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We show that the unit ball of the n-dimensional Lorentz space `w,q is not an intersection body for q > 2 and n ≥ 5.

We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B^s_{p,q}(\R^n)$ and $F^s_{p,q}(\R^n)$ for $s=\sigma_p=n\max(\frac 1p-1,0)$. These results may be also reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}(\R^n)$. We close several open problems outlined already by H. Triebel in [H. Triebel, The structure of functions, Birkh\"auser, Basel, 2001.] and explicitly formulated by D. D. Haroske in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall / CRC, Boca Raton, 2007.].

We show that the Lorentz sequence spaces d(ω,p) with 0 0 have unique unconditional basis. This completely settles the question of uniqueness of unconditional basis in Lorentz sequence spaces, and solves a problem raised by Popa in 1981 and Nawrocki and Ortynski in 1985.

In this paper, Strichartz estimates for the solution of the Schrödinger evolution equation are considered on a mixed normed space with Lorentz norm with respect to the time variable.

We consider the Schrödinger equation in three-dimensional space with small potential in the Lorentz space L 3 / 2 , ∞ L^{3/2,\infty } and we prove Strichartz-type estimates for the solution to this equation. Moreover, using Cook’s method, we prove the existence of the wave operator. In the last section we prove the equivalence between the homogeneous Sobolev spaces H ˙ s \dot {H}^s and H ˙ V s \dot {H}^s_V in the case 0 ≤ s > 3 2 0 \leq s > \frac {3}{2} .

Recently A. D. Ionescu (2000) established the endpoint estimate for the Kunze-Stein phenomenon, which states that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L 2,1 (G)*L 2,1 (G)⊆ L 2, ∞(G). In this paper, we will prove the corresponding result for the Jacobi transform. Our method is analytical, in which we do not use the structure of Lie groups.

We show that if X X is a rearrangement invariant space on [ 0 , 1 ] [0, 1] that is an interpolation space between L 1 L_{1} and L ∞ L_{\infty } and for which we have only a one-sided estimate of the Boyd index α ( X ) > 1 / p , 1 > p > ∞ \alpha (X) > 1/p, 1 > p > \infty , then X X is an interpolation space between L 1 L_{1} and L p L_{p} . This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type ( 1 , 1 ) (1, 1) and weak type ( p , p ) , 1 > p > ∞ (p, p), 1 > p > \infty .

We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension n > 2. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function f, ||Mf|| n', ∞≤C n ||f|| n',1 , n'=. The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.

We extend Federer’s co-area formula to mappings f f belonging to the Sobolev class W 1 , p ( R n ; R m ) W^{1,p}(\mathbb {R}^n;\mathbb {R}^m) , 1 ≤ m > n 1 \le m > n , p > m p>m , and more generally, to mappings with gradient in the Lorentz space L m , 1 ( R n ) L^{m,1}(\mathbb {R}^n) . This is accomplished by showing that the graph of f f in R n + m \mathbb {R}^{n+m} is a Hausdorff n n -rectifiable set.

A description of the Köthe dual of the Orlicz–Lorentz space Λ φ , w \Lambda _{\varphi , w} generated by an Orlicz function φ \varphi and a regular weight function w w is presented. It is also shown that in the case of separable Orlicz–Lorentz spaces the regularity condition on w w is necessary and sufficient for the coincidence of the Banach dual space with the described Köthe dual space.

We study ane hyperspheres M with constant sectional curvature (with respect to the ane metric h). A conjecture by M. Magid and P. Ryan states that every such ane hypersphere with nonzero Pick invariant is anely equivalent to either where the dimension n satises n =2 m 1o rn =2 m .U p to now, this conjecture was proved if M is positive denite or if M is a 3-dimensional Lorentz space. In this paper, we give an armative answer to this conjecture for arbitrary dimensional Lorentzian ane hyperspheres.

For the classical Hardy-Littlewood maximal function Mf , a well known and important estimate due to Herz and Stein, gives the equivalence (Mf)∗(t) ∼ f∗∗(t). In the present note, we study the validity of analogous estimates for maximal operators of the form Mp,qf(x) = sup x∈Q ‖fχQ‖p,q ‖χQ‖p,q , where ‖.‖p,q denotes the Lorentz space L(p, q)-norm.

We prove that every separable Banach lattice is lattice isometric to a closed sublattice of the Banach envelope of Weak L 1 . L^{1}.

Let Tf(x)=sup R>0 |S R f(x)| where SiR is the spherical partial sum operator. We show that T is bounded from the Lorentz space L pi,1 (R n ) into L pi, ∞(R n ), i=0, 1 when acting on radial functions and where p 0 = 2n/n+&, P 1 =2n/n-1

If ϕ ( t ) = t 1 / p ( p > 1 ) \phi (t) = {t^{1/p}}(p > 1) and ( X , S , μ ) (X,\mathcal {S},\mu ) is a completely nonatomic finite measure space, then the dual of the Lorentz space N ϕ {N_\phi } is denoted by M ϕ {M_\phi } and the closure of the simple functions in M ϕ {M_\phi } by M ϕ 0 M_\phi ^0 . It is known that ( M ϕ 0 ) ∗ = N ϕ {(M_\phi ^0)^ * } = {N_\phi } . In this note we show that given a positive number β > 1 \beta > 1 it is possible to construct a set of contractive embeddings of ( l ∞ / c 0 ) ({l_\infty }/{c_0}) into ( M ϕ / M ϕ 0 ) ∗ {({M_\phi }/M_\phi ^0)^ * } , each of which is bounded below by M = M ( β ) → 1 as β → 0 + M = M(\beta ) \to 1\;{\text {as}}\;\beta \to {{\text {0}}^ + } . The union of the ranges of these embeddings is a total set in ( M ϕ / M ϕ 0 ) ∗ {({M_\phi }/M_\phi ^0)^ * } .

The Banach envelope and a representation of the topological dual of the weak-lp sequence spaces which involves the Lorentz sequence spaces are computed. The local convexity of weak-Lp spaces is studied also. w-Lp spaces are function spaces which are closely related to Lp spaces. They were introduced in analysis when it was observed that several important operators such as the Hardy-Littlewood maximal function and the Hilbert transform map Lp into Lp for p > 1 but they do not map L1 into L1 and rather satisfy the weak condition: I{x: (Tf)(x) > y} a})4/P o w-Lp space and its topological dual avoiding the case when the measure is atomic-have been studied by several authors (see Cwikel, Sagher, and Hunt [1, 2, 5, 7]). The Banach envelope and the topological dual of w-L1 are unknown (although the envelope norm is known), but their properties were studied by Cwikel and Fefferman, and Kupka and Peck (see [3, 4, 9]). Our purpose in this paper is to study the Banach envelope and the topological dual of w-Lp when the measure is atomic. Surprisingly, the topological dual turns out to be a classical Lorentz sequence space (see [10]); we also show that the Banach envelope is a known space studied by Garling [6]. For 0 v2 > > Vn > ... > 0, d(v, 1) denotes the Lorentz space of all sequences a = (an)=1 of real numbers such that 00 ||aI|v,1 = sup E Iawr(n) lvn < 00i 7T n=1 Received by the editors December 3, 1985 and, in revised form, March 5, 1986 and April 30, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46A20, 46A45. (?)1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Let ( X , Σ , μ ) (X,\Sigma ,\mu ) be a positive measure space, and ϕ \phi be a concave nondecreasing function on R + → R + {R^ + } \to {R^ + } with ϕ ( 0 ) = 0 \phi (0) = 0 . Let N ϕ ( R ) {N_\phi }(R) be the Lorentz space associated with the function ϕ \phi . In this paper a complete characterization of the extreme points of the unit ball of N ϕ ( R ) {N_\phi }(R) is provided. It is also shown that the space N ϕ ( R ) {N_\phi }(R) is not reflexive in all nontrivial cases, thus generalizing a result of Lorentz. Several analytical properties of spaces N ϕ ( R ) {N_\phi }(R) , and their abstract analogues N ϕ ( E ) {N_\phi }(E) , are obtained when E is a Banach space.

It is well known that if G G is a locally compact and amenable group then the Banach spaces of operators of weak type ( 2 , 2 ) (2,2) and of strong type ( 2 , 2 ) (2,2) commuting with the right translations on G G are the same. In contrast we show that if G G is a nonabelian free group then there exists an operator of weak type ( 2 , 2 ) (2,2) commuting with the right translations on G G which is not of strong type ( 2 , 2 ) (2,2) .

Characterizations are obtained for those pairs of weight functions w , υ w,\upsilon for which the Hardy operator T f ( x ) = ∫ 0 x f ( s ) d s Tf(x) = \int _0^x {f(s)\;ds} is bounded from the Lorentz space L r , s ( ( 0 , ∞ ) , υ d x ) {L^{r,s}}((0,\infty ),\upsilon \,dx) to L p , q ( ( 0 , ∞ ) , w d x ) , 0 > p , q , r , s ⩽ ∞ {L^{p,q}}((0,\infty ),w\,dx),0 > p,q,r,s \leqslant \infty . The modified Hardy operators T η f ( x ) = x − η T f ( x ) {T_\eta }f(x) = {x^{ - \eta }}Tf(x) for η \eta real are also treated.

The boundedness properties of the generalized Hankel conjugate transformations H λ {H_\lambda } on certain weighted Lebesgue spaces are studied. These are used to establish a boundedness criterion for the H λ {H_\lambda } on the more general class of rearrangement invariant spaces. The positive operators in terms of which the criterion is given are used to construct pairs of spaces between which the H λ {H_\lambda } are continuous; in particular, a natural analogue of a well-known result of Zygmund concerning the classical conjugate function operator is obtained for the H λ {H_\lambda } .

Hardy’s well-known inequality relating the norm of a function and the norm of its integral mean in the Lebesgue spaces L p ( μ ) , d μ ( t ) = t σ − 1 d t {L^p}(\mu ),d\mu (t) = {t^{\sigma - 1}}dt , is extended to the class of rearrangement invariant spaces X ( μ ) X(\mu ) . These spaces include, for example, the L p ( μ ) {L^p}(\mu ) , the Lorentz and the Orlicz spaces. As an application, necessary and sufficient conditions are obtained for an operator related to the Laplace transform to be bounded as a linear operator between rearrangement invariant spaces of μ \mu -measurable functions.

If ( X , S , μ ) (X,S,\mu ) is a finite, completely nonatomic measure space and ϕ ( t ) = t 1 / p ( p > 1 ) \phi (t) = {t^{1/p}}(p > 1) then the Lorentz space N ϕ {N_\phi } is the dual space of the closed span of simple functions in M ϕ ( = N ϕ ∗ ) {M_\phi }( = N_\phi ^ * ) .

It can be shown [3, Chapter V, §6.3] that the space Ln\ cannot be replaced by any larger Lorentz space and in this sense the theorem is sharp. In this note, we shall prove Stein's Theorem using elementary principles. This should be compared with the proofs [2 and 1] which use more sophisticated Fourier analytic methods. Since this is a local theorem, we may assume that O = Qr, is a cube in Rn. Recall that a function yj is in Ln\ if and only if its decreasing rearrangement tp* (see [4, p. 189]) satisfies

In the present paper, examples are given to show that the convolution theorem, which is the L(p,q) analogue of Young's inequality for the LI spaces, is best possible. This result is then used to obtain a theorem about bounded linear translation invariant operators between two L(p, q) spaces. Introduction. Let G be a noncompact, locally compact group and let LP=LP(G), l?p< oo, denote the Lebesgue spaces associated with G. N. W. Rickert [5] has proved that the convolution of two LP functions need not exist. In this paper we extend this result to the L(p, q)(G) spaces. For our purposes this is done first where the underlying group is one of the Euclidean n spaces R", R1=R and the convolution of two measurable functionsf and g defined on Rn is ordinary convolution. That is, f * g(y) = f (y x)g(x) dx, y E R . This result is then used to obtain a theorem about bounded linear translation invariant operators between two L(p, q) spaces. For a discussion of the L(p, q) spaces the reader is referred to R. A. Hunt [4]. 1. Let Rn, R1=R be the Euclidean n spaces and let L(p, q)=L(p, q)(Rn) be a Lorentz space for which feL(p, q), implies f is locally integrable. Namely, we will consider only those L(p, q) spaces for the particular choice of indices I