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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 \(M(\psi)\), \(f_{t,\mathcal{L}}(x) =f_{\mathbf{P},\mathcal{L}}(x)\).

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
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