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$BMO_ψ$-spaces and applications to extrapolation theory

100%

Geiss S.

Studia Mathematica

1997

tom 122

nr 3

235-274

We investigate a scale of $BMO_ψ$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BMO_ψ$-$L_∞$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

A remark on extrapolation of rearrangement operators on dyadic $H^{s}$, 0 < s ≤ 1

51%

Geiss S. , Müller P. , Pillwein V.

Studia Mathematica

2005

tom 171

nr 2

196-205

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator $T_{s}$, 0 < s < 2, to be the linear extension of the map $(h_{I})/(|I|^{1/s}) ↦ (h_{τ(I)})(|τ(I)|^{1/s})$, where $h_{I}$ denotes the $L^{∞}$-normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that $T_{s₀}$ is bounded on $H^{s₀}$, then for all 0 < s < 2 the operator $T_{s}$ is bounded on $H^{s}$.