Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

The eigenvalue problem for the Laplacian equations

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω, u = 0, x ∈ (δ)Ω, where Ω (∩) Rn is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T.Yau et al.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ^* be such that δ^*δL(0,1). Define , where δ_n(0,1) and limδ_n=0. Let {α_n} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x₀,uK, define a sequence {x_n}K by x_n+1=α_nu+(1−α_n)S_nx_n. Then, {x_n} converges strongly to a fixed point of T.     

Strong characterizing sequences for subgroups of compact groups

In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R/Z generated by finitely many independent irrationals, then there is an infinite subset AZ which characterizes Γ in the sense that for γR/Z we have ∑_aAaγ<∞ if and only if γΓ. Here we consider a general compact metrizable Abelian group G instead of R/Z, and we characterize its finitely generated free subgroups Γ by subsets AG^*, where G^* is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.     

Inner Functions and Cyclic Composition Operators on H(Bn)

The present paper shows that the algebra generated by {C|  Aut(B_n)} is cyclic on H²(B_n), and any nonconstant function f  H²(B_n) is a cyclic vector of . In addition, the hypercyclic and cyclic composition operators will be discussed.     

Best Approximation by Normal and Conormal Sets

The aim of the present paper is to develop a theory of best approximation by elements of so-called normal sets and their complements—conormal sets—in the non-negative orthant ^I₊ of a finite-dimensional coordinate space ^I endowed with the max-norm. A normal (respectively, conormal) set arises as the set of all solutions of a system of inequalities f_α(x)0 (αA), x ^I₊ (respectively, f_α(x)0 (αA), x ^I₊), where f_α is an increasing function and A is an arbitrary set of indices. We consider these sets as analogues (in a certain sense) of convex sets, and we use the so-called min-type functions as analogues of linear functions. We show that many results on best approximation by convex and reverse convex sets and corresponding separation theory (but not all of them) have analogues in the case under consideration. At the same time there are no convex analogues for many results related to best approximation by normal sets.     

Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms

Let m and n be positive integers with n2 and 1mn−1. We study rearrangement-invariant quasinorms _R and _D on functions f: (0, 1)→ such that to each bounded domain Ω in ⁿ, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality _R(u*(|Ω| t))C_D(|^mu|* (|Ω| t)), uC^m₀(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which _D need not be rearrangement-invariant, _R(u*(|Ω| t))C_D((d/dt) ∫_{{x ⁿ : |u(x)|>u*(|Ω| t)}} |(u)(x)| dx), uC¹₀(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that _R cannot be replaced by an essentially larger quasinorm and _D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger.     

On the similarity of dual operators

Let k be a field with an involution σ and a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that ΛEnd(V) and Λ^*End(W) are dual operators. We show that if Λ and Λ^* are similar, then Λ^*=Λ^-1, where :V→W is Hermitian.     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|_n|²dσ}_n0, denoted by Lim(σ). Here {_n}_n0 are orthogonal polynomials in L²(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m( )=1) Lebesgue measure on . The second subset Mar( ) consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar( ) iff there are >0 and l>0 such that sup{|a_n+j|:0jl}>, n=0,1,2,…,{a_n} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ with {m}Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σMar( )). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zⁿ=λ, λ , n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zⁿ, n=1,2,…, of a closed arc J (including J= ) with removed open concentric arc J₀ (including J₀=). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).     

Regularity results for stable-like operators

For α[1,2) we consider operators of the form

and for α(0,1) we consider the same operator but where the f term is omitted. We prove, under appropriate conditions on A(x,h), that any solution u to will be in C^α+β if fC^β.     

Operators Which Do Not Have the Single Valued Extension Property

In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T  L(X) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ_o  C.     

-regularity of the Aronsson equation in

For a nonnegative, uniformly convex HC²(R²) with H(0)=0, if uC(Ω), ΩR², is a viscosity solution of the Aronsson equation (1.7), then uC¹(Ω). This generalizes the C¹-regularity theorem on infinity harmonic functions in R² by Savin [O. Savin, C¹-regularity for infinity harmonic functions in dimensions two, Arch. Ration. Mech. Anal. 176 (3) (2005) 351–361] to the Aronsson equation.