Distributions and Microlocal Methods

Quantum fields and Green functions are often singular at coincident points. Distributions make such objects precise through their action on test functions, while wavefront sets record the directions of their singularities. This chapter explains which derivatives, Fourier transforms, products, and extensions are defined.

Enter this chapter

Begin with Test-Function Spaces, Distributions, Support, and Convergence to replace pointwise pictures by test-function pairings. The wavefront-set route is useful when products or restrictions to a submanifold become the central issue.

Topics

1. Test-Function Spaces, Distributions, Support, and Convergence — Defines generalized functions through continuous linear tests and their support.
2. Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards — Derives localized sources, weak derivatives, and permitted changes of variables.
3. Tempered Distributions and Fourier Calculus — Extends Fourier transformation to singular and polynomially growing objects.
4. Distributional Kernels and Distributions on Manifolds — Describes operators by kernels and formulates distributions independently of coordinates.
5. Products, Scaling Degree, and Extensions of Singular Distributions — Separates allowed products from extensions across coincidence singularities.
6. Singular Support and Wavefront Sets — Records singular positions and directions to test products and pullbacks.

How the ideas fit together

Differentiating a distribution is straightforward because the derivative can be transferred to a test function. Multiplication is different: two singular factors need not define a product. A wavefront set records both a singular point and its nonzero cotangent directions; a sufficient product criterion excludes opposite singular directions at the same point. Its failure does not prove that every possible product prescription is impossible, but it removes this canonical guarantee. See Brouder, Dang, and Hélein 2014, Theorem 13.

Extension is a separate problem. A distribution already defined away from coincidence may admit several extensions differing by terms supported at coincidence. Choosing one is additional information, not ordinary pointwise multiplication.

Review the chapter

Why does differentiating the step distribution give a delta rather than zero?

Solution

For a compactly supported smooth test function $f$, the derivative acts as $-\int_0^\infty f'(x)\,\mathrm dx=f(0)$. The jump survives integration by parts even though the ordinary derivative vanishes away from zero.

Where to go next

Continue to Differential Equations and Green Operators to solve equations with distributional sources.

References

• Christian Brouder, Nguyen Viet Dang, and Frédéric Hélein, “A Smooth Introduction to the Wavefront Set,” Journal of Physics A: Mathematical and Theoretical 47 (2014), 443001. DOI. Open PDF.