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Are Generalized Lorentz “Spaces” Really Spaces? PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 12, Pages 3615{3625 S 0002-9939(04)07477-5 Article electronically published on July 20, 2004 ARE GENERALIZED LORENTZ \SPACES" REALLY SPACES? MICHAEL CWIKEL,´ ANNA KAMINSKA,´ LECH MALIGRANDA, AND LUBOSPICKˇ (Communicated by N. Tomczak-Jaegermann) Abstract. We show that the Lorentz space Λp(w) need not be a linear set for certain \non-classical" weights w. We establish necessary and sufficient conditions on p and w for this situation to occur. 1. Introduction and statement of main results Let f be a real-valued measurable function on R. We define the distribution function of f by f∗(λ)=jfx 2 R : jf(x)j >λgj for each λ>0, (where |·| denotes Lebesgue measure). The non-increasing rearrangement of f is defined by ∗ (1.1) f (t)=inffs>0:f∗(s) ≤ tg ;t2 [0; 1): We further denote Z 1 t f ∗∗(t)= f ∗(s) ds; t 2 (0; 1): t 0 When w is a non-negative measurable function on (0; 1) that is not identically zero, we say that w is a weight. Note that our definition here (which is consistent with the usage in [16], and in essentially all subsequent papers cited below) differs from another frequently used definition in the context of many kinds of function spaces, which requires w to be strictly positive. R 2 1 t Definition 1.1. Let p (0; ), let w be a weight, and let W (t)= 0 w(s) ds. Suppose that W (t) < 1 for all t>0. We define four types of function spaces on Received by the editors January 21, 2003 and, in revised form, July 16, 2003. 2000 Mathematics Subject Classification. Primary 46E30, 46B42. Key words and phrases. Lorentz spaces, Marcinkiewicz spaces, Lorentz-Orlicz spaces, weights, rearrangement. The first named author was supported by the Dent Charitable Trust|Non-Military Research Fund and by the Fund for Promotion of Research at the Technion. The second named au- thor was supported by project no. SMK{2136 of the Kempe Foundation in Sweden. The third named author was supported by the Swedish Natural Science Research Council (NFR){grant M5105-20005228/2000. The fourth named author was supported by grant no. 201/01/0333 of the Grant Agency of the Czech Republic and by grant no. MSM 113200007 of the Czech Ministry of Education. c 2004 American Mathematical Society 3615 3616 M. CWIKEL,A.KAMI´ NSKA,´ L. MALIGRANDA, AND L. PICK R by 8 n R o p 1 ∗ p 1=p >Λ (w)=f measurable : kfk p := (f (t)) w(t) dt < 1 ; > Λ (w) 0 < p;1 ∗ 1=p Λ (w)=f measurable : kfk p;1 := sup f (t)W (t) < 1 ; n Λ (w) R 0<t<1 o > p 1 ∗∗ p 1=p >Γ (w)=f measurable : kfkΓp(w) := (f (t)) w(t) dt < 1 ; > 0 : 1 ∗∗ p; k k 1 1=p 1 Γ (w)=f measurable : f Γp; (w) := sup0<t<1 f (t)W (t) < : The spaces Λp(w) were introduced by Lorentz in 1951 in [16]. Spaces whose norms involve f ∗∗ appeared explicitly for the first time in Calder´on's paper [5]. In [22] Sawyer considered Γp(w) for arbitrary weights, in connection with the study of certain duality problems concerning the spaces Λp(w). The spaces Λp;1(w)andΓp;1(w) are called Marcinkiewicz spaces or weak Lorentz spaces (see [15], [11]; see also [8], [9], [10], [6] and [7]). Lorentz [16] proved that, for p ≥ 1, kfkΛp(w) is a norm if and only if w is non- increasing. The class of weights for which kfkΛp(w) is merely equivalent to a Banach norm is however considerably larger. In fact it consists of all those weights w which, for some C and all t>0, satisfy Z Z 1 t (1.2) tp x−pw(x) dx ≤ C w(x) dx when p 2 (1; 1) t 0 ([22, Theorem 4], see also [1]), or Z Z 1 t C s (1.3) w(x) dx ≤ w(x) dx for 0 <s≤ t when p =1 t 0 s 0 ([6, Theorem 2.3]). It is also known that, for p 2 [1; 1), the space Λp(w)is equivalent to a Banach space if and only if the Hardy{Littlewood maximal operator acts boundedly from Λp(w)toΛp;1(w) ([9, Theorem 3.3], [10, Theorem 3.9], [6]). Furthermore, for p>1, this is equivalent to Λp(w)=Γp(w) ([22]). In [11, Th. 1.1] (see also [8, Corollary 2.2], [13, p. 6]) it was observed that the functional kfkΛp(w); 0 <p≤∞, does not have to be a quasinorm. It was shown that it is a quasinorm if and only if the function W satisfies the ∆2-condition, i.e., (1.4) W (2t) ≤ CW(t)forsomeC>1andallt 2 (0; 1): There are a number of natural and important function spaces which are not normable, such as weak L1 and Hp and Lp and Lp;q for p<1. They exhibit various different behaviours with respect to the existence of non-trivial dual spaces. In this note we encounter different, perhaps more exotic behaviour in the case of generalized Lorentz spaces. It can happen that the sum of two functions in the \space" is not in the space. Obviously f 2 Λp(w) implies that λf 2 Λp(w) for every λ 2 R. However in general, perhaps surprisingly, Λp(w) is not a linear space. In this note we shall give necessary and sufficient conditions for Λp(w) to be a linear space. In fact, other function spaces that are not closed with respect to addition some- times appear quite naturally in analysis.R One example going back to the 1930's ' is the Orlicz class L = ff measurable : R '(jf(t)j)dt < 1g, which is linear if and only if ' satisfies the ∆2-condition (see [19], where this was proved even in the case where ' is only a Baire function).P The spaces of functions of bounded f ! R n j − j 1g '-variation BV' = f :[a; b] :supP k=1 '( f(xk) f(xk−1) ) < ,where the supremum is taken over all partitions P of [a; b], are linear spaces if and only ARE GENERALIZED LORENTZ \SPACES" REALLY SPACES? 3617 if ' satisfies the ∆2-condition near zeroR (see [20]; cf. also [21]). The Wiener spaces 1 T p 1=p f !1 j j 1g ([23]) f measurable : limT ( 2T −T f(t) dt) < , sometimes also called the spaces of bounded power signals, are not linear spaces ([18]). Note that if in- stead of \limit" we take either \limit superior" or \supremum", the corresponding spaces will be linear (they are classical Besicovitch-Marcinkiewicz or Stepanoff-Weyl 1 ∗ spaces). The \weak L "spaceW = ff measurable : f is finite and kfkW := ∗∗ − ∗ 1g supt>0[f (t) f (t)] < of Bennett, DeVore and Sharpley ([3]; see also [4]) and the space Wp, determined by the \norm" ! 1 Z p 1 f ∗( t ) − f ∗(t) p k k 2 1 f Wp := dt < ; 0 t defined recently in [17] in connection with sharp limiting Sobolev embeddings, are also examples of spaces that are not linear. The latter two spaces are of special interest in interpolation theory. The space L(1;q) of [2] is yet another example, related to the two preceding ones. Before formulating and proving our main result it is convenient to present a lemma which will be needed later. Lemma 1.2. Suppose that w and v are non-negative measurable functions on 1 2 (0; ) and a is a positive number such that w(tR)=0for a.e. t R(0;a) and ≥ t t w(t)=v(t) for all t a, and the functions V (t)= 0 v(x)dx and W (t)= 0 w(x)dx are finite for all t>0. Let f : R ! R be a measurable function. Then the following are equivalent: (i) f 2 Λp(w). (ii) min fλ, jfjg 2 Λp(v) for some positive number λ. (iii) min fλ, jfjg 2 Λp(v) for every positive number λ. Proof. Clearly we can assume without loss of generality that f is non-negative. It follows easily from the definitions of the non-increasing rearrangement and the distribution function that, for each constant λ>0, the non-increasing rearrange- ment of the function minff;λg satisfies ∗ (1.5) (minff;λg) (t)=minff ∗(t),λg for each t>0: Now suppose that f 2 Λp(w). If f ∗(a) > 0wechooseλ = f ∗(a). If f ∗(a)=0 then we can choose λ however we wish, for example λ = 1. Note that in this latter case we have f ∗(t) = 0 for all t 2 [a; 1). In both cases we have Z Z Z 1 a 1 (minff ∗(t),λg)p v(t)dt ≤ λpv(t)dt + (f ∗(t))p w(t)dt 0 0 a ≤ p k kp 1 λ V (a)+ f Λp(w) < : This shows that (i) implies (ii). p Now suppose minff;λ0g2Λ (v)forsomeλ0 > 0. Then obviously minff;λg2 p ∗ Λ (v) for all λ 2 (0,λ0]. For each λ>λ0,letA = ft>0:f (t) >λ0g and B =(0; 1)nA.Thenwehave Z 1 k f gkp f ∗ gp min f;λ Λp(v) = min f (t),λ v(t)dt Z0 Z p ∗ p ≤ λ v(t)dt + minff (t),λ0g v(t)dt: A B 3618 M. CWIKEL,A.KAMI´ NSKA,´ L. MALIGRANDA, AND L.
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