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Extra info for Distributions and partial differential equations on superspace
1. Let A = G ~ . Then the supereztension of a distribution u E S(R") is unique. 1. /n the space of distributions S(R~ 'm, L c) there exists a fundamental solution for every linear differential operator with constant coefficients whose symbol is a scalar nondegenerate polynomial with a nilpotent soul. 7 show, a scalar degenerate linear differential operator may have no fundamental solution in S(R~ 'm, L c) and in V(R~ 'm, Lr Moreover, the following more general result is valid. 2. Let p be a linear differential operator with constant coefficients whose symbol is scalar degenerate, Then there are no fundamental solutions of this operator in the spaces of distributions s(Ru ~,m ,L c) ~lr/l and ~D(Ru ,LC).
Note that the sets Cm are not linear spaces, they are not even convex, and not bounded. We shall now construct a new CSA L/ = ht0@5(1, by setting blo = K e @ A / ' , b/z = A1. This is a pseudotopologicM CSA in which the soul is nilpotent, and the annihilator of the odd part is trivial, provided that • = 0. Recall that a set G in a pseudotopological linear space (X, -c) is said to be bounded if OG $ O, where O is the filter in the field K whose basis consists of neighborhoods of zero in the field K.
2. The operations of direct product and convolution of distributions are supercommutative [#,u}| = # | v- ( - i ) b'll~Iv | ~ = O; [~. v}. = ~ . I~,. ~ = O; 847 P r o o f . Let us check, for example, that the supercommutator [#, u}| vanishes. Consider the case ]#] = It,] = 1. It suffices to prove that ker[p, u}@ contains all functions q~(x, y, e, ~) = g(x, o)f(y, ~), where g and f are homogeneous elements. t,f)) = (-I) = (-t)"' (v, (#,gf}} , where a = If I(1 - Igl), ~= = ~ + (1 - I g l ) ( 1 - Ifl), ~3 = ~e + Igl(1 - I f l ) , ~4 = ~3 + Ifl Igl.
Distributions and partial differential equations on superspace by Khrennikov