For all $\epsilon>0$ there exists $f,g$ such that $\|f\ast g\|_p>(1-\epsilon)\|f\|_1\|g\|_p$.

Question

Let $f\in L^1(\mathbb{R})$ and $g\in L^p(\mathbb{R})$, for $1\leq p\leq\infty$. A well-known is result, called Young's Inequality is that $$\|f\ast g\|_p\leq\|f\|_1\|g\|_p,$$ where $$(f\ast g)(x)=\int_{-\infty}^\infty f(x-y)g(y)\,dy$$ is the convolution of $f$ and $g$. Now, the problem that I want to solve is to show we can "get close" to equality in the following precise sense:

For all $\epsilon>0$, there exists $f\in L^1(\mathbb{R})$ and $g\in L^p(\mathbb{R})$ such that $$\|f\ast g\|_p>(1-\epsilon)\|f\|_1\|g\|_p.$$

I am having a hard time trying to get functions satisfying this. Do you have any idea of how to approach this?

Answer 1

This question Limit of convolution should give a solution to your problem. In your notation, consider the sequence $g_n= (2n)^{-1/p} \chi_{(-n,n)}$ and $f$ is any positive integrable function.