Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

Solutions of non-periodic super-quadratic Dirac equations

This paper is concerned with solutions to the Dirac equation: −i∑αkk_∂u+aβu+M(x)u=Ru(x,u). Here M(x) is a general potential and R(x,u) is a self-coupling which is super-quadratic in u at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we construct linking levels of the variational functional ΦM such that the minimax value cM based on the linking structure of ΦM satisfies , where is the least energy of the “limit equation”. Thus we can show the c(C)-condition holds true for all and consequently obtain one least energy solution to the Dirac equation.     

Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation , where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family of closed operators on H such that (whenever eiλtx∈D(B)). For a closed subset Λ⊂R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f∈AP(R,H) such that σ(f)⊆Λ, there exists a unique mild solution u∈AP(R,H) of Eq. (∗) such that σ(u)⊆Λ; (ii) is invertible for all λ∈Λ and .     

Lim's theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀x∈E, ∀α∈[0,1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.     

Local “superlinearity” and “sublinearity” for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems −Δpu=fλ(x,u), , where Ω is a bounded domain in RN, N>p, and λ>0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq+b(x)ur, where 0?q<p−1<r?p^∗−1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in and , a C1,α estimate for equations of the form −Δpu=h(x,u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.     

Majorization of regular measures and weights with finite and positive critical exponent

For the sets , 1?p<∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp(R,dμ), we establish a majorization principle of their “boundaries,” i.e. for every there exists such that dμ/dν?1. A corresponding principle holds for the sets , p>0, of non-negative upper semi-continuous on R functions (weights) w such that P is dense in the space : For every there exists such that w?ω.     

On a Sobolev-type inequality related to the weighted p-Laplace operator

In this paper we deal with some Sobolev-type inequalities with weights that were proved by Maz'ya in [V.G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin, 1980] and by Caffarelli, Kohn and Nirenberg in [L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weight, Compos. Math. 53 (1984) 259-275]. For integers 1?k?N denote points ξ∈RN=Rk×RN−k as pairs (x,y). Let p∈(1,N), q∈(p,p^∗] and assume . Then there exists c>0 such that

     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

Generalized homothetic biorders

In this paper, we study the binary relations R on a nonempty N^∗-set A which are h-independent and h-positive (cf. the introduction below). They are called homothetic positive orders. Denote by B the set of intervals of R having the form [r,+∞[ with 0<r≤+∞ or ]q,∞[ with q∈Q_≥0. It is a Q_>0-set endowed with a binary relation > extending the usual one on R_>0 (identified with a subset of B via the map r?[r,+∞[). We first prove that there exists a unique map Φ_R:A×A→B such that (for all and all ) we have Φ(mx,ny)=mn⁻¹⋅Φ(x,y) and . Then we give a characterization of the homothetic positive orders R on A such that there exist two morphisms of N^∗-sets satisfying . They are called generalized homothetic biorders. Moreover, if we impose some natural conditions on the sets u₁(A) and u₂(A), the representation (u₁,u₂) is “uniquely” determined by R. For a generalized homothetic biorder R on A, the binary relation R₁ on A defined by is a generalized homothetic weak order; i.e. there exists a morphism of N^∗-sets u:A→B such that (for all ) we have . As we did in [B. Lemaire, M. Le Menestrel, Homothetic interval orders, Discrete Math. 306 (2006) 1669-1683] for homothetic interval orders, we also write “the” representation (u₁,u₂) of R in terms of u and a twisting factor.     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Boundedness of singular integral operators with variable kernels

Let K be a generalized Calderón-Zygmund kernel defined on Rn×(Rn?{0}). The singular integral operator with variable kernel given by

     

Weighted-Hardy functions with Hadamard gaps on the unit ball

We prove that an analytic function f on the unit ball B with Hadamard gaps, that is, (the homogeneous polynomial expansion of f) satisfying n_k+1/n_k?λ>1 for all k∈N, belongs to the space if and only if . Moreover, we show that the following asymptotic relation holds . Also we prove that lim_r→1(1-r²)^α‖Rf_r‖_p=0 if and only if . These results confirm two conjectures from the following recent paper [S. Stevi?, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)].     

A note on path kernels and partitions

The detour order of a graph G, denoted by τ(G), is the order of a longest path in G. A subset S of V(G) is called a P_n-kernel of G if τ(G[S])≤n−1 and every vertex v∈V(G)−S is adjacent to an end-vertex of a path of order n−1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that τ(G[A])≤a and τ(G[B])≤b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a P_n+1-kernel for every . Furthermore, if τ(G)=a+b, 1≤a≤b, and G has girth greater than , then G has an (a,b)-partition.     

On the signed edge domination number of graphs

Let be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n≥2), . In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that . Also for every two natural numbers m and n, we determine , where K_m,n is the complete bipartite graph with part sizes m and n.     

Weighted norm inequalities of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.     

On localization properties of Fourier transforms of hyperfunctions

In [A.G. Smirnov, Fourier transformation of Sato's hyperfunctions, Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space U(Rk) which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on Rk. It was shown that all Gelfand-Shilov spaces (α>1) of analytic functionals are canonically embedded in U(Rk). While the usual definition of support of a generalized function is inapplicable to elements of and U(Rk), their localization properties can be consistently described using the concept of carrier cone introduced by Soloviev [M.A. Soloviev, Towards a generalized distribution formalism for gauge quantum fields, Lett. Math. Phys. 33 (1995) 49-59; M.A. Soloviev, An extension of distribution theory and of the Paley-Wiener-Schwartz theorem related to quantum gauge theory, Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of and U(Rk) is studied. It is proved that an analytic functional is carried by a cone K⊂Rk if and only if its canonical image in U(Rk) is carried by K.     

Fourier analysis on domains in compact groups

Let Ω be a measurable subset of a compact group G of positive Haar measure. Let be a non-negative function defined on the dual space and let L²(μ) be the corresponding Hilbert space which consists of elements (ξπ)_π∈suppμ satisfying , where ξπ is a linear operator on the representation space of π, and is equipped with the inner product: . We show that the Fourier transform gives an isometric isomorphism from L²(Ω) onto L²(μ) if and only if the restrictions to Ω of all matrix coordinate functions , π∈suppμ, constitute an orthonormal basis for L²(Ω). Finally compact connected Lie groups case is studied.     

Resolvable group divisible designs with block size four and general index

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hⁿ and general index λ. The necessary conditions for the existence of such a design are n≥4, and . These necessary conditions are shown to be sufficient for all λ≥2, with the definite exceptions of (λ,h,n)∈{(3,2,6)}∪{(2j+1,2,4):j≥1}. The known existence result for λ=1 is also improved.