Hardy and Bellman operators in spaces connected with $H(\mathbb T)$ and $BMO(\mathbb T)$

S. S. Volosivets ; B. I. Golubov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2008), pp. 4-13

Résumé

Assume that $1\leq p\infty$ and the function $f\in L^p[0,\pi]$ has the Fourier series $\sum\limits^\infty_{n=1}a_n\cos nx$. According to Hardy, the series $\sum\limits^\infty_{n=1}n^{-1}\sum\limits^n_{k=1}a_k\cos nx$ is the Fourier series of a certain function $\mathcal H(f)\in L^p[0,\pi]$. But if $1 p\le \infty$ and $f\in L^p[0,\pi]$, then the series $\sum\limits^\infty_{n=1}\sum\limits^\infty_{k=n}k^{-1}a_k\cos nx$ is the Fourier series of a certain function $\mathcal B(f)\in L^p[0,\pi]$. Similar assertions are true for sine series. This allows one to define the Hardy operator $\mathcal H$ on $L^p(\mathbb T)$, $1\le p\infty$, and to define the Bellman operator $\mathcal B$ on $L^p(\mathbb T)$, $1 p\le\infty$. We prove that the Bellman operator boundedly acts in $VMO(\mathbb T)$, and the Hardy operator maps a certain subspace $C(\mathbb T)$ into $VMO(\mathbb T)$. We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces $H(\mathbb T)$, $L(\mathbb T)$, $VMO(\mathbb T)$ with respect to the Hardy and Bellman operators.

Détail

Keywords: Hardy transform, BMO, VMO, majorant of modulus of continuity.
Mots-clés : Bellman transform