Multiplication of distributions

multiplication of generalized functions

Let $\Omega$ be an open subset of ${\bf R} ^ { n }$. Following L. Schwartz [a7], a distribution, or generalized function, $u \in \mathcal{D} ^ { \prime } ( \Omega )$ can be multiplied by a smooth function $f \in C ^ { \infty } ( \Omega )$, the result being defined by its action on a test function $\varphi \in \mathcal D ( \Omega )$: $\langle f u , \varphi \rangle = \langle u , f \varphi \rangle$. The example of

\begin{equation*} 0 = ( \delta ( x ) x ) \operatorname { vp } \frac { 1 } { x } \neq \delta ( x ) \left( x \operatorname{vp} \frac { 1 } { x } \right) = \delta ( x ) \end{equation*}

shows that this product is not associative ($\delta ( x )$ denotes the Dirac measure, $\operatorname{vp} \frac { 1 } { x }$ the principal value distribution, cf. Generalized function; Generalized functions, product of). There are further limitations on defining products of distributions. Schwartz [a6] proved that whenever an associative differential algebra $( \mathcal{A} , \partial , \circ )$ contains $\mathcal{D} ^ { \prime } ( \Omega )$, the operations $( \partial , \circ )$ in $\mathcal{A}$ cannot simultaneously be faithful extensions of the distributional derivatives and the pointwise product of continuous functions. Thus, a multiplication of distributions can either be defined by imbedding the space of distributions into algebras, but giving up one or the other of the consistency properties above, or else can be defined only on subspaces of $\mathcal{D} ^ { \prime } ( \Omega )$ or for certain individual distributions.

The first approach is summarized under the heading generalized function algebras. By common usage of the term, "multiplication of distributions" refers to the second approach. Here again one may distinguish multiplier theory (multiplication as a continuous bilinear mapping on linear topological subspaces of $\mathcal{D} ^ { \prime } ( \Omega )$) and methods producing individual distributional products (without continuity at large of the operations).

Multiplier theory.

Typical examples are provided by the continuous multiplication mapping on the spaces of integration theory $( f , g ) \rightarrow f g : L ^ { p } ( \Omega ) \times L ^ { q } ( \Omega ) \rightarrow L ^ { 1 } ( \Omega )$, $1 / p + 1 / q = 1$, or the Sobolov spaces $H ^ { s } ( \Omega )$ (cf. also Sobolev classes (of functions)), which form an algebra when $s > n / 2$. By duality, a multiplication mapping $H ^ { s } ( \Omega ) \times H ^ { - s } ( \Omega ) \rightarrow H ^ { - s } ( \Omega )$ can be defined. For multiplier theory in Sobolev–Besov spaces, see [a8].

Another example arises from the convolution algebra $\mathcal{S} _ { \Gamma } ^ { \prime } ( \mathbf{R} ^ { n } )$ of tempered distributions with support in an acute cone $\Gamma \subset {\bf R} ^ { n }$. The inverse image of $\mathcal{S} _ { \Gamma } ^ { \prime } ( \mathbf{R} ^ { n } )$ under the Fourier transform $F$ is the algebra of retarded distributions, on which the product, defined by $u v = F ^ { - 1 } ( F u ^ { * } F v )$, is a sequentially continuous bilinear mapping.

Individual distributional products.

Product mappings will be defined on certain subsets $\mathcal{M} ( \Omega ) \subset \mathcal{D} ^ { \prime } ( \Omega ) \times \mathcal{D} ^ { \prime } ( \Omega )$ with values in $\mathcal{D} ^ { \prime } ( \Omega )$. The product will be bilinear, when applicable, commutative and partially associative: If $( u , v ) \in \mathcal{M} ( \Omega )$ and $f \in C ^ { \infty } ( \Omega )$, then both $( f u , v )$ and $( u , f v )$ belong to $\mathcal{M} ( \Omega )$ and $( f u ) v = u ( f v ) = f ( u v )$. With these properties, localization is possible, that is, the product mapping is uniquely defined by its restrictions to open neighbourhoods of points in $\Omega$. Equivalently, it suffices to define the products $( \varphi u ) ( \varphi v )$ for every $\varphi \in \mathcal D ( \Omega )$ to specify $uv$. The following definitions are instances of such products of increasing generality.

a) $\mathcal{M}_ { 1 } ( \mathbf{R} ^ { n } ) = \{$pairs of distributions with disjoint singular support. This is the localized version of the product of a distribution and a smooth function. Note that $( \delta ( x ) , \text { vp } 1 / x ) \notin \mathcal M _ { 1 } ( \mathbf R )$.

b) $\mathcal{M} _ { 2 } ( \mathbf{R} ^ { n } ) = \{$pairs of distributions $( u , v )$ such that the $\mathcal{S} ^ { \prime }$-convolution of $F ( \varphi u )$ and $F ( \varphi v )$ exists for all $\varphi \in {\cal D} ( {\bf R} ^ { n } ) \}$. The definition of the $\mathcal{S} ^ { \prime }$-convolution is a generalization of the convolution in $\mathcal{S} _ { \Gamma } ^ { \prime } ( \mathbf{R} ^ { n } )$ not requiring the support property, see [a3]. The product is defined locally by $( \varphi u ) ( \varphi v ) = F ^ { - 1 } ( F ( \varphi u ) ^ { * } F ( \varphi v ) )$. The product of retarded distributions is a special case, as is the wave front set criterion of L. Hörmander [a4] (cf. also Wave front): If for all $( x , \xi ) \in \mathbf{R} ^ { n } \times S ^ { n - 1 }$, $( x , \xi ) \in \operatorname {WF} ( v )$ implies $( x , - \xi ) \notin \operatorname{WF} ( u )$, then $( u , v )$ belongs to $\mathcal{M} _ { 2 } ( \mathbf{R} ^ { n } )$.

c) Regularization and passage to the limit. A strict delta-net is a net (cf. also Net (directed set)) of test functions $( \rho _ { \varepsilon } ) _ { \varepsilon > 0 } \subset \mathcal{D} ( \mathbf{R} ^ { n } )$ such that the supports of the functions $\rho _ { \varepsilon }$ shrink to $\{ 0 \}$ as $\varepsilon \rightarrow 0$, $\int \rho _ { \varepsilon } ( x ) d x = 1$ and $\int | \rho _ { \varepsilon } ( x ) | d x$ is bounded independently of $\varepsilon$. A model delta-net is a net of the form $\rho _ { \varepsilon } ( x ) = \varepsilon ^ { - n } \rho ( x / \varepsilon )$ with $\rho \in \mathcal{D} ( \mathbf{R} ^ { n } )$ fixed. Then

$\mathcal{M} _ { 3 } ( \mathbf{R} ^ { n } ) = \{$pairs of distributions $( u , v )$ such that $\operatorname { lim } _ { \varepsilon \rightarrow 0 } ( u ^ { * } \rho _ { \varepsilon } ) ( v ^ { * } \sigma _ { \varepsilon } )$ exists for all strict delta-nets $( \rho _ { \varepsilon } ) _ { \varepsilon > 0 }$ and $( \sigma _ { \varepsilon } ) _ { \varepsilon > 0 } \}$;

$\mathcal{M} _ { 4 } ( \mathbf{R} ^ { n } ) = \{$pairs of distributions $( u , v )$ such that $\operatorname { lim } _ { \varepsilon \rightarrow 0 } ( u ^ { * } \rho _ { \varepsilon } ) ( v ^ { * } \rho _ { \varepsilon } )$ exists for all model delta nets $( \rho _ { \varepsilon } ) _ { \varepsilon > 0 }$ and does not depend on the net chosen. The product of $u$ and $v$ is defined by the respective limit. Various other classes of delta nets are in use as well.

d) Harmonic regularization. Every distribution $u \in \mathcal{D} ^ { \prime } ( \mathbf{R} ^ { n } )$ can be represented as the boundary value as $\varepsilon \rightarrow 0$ of a harmonic function $u ( x , \varepsilon )$ in the variables $( x , \varepsilon ) \in \mathbf{R} ^ { n } \times ( 0 , \infty )$, obtained by convolution with the Poisson kernel (locally; cf. also Poisson integral). Then

$\mathcal{M} _ { 5 } ( \mathbf{R} ^ { n } ) = \{$pairs of distributions $( u , v )$ such that $\operatorname { lim } _ { \varepsilon \rightarrow 0 } u ( \, \cdot\, , \varepsilon ) v ( \, \cdot \, , \varepsilon )$ exists. The product by analytic regularization in dimension $n = 1$ is a special case.

It holds that ${\cal M} _ { i } ( {\bf R} ^ { n } ) \subset {\cal M} _ { i + 1 } ( {\bf R} ^ { n } )$ for all $i$, and the products coincide when they exist, see [a1], [a5]. Every inclusion is strict. The products defined in multiplier theory are special cases of $\mathcal{M} _ { 3 }$. A short review of further definitions, which may produce results not consistent with $\mathcal{M} _ { 5 }$, can be found in [a5].

The products $\mathcal{M} _ { 1 }$–$\mathcal{M} _ { 5 }$ can be used to define restrictions of distributions to submanifolds or to compute convolutions, for example. Generally (with exceptions), they cannot be used to define multiplications arising in non-linear partial differential equations because they are not stable with respect to perturbations, due to lack of continuity. In non-linear partial differential equations, either generalized function algebras or multiplier theory are applicable. A typical example for the latter is a conservation law like $\partial _ { t } u ( x , t ) + \partial _ { x } ( u ^ { m } ( x , t ) ) = 0$ where the multiplication is done in $L^{\infty}$ and the derivatives are computed in ${\cal D} ^ { \prime }$.

Related to multiplier theory, introduced to derive estimates in non-linear (pseudo-)differential equations, is the paraproduct of J.M. Bony [a2]. Given $v \in L ^ { \infty } ( \mathbf{R}^ { n } )$ with compact support, the paramultiplication by $v$ is a linear operator $T_\nu$ mapping the Sobolev space $H ^ { s } ( {\bf R} ^ { n } )$ into itself for any $s \in \mathbf{R}$. The paraproduct does not reproduce the pointwise product (when defined by multiplier theory, for example) but serves to control non-linear terms up to some more regular deviation. For example, if $u$, $v$ belong to $H ^ { s } ( {\bf R} ^ { n } )$ with $s > n / 2$, then $u v - ( T _ { u } v + T _ { v } u ) \in H ^ { r } ( \mathbf{R} ^ { n } )$ for every $r < 3 n / 2$.

See also Generalized function algebras.

References

[a1]	V. Boie, "Multiplication of distributions" Comment. Math. Univ. Carolinae , 39 (1998) pp. 309–321
[a2]	J.M. Bony, "Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires" Ann. Sci. École Norm. Sup. Sér. 4 , 14 (1981) pp. 209–246
[a3]	P. Dierolf, J. Voigt, "Convolution and $S ^ { \prime }$-convolution of distributions" Collect. Math. , 29 (1978) pp. 185–196
[a4]	L. Hörmander, "Fourier integral operators I" Acta Math. , 127 (1971) pp. 79–183
[a5]	M. Oberguggenberger, "Multiplication of distributions and applications to partial differential equations" , Longman (1992)
[a6]	L. Schwartz, "Sur l'impossibilité de la multiplication des distributions" C.R. Acad. Sci. Paris , 239 (1954) pp. 847–848
[a7]	L. Schwartz, "Théorie des distributions" , Hermann (1966) (Edition: nouvelle)
[a8]	H. Triebel, "Theory of function spaces" , Birkhäuser (1983)