# Application of Sobolev's embedding

I was reading the famous paper of Kenig, Ponce and Vega about Oscillatory integrals and in page 8 they say that a direct application of Sobolev's embedding provide the following inequality: for $$u(t,x):\mathbb{R}^2\to\mathbb{R}$$ we have $$\qquad \qquad \Vert u(t,x)\Vert_{L^8(\mathbb{R}^2)}\leq c\big\Vert\vert D_x^\gamma\vert\,\vert D_t^\gamma\vert u(t,x)\big\Vert_{L^6(\mathbb{R}^2)}, \qquad \qquad (*)$$ where $$\gamma=\tfrac{1}{24}$$. Here, the operator $$D_x$$ and $$D_t$$ are given in terms of their Fourier symbol $$\vert \xi\vert$$ and $$\vert\tau\vert$$. Now, in order to explain my doubt let me recall Sobolev embedding: Let $$k>\ell$$ and $$1\leq p satisfying: $$\dfrac{1}{p}-\dfrac{k}{n}=\dfrac{1}{q}-\dfrac{\ell}{n}.$$ Then, $$W^{k,p}(\mathbb{R}^n)\subset W^{\ell,q}(\mathbb{R}^n)$$. Hence, I tried to apply the latter embedding with $$n=2$$, $$p=6$$, $$q=8$$, $$\ell=0$$ from where I got $$k=\tfrac{1}{12}$$, what (if I am not wrong) means $$\Vert u(t,x)\Vert_{L^8(\mathbb{R}^2)}\leq c\Vert u(t,x)\Vert_{W^{1/12,6}(\mathbb{R}^2)}.$$ Now, my problem is that I don't know how to use the latter inequality in order to obtain $$(*)$$. For me it seems that somehow the factor $$k=\tfrac{1}{12}$$ is distributed into two terms of size $$\tfrac{k}{2}=\gamma=\tfrac{1}{24}$$ (related to these two factors $$\vert D_x^\gamma\vert$$ and $$\vert D_t^\gamma\vert$$ in $$(*)$$), but it is not clear to me how to rigorously do that.

• Your second inequality is weaker than $(*)$, I don't think you can use it to prove that. Aug 25, 2020 at 7:37
• @LL3.14 I have the same feeling, however, that is argument Kenig-Ponce-Vega give. I am not sure how they use Sobolev's embedding, but they explicitly wrote inequality $(*)$ for general functions (has nothing to do with a solution of any PDE) (of course, whenever both sides of the ineq make sense).
– W2S
Aug 25, 2020 at 9:37