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This tag is used if a reference is needed in a paper or textbook on a specific result.

4 votes
1 answer
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Reference or proof of a theorem of L. Fejér on summability of Fourier series

In the article "Ensembles exceptionnels" by A. Beurling, the author cites the following theorem of Fejér: Suppose that a $2\pi$ periodic function $ f $, Lebesgue integrable in $(0,2\pi)$ satisfies the …
5 votes
Accepted

Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?

It is not neccesary in general that $\varphi \in L^\infty(\mathbb{D})$, but it is necessary and sufficient that in a certain sense $\varphi$ must be bounded ``on average in the hyperbolic sense''. The …
ash's user avatar
  • 151
4 votes

$L^p$ 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 $D_1 …
an_ordinary_mathematician's user avatar
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 Riesz …
an_ordinary_mathematician's user avatar