NOTES ON DISTRIBUTIONS MATH 565, FALL 2017 1. Introduction and examples The notion of a distribution (also called generalized functions) was introduced to make sense of operations such as the delta function, which in some sense we very much want to be a function, but we cannot apply the usual rules of integration and differentiation to it. n 1 Given a nonempty open set Ω ⊂ R , consider Cc (Ω) as the collection of smooth compactly supported functions φ :Ω ! C satisfying supp(φ) ⊂ Ω. In the theory of distributions, such functions are also called test functions and very often the notation D(Ω) is used alternatively, so 1 treat it as the same as Cc (Ω). It is not hard to check that D(Ω) is always vector space but it is not a trivial task to endow D(Ω) with a topology for which D(Ω) will be complete in the sense that every Cauchy sequence converges. It turns out this can be done by passing to the theory of topological vector spaces, vector spaces whose topology is not necessarily induced by a norm (or even a metric!). Nonetheless, good sense can be made of convergence in the following way. Definition 1.1. A sequence fφjg in D(Ω) is said to converge to a function φ 2 D(Ω) if there exists a compact set K ⊂ Ω such that supp(φj) ⊂ K for every j 2 N and α α α α (1.1) lim k@ φj − @ φk1 = lim sup j@ φj(x) − @ φ(x)j = 0; for every multi-index α: j!1 j!1 x2Ω Note that since supp(φj) ⊂ K for each j, we must have supp(φ) ⊂ K as well and hence we could replace supx2Ω by supx2K here. Definition 1.2. A distribution on Ω is a linear map u : D(Ω) ! C satisfying the following: for each compact K ⊂ Ω, there exists a constant CK and an integer NK ≥ 0 such that X α (1.2) jhu; φij ≤ CK k@ φk1: jα|≤NK The space of distributions over Ω is denoted by D0(Ω), it is a vector space under the natural addition and scalar multiplication operations 0 (1.3) hc1u1 + c2u2; φi := c1hu1; φi + c2hu2; φi uj 2 D (Ω); cj 2 C; j = 1; 2: Note that our definition of convergence means that u is sequentially continuous in that if φj ! φ in the sense of (1.1), then limj!1hu; φji = hu; φi: Since u is a linear map u : D(Ω) ! C, one might say that the notation for the action of u on a test function φ should be u(φ) rather than hu; φi, but there are advantages to viewing u as a \pairing" with test function. 1 0 0 Definition 1.3. A sequence fukgk=1 ⊂ D (Ω) converges to u in the topology of D (Ω) if lim huk; φi = hu; φi for every φ 2 D(Ω): k!1 Remark 1.4. Note that if φk or uk instead depends on a parameter p lying in a metric space with limit point p0, then we can define limp!p0 φp and limp!p0 up in the natural way by saying that these limits are φ and u respectively if and only if the limit of every sequence pk ! p as k ! 1 satisfies φpk ! φ and upk ! u respectively. 1 2 NOTES ON DISTRIBUTIONS 1.1. Examples. n Example 1.5. Given an open set Ω ⊂ R , define the locally integrable functions on Ω as those which are integrable over any compact set K ⊂ Ω: 1 1 n Lloc(Ω) := ff measurable on Ω: f1K 2 L (R ) for any compact K ⊂ Ωg: 1 R Given any f 2 Lloc(Ω), f defines a distribution via hf; φi := Ω f(x)φ(x)dx. To see that this operation satisfies (1.2), observe that if supp(φ) ⊂ K, then φ = φ1K and hence Z Z Z jhf; φij = f(x)1K (x)φ(x)dx = f(x)φ(x)dx ≤ jf(x)jdx kφk1 Ω K K R by H¨older'sinequality. Hence (1.2) is satisfied with NK = 0 and CK = K jf(x)jdx. The previous example is of fundamental importance in the theory of distributions. Frequently we will conflate the idea of a locally integrable function with the distribution it defines. n Example 1.6. Given any open Ω ⊂ R containing the origin, the Dirac distribution δ is defined by hδ; φi = φ(0). It is nearly trivial that jhδ; φij ≤ kφk1 and hence δ satisfies the inequality (1.2) with CK = 1 and NK = 0. R −n Moreover, if is integrable with (x)dx = a 2 C, define (x) = (x/). Treating as a distribution as in the previous example, the converge to aδ in the sense of distributions: Z Z Z −n lim h ; φi = lim (x/)φ(x)dx = lim (x)φ(x)dx = φ(0) (x)dx = haδ; φi: !0+ !0+ !0+ where the penultimate identity follows from a standard application of the dominated convergence theorem. In the special case a = 1, this rigorously establishes the common non-rigorous definition of the Dirac distribution as being a limit of functions which are increasingly peaked at the origin. Example 1.7. Let Ω = (0; 1) ⊂ R. Let u be the linear map on D(Ω) defined by 1 X hu; φi = nφ (1=n) : n=1 That u is indeed well-defined is implicit in the ensuing observation establishing (1.2) for this dis- tribution. Suppose K ⊂ (0; 1) is compact. Then d(K; (−∞; 0]) > 0, so there exists N 2 N large such that K ⊂ [1=N; 1) and hence for any φ supported in K, N N N X X X N(N + 1) jhu; φij = nφ (1=n) ≤ njφ (1=n) j ≤ kφk n = kφk : 1 2 1 n=1 n=1 1 N(N+1) Therefore (1.2) is satisfied with CK = 2 and NK = 0. Note that u does not define a distribution on R. n Example 1.8. Let Ω ⊂ R and let S be any parameterized k-surface with surface measure dS. Then Z hu; φi := φdS; S is easily verified to define a distribution. In particular, if supp(φ) ⊂ K, then the constants in (1.2) R 1 n−1 can be taken as NK = 0 and CK = K\S 1 dS. Cases of special interest will be when S = S is the unit sphere or more generally, S is quadratic surface. P1 (n) Exercise 1.9. Check that hu; φi := n=1 nφ (n) defines a distribution on R satisfying (1.2) for (n) any compact K ⊂ R (where φ is the n-th derivative of φ). 1 R There is a slight technicality here in that its not obvious that K\S 1 dS is always well-defined, but it is possible to enlarge K so that the integral is indeed well-defined. NOTES ON DISTRIBUTIONS 3 2. Operations on distributions 1 2.1. Multiplication by a smooth function. As we saw, any function f 2 Lloc(Ω) defines a distribution via Z hf; φi = f(x)φ(x) dx: If 2 C1(Ω), we of course define the product of and f as simply the function ( f)(x) = (x)f(x), its corresponding action on D(Ω) is then Z Z h f; φi = (x)f(x)φ(x) dx = f(x) (x)φ(x) dx = hf; φi: Indeed, the product φ defines a test function and supp( φ) ⊂ supp(φ), so the right hand side here is meaningful. Definition 2.1. Given u 2 D0(Ω) and 2 C1(Ω), the product u is defined to be the distribution h u; φi := hu; φi: Note that this does indeed define a distribution in that it satisfies (1.2): if K ⊂ Ω is compact, and C , N are the constants satisfied by u in (1.2), then sup sup j@α (x)j is finite, and K K jα|≤NK x2K hence for supp(φ) ⊂ K, the Liebniz rule implies the existence of a constant M depending on this quantity such that X α X α jh u; φij = jhu; φij ≤ CK k@ ( φ)k1 ≤ MCK k@ φk1: jα|≤NK jα|≤NK Hence u also satisfies (1.2), with the constant MCK replacing CK . k β 1 2.2. Differentiation of a distribution. Now suppose f 2 C (Ω) is such that @ f 2 Lloc(Ω) for every multi-index jβj ≤ k. Then we would have that for any fixed jβj ≤ k, @βf defines a distribution via Z h@βf; φi = @βf(x)φ(x) dx: However, an integration by parts shows that Z Z h@βf; φi = @βf(x)φ(x) dx = (−1)jβj f(x)@βφ(x) dx = (−1)jβjhf; @βφi: Indeed, @βφ defines a test function and hence we are led to make the following definition: Definition 2.2. Given u 2 D0(Ω) and a multi-index β, the partial derivative @βu is defined as h@βu; φi := (−1)jβjhu; @βφi: Once again there is the matter of checking that this operation actually defines a distribution on Ω, so suppose K ⊂ Ω is any compact subset and that CK ;NK are the constants satisfied by u in (1.2). Then β β X α β X α jh@ u; φij = jhu; @ φij ≤ CK k@ (@ φ)k1 ≤ CK k@ φk1: jα|≤NK jα|≤NK +jβj Example 2.3. Let H(x) be the Heaviside function on R ( 1; if x ≥ 0 H(x) = 0; if x < 0: 4 NOTES ON DISTRIBUTIONS R 1 R 1 Since H is locally integrable, hH; φi = −∞ H(x)φ(x)dx = 0 φ(x)dx defines a distribution on R.