Convolution Inequality: Conditions for Equality

[StatusX]

Given f,g in L^1(R), I'm given the inequality:

$$||f * g||_1 \leq ||f||_1 ||g||_1$$

And now I'm supposed to derive the conditions for equality to hold. I keep going around in circles. So far, I can show equality holds iff there is some real function $\phi(x)$ such that $e^{i \phi(x+y)} f(x) g(y) \geq 0$ holds almost everywhere, but I can't easily translate this into conditions on f and g. If f and g were both almost everywhere non-zero, this would just say they need to have linear phases with the same slope, but this doesn't help when they are zero on sets of positive measure.

For the case when f is real and g is non-negative, equality holds iff the support of $f^+*g$ and $f^-*g$ don't intersect, where $f^\pm$ are the positive and negative parts of f (I use the fact that convolutions of L^1 functions are continuous). The best I can do with this is say it is equivalent to there being some x such that the support of g(y) intersects both the support of $f^+(x-y)$ and that of $f^-(x-y)$ in sets of positive measure. Is this the best I can do? And how would I extend it to the general case? Like I said, I'm not getting anywhere.

 

[StatusX]

I'm doing other problems now and I keep coming back to this same related question: when does |f*g(x)|=|f|*|g|(x) hold almost everywhere. This is all I need to know, but I can't figure it out. It seems like I'm overlooking something really simple. Can anyone help me out?

 

[tacman]

The conditions for equality in the convolution inequality can be derived by considering the properties of convolutions and the properties of L^1 functions. Let's start by considering the case when f and g are both non-negative functions.

First, note that for any non-negative function f and g, the convolution f*g is a non-negative function. This is because the value of the convolution at any point x is given by the integral of f(y)g(x-y) over all y, and since both f and g are non-negative, the integrand is non-negative for all y. Therefore, we can write the convolution inequality as:

||f * g||_1 = \int_{-\infty}^{\infty} |f*g(x)| dx \leq \int_{-\infty}^{\infty} f(x)g(x) dx = ||f||_1 ||g||_1

Now, for equality to hold, we need to have |f*g(x)| = f(x)g(x) for almost all x. This means that the integrand in the first integral must be equal to the integrand in the second integral almost everywhere. Since we know that both f and g are non-negative, this means that we must have f(y)g(x-y) = f(x)g(y) for almost all x and y. This is equivalent to saying that there exists a real function \phi(x) such that e^{i\phi(x+y)}f(x)g(y) = f(x)g(y) for almost all x and y.

Now, let's consider the case when f and g are not necessarily non-negative. In this case, we can write f = f^+ - f^- and g = g^+ - g^- where f^+ and g^+ are the positive parts of f and g, and f^- and g^- are the negative parts. We can then rewrite the convolution inequality as:

||f * g||_1 = ||f^+ * g^+ - f^- * g^-||_1 \leq ||f^+ * g^+||_1 + ||f^- * g^-||_1 = ||f^+||_1 ||g^+||_1 + ||f^-||_1 ||g^-||_1 = ||f||_1 ||g||_1

Now, for equality to hold, we need to