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Additional info for Foundations of the Classical Theory of Partial Differential Equations
Sample text
2). 6. Multiplication of a Distribution by a Smooth Function. Linear Differential Operators in Spaces of Distributions (cf. Schwartz 195G1951; Shilov 1965; HSrmander 1983-1985). In accordance with the general principle stated at the beginning of Sect. 5, multiplication of a distribution f E D’(O) by a smooth function a E C-(0) is defined by the formula (af,d = (f,w), W-Q where cp E D(0). The question naturally arises: by which smooth functions a can any tempered distribution be multiplied without going outside the class of tempered distributions?
7) will be the same, so that, changing sign if necessary, we arrive at an equation whose principal part at the point 53. 1). For a hyperbolic equation the principal part at the point 2 in the canonical form will be as in the wave equation in IX”, and for a parabolic equation the principal part will become the Laplacian on n - 1 variables in Iw”. 7) in a whole region, as opposed to a single point, by the transformation just described, even if the equation is of constant type. 4). Under changes of variables all the invariants of the Riemannian metric (for example the sectional curvature) are preserved.
This is done by the standard methods of the theory of topological vector spaces. There are several ways of introducing a topology. The most important for us is the weak topology defined by the seminorms where E = D(n), E(n), or S(P), and E’ is the corresponding conjugate space of distributions. In the majority of cases one can use weak convergence, which we shall refer to as simply convergence, instead of this topology. Weak convergence is defined as follows: if (fk Ik = 1,2,3,. . } is a sequence of functionals of E’, we shall write that fk + f if (fk, ‘p) + (f, ‘p) for any cp E E.