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Lebesgue’s differentiation theorems in r.i. quasi-Banach spaces and Lorentz spaces \(\Gamma_{p,w}\). (English) Zbl 1246.46026

Let \(0 < \alpha \leq \infty\), \(L_0\) be the space of (equivalence classes of) measurable real functions on \([0, \alpha)\). For \(f \in L_0\), denote \(f^{**}(t)= \frac{1}{t}\int_0^t f^*(s)ds\), where \(f^*\) is the decreasing rearrangement of \(f\). Let \(0 < p < \infty\) and \(w \in L_0\) be a non-negative weight function. The Lorentz space \(\Gamma_{p,w}\) is the space of those \(f \in L_0\) for which \(\int (f^{**}(t))^p w(t)dt < \infty\). The quasinorm on \(\Gamma_{p,w}\) is defined as \(\|f\|_{\Gamma_{p,w}}= \left(\int (f^{**}(t))^p w(t)dt\right)^{1/p}\).
A quasi-Banach function space \(E\) on \((0, \alpha)\) is said to have the Lebesgue differentiation property (LDP) if for every \(f \in E\) \[ \lim_{\epsilon \to 0} \frac{\|(f - f(t))\chi_{[t-\epsilon, t+\epsilon]}\|_E}{\|\chi_{[t-\epsilon, t+\epsilon]}\|_E} = 0 \] for almost all \(t \in [0, \alpha)\).
The authors prove the LDP for order continuous rearrangement-invariant quasi-Banach function spaces on \((0, \infty)\) that satisfy a lower \(\phi\)-estimate for \(\|\cdot\|_E\). Classes of Lorentz space \(\Gamma_{p,w}\) that do have or do not have the LDP are presented.
Apart from that the authors give another type of generalization of the Lebesgue differentiation theorem that establishes conditions for pointwise convergence of the best or extended best constant approximants in Lorentz spaces. It is also shown that the extended best constant approximant operator \(T_{(p,A)}\) assumes a unique constant value for a function \(f \in \Gamma_{p-1,w}, 1 < p < \infty\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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