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Singular symmetric functionals and Banach limits with additional invariance properties. (English. Russian original) Zbl 1075.46028
Izv. Math. 67, No. 6, 1187-1212 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 6, 111-136 (2003).
The authors exhibit a Banach limit $$\mathcal{L}$$ with certain additional invariance properties, $$\mathcal{L}(Tx) =\mathcal{L} (Hx) =\mathcal{L}(V_{n}x) =\mathcal{L}(x)$$. Here the $$T$$ is the translation operator and $$H$$ is the Hardy or Cesàro operator and the $$V_{n}$$’s are the dilation operators.
Let $$\Omega _{0}$$ be the set of concave non-negative functions $$\psi$$ on $$[0,\infty)$$ such that $$\psi (+0) =\lim_{t\downarrow 0}\psi (t) =0$$ and suppose that $$\lim_{t\rightarrow 0}\frac{\psi (2t)}{\psi (t)}=1$$. In section 5 of the paper, the authors use the above Banach limit to show that for any monotone sequence $$t_{n}\rightarrow 0$$ such that $$\lim_{n\rightarrow \infty }\frac{t_{n+1}}{ t_{n}}=r$$, $$0<r<1$$, if $$\mathbf{P}=\{2^{-n}\}_{n=1}^{\infty}$$, the associated singular symmetric functions satisfy, for all $$x$$ in the Marcinkiewicz space $$M(\psi)$$, $$f_{t,\mathcal{L}}(x) =f_{\mathbf{P},\mathcal{L}}(x)$$.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46L52 Noncommutative function spaces
##### Keywords:
singular symmetric functionals; Banach limits
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