Today I thought that I had a new proof of the adjoint Fourier transform for the sphere in three dimensions; but it turns out it was wrong. I would like to record it here, which I hope is useful at some point.

First let us define the notion of Fourier transform. Let

Then the Plancherel theorem reads,

From this definition the convolution of two functions behaves under the Fourier transform like

Now let us begin the argument. By Plancherel,

In view of this, we may assume that .

We write the integrand out,

(1) By Cauchy-Schwarz inequality, we have

We observe that

Then

(2)Then by Minkowsik inequality,

Note that the best constant for Young’s inequality is .

(I orginially thought that . However, it is wrong since or , which can be seen from the asympotics for large in . Incidentally, today I found that Terry remarked on his blog that the Dirac mass by a trick of epsilon regularization (another way is to use Pancherel theorem and observe that ).)

At this point, there is no need to read on since the following is based on Step 2. I worte it down before I realized the mistake; so I backed it up as my notes.

{Then we can conclude that

}

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