By Khrennikov

Similar mathematics books

Leopold vintage Library is overjoyed to post this vintage publication as a part of our huge assortment. As a part of our on-going dedication to offering price to the reader, we now have additionally supplied you with a hyperlink to an internet site, the place you'll obtain a electronic model of this paintings at no cost. a few of the books in our assortment were out of print for many years, and consequently haven't been available to most people.

Extra info for Distributions and partial differential equations on superspace

Sample text

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.