Posted by: Shuanglin Shao | October 1, 2008

Readings on sharp Strichartz inequalities for the Schrodinger equation

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

\|e^{it\Delta}f\|_{L_{t,x}^{2+4/d}(\Bbb R\times \Bbb R^d)}\le C\|f\|_{L^2_x(\Bbb R^d)}

where e^{it\triangle}f is the solution to the following free Schrodinger equation

iu_t+\triangle u=0

with initial data u(0,x)=f(x), where u(t,x):\Bbb R\times {\Bbb R}^d\to \Bbb C.

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

C_{p,q}:=\sup_{f\neq 0} \frac {\|e^{it\Delta}f\|_{L^q_sL^r_x(\Bbb R\times {\Bbb R}^d)}}{\|\|_{L^2_x({\Bbb R}^d)}}

when q=r=2+4/d in lower dimensions d=1,2.  They first built an interesting representation formula for the Strichartz norm \|e^{it\Delta}f\|^{2+4/d}_{L^{2+4/d}(\Bbb R\times {\Bbb R}^d)} as follows,

If d=1, then

\int\int_{\Bbb R} |e^{it\Delta}f|^6dxdt=\frac 1{\sqrt{12}}\langle f\bigotimes f\bigotimes f, P_1(f\bigotimes f\bigotimes f)\rangle_{L^2_x({\Bbb R}^3)},

where f\bigotimes g denotes the usual tensor product and P_1:L^2_x({\Bbb R}^3)\to L^2_x({\Bbb R}^3) denotes the orthogonal projection to the closed linear subspace which consists of functions invariant under rotations of {\Bbb R}^3 which keep the (1,1,1) direction fixed.

If d=2,

\int\int_{{\Bbb R}^2} |e^{it\Delta}f|^4dxdt=\frac 14\langle f\bigotimes f, P_2(f\bigotimes f)\rangle_{L^2_x({\Bbb R}^4)},

P_2:L^2_x({\Bbb R}^4)\to L^2_x({\Bbb R}^4) denotes the orthogonal projection to the closed linear subspace which consists of functions invariant under rotations of {\Bbb R}^4 which keep the (1,0,1,0) and (0,1,0,1) directions fixed.

They proved them in a simple way.  we only look at the d=1 case and its explicit maximizers. By using the definition of the delta function \delta(\xi)=(2\pi)^{-1}\int e^{-ix\xi}dx, they expressed the left side \int\int_{\Bbb R} |e^{it\Delta}f|^6dxdt as

\frac 1{2\pi}\int \int_{{\Bbb R}^3}\delta((1,1,1)\cdot(\eta-\zeta))\delta(|\eta|^2-|\zeta|^2)\overline{f\bigotimes f\bigotimes f(\eta)}f\bigotimes f\bigotimes f(\zeta)d\eta d\zeta.

This leads to define the following symmetric linear operator: for G\in \mathcal{C}_0^\infty({\Bbb R}^3),

A_1(G)(\eta):=\frac 1{2\pi}\int_{{\Bbb R}^3}G(\zeta)\delta((1,1,1)\cdot(\eta-\zeta))\delta(|\eta|^2-|\zeta|^2)d\zeta

Then they showed that A_1 is a bounded operator with operator bound \frac 1{\sqrt{12}} on \mathcal{C}_0^\infty({\Bbb R}^3)  by showing the measure m_{\eta} (d\zeta):=\frac {\sqrt{3}}{\pi}\delta((1,1,1)\cdot(\eta-\zeta))\delta(|\eta|^2-|\zeta|^2) is a probability measure on {\Bbb R}^3 for almost every \eta\in {\Bbb R}^3.  Finally  they extended A_1 onto the whole L^2_x({\Bbb R}^3) and showed that A_1 is a multiple of the orthogonal projection P_1.  Now they obtain the representation formula for the Strichartz norm.

Once having representation formula, they obtained that

\|e^{it\Delta}f\|^6_{L^6_{t,x}(\Bbb R\times {\Bbb R}^3)}\le \frac 1{\sqrt{12}}\langle f\bigotimes f\bigotimes f, P_1(f\bigotimes f\bigotimes f)\rangle_{L^2_x({\Bbb R}^3)},

which gives C_{6,6}\le \frac 1{\sqrt{12}}. Also they saw that in order to have equality, the function f\bigotimes f\bigotimes f must lie in the rangle of P_1, that is to say, it is invariant under rotations of {\Bbb R}^3 which fix (1,1,1). Obviously, all functions of the form

Ae^{(-\lambda+i\mu )x^2+cx}

with \lambda >0, \mu\in \Bbb R, A\in \Bbb C, 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 f is a maximizer for the Strichartz inequality (then f\bigotimes f\bigotimes f is in the range of P_1). Then Q_t(f) never vanishes for all $t>0$. Here Q_t(f) is the convolution of f with the approximation to identity Q_t(f):=\frac {1}{(2\pi t)^{1/2}}\int e^{-\frac{|x-y|^2}{2t} }f(y)dy.

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

The fact “2” above is actually the following general theme: if f is differentialable and never vanishes and f\bigotimes f\bigotimes f is in the range of P_1, then f is a Gausssian, which is proven by working out the explicit forms of the rotations M(\theta) in {\Bbb R}^3  which keep (1,1,1) invariant and use h(\eta)=h(M(\theta)\eta) for all differentiable functions invariant under rotations.  At this step, we note that f\bigotimes f\bigotimes f is a product of one dimensional function, which we need in the one dimensional argument. In higher dimensions, for f\in L^2_x({\Bbb R}^d) and f\bigotimes \cdots\bigotimes f invariant under rotations which keep (e_i,\cdots, e_i)_{1\le i\le d} with standard basis e_i\in {\Bbb R}^d, we will have to prove that f is a product of  one dimensional functions.

Recently, Carneiro generaized this argument to a prove a sharp form of  “Strichartz inequalities” as follows, for k\in \mathcal{Z}, k\ge 2 and (d,k)\neq (1,2),  

\|u(t,x)\|_{L^{2k}_tL^{2k}_x(\Bbb{R}\times \Bbb{R}^d)}\le \left(C_{d,k}\int_{\Bbb{R}^{dk}} |\hat{F}(\eta)|^2K(\eta)^{\frac {d(k-1)-2}{2}}\right)^{1/2k}d\eta

with

C_{d,k}=[2^{d(k-1)-1}k^{d/2}\pi^{(d(k-1)-2)/2}\Gamma(\frac {d(k-1)}{2})]^{-1}

and F(\eta)=f(\eta_1)\cdots f(\eta_d) with \eta_i\in \Bbb{R}^d for 1\le i\le d and the kernel K(\eta)=\frac{1}{k}\sum_{1\le i<j\le k}|\eta_i-\eta_j|^2.

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

How to find K 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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