Skip to main content
MathOverflow

Return to Question

Corrected typo
Source Link
Víctor
Víctor
  • 123
  • 4

Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form

$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\dot{H}^{1/2}(D)}. $$$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\dot{H}^{1/2}(D)}^2. $$

Many thanks.

Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form

$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\dot{H}^{1/2}(D)}. $$

Many thanks.

Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form

$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\dot{H}^{1/2}(D)}^2. $$

Many thanks.

Source Link
Víctor
Víctor
  • 123
  • 4

Controlling the tensor product of functions in $H^1$ with lower derivatives

Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form

$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\dot{H}^{1/2}(D)}. $$

Many thanks.