Generalized difference sequence spaces and their dual spaces

The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785-791]. In this paper we compute pα- and N-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀, and compute β- and γ-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀.     

On some new generalized sequence spaces

In this paper we define the sequence sets ?_∞(u,Δ²,p), c(u,Δ²,p) and c₀(u,Δ²,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?_∞(u,Δ²,p),?_∞).     

On the m order difference sequence space of generalized weighted mean and compact operators

In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space ?(u, v, p; Δ^(m)), which consist of the sequences whose generalized weighted Δ^(m)-difference means are in the linear space ?(p) defined by I.J. Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from ?(u, v, p, Δ^(m)) to ?_∞, c and c₀. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space ?_p(u, v, Δ^(m))(1 ≤ p < ∞).     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

The ball generated property in operator spaces

The Ball Generated Property (BGP) was introduced by Corson and Lindenstrauss and subsequently analysed in detail by Godefroy and Kalton. In this work, the (BGP) is studied in spaces of operators. It is shown that (BGP) is stable under c₀ and l_p-sums for 1 < p < ∞ and a characterization is provided for C(K, X)-spaces with (BGP). A similar characterization is obtained for L(X, C(K))-spaces. (BGP) is shown to be stable under injective tensor products.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(Xd) to Lp(Xd) for all d/(d+α)<p?∞ and, consequently, is of weak type (1,1), where 0<α?1 is depending only on θ and X=R or X=T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f∈L₁(Xd) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(Xd) and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.     

New oscillation criteria for linear matrix Hamiltonian systems

Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t₀,∞), (−∞<t₀). These results are sharper than some previous results even for self-adjoint second order matrix differential systems.     

On equivalence of simple closed curves in flat surfaces

By explicit constructions,we give direct proofs of the following results: for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX(α) = lX(β),extX(α) = extX(β) and lq(α) = lq(β),where lX(·),extX(·) and lq(·) are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.     

Banach spaces embedding intoL 0

Our main result in this paper is that a Banach spaceX embeds intoL ₁ if and only ifl ₁(X) embeds intoL ₀; more generally if 1≦p<2,X embeds intoL _p if and only ifl _p(X) embeds intoL ₀. Research supported by NSF grant MCS-8301099.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.