Distributions enlarge functions so differentiation is always defined and limits can be differentiated without pointwise regularity. They are the natural language for weak PDE solutions, fundamental solutions, and singular sources.
Test functions and distributions
Definition. Let \(\Omega\subset\mathbb R^d\) be open. \(\mathcal D(\Omega)=C_c^\infty(\Omega)\) carries the topology in which \(\varphi_n\to0\) when all supports lie in one compact set and every derivative converges uniformly to zero. A distribution is a continuous linear functional \[ T:\mathcal D(\Omega)\to\mathbb R \quad\text{or }\mathbb C. \]
Every \(f\in L_{\mathrm{loc}}^1(\Omega)\) defines \[ \langle T_f,\varphi\rangle =\int_\Omega f(x)\varphi(x)\,\mathrm{d}x. \] The Dirac mass is \(\langle\delta_a,\varphi\rangle=\varphi(a)\) and is not represented by an ordinary locally integrable function.
Distributional derivatives
Definition. For a multi-index \(\alpha\), \[ \langle D^\alpha T,\varphi\rangle =(-1)^{|\alpha|} \langle T,D^\alpha\varphi\rangle. \] This definition is forced by integration by parts and has no boundary term because \(\varphi\) is compactly supported.
Example (jump derivative). For the Heaviside function \(H=\mathbf1_{(0,\infty)}\), \[ \langle H',\varphi\rangle =-\int_0^\infty\varphi'(x)\,\mathrm{d}x =\varphi(0) =\langle\delta_0,\varphi\rangle. \] Thus \(H'=\delta_0\) in distributions.
Convergence and differentiation
Proposition. If \(T_n\to T\) in \(\mathcal D'(\Omega)\), meaning \(\langle T_n,\varphi\rangle\to\langle T,\varphi\rangle\) for every test function, then \[ D^\alpha T_n\to D^\alpha T. \]
Proof. \[ \langle D^\alpha T_n,\varphi\rangle =(-1)^{|\alpha|}\langle T_n,D^\alpha\varphi\rangle \to (-1)^{|\alpha|}\langle T,D^\alpha\varphi\rangle. \]
This continuity is why weak limits pass through linear differential operators even when classical derivatives do not converge.
Convolution and mollification
Let \(\rho\in C_c^\infty\), \(\rho\ge0\), \(\int\rho=1\), and \(\rho_\varepsilon(x)=\varepsilon^{-d}\rho(x/\varepsilon)\). For \(T\in\mathcal D'\), \[ T*\rho_\varepsilon \] is smooth on the interior where convolution is defined, and \[ D^\alpha(T*\rho_\varepsilon) =(D^\alpha T)*\rho_\varepsilon =T*D^\alpha\rho_\varepsilon. \] Moreover \(T*\rho_\varepsilon\to T\) in distributions. For \(f\in L^p_{\mathrm loc}\), mollification also converges in local \(L^p\) for \(1\le p<\infty\).
Fundamental solutions
Definition. A fundamental solution of a constant-coefficient operator \(L\) is a distribution \(E\) satisfying \[ LE=\delta_0. \] Then \(u=E*f\) formally solves \(Lu=f\), because \[ L(E*f)=(LE)*f=\delta_0*f=f. \]
For \(d\ge3\), the Laplacian has \[ \Phi(x)=\frac1{(2-d)|S^{d-1}|}|x|^{2-d}, \qquad -\Delta\Phi=\delta_0 \] under the chosen sign convention. The singularity is exactly what produces the point source.
Weak derivatives and Sobolev regularity
A locally integrable function \(g\) is the weak derivative \(D^\alpha f\) if the distributional derivative of \(T_f\) is \(T_g\). Sobolev spaces require these weak derivatives to belong to \(L^p\), turning distribution theory into quantitative function-space regularity.