Microlocal analysis studies singularities using both position and frequency direction. It asks not only where a function or distribution fails to be smooth, but which directions of oscillation reveal that failure. This extra information helps explain wave propagation, elliptic regularity and why an imaging system may recover some edges more reliably than others.
The gold standard is to distinguish a genuine mathematical singularity from a large but smooth oscillation, and to specify the operator before predicting what happens to either. This guide develops the wavefront set through explicit examples, then uses symbols, transport equations and imaging geometry to show what the theory can establish and where its assumptions matter.
Use the Gold Standard master, Education parent and How X Works for the wider collection. The existing Distribution Theory guide supplies the foundation for impulses and weak derivatives. Spectral Geometry explores the complementary question of how global geometry affects modes and eigenvalues.
Choose your microlocal analysis route
Chapters 1–3: Locate singularities and test Fourier decay
Chapters 4–6: Calculate directions and symbols
Chapters 7–9: Use operators and follow fronts
Chapters 10–12: Check smoothing, products and imaging
Chapter index
Locate singularities and test Fourier decay · Chapters 1–3
1. Add direction to the location of a singularity
2. Use distributions without treating impulses as ordinary values
Calculate directions and symbols · Chapters 4–6
4. Calculate the wavefront of an impulse and a line
Use operators and follow fronts · Chapters 7–9
7. Understand pseudodifferential operators and parametrices
Check smoothing, products and imaging · Chapters 10–12
10. Separate propagation from smoothing
Transfer and continue · Chapters 13–14
CHAPTER 1 OF 14 · Locate singularities and test Fourier decay
1. Add direction to the location of a singularity
A map of singular points is useful but incomplete. Consider a sharp straight edge in a two-dimensional image. Crossing the edge changes the intensity abruptly, while moving along the edge may reveal no abrupt change. The location of the edge does not by itself record this directional distinction.
The singular support of a distribution is the set of points near which it cannot be represented by a smooth function. Here smooth means infinitely differentiable. This definition includes jumps, corners and impulses, but it also includes less visually obvious failures of differentiability. A graph can look almost flat while still failing the mathematical smoothness test.
The wavefront set refines singular support by attaching nonzero covectors to singular locations. In Euclidean coordinates, write a pair as (x,ξ): x is position and ξ is a frequency covector. Covectors are natural because phases such as x·ξ pair positions with frequencies, and phase differentials transform correctly when coordinates change.
What the extra information does
For a jump across a smooth curve, the singular covectors are normal to the curve. They are not the directions in which one walks along the curve. For a point impulse, every nonzero frequency direction is singular. Both objects are sharply localised, but their directional structures differ.
This distinction matters for an inverse problem. A measurement system might be sensitive to one family of normal directions and weakly sensitive to another. Calling the entire image either visible or invisible would throw away information that the acquisition geometry can actually provide.
The wavefront set is a qualitative description of lack of smoothness. It does not, by itself, supply edge contrast, jump size or a numerical error bar. Multiplying a nonzero singular distribution by a nonzero constant preserves its wavefront set while changing its amplitude. Later chapters will separate that qualitative information from quantitative regularity and measurement accuracy.
CHAPTER 2 OF 14 · Locate singularities and test Fourier decay
2. Use distributions without treating impulses as ordinary values
A distribution acts on smooth compactly supported test functions. The Dirac delta at the origin acts by δ₀(φ)=φ(0). This definition makes sense without pretending that delta is an ordinary function with an infinite value at one point and zero elsewhere.
For an ordinary locally integrable function u, the associated distribution acts by integration against φ. Distributional differentiation is defined by moving the derivative onto the test function with a minus sign. Thus the derivative of a one-dimensional step is a delta distribution, even though its classical derivative does not exist at the jump.
An explicit derivative check
Let H(x) equal zero for negative x and one for positive x. Its value at zero does not affect the associated distribution. For a test function φ, H′(φ)=−∫₀∞φ′(x)dx=φ(0), because φ vanishes outside a bounded interval. Therefore H′=δ₀.
Now let u(x)=3H(x−2). The same calculation gives u′=3δ₂. Its singular support is the single point x=2. The coefficient three changes the strength of the jump and its derivative, but not the position or the nonzero frequency directions of the singularity.
In two dimensions, H(x₁) describes a jump across the line x₁=0. Differentiating in x₁ produces a delta supported on that line. Differentiating in x₂ gives zero. This elementary calculation already predicts that the singular structure is normal to the line rather than tangent to it.
The Distribution Theory article develops these operations further. The essential discipline here is to state what action is defined. Differentiation always extends to distributions; unrestricted multiplication of two arbitrary distributions does not. A symbolic manipulation that is harmless for smooth functions can therefore require an additional theorem in the singular setting.
CHAPTER 3 OF 14 · Locate singularities and test Fourier decay
3. Localise before studying Fourier decay
Use the Fourier convention û(ξ)=∫exp(−ix·ξ)u(x)dx when the integral is meaningful, and extend it to appropriate distributions. A smooth compactly supported function has a Fourier transform that decreases faster than every inverse power of |ξ|. Repeated integration by parts explains the mechanism: derivatives transfer to the smooth amplitude while inverse powers of frequency appear.
The compact support condition removes boundary terms in that argument. An abrupt crop is different. Multiplying smooth data by a hard rectangular window introduces jumps at the window boundary, and those artificial jumps generate slow Fourier tails. A spectral plot can therefore describe the crop as well as the original object.
The local test
Choose a smooth cutoff χ supported near a point x₀ and equal to one in a smaller neighbourhood. Examine the Fourier transform of χu. To declare a direction ξ₀ regular, require rapid decay throughout an open cone around ξ₀: for every nonnegative integer N, a constant C_N bounds the transform by C_N(1+|ξ|)^−N within that cone.
The constants may depend on N and the cutoff. The condition is not one fitted slope on one finite plot. It demands all inverse-power decay estimates in a directional neighbourhood. A single ray on which cancellation happens accidentally is not enough to establish microlocal regularity.
If a suitable cutoff and cone exist, (x₀,ξ₀) is absent from the smooth wavefront set. Otherwise it belongs to the wavefront set. Zero frequency is excluded: the definition concerns high-frequency behaviour along directions, not what happens at the origin of frequency space.
Localisation prevents a singularity far away from being attributed to x₀. Directional restriction prevents singular behaviour in one orientation from being assigned indiscriminately to all orientations. Those two restrictions explain the name microlocal more concretely than the vague phrase “analysis on very small scales.” It is locality in position together with locality in frequency direction.
A windowing experiment with an exact answer
Take the perfectly smooth constant function one and observe it only on the interval [−a,a], using a hard cutoff. The recorded function is the indicator of that interval. Its Fourier transform is 2sin(aξ)/ξ for ξ≠0, with limiting value 2a at zero. The inverse-frequency envelope comes from the two artificial endpoint jumps, not from a defect in the original constant function.
There are many frequencies where the numerator vanishes. Sampling only those frequencies could suggest extremely fast decay, even though nearby frequencies retain the inverse-frequency envelope. This is a concrete reason to require estimates throughout a cone rather than on a conveniently chosen sequence of points. In one dimension a positive cone is simply the positive frequency half-line, and the same warning applies.
Replace the hard cutoff by a smooth compactly supported window. The product with the constant function is now smooth and compactly supported, so repeated integration by parts yields rapid decay. The experiment has changed the mathematical regularity of the recording boundary while leaving the central constant signal intact.
For sampled data, compare several windows and several sampling spacings. A feature that follows the crop boundary is suspect; a feature that persists at the same interior location deserves further investigation. Neither observation alone proves a wavefront-set theorem, but these checks prevent a routine processing artifact from being presented as an intrinsic singularity. Keep the unwindowed acquisition details as well as the processed spectrum, so another reader can distinguish the original signal from the operations used to inspect it.
CHAPTER 4 OF 14 · Calculate directions and symbols
4. Calculate the wavefront of an impulse and a line
For a point delta δ₀ in Euclidean space, multiplication by a cutoff χ with χ(0)=1 leaves δ₀ unchanged. Its Fourier transform is the constant one in the convention used here. It does not decay along any nonzero frequency direction. Thus the wavefront set consists of all pairs (0,ξ) with ξ≠0.
Away from the origin, choose a cutoff whose support misses zero. The localised distribution vanishes, so there is no wavefront there. This combines the two parts of the definition: localisation determines the position, and Fourier decay determines the directions above that position.
A line-supported example
In two dimensions, consider u=δ(x₁), constant in the x₂ variable. Localise with a product cutoff χ₁(x₁)χ₂(x₂), with χ₁(0)=1. Its Fourier transform is the one-dimensional Fourier transform of χ₂ evaluated at ξ₂ and is independent of ξ₁.
Along directions with a nonzero tangential frequency component ξ₂, the smooth cutoff produces rapid decay within a sufficiently narrow cone. Along the normal directions (ξ₁,0), with ξ₁ nonzero, that mechanism does not supply decay in the growing normal frequency. The wavefront is therefore normal to the line x₁=0.
For a jump H(x₁), the wavefront has the same normal directions along the same line, although the singularity is weaker than a delta layer. Differentiation in x₁ strengthens the singularity without changing its normal orientation. This shows why the smooth wavefront set does not encode the complete severity of a defect.
For a tilted line x₁+2x₂=0, a normal covector is (1,2). The nonzero multiples a(1,2), for a≠0, describe its conormal directions. A tangent vector is (2,−1), and their pairing is 1·2+2·(−1)=0. That zero pairing is the coordinate calculation behind the word normal in this example.
CHAPTER 5 OF 14 · Calculate directions and symbols
5. Distinguish high frequency from nonsmoothness
The function sin(1000x) is smooth. Its ordinary smooth wavefront set is empty, despite its rapid oscillation. Multiplying it by a smooth compactly supported cutoff still gives a smooth compactly supported function, whose Fourier transform eventually decreases rapidly. Large finite frequency is not the same condition as failure of smoothness.
The distinction becomes more subtle for a family u_h(x)=a(x)exp(iφ(x)/h) as h tends to zero. Each member may be smooth, while the family has increasingly rapid oscillation. Semiclassical analysis records the high-frequency behaviour of such families using a small parameter h and appropriately defined semiclassical wavefront sets.
A phase-gradient calculation
Take φ(x₁,x₂)=3x₁+4x₂. Its differential is the covector (3,4). The oscillatory factor exp(iφ/h) has ordinary frequency vector (3/h,4/h), while the corresponding semiclassically scaled covector is (3,4). Its Euclidean magnitude is five and its unit direction is (3/5,4/5).
Level sets of the phase satisfy 3x₁+4x₂=constant. Their tangent vectors pair to zero with (3,4). The phase gradient describes the orientation of oscillation across those level sets, rather than motion along them. A propagation law is still needed before that information can be converted into a trajectory through space.
Changing h from 0.1 to 0.05 doubles the ordinary frequency magnitude from 50 to 100. It does not change the scaled direction or make either fixed function nonsmooth. Mixing the classical and semiclassical meanings of wavefront would turn this straightforward observation into a false contradiction.
In a numerical discussion, state whether the object is one distribution, a parameter-dependent family or sampled data. A finite image cannot literally display every asymptotic frequency. It can provide evidence about a model, but a finite Fourier plot does not prove the full wavefront-set condition. This limitation matters whenever sharp-looking oscillations are described as mathematical singularities.
CHAPTER 6 OF 14 · Calculate directions and symbols
6. Read a differential operator through its symbol
Set D_j=(1/i)∂/∂x_j. Acting on exp(ix·ξ), D_j returns ξ_j times the same exponential. A differential operator expressed in the D_j variables therefore has a polynomial symbol in ξ. Its principal symbol keeps the terms of highest differential order and describes the leading high-frequency behaviour.
For P=−Δ, the principal symbol is |ξ|². It is nonzero for every nonzero real ξ, which is the basic ellipticity condition in this scalar example. For P=D₁, the symbol is ξ₁ and vanishes when ξ₁=0. The operator therefore behaves differently in different frequency directions.
A directional test with derivatives
Consider u=H(x₂). Its singular covectors are nonzero multiples of dx₂. Applying D₁ gives zero, so D₁u is smooth even though u is not. There is no conflict with elliptic regularity: the principal symbol ξ₁ vanishes precisely at those singular covectors.
Applying D₂ instead differentiates across the jump and produces a nonzero multiple of δ(x₂). The operator detects the singularity because it acts in the normal direction. This simple comparison shows why one cannot conclude that u is smooth from the smoothness of Pu without examining the symbol.
For P=D₁²+4D₂², the symbol is ξ₁²+4ξ₂². It is positive at every nonzero real covector, so the operator is elliptic. By contrast, D₁²−D₂² has symbol ξ₁²−ξ₂², which vanishes along ξ₁=±ξ₂. Those characteristic directions allow a different kind of singular behaviour.
A sign convention changes formulas but not the underlying need to check them consistently. If a source uses ordinary derivatives instead of D_j, factors of i and minus signs appear. Derive the symbol by acting on an exponential once, rather than relying on memory. Lower-order terms also matter for many quantitative questions, even when they do not change the principal characteristic set.
CHAPTER 7 OF 14 · Use operators and follow fronts
7. Understand pseudodifferential operators and parametrices
Differential symbols are polynomials in frequency. Pseudodifferential operators extend this construction to suitable smooth symbols with controlled growth and derivative estimates. A symbol may depend on both x and ξ, allowing an operator to respond differently at different locations and frequency directions.
For a symbol depending only on ξ, the construction is a Fourier multiplier: transform, multiply by a(ξ), then inverse transform. The multiplier (1+|ξ|²)^−1, for example, is the inverse of 1−Δ on the usual whole-space Fourier model. It suppresses high frequencies by roughly two powers.
What two powers of improvement mean
At frequency magnitude ten, this multiplier is 1/101. At magnitude one hundred, it is 1/10001. The suppression increases with frequency, but it is polynomial rather than faster than every power. The operator improves Sobolev regularity by two orders in the standard setting; it is not an infinitely smoothing operator.
This distinction prevents a common mistake. “Smoothing” in casual language may mean reducing roughness, whereas a smoothing operator in the classical calculus has a smooth kernel and removes the relevant distributional singularities. Finite regularity gain need not make an initially singular distribution infinitely differentiable.
An elliptic parametrix is an inverse modulo a smoothing error in the relevant setting. Schematically QP=I+R, with R smoothing, and similarly on the other side where appropriate. Applying this identity to Pu=f expresses u in terms of Qf and a smooth remainder. That is the mechanism connecting ellipticity with regularity.
The parametrix is not automatically a global exact inverse. Kernels, compatibility conditions, boundaries and behaviour at infinity can obstruct global solvability. Its purpose here is to recover local high-frequency information. The fact that local singularities can be controlled does not imply that every forcing term has a unique global solution under every boundary condition.
CHAPTER 8 OF 14 · Use operators and follow fronts
8. Apply elliptic regularity with the correct scope
For a suitable differential or pseudodifferential operator P, microlocal elliptic regularity says that singularities of u away from the characteristic set must be accounted for by singularities of Pu. A useful inclusion is WF(u)⊆WF(Pu)∪Char(P). Where P is elliptic, u cannot hide a singularity that disappears from Pu.
The complementary pseudolocal principle says that these operators do not create new smooth-wavefront singularities at unrelated position-frequency pairs, under the usual support and symbol hypotheses. Together, the two statements explain why elliptic operators preserve the location and directions of smooth singularities while changing their strength.
A one-dimensional fundamental solution
Let u(x)=exp(−|x|)/2. Away from zero, u″=u. Its first derivative jumps from 1/2 on the left of zero to −1/2 on the right, a jump of −1. Distributionally, u″=u−δ₀. Consequently (1−d²/dx²)u=δ₀.
The solution is continuous but has a corner at zero. The forcing is a point impulse. The elliptic inverse has improved regularity, yet the singular location remains. This explicit calculation gives a concrete counterexample to the claim that every elliptic inverse turns arbitrary forcing into a smooth function.
If the forcing is smooth on an open interval, elliptic regularity makes the solution smooth there. It is a local statement: a singular source elsewhere does not automatically destroy smoothness throughout the domain. At boundaries, the domain and boundary conditions must also enter the analysis; an interior theorem alone does not settle boundary regularity.
When reading a numerical output, ask whether the apparent roughness occurs where the forcing is singular, where the coefficients lack the assumed smoothness, or near a boundary that has not been treated. Each possibility calls for a different repair. Invoking the word elliptic without checking those conditions is not enough to diagnose an error.
CHAPTER 9 OF 14 · Use operators and follow fronts
9. Follow a transported jump exactly
The transport equation u_t+c u_x=0 with constant c has solutions u(t,x)=f(x−ct). The initial profile moves at speed c without changing shape. If f has a jump, that jump travels along a straight line. This gives an explicit model of propagation of singularities without requiring the full general theory.
Take f(x)=H(x−x₀). Then u(t,x)=H(x−ct−x₀). The singular line is x=ct+x₀. Its conormal covectors in coordinates (t,x) are nonzero multiples of (−c,1), because the defining phase is x−ct−x₀.
Original trajectory calculation
Set c=2 and x₀=−3 in consistent units. At time t=4, the jump is at x=5. A normal covector is (−2,1), whereas a tangent vector to the trajectory is (1,2). Their pairing is −2·1+1·2=0. The covector and trajectory direction have distinct roles.
Using P=D_t+cD_x, the principal symbol is p=τ+cξ. Substituting the conormal pair (τ,ξ)=a(−c,1) gives p=0. Thus the jump lies in characteristic directions, exactly where elliptic regularity does not force it to disappear.
Hamilton’s equations for this symbol give dt/ds=1, dx/ds=c and constant τ and ξ. Their projection reproduces the same straight trajectory. This agreement links the elementary solution formula to the phase-space language used in more general propagation theorems.
If c changes to −2, the profile moves in the opposite direction and the conormal becomes proportional to (2,1). A diagram remembered from the positive-speed case is insufficient. Recalculate the phase and symbol. The simple sign check is especially valuable when different authors use opposite Fourier conventions or write the transport equation with a minus sign.
CHAPTER 10 OF 14 · Check smoothing, products and imaging
10. Separate propagation from smoothing
Wave equations and heat equations treat high frequencies differently. For the constant-speed one-dimensional wave equation, a travelling profile f(x−ct) is a solution in the distributional sense. A sharp front can persist and move. For the heat equation on the whole line, the positive-time heat kernel spreads a point impulse into a smooth Gaussian.
For diffusivity κ>0, the heat kernel is G(t,x)=(4πκt)^−1/2 exp(−x²/(4κt)) for t>0. Its Fourier multiplier is exp(−κtξ²), which decays faster than every inverse power for fixed positive time. This contrasts with the polynomial multiplier of an elliptic inverse in Chapter 7.
Original spreading calculation
The Gaussian has variance 2κt. With κ=0.5 and t=4, its standard deviation is two in the chosen length units. At t=1, the standard deviation is one. Four times as much time doubles the characteristic width, rather than quadrupling it.
At t=4 and κ=0.5, the kernel at x=2 is exp(−1/2) times its value at the origin, approximately 0.607 of the peak. This ratio follows from the exponent and does not depend on the normalising prefactor. It provides a straightforward check of a plotted heat profile.
The initial delta has nonempty wavefront set, but each fixed positive-time Gaussian is smooth. This does not contradict propagation results for real-principal-type operators: the hypotheses and mechanisms differ. One should not apply a wave-propagation slogan indiscriminately to every evolution equation.
General propagation theorems track singularities along bicharacteristics under precise assumptions on the principal symbol and forcing. Boundaries, glancing rays, rough coefficients and degenerate symbols need additional analysis. The constant-coefficient examples are reliable starting points, not permission to erase those complications in a variable medium or a bounded physical device.
CHAPTER 11 OF 14 · Check smoothing, products and imaging
11. Check products and restrictions before using them
Multiplication of arbitrary distributions is not generally defined. Wavefront information supplies a useful sufficient criterion: the product uv is canonically defined when no singular covector of u at a point is the negative of a singular covector of v at that same point. This prevents a problematic cancellation of high-frequency directions.
The criterion is sufficient, not necessary. If it fails, one cannot conclude that every possible product is meaningless. Ordinary bounded functions may still have a pointwise product. What fails is the ability to justify the operation solely through this general wavefront criterion.
Transverse and coincident singularities
In the plane, δ(x₁) and δ(x₂) have normal directions along different coordinate axes. A nonzero covector on one axis cannot be the negative of a covector on the other. Their product is the point delta δ(x₁)δ(x₂), acting on a test function by evaluation at the origin.
By contrast, trying to square the one-dimensional δ₀ encounters opposite covectors at the same point. The usual distribution theory supplies no canonical δ₀². An approximation by narrow smooth peaks does not automatically fix the problem: its square can diverge or depend on how the peaks are regularised.
The step function H provides the warning about necessity. Its wavefront contains both nonzero directions at the jump, so the criterion does not justify H·H. Nevertheless, as a locally bounded function, H²=H almost everywhere and defines a perfectly meaningful distribution. Different justifications answer different questions.
Restrictions to submanifolds require similar care. A transversality condition between the distribution’s wavefront and the submanifold’s normal covectors provides a standard route to a well-defined pullback. Evaluating a distribution “on a line” is not always as innocent as substituting coordinates into a smooth formula. Before restricting, identify which singular directions the chosen line encounters.
CHAPTER 12 OF 14 · Check smoothing, products and imaging
12. Explain directional visibility in imaging
Microlocal analysis is useful in imaging because an edge has both a location and a normal direction. An acquisition system does not necessarily measure every such pair equally well. The forward operator and its associated geometry determine how object singularities appear in the data and which of them can be recovered stably by a reconstruction procedure.
For the ideal two-dimensional Radon transform, define Rf(θ,s) as the integral of f over the line x·θ=s, where θ is a unit normal. The one-dimensional Fourier transform in s samples the two-dimensional Fourier transform of f along ξ=σθ. This Fourier-slice relation makes the role of angle explicit.
A limited-angle example
Suppose measured normals θ range only between −20 and 20 degrees relative to the positive x₁ direction. Since σ can be positive or negative, the sampled frequency lines include the opposite directions as well. Normals near the x₁ axis are represented; normals near the x₂ axis are not directly represented by that angular range.
A vertical edge x₁=constant has horizontal normal and is favourably oriented for those measurements. A horizontal edge x₂=constant has vertical normal and is not covered in the same way. This statement uses the convention that θ labels line normals. If an instrument reports line directions instead, the angles must be rotated by ninety degrees before applying the interpretation.
Missing angular information is not restored merely by drawing a visually plausible continuation. Priors or regularisation may supply a useful estimate, but the result should be distinguished from information stably determined by the measured data. In some ideal classes uniqueness and stability are separate questions; “not directly visible” need not mean a universal proof of nonuniqueness.
MIT’s microlocal imaging notes develop how imaging operators map singularities. For a real system, add its aperture, frequency band, noise and modelling assumptions. A qualitative visibility statement cannot replace a quantitative uncertainty estimate for a reconstructed feature.
Consider the tilted jump u(x₁,x₂)=H(x₁+2x₂). Its singular support is the line x₁+2x₂=0, and its wavefront directions there are nonzero multiples of (1,2). A colleague proposes applying the tangential derivative T=2∂₁−∂₂ to make the edge more visible.
Differentiate distributionally: ∂₁u=δ(x₁+2x₂) and ∂₂u=2δ(x₁+2x₂). Therefore Tu=2δ−2δ=0. The derivative misses the edge because its vector is tangent to the singular line. The output is smooth while the input is singular, consistent with the symbol vanishing on the conormal directions.
Repair the chosen operation
Use the normal derivative N=∂₁+2∂₂ instead. Then Nu=δ+4δ=5δ(x₁+2x₂). This operation detects and strengthens the singularity. The factor five is the squared Euclidean length of the normal vector (1,2), given this unnormalised derivative convention.
If a unit normal derivative were used, divide N by √5. Its result would be √5δ(x₁+2x₂), not five times the delta. Normalisation affects amplitude even when the wavefront directions stay the same. Record the convention before comparing two computations.
Now replace H with sin(50(x₁+2x₂)). The same directional differentiation identities remain useful, but the function is smooth and has empty ordinary wavefront set. It is incorrect to carry the jump’s wavefront claim over just because the level sets have the same orientation.
Finally, blur the jump with a smooth Gaussian of fixed positive width. The resulting function is smooth. A steep gradient may remain, and recovering the original edge from noisy blurred data can be unstable, but steepness does not reintroduce an ordinary singularity. These three variations test separate skills: identify the conormal, distinguish strength from direction, and distinguish a singular object from a smooth approximation.
CHAPTER 14 OF 14 · Transfer and continue
14. Build a dependable learning and checking route
Does the wavefront set tell me how large an edge is?
No. It records where smoothness fails and in which frequency directions. Nonzero amplitude scaling leaves it unchanged. Sobolev wavefront sets and related quantitative estimates can refine regularity information, while measurement uncertainty and amplitude require further data and analysis.
Is every oscillation a singularity?
No. A smooth sinusoid has empty ordinary smooth wavefront set, regardless of its fixed frequency. A family whose frequency grows as a parameter tends to zero belongs to a different, semiclassical question. State which object and which notion of wavefront are being used.
Does ellipticity guarantee a unique solution?
No. Interior microlocal regularity and global solvability are different issues. An elliptic parametrix controls singularities modulo smooth errors, but kernels, boundary conditions and global compatibility can still matter. A successful local regularity argument is not an automatic existence-and-uniqueness theorem.
What should I learn first?
Learn Fourier transforms, integration by parts and weak derivatives. Verify the step-to-delta calculation, then compute localised Fourier transforms for a point impulse and a straight edge. Next study symbol estimates and elliptic parametrices before tackling general propagation theorems. The explicit transport solution is a useful bridge between elementary PDE and Hamiltonian language.
For primary mathematical reading, consult Richard Melrose’s pseudodifferential-operator chapter, Shiqi Ma’s microlocal-analysis notes and András Vasy’s overview of microlocal applications. These sources have different scopes; an introductory example should not be mistaken for the full assumptions of an advanced theorem.
Keep a working record of the distribution, coordinate convention, cutoff, frequency cone, operator, principal symbol, characteristic set and proposed conclusion. For numerical data, add sampling, windowing, blur, noise and acquisition angles. A trustworthy argument makes clear which conclusion is exact, which is asymptotic and which is an empirical estimate.
Continue through Distribution Theory to strengthen the weak-derivative foundation, or Spectral Geometry to connect local operator behaviour with global modes. Return to the Gold Standard master when the next question belongs to a different subject.