## New answers tagged sobolev-spaces

4
votes

Accepted

### $L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

I think similar questions always translate in the $L^p$ boundedness of a Fourier multiplier. In this case you want a Fourier multiplier which "exchanges the operator $D_1D_2D_3$ with the operator ...

6
votes

Accepted

### $L^p$ domination of mixed partial derivatives by the unmixed ones?

Since you tagged reference-request:
In the PDE/harmonic analysis literature this is a consequence of the Calderon-Zygmund Inequality, it is one of the main tools for studying elliptic regularity ...

3
votes

### $L^p$ domination of mixed partial derivatives by the unmixed ones?

It should be true for $p>1$. For a function $\varphi$ in the Schwartz class it holds that \begin{equation}
D_1D_2 \varphi(x) = -R_1 R_2 \Delta \varphi(x),
\end{equation}
where $R_1, R_2$ are the ...

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