎

Lemma 1.1 now gives a global coefficient with exactly the hypotheses of the operator estimate.

3. The right frequency cutoff gives an ordinary operator

Let \(q_e\) be the extension just constructed. Smoothly regularize it at low transverse frequency and choose a transverse order-one quantization \(Q(t)\). Its principal symbol is \(q_e\); a complex order-zero term is allowed. Put

The full operator theorem gives

The full left symbol \(q_l(t,x,\eta)\) of \(Q(t)\) is uniformly transverse \(S^1_{1,0}\), with all time derivatives bounded in that class. Its frequency scale is \(\langle\eta\rangle\). To obtain the ordinary full scale \(\langle(\tau,\eta)\rangle\), choose \(b(\tau,\eta)\in S^0_{1,0}\), smooth and homogeneous of degree zero at high frequency, such that

It vanishes near the origin. Such a cutoff is obtained from a smooth angular cutoff and a radial cutoff. Write

Lemma 3.1. The operator \(\mathcal G\) is an ordinary full order-one operator. Its exact left symbol is

Its principal symbol agrees with (2.5) near the marked covector.

Proof. In left quantization a Fourier multiplier on the right multiplies the symbol exactly. This gives (3.5), with no composition remainder.

On the high-frequency support of \(b\), including the supports of its derivatives,

A term in a full frequency derivative of (3.5) differentiates \(b\), \(q_l\), or the linear factor \(\tau\). The bounds for those factors and (3.6) give

At bounded frequency all derivatives are uniformly bounded. This proves the ordinary symbol assertion. The principal part of \(q_l\) is \(q_e\), which equals \(q\) on the smaller germ, and \(b=1\) there at high frequency. Thus the principal symbol agrees with (2.5). ∎

Plancherel and the same frequency support give

The last step applies (3.2) to \(\mathcal Bu\), which is Schwartz, and uses the boundedness of the Fourier multiplier.

We can replace \(\mathcal B,\mathcal G\) by proper operators \(\mathcal B_p,\mathcal G_p\) with the same respective principal symbols. Here the ordinary proper kernel construction has bounded errors in the required global norms. To see this explicitly, multiply each kernel by a fixed smooth function of the difference of its base variables, equal to one near the diagonal and compactly supported. Outside that diagonal neighborhood, repeated frequency integration by parts gives, for every fixed base derivative and every \(N\),

Choose enough frequency derivatives to make the differentiated symbol integrable: its order is at most \(1+|\alpha|-M\), so \(M>n+1+|\alpha|\) suffices. Further integrations give the requested decay power. All symbol bounds are uniform in the base variables. The integral kernel bound with \(N>n\) is uniform on \(L^2\). Applying it to every required base derivative shows that the off-diagonal error sends \(L^2\) to each fixed integer Sobolev space, hence to \(H^a\). Consequently (3.8) gives

The proper cutoff \(\mathcal B_p\) is elliptic at the marked covector. If \(\mathcal G_p\) has a lower symbol without a homogeneous expansion, choose a proper classical representative \(\mathcal G_c\) of its principal symbol. Their difference has ordinary order zero and is \(L^2\) bounded. Equation (3.10) holds with \(\mathcal G_c\) in place of \(\mathcal G_p\).

The full estimate-to-regularity theorem in the conic-gain prerequisite now gives loss

for \(\mathcal G_c\) at the marked covector, for every distribution and every real datum Sobolev index.

4. Return to the original operator and retain the sharp loss

Theorem 4.1. Let \(P\) be a properly supported ordinary scalar pseudodifferential operator of real order \(m\), with a smooth degree-\(m\) homogeneous principal representative modulo order \(m-1\). Suppose that representative satisfies \((P)\) on a conic neighborhood of a nonzero characteristic covector \(c\), and \(\kappa(c)=k<\infty\). Then \(k\) is even and at least two. For every \(s\in\mathbb R\) and every distribution,

The loss \(k/(k+1)\) is optimal. Every larger loss strictly below one also holds.

Proof. First take \(m=1\). Lemma 2.2 gives the exact homogeneous normalization. Proposition 2.3 and Corollary 2.4 give its nonnegative coefficient and positive pure time jet. Lemma 1.1 extends it with uniform finite time type. Section 3 proves a full conic estimate for an ordinary proper operator with the normalized principal symbol.

Principal-symbol invariance from the conic-gain prerequisite removes arbitrary complex order-zero differences. Its weak-test proof uses only ordinary mapping orders: an order-zero difference sends the lower \(H^s\) input of the order-one weak test to the \(H^s\) datum space. It therefore also handles a remainder with no homogeneous expansion. The weak-test equivalence then reaches every distribution. Homogeneous graph transport carries the loss back through the exact canonical map, and elliptic multiplier invariance removes the normalization factor.

For general \(m\), choose a positive elliptic degree-one weight \(\rho\). Multiply \(P\) by a proper elliptic operator with principal symbol \(\rho^{1-m}\). Its principal degree is one; bracket depth and \((P)\) are unchanged by the nonzero multiplier. Its datum index is

The degree-one conclusion adds \(1/(k+1)\) to that index, giving exactly (4.1). Equivalently the real-order multiplier theorem preserves the loss.

The necessary finite-type bound from the crossing-direction prerequisite says that a strict loss \(\delta\) at this point must satisfy

This proves optimality. A larger strict loss asks for a lower output Sobolev index, so it follows by Sobolev inclusion. ∎

At a noncharacteristic covector, the ordinary conic inverse instead gives the elliptic gain \(m\). At a characteristic point the sign condition alone does not give a finite gain: finite bracket depth is a separate hypothesis.

5. Exercises with complete solutions

Exercise 1 — a nonflat cutoff contact, 8 points. On a small time interval let \(q/r=t^6\) and \(\theta=1-t^4\). Calculate the first nonzero jet of the positive extension (1.3) at zero. Explain why differentiating the cutoff cannot destroy all jets through order six.

Solution. The interval is small enough that \(0\leq\theta\leq1\). Here

Its first nonzero derivative is the fourth, equal to \(4!=24\). The original sixth coefficient remains positive too. More generally, at a zero with \(\theta=1\), the first nonzero coefficient of each nonnegative summand is positive and even. The earlier coefficient survives, or equal-degree coefficients add, exactly as in Lemma 1.1. A cutoff equal to one only at the marked point still has this property.

Exercise 2 — a moving zero set, 10 points. Let \(h(x,\omega),v(x,\omega)\) be real smooth functions near a marked parameter, with both equal to zero there. On a product neighborhood put

Find the pure time jet, prove the hypotheses of Lemma 1.1, and determine the bracket depth of \(\tau+iq\) at the marked covector.

Solution. The coefficient is nonnegative and homogeneous of degree one. Its fourth time derivative is exactly \(24|\eta|\), independently of the parameters. A compactly supported angular and base cutoff therefore gives a global extension with (1.4), with \(k=4\). The local zeros are precisely \(t=h\) and \(v=0\); they need not form a smooth submanifold.

For the bracket depth use the one-direction weighted theorem. Give \(t\) weight one, \(\tau\) weight four and every centered transverse coordinate weight \(5/2\). Since \(h,v\) vanish at the marked parameter, \(h\) has weight at least \(5/2\) and \(v^2\) has weight at least five. The weight-four imaginary part is \(|\eta_0|t^4\); variations of \(|\eta|\) contribute higher weight. The leading-bracket theorem makes every word through four leaves vanish, and

Thus \(\kappa=4\). The gain of an order-one operator is \(1/5\), and its optimal loss is \(4/5\).

Exercise 3 — all complex multipliers, 12 points. Prove that every symbol \(p=f+iq\), with \(f,q\) real smooth and \(q\geq0\) on the whole working neighborhood, satisfies condition \((P)\). Apply this to \(p=\tau+i\eta t^6\) on \(\eta>0\), and determine its type and optimal loss at \(t=\tau=0\).

Solution. Let \(a=A+iC\) be any smooth nonzero multiplier, and along a nonconstant characteristic curve put

The algebraic identity

holds on that curve, even when \(A=0\). If \(G=0\), then \(ap=0\), so \(f=q=0\). Nonnegativity gives \(dq=0\) there. Hence \(H_F=A H_f\) at that point. If also \(A=0\), the Hamilton field vanishes, and uniqueness makes the entire curve constant, a contradiction.

Thus \(A,G\) never vanish together on a nonconstant curve. By (5.2), \(A+G\) cannot be zero: its zero with \(AG\geq0\) would force both to be zero. Its sign is consequently constant on the connected time interval. Whenever \(G\ne0\), (5.2) makes its sign agree with that of \(A+G\). Therefore \(G\) has no change of sign in either direction. This proves the full condition for arbitrary variable phases and moduli.

For the model, every nontrivial bracket is a time derivative of \(\eta t^6\). Brackets between such derivatives are zero because they are independent of the conjugate transverse position. All values through six leaves vanish, and

Thus \(k=6\), the gain is \(1/7\), and the optimal loss is \(6/7\).

Exercise 4 — reflect time and track a real order, 8 points. Start with \(p=\tau-i\eta t^8\) on \(\eta>0\). Carry out the sign choice following Proposition 2.3. If the original operator has order \(m=7/3\) and its datum is \(H^{-2}\), determine the final solution index.

Solution. The map \((t,\tau)\mapsto(-t,-\tau)\) preserves \(d\tau\wedge dt\). After substitution the symbol is \(-\tau-i\eta t^8\); multiplication by \(-1\) gives \(\tau+i\eta t^8\). The eighth derivative is \(8!\eta>0\). These operations preserve the loss and do not change \(D=-i\partial\).

The optimal loss is \(8/9\). Order reduction changes the datum index to \(-2+7/3-1=-2/3\). The normalized gain is \(1/9\), giving

This is also \(-2+7/3-8/9\), as required by (4.1).

Exercise 5 — compare the two frequency scales, 10 points. In one transverse dimension, show that \(\langle\eta\rangle\), viewed as a function of \((\tau,\eta)\), is not an ordinary full order-one symbol. Prove that \(b(\tau,\eta)\langle\eta\rangle\), with \(b\) as in (3.3), is such a symbol at every derivative order.

Solution. At \(\eta=0\),

The ordinary full order-one bound for that derivative would be \(C\langle(\tau,0)\rangle^{-1}\), which tends to zero as \(|\tau|\to\infty\). No fixed \(C\) works.

After multiplication by \(b\), the high-frequency supports of \(b\) and its derivatives satisfy (3.6). In each Leibniz term, a total of \(j\) frequency derivatives contributes the factor \(\langle(\tau,\eta)\rangle^{-j}\): derivatives of \(b\) use the full scale, and derivatives of \(\langle\eta\rangle\) use a comparable scale. The result has size at most \(C_j\langle(\tau,\eta)\rangle^{1-j}\). This proves all orders; bounded frequencies are smooth. It also explains why the exact product \(L\mathcal B\) in (3.4) is the appropriate ordinary operator.

Exercise 6 — one orientation allows odd type, 10 points. On \(\eta>0\), compare \(p=\tau+i\eta t^5\) and its conjugate. Determine their time-line orientations and explain why neither satisfies \((P)\). Does Theorem 4.1 decide the analytic sufficiency of the favorable odd model?

Solution. On the increasing time line, \(\eta t^5\) goes from negative to positive, so \(p\) has the favorable orientation \((\overline\Psi)\). The conjugate has the positive-to-negative crossing, with standard \((\Psi)\) orientation. The complete multiplier assertions for these orientations are given by the even-factor comparison theorem in the mixed-polynomial prerequisite, with \(s=0\) and monotone quotient \(t^5\eta\).

Each symbol changes sign on the line for the multiplier \(a=1\), so each fails \((P)\). Both have bracket depth five. Theorem 4.1 requires both orientations and therefore does not apply. The favorable odd model has its separate gain \(1/6\), proved in Finite type and the sign of the symbol. General sufficiency with only the appropriate oriented neighborhood condition requires the more general analytic argument.

References

• [L] Nicolas Lerner, Semi-classical estimates for non-selfadjoint operators, Asian Journal of Mathematics 11 (2007), 217–250. Open author's manuscript. Finite-type estimates, condition \((P)\) and oriented symbol conditions. Lerner's Fourier convention uses \(2\pi\); ours is \(D=-i\partial\).
• [E] Lawrence C. Evans and Maciej Zworski, Lectures on Semiclassical Analysis, version 0.2. Open lecture notes. Symplectic transformations, symbol calculus and microlocal estimates.
• [H] Lars Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators, Springer, 2009 reprint. Publisher's record. Finite bracket type and subellipticity.

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