Is there a way to solve this convolution inequality?

[amirmath]

Dear friends,

I am interesting to find some functions g satisfying the following convolution inequality

(g\astv)(t)\leqv(t)

for any positive function v\inL^{1}[0,T] and * denotes the convolution between g and v.

 

[Undecided Guy]

Dear friends,

I am interesting to find some functions g satisfying the following convolution inequality

(g\astv)(t)\leqv(t)

for any positive function v\inL^{1}[0,T] and * denotes the convolution between g and v.

The way you've worded the statement, it's not possible. Suppose that v \in L^1[0, T] satisfies 0 < (g*v)(0) < v(0). Let v'(t) = v(t) for all t other than 0 and v'(0) = .5(g*v)(0). Then v = v' in the sense of L1, but (g*v)(0) > v'(0).