Points of monotonicity in Musielak--Orlicz function spaces endowed with the Orlicz norm
Copyright policy )Points of monotonicity in Musielak--Orlicz function spaces endowed with the Orlicz norm
Let \((X,\|\cdot\|,\leq)\) be a Banach lattice, let \(X^+\) denote the positive cone in \(X\) and let \(S(X)\) be the unit sphere of \(X\). A point \(x\in S(X^+)\) is said to be upper (lower) monotone if for any \(y\in X^+\backslash\{0\},\) (any \(y\in X^+\backslash \{0\}, y\leq x)\) there holds \(\|x+y\|>1,(\|x-y\|0\right\}, \] endowed with the Amemiya-Orlicz norm. The authors obtain characterizations of the points \( x\in S((L_M^0)^+)\) which are UM and LM points, respectively. Analogous results are obtained for so called upper (lower) locally uniformly monotone points of \(S((L_M^0)^+).\) For example: \(x\in S((L_M^0)^+)\) is an UM point if and only if \(K(x)=\emptyset\) or \(kx(t)\geq\sup\{u\geq 0|M(t,u)=0\},\mu\)-a.e. for any \(k\in K(x),\) whenewer \(K(x)\neq\emptyset.\) Here \(K(x)\) is a suitable set of real numbers associated to \(x,\) which depends of \(M.\)
Musielak-Orlicz space, Geometry and structure of normed linear spaces, lower local uniform monotonicity, points of lower monotonicity, upper local uniform monotonicity, condition delta two, Orlicz norm, points of upper monotonicity, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), points of lower (upper) local uniform monotonicity
Musielak-Orlicz space, Geometry and structure of normed linear spaces, lower local uniform monotonicity, points of lower monotonicity, upper local uniform monotonicity, condition delta two, Orlicz norm, points of upper monotonicity, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), points of lower (upper) local uniform monotonicity
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