Nonsquareness in Musielak-Orlicz-Bochner Function Spaces

and Applied Analysis 3 Proposition 1.2. Function σ t is μ-measurable. Proof. Pick a dense set {ri}i 1 in 0,∞ and set Bk { t ∈ T : M ( t, 1 2 rk ) 1 2 M t, rk } , qk t rkχBk t k ∈ N . 1.7 It is easy to see that for all k ∈ N, σ t ≥ qk t μ-a.e on T . Hence, supk≥1qk t ≤ σ t . For μ-a.e t ∈ T , arbitrarily choose ε ∈ 0, σ t . Then, there exists rk ∈ σ t − ε, σ t such that M t, 1/2 rk 1/2 M t, rk , that is, qk t ≥ rk > σ t − ε. Since ε is arbitrary, we find supk≥1qk t ≥ σ t . Thus, supk≥1qk t σ t . It is easy to prove the following proposition. Proposition 1.3. For any α ∈ 0, 1 , if u t ≤ σ t , then M t, αu t αM t, u t . Proposition 1.4. For any α ∈ 0, 1 , if u t < σ t < v t , then M t, αu t 1 − α v t < αM t, u t 1 − α M t, v t . Proposition 1.5. For any α ∈ 0, 1 , if σ t < v t , thenM t, αv t < αM t, v t . Definition 1.6 see 2 . We say that M t, u satisfies condition Δ M ∈ Δ if there exist K ≥ 1 and a measureable nonnegative function δ t on T such that ∫T M t, δ t dt < ∞ and M t, 2u ≤ KM t, u for almost all t ∈ T and all u ≥ δ t . First, we give some results that will used in the further part of the paper. Lemma 1.7 see 2 . SupposeM ∈ Δ. Then ρM u 1 ⇔ ‖u‖ 1. Lemma 1.8 see 9 . Let LM X be Musielak-Orlicz-Bochner function spaces, then, if K u φ, one has ‖u‖ ∫ T A t · ‖u t ‖dt, whereA t limu→∞ M t, u /u . 2. Main Results Theorem 2.1. LM X is nonsquare space if and only if a M ∈ Δ, b for any u, v ∈ S LM X , one has μ{t ∈ suppu ∩ suppv : ‖u t ‖ ‖v t ‖ > 2e t } > 0 or μ {t ∈ T : ‖u t ‖ > σ t } ∪ {t ∈ T : ‖v t ‖ > σ t } > 0, c X is nonsquare space. In order to prove the theorem, we give a lemma. Lemma 2.2. Let X be nonsquare space, then for any x, y / 0, one has ‖x‖ ∥y∥ −min∥x y∥,∥x − y∥ > 0. 2.1 4 Abstract and Applied Analysis Proof. For any x, y / 0, without loss of generality, we may assume ‖x‖ ≤ ‖y‖. Since X is nonsquare space, we have ‖x‖ ‖x‖ ‖x‖ ∥ ∥ ∥ ∥ ∥ ‖x‖ ∥ ∥y ∥ ∥ y ∥ ∥ ∥ ∥ ∥ > min {∥ ∥ ∥ ∥ ∥ x ‖x‖ ∥ ∥y ∥ ∥ y ∥ ∥ ∥ ∥ ∥ , ∥ ∥ ∥ ∥ ∥ x − ‖x‖ ∥ ∥y ∥ ∥ y ∥ ∥ ∥ ∥ ∥ } . 2.2 Therefore, by 2.2 , we obtain ∥ ∥x y ∥ ∥ ≤ ∥ ∥ ∥ ∥ ∥ x ‖x‖ ∥ ∥y ∥ ∥ · y ∥ ∥ ∥ ∥ ∥ ( 1 − ‖x‖ ∥ ∥y ∥ ∥ ) · ∥y∥ < ‖x‖ ‖x‖ ∥ ∥y ∥ ∥ − ‖x‖ ‖x‖ ∥y∥ 2.3