Since last weekend, I have read several papers due to Bennett-Bez-Carbery-Hundertmark, Carneiro, Foschi, Hundertmark-Zharnitsky and on the sharp Strichartz inequality

where is the solution to the following free Schrodinger equation

with initial data , where .

I am happy that the latest two papers cited mine on the existence of a maximiser for the nonendpoint Strichartz inequality. For my own benefits, I would like to record here the main idea of their proofs respectively.

I choose to start with Hundertmark-Zharnitsky’s paper, “on sharp Strichartz inequalities in low dimensions”(IMRN) since this paper introduces a beatiful argument which seems to be of general interest (at least essential to papers by Carneiro and Bennett-Bez-Carbery-Hundertmark). In this paper, Hundertmark-Zharnitsky obtained the sharp value for

when in lower dimensions . They first built an interesting representation formula for the Strichartz norm as follows,

If , then

where denotes the usual tensor product and denotes the orthogonal projection to the closed linear subspace which consists of functions invariant under rotations of which keep the direction fixed.

If ,

denotes the orthogonal projection to the closed linear subspace which consists of functions invariant under rotations of which keep the and directions fixed.

They proved them in a simple way. we only look at the case and its explicit maximizers. By using the definition of the delta function they expressed the left side as

This leads to define the following symmetric linear operator: for ,

Then they showed that is a bounded operator with operator bound on by showing the measure is a probability measure on for almost every . Finally they extended onto the whole and showed that is a multiple of the orthogonal projection . Now they obtain the representation formula for the Strichartz norm.

Once having representation formula, they obtained that

which gives . Also they saw that in order to have equality, the function must lie in the rangle of , that is to say, it is invariant under rotations of which fix . Obviously, all functions of the form

with , i.e., Gausssians, are maximisres.

In the second part of their paper, they managed to show that the Gausssians turn out to be the only maximizers in the following three steps,

“1”. Assume is a maximizer for the Strichartz inequality (then is in the range of ). Then never vanishes for all $t>0$. Here is the convolution of with the approximation to identity .

“2”. never vanishes and is differentiable, then is a Gausssian. hence as a limit, is a Gausssian too.

The fact “2” above is actually the following general theme: if is differentialable and never vanishes and is in the range of , then is a Gausssian, which is proven by working out the explicit forms of the rotations in which keep invariant and use for all differentiable functions invariant under rotations. At this step, we note that is a product of one dimensional function, which we need in the one dimensional argument. In higher dimensions, for and invariant under rotations which keep with standard basis , we will have to prove that is a product of one dimensional functions.

Recently, Carneiro generaized this argument to a prove a sharp form of “Strichartz inequalities” as follows, for and ,

with

and with for and the kernel .

This inequality is sharp if and only if is a Gaussian.

How to find is mysterious for me here.

(I leave the discussion on Foschi’s and Bennett-Bez-Carbery-Hundertmark’s papers to a later time)

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