Abstract
We give criteria for local uniform non-squareness of Orlicz–Lorentz function spaces \(\varLambda _{\varphi ,\omega }\) equipped with the Luxemburg norm, where the widest possible class of convex Orlicz functions is admitted. As immediate consequences, criteria for local uniform non-squareness of Orlicz function spaces \(L^{\varphi }\), which complete the already known results, are deduced.
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1 Preliminaries
We say that a Banach space X is non-square if for any x and y from S(X) (the unit sphere of X), we have \(\min (\Vert \frac{x-y}{2}\Vert ,\Vert \frac{x+y}{2}\Vert )<1\). A Banach space X is said to be locally uniformly non-square if for any \(x\in S(X)\) there exists \(\delta =\delta (x)\in (0,1)\) such that \(\min (\Vert \frac{x-y}{2}\Vert ,\Vert \frac{x+y}{2}\Vert )\le 1-\delta \) for any \(y\in B(X)\) (the unit ball of X). Finally, we say that a Banach space X is uniformly non-square if there exists \(\delta \in (0,1)\) such that \(\min (\Vert \frac{x-y}{2}\Vert ,\Vert \frac{x+y}{2}\Vert )\le 1-\delta \) for any \(x, y\in B(X)\).
Recall that uniform non-squareness of Banach spaces was defined by James as a geometric property which implies super-reflexivity (see [9, 10]). Hence, knowing that a Banach space possesses this property, even without any characterization of its dual space, we can infer that it is super-reflexive, and so reflexive. Additionally, as García-Falset, Llorens-Fuster and Mazcu\(\tilde{\mathrm {n}}\)an-Navarro have shown, uniformly non-square Banach spaces possess the fixed-point property (see [5]). Therefore, it is natural and interesting to ask about criteria of non-squareness of various well-known classes of Banach spaces. The purpose of this paper is to complete the study started in [3] (see also [2]).
Let \(L^{0}=L^{0}([0,\gamma ))\) be the space of all (equivalence classes of) Lebesgue measurable real-valued functions defined on the interval \([0,\gamma )\), where \(\gamma \le \infty \). For any \(x,y\in L^{0}\), we write \(x\le y\) if \(x(t)\le y(t)\) almost everywhere with respect to the Lebesgue measure m on the interval \([0,\gamma )\).
Given any \(x\in L^{0}\), we define its distribution function \(\mu _{x}:[0,+\infty )\rightarrow [0,\gamma ]\) by
(see [1, 14, 15]) and the non-increasing rearrangement \(x^{*}:[0,\gamma )\rightarrow [0,\infty ]\) of x as
(under the convention \(\inf \emptyset =\infty \)). We say that two functions \(x,y\in L^{0}\) are equimeasurable if \(\mu _{x}(\lambda )=\mu _{y}(\lambda )\) for all \(\lambda \ge 0\). Then we obviously have \(x^{*}=y^{*}\).
A Banach space \(E=(E,\le ,\Vert \cdot \Vert )\), where \(E\subset L^{0}\), is said to be a Köthe space if the following conditions are satisfied:
\(\mathrm{(i)}\) if \(x\in E\), \(y\in L^{0}\) and \(|y|\le |x|\), then \(y\in E\) and \(\Vert y\Vert \le \Vert x\Vert \),
\(\mathrm{(ii)}\) there exists a function x in E that is strictly positive on the whole \([0,\gamma )\).
Recall that the Köthe space E is called a symmetric space if E is rearrangement invariant in the sense that if \(x\in E\), \(y\in L^{0}\) and \(x^{*}=y^{*}\), then \(y\in E\) and \(\Vert x\Vert =\Vert y\Vert \). For basic properties of symmetric spaces we refer to [1, 14, 15].
In the whole paper \(\varphi \) will denote an Orlicz function, that is, \(\varphi :[-\infty ,\infty ]\rightarrow [0,\infty ]\) (our definition is extended from R into \(R^{e}\) by assuming \(\varphi (-\infty )=\varphi (\infty )=\infty \)) and \(\varphi \) is convex, even, vanishing and continuous at zero, left continuous on \((0,\infty )\) and not identically equal to zero on \((-\infty ,\infty )\). Let us denote
and
Let us note that if \(a_{\varphi }>0\), then \(\beta =a_{\varphi }\), while the left-continuity of \(\varphi \) on \((0,\infty )\) is equivalent to the fact that \(\lim _{u\rightarrow (b_{\varphi })^{-}}\varphi (u)=\varphi (b_{\varphi })\).
Recall that an Orlicz function \(\varphi \) satisfies the condition \(\varDelta _{2}\) for all \(u\in {\mathbb {R}}\) (\(\varphi \in \varDelta _{2}({\mathbb {R}})\), for short) if there exists a constant \(K>0\) such that the inequality
holds for any \(u\in {\mathbb {R}}\) (then we have \(a_{\varphi }=0\) and \(b_{\varphi }=\infty \)). Analogously, we say that an Orlicz function \(\varphi \) satisfies the condition \(\varDelta _{2}\) at infinity (\(\varphi \in \varDelta _{2}(\mathbb {\infty })\), for short) if there exist a constant \(K>0\) and a constant \(u_{0}\ge 0\) such that \(\varphi (u_{0})<\infty \) and inequality (2) holds for any \(u\ge u_{0}\) (then we obtain \(b_{\varphi }=\infty \)).
For any Orlicz function \(\varphi \) we define its complementary function in the sense of Young by the formula
for all \(u\in {\mathbb {R}}\). It is easy to show that \(\psi \) is also an Orlicz function.
Let \(\omega :[0,\gamma )\rightarrow {\mathbb {R}}_{+}\) be a non-increasing and locally integrable function, which will be called a weight function. Define
Recall that Orlicz–Lorentz spaces—a natural generalization of Orlicz and Lorentz spaces—were introduced at the beginning of the 1990s [11, 12, 16,17,18]. These investigations spurred further investigations of Orlicz–Lorentz spaces.
Given an Orlicz function \(\varphi \) and a weight function \(\omega \), we define a convex modular \(I_{\varphi ,\omega }:L^{0}\rightarrow {\mathbb {R}}_{+}^{e}=[0,\infty ]\) by the formula
The modular space
generated by the modular \(I_{\varphi ,\omega }\), is called the Orlicz–Lorentz function space. Since the modular unit ball \(\{x\in L^{0}:I_{\varphi ,\omega }(x)\le 1\}\) is an absolutely convex and absorbing subset of \(\varLambda _{\varphi ,\omega }([0,\gamma ))\), its Minkowski functional \(\Vert \cdot \Vert _{\varphi ,\omega }\), defined by
is a seminorm in \(\varLambda _{\varphi ,\omega }([0,\gamma ))\). It is easy to check that, thanks to the condition \(\varphi (u)\rightarrow \infty \) as \(u\rightarrow \infty \), it is a norm in \(\varLambda _{\varphi ,\omega }([0,\gamma ))\), called the Luxemburg norm.
In our investigations we apply the results concerning the monotonicity properties of Lorentz function spaces first presented in [8] and [4]. Let us recall that a Lorentz function space \(\varLambda _{\omega }\) is defined by the formula
A Banach lattice \(E=(E,\le ,\Vert \cdot \Vert )\) is said to be locally uniformly monotone, whenever for any \(x\in (E)_{+}\) (the positive cone of E) with \(\Vert x\Vert =1\) and any \(\varepsilon \in (0,1)\) there is \(\delta (x, \varepsilon )\in (0,1)\) such that the conditions \(0\le y\le x\) and \(\Vert y\Vert \ge \varepsilon \) imply that \(\Vert x-y\Vert \le 1-\delta (x,\varepsilon )\).
Theorem 1
([4], Proposition 4.1) The Lorentz function space \(\varLambda _{\omega }\) is locally uniformly monotone if and only if \(\omega \) is positive on \([0,\gamma )\) and \(\int _{0}^{\gamma }\omega (t)dt=\infty \) whenever \(\gamma =\infty \).
In our investigations we will also apply Lemma 1. It is proved analogously as in the case of Orlicz spaces using the convexity of the modular \(I_{\varphi ,\omega }\) (cf. also [13] for a more general case).
Lemma 1
Suppose that an Orlicz function \(\varphi \) satisfies a suitable condition \(\varDelta _{2}\), that is, \(\varphi \in \varDelta _{2}({\mathbb {R}})\) if \(\gamma =\infty \) and \(\int _{0}^{\infty }\omega (t)dt=\infty \), and \(\varphi \in \varDelta _{2}(\mathbb \infty )\) otherwise.
Then for any \(\varepsilon \in (0,1)\) there exists \(\delta =\delta (\varepsilon )\in (0,1)\) such that \(\Vert x\Vert \le 1-\delta \) for any \(x\in \varLambda _{\varphi ,\omega }\) whenever \(I_{\varphi ,\omega }(x)\le 1-\varepsilon \).
In particular, for any \(x\in \varLambda _{\varphi ,\omega }\), we then have \(\Vert x\Vert =1\) if and only if \(I_{\varphi ,\omega }\left( x\right) =1\).
2 Results
We start with the following
Theorem 2
Let \(\gamma =\infty \). Then the Orlicz–Lorentz function space \(\varLambda _{\varphi ,\omega }\) is locally uniformly non-square if and only if \(\int _{0}^{\infty }\omega (t)dt=\infty \), \(\varphi \in \varDelta _{2}({\mathbb {R}})\) and \(\varphi (\frac{u}{2})<\frac{1}{2}\varphi (u)\) for every \(u>0\).
Proof. Sufficiency. Let us fix \(x\in S(\varLambda _{\varphi ,\omega })\). Since \(\varphi \in \varDelta _2({\mathbb {R}})\), by Lemma 1 it is enough to show that there exists \(\delta \in (0, 1)\) such that \(\min \left( I_{\varphi ,\omega }\left( \frac{x-y}{2}\right) , I_{\varphi ,\omega }\left( \frac{x+y}{2}\right) \right) \le 1-\delta \) for every \(y\in B\left( \varLambda _{\varphi ,\omega }\right) \). There exist \(t_1, t_2\) such that \(0\le t_1<t_2<\infty \), \(0<x^{*}(t_2)\le x^{*}(t_1)<\infty \) and
for some \(\xi \in (0, 1]\). Without loss of generality we may assume that \(x^{*}(s)>x^{*}(t)>x^{*}(w)\) for all \(s\in (0, t_1), t\in (t_1, t_2)\) and \(w>t_2\) if \(t_1>0\), and \(x^{*}(t)>x^{*}(w)\) for all \(t\in (t_1, t_2)\) and \(w>t_2\), otherwise. By [14], property \(7^{\circ }\), page 64 there exist sets \(e_{t_1}\) and \(e_{t_2}\) with \(m(e_{t_1})=t_1\), \(m(e_{t_2})=t_2\),
(in the case \(t_1=0\) we have \(e_{t_1}=\emptyset \)), whence it follows that \(e_{t_1}\subsetneqq e_{t_2}\). Moreover, by the properties of the function \(\varphi \), we get
for \(u\in \left[ x^{*}\left( t_1\right) , x^{*}\left( t_2\right) \right] \), where \(\eta _x \in (0, 1)\). For any \(y\in B\left( \varLambda _{\varphi , \omega }\right) \) we define
By the inequality (5), we obtain \(I_{\varphi , \omega } \left( x \chi _{e_{t_2}{\setminus } e_{t_1}}\right) \ge \xi \), whence \(I_{\varphi , \omega } \left( x \chi _{B_1}\right) \ge \frac{\xi }{2}\) or \(I_{\varphi , \omega } \left( x \chi _{B_2}\right) \ge \frac{\xi }{2}\). Let us assume that the first inequality holds. Then we have
for m-a.e. \(t \in B_1\). Consequently,
By the local uniform monotonicity of the Lorentz space \(\varLambda _{\omega }\) we get
where \(\delta \left( \varphi \circ x,\frac{\xi \cdot \eta _{x}}{2}\right) \) is a constant from the definition of local uniform monotonicity of the Lorentz space \(\varLambda _{\omega }\) corresponding to \(\varphi \circ x\) and \(\varepsilon =\frac{\xi \cdot \eta _{x}}{2}\). In the second case, that is if \(I_{\varphi ,\omega }\left( x\chi _{B_2}\right) \ge \frac{\xi }{2}\), proceeding analogously we can prove that \(I_{\varphi ,\omega }\left( \frac{x+y}{2}\right) \le 1-\delta \left( \varphi \circ x,\frac{\xi \cdot \eta _x}{2}\right) \).
Necessity. If \(\int _{0}^{\infty }\omega \left( t\right) dt<\infty \) or \(\varphi \notin \varDelta _{2}({\mathbb {R}})\), then \(\varLambda _{\varphi ,\omega }\) contains an order isometric copy of \(l^{\infty }\) (see [11, Theorem 2.4]). Suppose now that \(\int _{0}^{\infty }\omega \left( t\right) dt=\infty \), \(\varphi \in \varDelta _{2}({\mathbb {R}})\) and there exists \(u_0>0\) such that \(\varphi \left( \frac{1}{2}u_0\right) =\frac{1}{2}\varphi \left( u_0\right) \). Then \(\varphi \left( \frac{1}{2}u\right) =\frac{1}{2}\varphi \left( u\right) \) for any \(u\in \left[ 0, u_0\right) \). Moreover, we can find \(t_0>0\) such that \(\int _{0}^{t_0}\varphi \left( u_0\right) \omega \left( t\right) dt=1\) and \(t_n\) such that \(\int _{0}^{t_n}\varphi \left( u_n\right) \omega \left( t\right) dt=1\) for all \(n\in {\mathbb {N}}\), where \(u_n=\frac{1}{2^n}u_0\). Taking
we get \(I_{\varphi , \omega }\left( x\right) =I_{\varphi , \omega }\left( y_n\right) =1\) for all \(n\in {\mathbb {N}}\). Since
we get
Therefore
Consequently
which means that the space \(\varLambda _{\varphi ,\omega }\) is not locally uniformly non-square.
Theorem 3
Let \(\alpha =\gamma <\infty \) and \(\int _{0}^{\gamma }\varphi (\beta )\omega (t)dt<1\). Then the following conditions are equivalent:
- (i)
\(\varphi \in \varDelta _2(\infty )\),
- (ii)
the Orlicz–Lorentz space \(\varLambda _{\varphi ,\omega }\) is non-square,
- (iii)
the Orlicz–Lorentz space \(\varLambda _{\varphi ,\omega }\) is locally uniformly non-square.
Proof
The implication (iii)\(\Rightarrow \)(ii) is obvious. The implication (ii)\(\Rightarrow \)(i) has been proved in Theorem 2.2 in [3]. We shall show (i)\(\Rightarrow \)(iii). Let us take an arbitrary \(x\in S\left( \varLambda _{\varphi ,\omega }\right) \). By the assumption that \(\int _{0}^{\gamma }\varphi (\beta )\omega (t)dt<1\) and \(\varphi \in \varDelta _2(\infty )\), we get
Hence, there exist \(t_1, t_2\) such that \(0\le t_1 < t_2 \le t_0\), \(\beta<x^{*}(t_2)\le x^{*}(t_1)<\infty \) and
Proceeding analogously as in the proof of sufficiency in Theorem 2, we can find \(\delta >0\) depending only on x such that \(\min \left( \left\| \frac{x-y}{2}\right\| _{\varphi , \omega }, \left\| \frac{x+y}{2}\right\| _{\varphi , \omega }\right) \le 1-\delta \) for every \(y\in B\left( \varLambda _{\varphi , \omega }\right) \).
Theorem 4
Let \(\alpha =\gamma <\infty \) and \(\int _{0}^{\gamma }\varphi (\beta )\omega (t)dt\ge 1\). Then the Orlicz–Lorentz function space \(\varLambda _{\varphi ,\omega }\) is locally uniformly non-square if and only if \(\varphi \in \varDelta _{2}(\infty )\), \(\psi \in \varDelta _{2}(\infty )\), where \(\psi \) denotes the complementary function of \(\varphi \) in the sense of Young (see formula (3)), and \(\int _{0}^{\gamma _{0}/2}\varphi (\beta )\omega (t)dt<1\).
Proof
Sufficiency. Since \(\varphi \in \varDelta _{2}(\infty )\), by Lemma 1, it is sufficient to prove the corresponding inequalities for the modular.
Case 1. Suppose that \(\gamma _{0}>0\). Let us take an arbitrary \(x\in S\left( \varLambda _{\varphi ,\omega }\right) \). If
then we proceed analogously as in the proof of the implication (i)\(\Rightarrow \)(iii) in Theorem 3.
Let us now assume that inequality (6) does not hold or, equivalently, \(x^{*}(0)\le \beta \). Define
By the assumptions we get \(c\in \left( 0, \frac{1}{8}\right) \). Let \(u_0\) be such that
We have \(u_0>\beta \) and, by the condition \(\psi \in \varDelta _2(\infty )\), there exists \(\eta =\eta (u_0)\in (0, 1)\) such that
for \(u\ge u_0\). Moreover, in virtue of \(\varphi \in \varDelta _2(\infty )\), for \(K=\frac{2-\eta }{2(1-\eta )}\) we can find \(\tau \in (0, 1)\) such that
for \(u \ge u_0\). Thus, for \(u\ge u_1\), where \(u_1=\max \left( u_0, \frac{\beta }{\tau }\right) \), we get
In virtue of (8) and (9), we obtain
for \(u\ge u_1\). Let us take an arbitrary \(y\in S\left( \varLambda _{\varphi ,\omega }\right) \) and define
Case 1.1. Suppose
Define
We have \(\left| y(t) \right| \ge u_1\) for m-a.e. \(t\in f(u_1+\beta )\) and \(e(u_1+2\beta )\subset f(u_1+\beta )\). Moreover, in virtue of (10), we obtain
for m-a.e. \(t\in f(u_1+\beta )\), whence
and in consequence
Define
By [3, Remark 1.1], we get
By (12), (13) and (14) we obtain
Since the function
is equimeasurable with \(y\chi _{([0,\gamma ){\setminus } e(u_1+2\beta )}\), by Hardy-Littlewood inequality, we have
Simultaneously, by
we get
whence
Eventually, by (12), (15), (16) and (17), we have
Case 1.2. Let now
and
Let \(t_1\) and \(t_2\) be such that the following equalities hold:
We have \(0<t_1<t_2<\gamma \). Moreover, by (19) and (20), we get
whence, by the definition of \(t_{1}\), we obtain \(m(e(u_0))\ge t_1\). Define
We have \(m\left( A_{y}^{1}\right) \ge \frac{t_1}{2}\) or \(m\left( A_{y}^{2}\right) \ge \frac{t_1}{2}\). Let us assume that the first inequality holds. Then, by inequality (8),
for m-a.e. \(t\in A_{y}^{1}\). Moreover,
for m-a.e. \(t\in [0, \gamma ){\setminus } A_{y}^{1}\). Hence
where \(\nu =\min \left( \eta , 1-\frac{\varphi \left( \frac{u_0+\beta }{2}\right) }{\varphi (u_0)}\right) \). Since \(y(t)\ge u_0\) for m-a.e. \(t\in A_{y}^{1}\), we obtain
for the same t. It follows from the inequality \(0\le \varphi \circ y-\nu \varphi \circ y\chi _{A_{y}^{1}}\le \varphi \circ y\) that
Furthermore, it follows from (22) that for any \(t\in [0,\gamma )\) such that \(y^{*}(t)<\frac{u_0+\beta }{2}\) we have equality in (23). Simultaneously
Hence, by [14, property \(1^\circ \), p. 63] we obtain
Since \(I_{\varphi ,\omega }(y)=1\), by the definitions of \(e(\frac{u_{0}+\beta }{2})\) and \(t_2\) (see formulas (11) and (21), respectively), we have \(m(e(\frac{u_{0}+\beta }{2}))\le t_{2}\). Hence, by the remark concerning inequality (23), we obtain
and in consequence
Therefore,
If \(m(A_{y}^{1})<\frac{t_1}{2}\), then \(m(A_{y}^{2})\ge \frac{t_1}{2}\), and proceeding analogously we obtain
Case 1.3. Suppose that
Let \(t_3\) and \(u_2\) be such that
and
By the definition of \(u_0\) (see (7)) we get \(\frac{1}{2}\gamma _{0}<t_3<\frac{3}{4}\gamma _{0}\). Simultaneously, since \(\int _{0}^{\gamma }\varphi (\beta )\omega (t)dt\ge 1\), we have \(0<u_2<\beta <u_0\). Define
By the inequalities \(x^{*}(0)\le \beta \) and (26) we obtain \(m(B_x)\ge t_3\) and \(m(B_y)\ge t_3\). Define
and
Case 1.3.1. Assume that \(m(C_{x,y}^{+})\ge \frac{t_4}{2}\). Then we have
for m-a.e. \(t\in C_{x,y}^{+}\). Indeed, if \(\vert x(t) \vert \ge \vert y(t)\vert \), then
On the other hand, if \(\vert x(t) \vert <\vert y(t)\vert \), we get
It is obvious that for \(t\in [0,\gamma ){\setminus } C_{x,y}^{+}\) we have
Thus,
Since
by the local uniform monotonicity of the Lorentz space \(\varLambda _\omega \), we get
where \(\delta \left( \varphi \circ x,\frac{u_2\zeta }{4\beta }\right) \) is a constant from the definition of the local uniform monotonicity of the Lorentz space \(\varLambda _\omega \) corresponding to \(\varphi \circ x\) and \(\varepsilon =\frac{u_2 \zeta }{4\beta }\).
Analogously, if \(m(C_{x,y}^{-})>\frac{t_4}{2}\), we have
Case 1.3.2. Finally assume that \(m(C_{x,y}^{+})<\frac{t_4}{2}\) and \(m(C_{x,y}^{-})<\frac{t_4}{2}\). The proof of this case is analogous to the proof of Case 2.2.2 in Theorem 2.4 in [3]. We shall present it for completeness’ sake. Let us define
where \(B_{x}\) and \(B_{y}\) are defined by formulas (27) and (28). We have \(C_x\cap C_y=\emptyset \) and
whence
Define
By [14, property \(7^{\circ }\), p. 64] there exist sets \(e_{\gamma _{0}}=e_{\gamma _{0}}\left( \frac{x+y}{2}\right) \) and \(e_{t_{5}}=e_{t_{5}}\left( \frac{x+y}{2}\right) \) such that \(m\left( e_{\gamma _{0}}\right) =\gamma _{0}\), \(m\left( e_{t_{5}}\right) =t_{5}\),
and
Without loss of generality we may assume that \(e_{\gamma _{0}}\subset e_{t_{5}}\). Denoting \(D_{\gamma _{0}}=e_{t_{5}}{\setminus } e_{\gamma _{0}}\) and \(D_{t_{5}}=\left[ 0, \gamma \right) {\setminus } e_{t_{5}}\), by [3, Remark 1.1], we obtain
By inequality (36) we have
Hence, without loss of generality, it may be assumed that \(m(C_{x}\cap D_{t_{5}})>a\). Then, since \(C_{x}\subset B_{x}\) we have \(m(\{t:|x|\chi _{D_{t_{5}}}(t)\ge u_{2}\})>a\) and thus \(\left( x\chi _{D_{t_5}}\right) ^{*}(a)\ge u_2\). If \(\left( x\chi _{e_{\gamma _{0}}}\right) ^{*}(\gamma _{0}-a)\le \frac{u_2}{4}\), then
where p is in fact the constant value of the derivative of \(\varphi \) on the interval \((0,\beta )\) (the interval of linearity of \(\varphi \)) and \(\omega _0=\omega (t)\) for any \(t\in (0, \gamma _{0})\). By the definition of \(\gamma _{0}\) we get \(\omega _0-\omega (t_5)>0\). Hence and by Hardy-Littlewood inequality (proceeding analogously as in formula (16)), we obtain
Suppose now that \(\left( x\chi _{e_{\gamma _{0}}}\right) ^{*}(\gamma _{0}-a)>\frac{u_2}{4}\). Since \(m(C_{x,y}^{+}\cup C_{x,y}^{-})<t_4<\frac{1}{8}\gamma _{0}\), we have \(m \left( \Big \{ t\in e_{\gamma _{0}}: \vert y(t) \vert >\frac{u_2}{4}\Big \}\right) \le t_4+a\le \frac{5}{32}\gamma _{0}\). Thus \(\left( y\chi _{e_{\gamma _{0}}}\right) ^{*}(\gamma _{0}-a)\le \frac{u_2}{4}\). Simultaneously,
Hence, by (35) we get
and we conclude that \((y\chi _{D_{t_{5}}})^{*}(a)\ge u_{2}\). Repeating the procedure from (37) and (38) for y we obtain
Summarizing Case 1, for any \(x\in S(\varLambda _{\varphi , \omega })\) such that \(x^{*}(0)\le \beta \), by (18), (24), (25), (31), (32) and (39) we have
Case 2. Let \(\gamma _{0}=0\). If \(x^{*}(0)>\beta \), we can proceed analogously as in the proof of Case 1. Assume now that \(x^{*}(0)\le \beta \). Let \(c=\frac{1}{8}\) and \(u_0>\beta \). Since \(\psi \in \varDelta _2(\infty )\), there exists \(\eta \in (0,1)\) such that inequality (8) holds for \(u\ge u_0\). We define \(u_1\), \(e(u_0)\) and \(e(u_1+2\beta )\) as in Case 1. If
then proceeding analogously as in Case 1.1 we obtain
Suppose now that
and
We define \(t_1\) and \(t_2\) by the equalities
and
We have \(0<t_1<t_2<\gamma \). Proceeding analogously as in Case 1.2, we get
where \(\nu =\min \left( \eta , 1-\frac{\varphi \left( \frac{u_0+\beta }{2}\right) }{\varphi (u_0)}\right) \).
Finally let us assume that
and define \(t_3\) and \(u_2\) by the equalities
and
It is obvious that \(0<u_2<\beta <u_0\). Defining the sets \(B_x\) and \(B_y\) as in Case 1.3 (see (27) and (28)), we obtain \(m(B_x)\ge t_3\) and \(m(B_y)\ge t_3\). By the assumption that \(\gamma _{0}=0\), we can find \(t_{\gamma }\) such that \(0<t_{\gamma }<t_3\) and \(\omega (t_{\gamma })>\omega (t_{3})\). Let now \(t_4:=\frac{t_{\gamma }}{4}<\frac{t_3}{4}\) and let \(C_{x,y}^{+}\) and \(C_{x,y}^{-}\) be defined by formulas (29) and (30). If \(m(C_{x,y}^{+})\ge \frac{t_4}{2}\) or \(m(C_{x,y}^{-})\ge \frac{t_4}{2}\), then, analogously as in Case 1.3.1, we obtain
where \(\zeta =\int _{0}^{t_4/2}\varphi \left( \frac{u_2}{4}\right) \omega (t)dt\) and \(\delta \left( \varphi \circ x,\frac{u_2\zeta }{4\beta }\right) \) is the constant from the definition of local uniform monotonicity of the Lorentz space \(\varLambda _{\omega }\) corresponding to \(\varphi \circ x\) and \(\varepsilon =\frac{u_2\zeta }{4\beta }\).
In the case when \(\max \left( m(C_{x,y}^{+}), m(C_{x,y}^{-})\right) <\frac{t_4}{2}\), we define the sets \(C_x\) and \(C_y\) as in Case 1.3.2 (see (33) and (34)). We have \(C_x\cap C_y=\emptyset \) and, moreover,
whence
Defining \(a=\frac{1}{32}t_{\gamma }<\frac{1}{32}t_3\) and putting \(t_{\gamma }\) in place of \(\gamma _{0}\) and \(t_3\) in place of \(t_5\), we can repeat the procedure from Case 1.3.2. We get
where p is the derivative of \(\varphi \) on the interval \((0,\beta )\).
Recapitulating Case 2, for any x such that \(x^{*}(0)\le \beta \), by (40), (41), (42) and (43) we get
Necessity. The necessity of conditions \(\varphi \in \varDelta _{2}(\infty )\) and \(\int _{0}^{\gamma _{0}/2}\varphi (\beta )\omega (t)dt<1\) has been shown in Theorem 2.2 of [3]. Suppose that \(\psi \notin \varDelta _{2}(\infty )\). Then there exists a sequence \((u_n)_{n=1}^{\infty }\) such that \(u_n \rightarrow \infty \) as \(n\rightarrow \infty \) and
for any \(n\in {\mathbb {N}}\). We can assume that \(u_1>2\beta \).
There exist \(t_0>0\) such that \(\int _{0}^{t_0}\varphi (\beta )\omega (t)dt=1\) and \(t_n>0\) such that \(\int _{0}^{t_n}\varphi (u_n)\omega (t)dt=1\) for all \(n\in {\mathbb {N}}\). Then we obtain \(t_n<t_0\) for all \(n\in {\mathbb {N}}\) and \(t_n\rightarrow 0\) as \(n\rightarrow \infty \). Define
For any \(n\in {\mathbb {N}}\) consider a straight line passing through the points \(\left( \frac{1}{2}u_n, \frac{n-1}{2n}\varphi (u_n)\right) \) and \(\left( u_n, \varphi (u_n)\right) \), given by the equation
It is easy to observe that for any \(u\in (0, u_n)\) the graph of the function \(\varphi \) lies above the graph of this line. Hence, we obtain
and
In consequence,
3 Applications to Orlicz function spaces
Note that in the case when \(\omega (t)=1\) for any \(t\in (0,\gamma )\) the corresponding Orlicz–Lorentz function spaces become the well-known Orlicz function spaces \(L^{\varphi }=L^{\varphi }([0,\gamma ))\). On the basis of the results presented in [3] and the foregoing part of the present paper we can easily deduce criteria for non-squareness properties of Orlicz function spaces \(L^{\varphi }\).
In the case when \(\gamma =\infty \), by Theorem 2.1 from [3] and Theorem 2, we get the following
Corollary 1
([7], Theorem 2) Let \(\gamma =\infty \). Then the following conditions are equivalent:
- 1.
\(\varphi \in \varDelta _2({\mathbb {R}})\) and \(\varphi (\frac{u}{2})<\frac{1}{2}\varphi (u)\) for any \(u>0\),
- 2.
the Orlicz space \(L^{\varphi }\) is non-square,
- 3.
the Orlicz space \(L^{\varphi }\) is locally uniformly non-square.
In turn, by Theorem 2.3 from [3], we have
Corollary 2
([6], Theorem 1.2.(i)) Let \(\gamma =\infty \). Then the Orlicz space \(L^{\varphi }\) is uniformly non-square if and only if \(\varphi \in \varDelta _2({\mathbb {R}})\) and \(\psi \in \varDelta _2({\mathbb {R}})\), where \(\psi \) denotes the complementary function of \(\varphi \) in the sense of Young (see formula (3)).
Let now \(\gamma <\infty \) and let \(\beta \) be defined by formula (1) (in the paper [3] the same constant was denoted by \(\delta \)). Then by Theorems 2.2 and 2.4 from [3] and Theorems 3 and 4 , we obtain
Corollary 3
Let \(\gamma <\infty \). Then the following assertions are true:
- 1.
([7], Theorem 3) The Orlicz space \(L^{\varphi }\) is non-square if and only if \(\varphi \in \varDelta _2(\infty )\) and \(\varphi (\beta )\cdot \gamma <2\).
- 2.
The Orlicz space \(L^{\varphi }\) is locally uniformly non-square if and only if \(\varphi \in \varDelta _2(\infty )\) and \((\varphi (\beta )\cdot \gamma <1\) or \(\varphi (\beta )\cdot \gamma \in [1,2)\) and \(\psi \in \varDelta _2(\infty ))\).
- 3.
[6], Theorem 1.2.(ii)) The Orlicz space \(L^{\varphi }\) is uniformly non-square if and only if \(\varphi \in \varDelta _2(\infty )\), \(\psi \in \varDelta _2(\infty )\) and \(\varphi (\beta )\cdot \gamma <2\).
Therefore, if \(\varphi (\beta )\cdot \gamma <1\), then the Orlicz space \(L^{\varphi }\) is locally uniformly non-square if and only if it is non-square, while if \(\varphi (\beta )\cdot \gamma \in [1,2)\), then the Orlicz space \(L^{\varphi }\) is locally uniformly non-square if and only if it is uniformly non-square.
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Foralewski, P., Kończak, J. Local uniform non-squareness of Orlicz–Lorentz function spaces. RACSAM 113, 3425–3443 (2019). https://doi.org/10.1007/s13398-019-00703-7
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DOI: https://doi.org/10.1007/s13398-019-00703-7
Keywords
- Local uniform non-squareness
- Non-squareness
- Uniform non-squareness
- Orlicz–Lorentz space
- Lorentz space
- Local uniform monotonicity