J. Math. Anal. Appl. 444 (2016) 1133–1154 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa ✩ Rademacher functions in Morrey spaces Sergei V. Astashkin a,b,∗, Lech Maligranda c a Department of Mathematics and Mechanics, Samara State University, Acad. Pavlova 1, 443011 Samara, Russia b Samara State Aerospace University (SSAU), Moskovskoye shosse 34, 443086, Samara, Russia c Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden a r t i c l e i n f o a b s t r a c t Article history: The Rademacher sums are investigated in the Morrey spaces Mp,w on [0, 1] for Received 20 January 2016 1 ≤ p < ∞ and weight w being a quasi-concave function. They span l2 space in Available online 14 July 2016 −1/2 2 Mp,w if and only if the weight w is smaller than log2 on (0, 1). Moreover, if Submitted by P.G. Lemarie-Rieusset t 1 <p < ∞ the Rademacher subspace Rp,w is complemented in Mp,w if and only if it R Keywords: is isomorphic to l2. However, the Rademacher subspace 1,w is not complemented Rademacher functions in M1,w for any quasi-concave weight w. In the last part of the paper geometric Morrey spaces structure of Rademacher subspaces in Morrey spaces Mp,w is described. It turns Korenblyum–Krein–Levin spaces out that for any infinite-dimensional subspace X of Rp,w the following alternative Marcinkiewicz spaces holds: either X is isomorphic to l2 or X contains a subspace which is isomorphic to Complemented subspaces c0 and is complemented in Rp,w. © 2016 Elsevier Inc. All rights reserved. 1. Introduction and preliminaries The well-known Morrey spaces introduced by Morrey in 1938 [20] in relation to the study of partial differential equations were widely investigated during last decades, including the study of classical operators of harmonic analysis: maximal, singular and potential operators—in various generalizations of these spaces. In the theory of partial differential equations, along with the weighted Lebesgue spaces, Morrey-type spaces also play an important role. They appeared to be quite useful in the study of the local behavior of the solutions of partial differential equations, a priori estimates and other topics. Let 0 <p < ∞, w be a non-negative non-decreasing function on [0, ∞), and Ωa domain in Rn. The Morrey space Mp,w = Mp,w(Ω) is the class of Lebesgue measurable real functions f on Ωsuch that ✩ Research of the first named author was partially supported by the Ministry of Education and Science of the Russian Federation. * Corresponding author. E-mail addresses: [email protected] (S.V. Astashkin), [email protected] (L. Maligranda). http://dx.doi.org/10.1016/j.jmaa.2016.07.008 0022-247X/© 2016 Elsevier Inc. All rights reserved. 1134 S.V. Astashkin, L. Maligranda / J. Math. Anal. Appl. 444 (2016) 1133–1154 ⎛ ⎞ 1/p ⎜ 1 ⎟ f =sup w(r) ⎝ |f(t)|p dt⎠ < ∞, (1) Mp,w | | 0<r<diam(Ω),x0∈Ω Br(x0) Br (x0)∩Ω where Br(x0)is a ball with the center at x0 and radius r. It is a quasi-Banach ideal space on Ω. The so-called | | ≤| | ∈ ∈ ≤ ideal property means that if f g a.e. on Ωand g Mp,w, then f Mp,w and f Mp,w g Mp,w . In 1/p particular, if w(r) =1then Mp,w(Ω) = L∞(Ω), if w(r) = r then Mp,w(Ω) = Lp(Ω) and in the case when 1/q w(r) = r with 0 <p ≤ q<∞ Mp,w(Ω) are the classical Morrey spaces, denoted shortly by Mp,q(Ω) (see [14, Part 4.3], [15,23] and [29]). Moreover, as a consequence of the Hölder–Rogers inequality we obtain monotonicity with respect to p, that is, →1 ≤ ∞ Mp1,w(Ω) Mp0,w(Ω) if 0 <p0 p1 < . C For two quasi-Banach spaces X and Y the symbol X → Y means that the embedding X ⊂ Y is continuous and fY ≤ CfX for all f ∈ X. It is easy to see that in the case when Ω =[0, 1] quasi-norm (1) can be defined as follows ⎛ ⎞ 1/p | | ⎝ 1 | |p ⎠ f Mp,w =sup w( I ) f(t) dt , (2) I |I| I where the supremum is taken over all intervals I in [0, 1]. In what follows |E| is the Lebesgue measure of a set E ⊂ R. The main purpose of this paper is the investigation of the behavior of Rademacher sums n Rn(t)= akrk(t),ak ∈ R for k =1, 2, ..., n, and n ∈ N k=1 in general Morrey spaces Mp,w. Recall that the Rademacher functions on [0, 1] are defined by rk(t) = sign(sin 2kπt), k ∈ N, t ∈ [0, 1]. By Rp,w we denote the subspace spanned by the Rademacher functions rk, k =1, 2, ... in Mp,w. The most important tool in studying Rademacher sums in the classical Lp-spaces and in general rear- rangement invariant spaces is the so-called Khintchine inequality (cf. [11, p. 10], [1, p. 133], [16, p. 66] and [4, p. 743]): if 0 <p < ∞, then there exist constants Ap, Bp > 0such that for any sequence of real numbers { }n ∈ N ak k=1 and any n we have n 1/2 n 1/2 | |2 ≤ ≤ | |2 Ap ak Rn Lp[0,1] Bp ak . (3) k=1 k=1 Therefore, for any 1 ≤ p < ∞, the Rademacher functions span in Lp an isomorphic copy of l2. Also, the subspace [rn]is complemented in Lp for 1 <p < ∞ and is not complemented in L1 since no complemented infinite dimensional subspace of L1 can be reflexive. In L∞, the Rademacher functions span an isometric copy of l1, which is uncomplemented. The only non-trivial estimate for Rademacher sums in a general rearrangement invariant (r.i.) space X on [0, 1] is the inequality n 1/2 2 RnX ≤ C |ak| , (4) k=1 S.V. Astashkin, L. Maligranda / J. Math. Anal. Appl. 444 (2016) 1133–1154 1135 where a constant C>0 depends only on X. The reverse inequality to (4) is always true because X ⊂ L1 and we can apply the left-hand side inequality from (3) for L1. Already in 1930, Paley and Zygmund [22] proved estimate (4) for X = G, where G is the closure of L∞[0, 1] in the Orlicz space LM [0, 1] generated by 2 the function M(u) = eu − 1. The proof can be found in Zygmund’s classical books (see [30, p. 134] or [31, p. 214]). Later on Rodin and Semenov [25] showed that estimate (4) holds if and only if G ⊂ X. This inclusion means that X in a certain sense “lies far” from L∞[0, 1]. In particular, G is contained in every Lp[0, 1] for p < ∞. Moreover, Rodin–Semenov [26] and Lindenstrauss–Tzafriri [17, pp. 134–138] proved that [rn]is complemented in X if and only if G ⊂ X ⊂ G, where G denotes the Köthe dual space to G. In contrast, Astashkin [3] studied the Rademacher sums in r.i. spaces which are situated very “close” to L∞. In such a case a rather precise description of their behavior may be obtained by using the real method of interpolation (cf. [10]). Namely, every interpolation space X between the spaces L∞ and G can K be represented in the form X =(L∞, G)Φ , for some parameter Φof the real interpolation method, and ∞ ≈{ }∞ K then k=1 akrk X ak k=1 F , where F =(l1, l2)Φ . Investigations of Rademacher sums in r.i. spaces are well presented in the books by Lindenstrauss–Tzafriri [17], Krein–Petunin–Semenov [13] and Astashkin [4]. At the same time, a very few papers are devoted to con- sidering Rademacher functions in Banach function spaces, which are not r.i. Recently, Astashkin–Maligranda [6] initiated studying the behavior of Rademacher sums in a weighted Korenblyum–Krein–Levin space Kp,w, for 0 <p < ∞ and a quasi-concave function w on [0, 1], equipped with the quasi-norm ⎛ ⎞ x 1/p ⎝ 1 | |p ⎠ f Kp,w =supw(x) f(t) dt (5) 0<x≤1 x 0 (cf. [12,18], [28, pp. 469–470], where w(x) =1). If the supremum in (2) is taken over all subsets of [0, 1] of measure x, then we obtain an r.i. counterpart of the spaces Mp,w and Kp,w, the Marcinkiewicz space (∗) Mp,w[0, 1], with the quasi-norm ⎛ ⎞ x 1/p 1 ∗ p ∗ ⎝ ⎠ f M ( ) =sup w(x) f (t) dt , (6) p,w 0<x≤1 x 0 where f ∗ denotes the non-increasing rearrangement of |f|. In what follows we consider only function spaces on [0, 1]. Therefore, the weight w will be a non-negative non-decreasing function on [0, 1] and without loss of generality we will assume in the rest of the paper that w(1) =1. Then, we have →1 (∗) →1 →1 →1 L∞ Mp,w Mp,w Kp,w Lp (7) because the corresponding suprema in (5), (2) and (6) are taken over larger classes of subsets of [0, 1]. (∗) −1/p ∞ Observe that if limt→0+ w(t) > 0, then Mp,w = Mp,w = L∞, and if sup0<t≤1 w(t) t < , then (∗) Mp,w = Mp,w = Lp with equivalent quasi-norms. To avoid these trivial cases, throughout the paper we will assume also that t1/p lim w(t) = lim inf =0. (8) t→0+ t→0+ w(t) In particular, the latter assumptions ensure that the second and the third inclusions in (7) are proper.