List of Distributions

Introduction

This article lists the distributions and the generalised functions the corpus introduces. A distribution is a continuous linear functional on a space of test functions, so that the objects of the calculus — differentiation, multiplication, convolution, Fourier transformation — are defined by duality, and a row below names one distribution, records the operations it supports and the sense in which it is a generalised function, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The objects are grouped by the layer that introduces them: the distribution spaces and their duality, the distinguished distributions — the delta, the Heaviside function, the principal value and the finite part — the operations and their limits, the fundamental solutions and kernels, the distributions supported on a surface, and the distributions of the $p$-adic, complex and hypercomplex grounds. The rows that record an operation which is not defined — the product of two distributions, the restriction without a transversality hypothesis, the Fourier transform of an untempered distribution — stand beside the distributions as non-examples.

The Distribution Spaces

Object	What it is dual to, and what that buys	Introduced in
a distribution $u \in \mathcal D'(\Omega)$	the test functions $\mathcal D(\Omega) = C_c^\infty(\Omega)$ under the LF topology; all derivatives and all multiplications by smooth functions are defined	Distributions and Fundamental Solutions
a distribution of finite order	the dual of $C^k_c$ for some $k$; the smallest $k$ is the order	Distributions and Fundamental Solutions
a compactly supported distribution $u \in \mathcal E'(\Omega)$	the dual of $C^\infty(\Omega)$; convolution with every distribution is defined	Distributions and Fundamental Solutions
a tempered distribution $u \in \mathcal S'(\mathbb R^n)$	the Schwartz class $\mathcal S(\mathbb R^n)$; the Fourier transform is defined	Distributions and Fundamental Solutions; Fourier Analysis on Euclidean Spaces
a distribution of function type	a locally integrable function $f$, acting by $\varphi \mapsto \int f\varphi$; the embedding of $L^1_{loc}$ into $\mathcal D'$	Distributions and Fundamental Solutions
a measure as a distribution	a Radon measure, acting by $\varphi \mapsto \int\varphi\,d\mu$; the continuous dual of $C_0$ is a subspace of $\mathcal D'$	Distributions and Fundamental Solutions; Locally Compact Groups and Haar Measure
a positive distribution	the distribution associated to the measure it defines by the Riesz representation theorem	Distributions and Fundamental Solutions
the Schwartz kernel $K \in \mathcal D'(Y \times X)$	the distribution representing a continuous operator $T : \mathcal D(X) \to \mathcal D'(Y)$; unique	The Schwartz Kernel Theorem

The Distinguished Distributions

Object	Its definition, and the operations it supports	Introduced in
the delta distribution $\delta$	$\langle\delta,\varphi\rangle = \varphi(0)$; differentiable, the convolution identity, Fourier transform $1$	Distributions and Fundamental Solutions
the translated delta $\delta_a$	$\langle\delta_a,\varphi\rangle = \varphi(a)$; the derivative of the Heaviside step at $a$	Distributions and Fundamental Solutions
the derivatives $\partial^\alpha\delta$	$\langle\partial^\alpha\delta,\varphi\rangle = (-1)^{\|\alpha\|}\partial^\alpha\varphi(0)$; the only distributions supported at a point are their finite combinations	Distributions and Fundamental Solutions
the Heaviside function $H$	$H(x) = \mathbf 1_{x>0}$ as a distribution; its derivative is $\delta$	Distributions and Fundamental Solutions
the principal value $\mathrm{pv}\,\frac1x$	$\langle \mathrm{pv}\,\frac1x,\varphi\rangle = \lim_{\epsilon\to0}\int_{\lvert x\rvert>\epsilon}\frac{\varphi(x)}{x}dx$; the distributional derivative of $\log\lvert x\rvert$, with Fourier transform $-\pi i\,\mathrm{sgn}$	Distributions and Fundamental Solutions; Fourier Analysis on Euclidean Spaces
the finite part $\mathrm{pf}\,\frac1{x^2}$	the Hadamard regularisation of the divergent integral; it supports a derivative and a convolution with the Hilbert kernel	Distributions and Fundamental Solutions
the delta distribution of a hypercomplex system	the identity of the convolution algebra of the split-complex, dual-number, quaternion or biquaternion plane	Split-Complex Harmonic Analysis; Dual-Numbers Harmonic Analysis; Quaternion Harmonic Analysis; Biquaternion Integration
the delta distribution on a locally compact group	the identity $\delta_e$ of the convolution algebra of measures, together with the Haar measure	Harmonic Analysis on Groups

Distributions Supported on a Surface

Object	Its definition, and the operations it supports	Introduced in
the surface delta $\delta_S$ (the single layer of unit density)	$\langle\delta_S,\psi\rangle = \int_S\psi\,dS$; equals $\lvert\nabla\varphi\rvert\,\delta\circ\varphi$ for a defining function $\varphi$ of $S$	Distributions on Surfaces, Layers, and Jump Conditions
the normal layer $n\,\delta_S$	the gradient of the Heaviside function of a defining function, $\nabla H(\varphi)$	Distributions on Surfaces, Layers, and Jump Conditions
the double layer $\partial_n\delta_S$	the normal derivative of the surface delta; a layer of order $1$, the density of a normal dipole sheet	Distributions on Surfaces, Layers, and Jump Conditions
the layer of order $k$, $\partial_n^{\,k}(g\delta_S)$	a distribution supported on $S$; every distribution supported on a hypersurface is locally a finite sum of layers	Distributions on Surfaces, Layers, and Jump Conditions
the singular part of the divergence, $(n\cdot[A])\,\delta_S$	the single layer produced by a discontinuous vector field; the surface charge density	Distributions on Surfaces, Layers, and Jump Conditions
the singular part of the rotor, $n\times[A]\,\delta_S$	the tangential single layer produced by a discontinuous vector field; the surface current	Distributions on Surfaces, Layers, and Jump Conditions
the jump formula, $\partial_j u = \{\partial_j u\} + [u]n_j\delta_S$	separates the classical derivative of a piecewise smooth function from the layer its jump creates	Distributions on Surfaces, Layers, and Jump Conditions

Operations and Their Limits

Object	The operation it supports, and the limit of that operation	Introduced in
differentiation of a distribution	$\langle u',\varphi\rangle = -\langle u,\varphi'\rangle$; defined for every distribution and of no higher order than the distribution	Distributions and Fundamental Solutions
multiplication by a smooth function	$\langle au,\varphi\rangle = \langle u,a\varphi\rangle$; defined for $a \in C^\infty$	Distributions and Fundamental Solutions
the product of two distributions	defined only when their wave front sets do not meet in a forbidden way; $\delta\cdot\delta$ and $\delta\cdot\mathrm{pv}\frac1x$ are not defined	Distributions and Fundamental Solutions; Microlocal Analysis
convolution with a test function	$u * \varphi$ is a smooth function, a mollification and approximation of $u$	Distributions and Fundamental Solutions
convolution of two distributions	defined when one has compact support; it is commutative only under that hypothesis	Distributions and Fundamental Solutions
the Fourier transform of a tempered distribution	$\langle\hat u,\varphi\rangle = \langle u,\hat\varphi\rangle$; exchanges differentiation and multiplication	Distributions and Fundamental Solutions; Fourier Analysis on Euclidean Spaces
restriction and pullback	defined under a transversality condition on the wave front set; the trace on a boundary needs it	Distributions and Fundamental Solutions; Microlocal Analysis; Distributions on Surfaces, Layers, and Jump Conditions
the wave front set $\operatorname{WF}(u)$	the refined singular support: the directions in which $u$ fails to be smooth; the product and the pullback are governed by it	Microlocal Analysis
the order and the singular support	the structure theorem: a distribution of finite order is a sum of derivatives of measures	Distributions and Fundamental Solutions

Fundamental Solutions, Parametrices and Kernels

Object	What it is, and the operations it supports	Introduced in
a fundamental solution $E$ of $P$	$P E = \delta$; it solves $Pu = f$ by $u = E * f$ when the convolution is defined	Distributions and Fundamental Solutions
the fundamental solution of the Laplacian	the Newtonian potential, $E = \lvert x\rvert^{2-n}/((2-n)\omega_n)$, for $n \geq 3$	Distributions and Fundamental Solutions; Potential Theory
the fundamental solution of the heat operator	the Gaussian kernel, giving the heat semigroup and the solution of the initial-value problem	Distributions and Fundamental Solutions; Semigroups and Evolution Equations
the fundamental solution of the Cauchy–Riemann and Dirac operators	the Cauchy kernel $1/(\pi z)$ and the Clifford kernel $x^{\natural}/\lvert x\rvert^{n+1}$	Regularity and the Cauchy–Riemann Operator; Dirac Differential Operators
a parametrix of an elliptic operator	a distribution $E$ with $PE = \delta + R$ for a smoothing remainder $R$; it gives elliptic regularity	Distributions and Fundamental Solutions; Pseudodifferential Operators
the Green function	the fundamental solution of a boundary-value problem, carrying the boundary conditions	Partial Differential Equations; Potential Theory
the smooth kernel	a kernel $K \in C^\infty(Y\times X)$; it defines a smoothing operator, and this is the converse direction of elliptic regularity	The Schwartz Kernel Theorem
the kernel of a pseudodifferential operator	a distribution on the diagonal whose singularities are encoded by the symbol	The Schwartz Kernel Theorem; Pseudodifferential Operators

Distributions on Other Grounds

Object	Its definition, and the operations it supports	Introduced in
a $p$-adic distribution	a continuous linear functional on $C(\mathbb Z_p,\mathbb Q_p)$; the Amice transform identifies it with a power series of bounded coefficients	p-adic Integration
a $p$-adic measure	a $p$-adic distribution of norm at most $1$; identified with $\mathbb Z_p[[T]]$, the Iwasawa algebra	p-adic Integration
the Bernoulli distribution	a $p$-adic distribution that is not a measure; its integral against a power of $x$ gives the Kubota–Leopoldt $p$-adic $L$-function	p-adic Integration
the distributional boundary value of a holomorphic function	the limit of $f(x+iy)$ as $y \to 0$; the Sokhotski–Plemelj formula for the jump	Complex Harmonic Analysis
the delta distribution on the dual numbers and the split complex plane	the identity for the convolution of the hypercomplex system	Dual-Numbers Harmonic Analysis; Split-Complex Harmonic Analysis

Operations That Are Not Defined

Object	The failure	Introduced in
the product $\delta\cdot\delta$ and $\delta\cdot\mathrm{pv}\frac1x$	no associative multiplication of distributions extends the product of functions; the wave front sets meet in a forbidden way	Distributions and Fundamental Solutions
"the delta function" as a function	there is no function $\delta$ with the defining property; the object exists only as a functional	Distributions and Fundamental Solutions
the derivative of a distribution times itself	not defined without a Hörmander product condition	Microlocal Analysis
the restriction of a distribution to a hypersurface	not defined without the transversality of the wave front set with the conormal	Microlocal Analysis
the Fourier transform of an arbitrary distribution	not defined: only the tempered distributions carry it, and the general distribution has no decay	Distributions and Fundamental Solutions
the convolution of two distributions without compact support	not defined in general; the supports must be compatible	Distributions and Fundamental Solutions
the integral of a distribution over an unbounded set	not defined: only the pairing with a test function is	Distributions and Fundamental Solutions

Summary

This list gathers the distributions of the corpus: the spaces $\mathcal D'$, $\mathcal E'$ and $\mathcal S'$ and the function-type, measure and positive distributions; the delta and its derivatives, the Heaviside function, the principal value and the finite part; the operations of differentiation, multiplication, convolution, Fourier transformation and restriction, with the limits each meets; the fundamental solutions, parametrices, Green functions and Schwartz kernels; and the $p$-adic, holomorphic and hypercomplex distributions. The closing table records the operations that the theory does not define.

Summary of Notation

The objects are named by their standard symbols; the tables use the following.

Symbol	Meaning
$\mathcal D$, $\mathcal E$, $\mathcal S$	test functions, smooth functions, Schwartz class
$\mathcal D'$, $\mathcal E'$, $\mathcal S'$	distributions, compactly supported and tempered distributions
$\langle u,\varphi\rangle$	the duality pairing
$\delta$, $\delta_a$, $\partial^\alpha\delta$	the delta distribution, its translate, its derivatives
$H$, $\mathrm{pv}\frac1x$, $\mathrm{pf}\frac1{x^2}$	Heaviside function, principal value, finite part
$\hat u$	the Fourier transform of a tempered distribution
$E$, $R$	a fundamental solution or parametrix, and the smoothing remainder
$\operatorname{WF}(u)$	the wave front set