Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

Symmetric block bases in finite-dimensional normed spaces

It is shown that for 1 ≦p < ∞, any basisC-equivalent to the unit vector basis ofl _p ⁿ has a (1 + ε)-symmetric block basis of cardinality proportional ton/logn. When 1 <p < ∞, an upper bound proportional ton log logn/logn is also obtained. These results extend results of Amir and Milman in [2].     

Greedy Bases for Besov Spaces

We prove that the Banach space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{\ell_{q}}, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1≤p≤∞ and 1<q<∞. Furthermore, the Banach spaces (?_n=1^￥l_pⁿ)_l1(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{1}}, with 1<p≤∞, and (?_n=1^￥l_pⁿ)_c0(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{c_{0}}, with 1≤p<∞, do not have a greedy basis. We prove as well that the space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{q}} has a 1-greedy basis if and only if 1≤p=q≤∞.     

SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES ——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 ＜ p ＜ 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 ＜ p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

On Lp-convergence of Grunwald interpolation

In this paper, the L_p-convergence of Grünwald interpolation G_n(f,x) based on the zeros of Jacobi polynomials J _n ^(α,β) (x)(−1<α,β<1) is considered. L_p-convergence (0<p<2) of Grünwald interpolation G_n(f,x) is proved for p·Max(α,β)<1. Moreover, L_p-convergence (p>0) of G_n(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.     

Weighted Weak Behaviour of Fourier-Jacobi Series

Let w(x) = (1 - x)^α (1 + x)^β be a Jacobi weight on the interval [-1, 1] and 1 < p < ∞. If either α > ?1/2 or β > ?1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators S_n are uniformly bounded from L^p,1 to L^p,∞, thus extending a previous result for the case that both α, β > ?1/2. For α, β > ?1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f→uS_n(u^?1f), where u is also Jacobi weight.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

Embedding Subspaces of Lp into 𝓁Np, 0 < p < 1

It is shown that if X is an n – dimensional subspace of L_p, 0 < p < 1, then there exists a subspace Y of 𝓁^N_p such that d(X, Y) ≤ 1 + ε and N ≤ C(ε, p)n(log n)³.     

Characterization for commutators of n -dimensional fractional Hardy operators

In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L ^p1 (ℝ ⁿ ) to L ^p2 (ℝ ⁿ ) if and only if b is a CṀO(ℝ ⁿ ) function, where 1/p ₁ − 1/p ₂ = β/n, 1 < p ₁ < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ ⁿ ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)     

Resolvent estimates for the linearized operator in L p associated with motion of compressible viscous fluids

We consider the resolvent problem for the linearized system of equations that describe motion of compressible viscous barotropic fluids in a bounded domain with the Navier boundary condition. This problem has uniquely a solution in [(W)\dot]¹_p ×(W²_p)ⁿ{\dot{W}^{1}_{p} \times (W^{2}_{p})^{n}} satisfying L _p estimates for any 1 < p < ∞. Moreover, resolvent estimates for the linearized operator of the above system in [(W)\dot]¹_p ×(L_p)ⁿ{\dot{W}^{1}_{p} \times (L_{p})^{n}} are established. Our main results yield clearly that the linearized operator is the infinitesimal generator of a uniformly bounded analytic semigroup on [(W)\dot]¹_p ×(L_p)ⁿ{\dot{W}^{1}_{p} \times (L_{p})^{n}}.     

Gelfand pairs,K-spherical means and injectivity on the Heisenberg group

We study the injectivity properties of the spherical mean value operators associated to the Gelfand pairs (H ⁿ,K), whereK is a compact subgroup ofU(n). We show that these spherical mean value operators are injective onL ^p Hⁿ) for 1≤p<∞. Forp=∞, these operators are not injective. Nevertheless, if the spherical meansf*μ _i overK-orbits of sufficiently many points (z _i,t _i) ∈H ⁿ vanish, we identify a necessary and sufficient condition on the points (z _i,t _i) which guaranteesf=0. ForK=U(n), this is equivalent to the condition for the two-radius theorem. Research supported by N.B.H.M. Research Grant, Govt. of India.     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

On an oppenheim-type conjecture for systems of quadratic forms

In this paper we establish dimension freeL ^p (ℝ ⁿ ,|x|^α) norm inequalities (1<p<∞) for the oscillation and variation of the Riesz transforms in ℝ ⁿ . In doing so we findA _p -weighted norm inequalities for the oscillation and the variation of the Hilbert transform in ℝ. Some weighted transference results are also proved. Partially supported by European Commission via the TMR network “Harmonic Analysis”.     

An approximation of a nonlinear integral equation driven by a function of boundedp-variation

The uniform distance between the solution of a nonlinear equation driven by a functionh with boundedp-variation and its Milstein-type approximation is estimated by δ _n v γ _p (n) v γ _p ² (n), where δ _n =max(t _k −t _k−1 ) is the maximum step size of the approximation on the interval [0,T], γ _p (n)=max υ _p ^1/p (h;[t _k-1,t _k ]), 1 <p < 2, and υ _p (h;[t _k-1,t _k ]) is thep-variation of the functionh on [t _k-1,t _k]. In particular, ifh is a Lipschitz function of order α, then the uniform distance has the bound δ _n ^α for δ_n <1. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius Technical University, Saulétekio 11, 2054 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 317–330, July–September, 1999.     

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

Moufang Loops of Odd Orderpq1 ··· qnr1 ··· rm

LetLbe a Moufang loop of odd orderpαqα11···qnαnwherepandqiare primes with 3 ≤ p < q1 < ··· < qnand αi ≤ 2. In this paper, we prove thatLis a group ifpandqiare primes with 3 ≤ p < q1 < ··· < qn: (i) α ≤ 3, or (ii) α ≤ 4,p ≥ 5.