Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: where N ≥ 3,λ₁,λ₂ ∈ [0,Λ_N), is the best Hardy constant. is the critical Sobolev exponent. a₁,a₂, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. with . By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a₁,a₂,b and λ₁,λ₂. Copyright © 2012 John Wiley & Sons, Ltd.     

Attractors for equations describing the flow of generalized Newtonian fluids

Initial boundary-value problems for the equations

A countably-valued sleeping stockbroker process

We exhibit a stationary countably-valued process {V _n } _– which is deterministic, but which is nondeterministic in the sense that whenever ...n _–2<n _–1<n ₀<n ₁<... are indices with no two consecutive, then is an independent process. This answers a question of Ref. 1.In addition, althoughnV _n is deterministic, its time-reversalnV _–n is not deterministic.Partly supported by grant DMS-9248571. Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, contract DAAL03-91-C-0027. 72604-3435@CompuServe.COM.     

Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry

We show that there is a close relation between standing-wave solutions for the FitzHugh-Nagumo system where 0<a<1/2 and δ γ=β ² ∈ (0,a), and the following combinatorial problem: (*) Given K points Q ₁ , . . . , Q _K ∈R ^N with minimum distance 1, find out the maximum number of times that the minimum distance 1 can occur. More precisely, we show that for any given positive integer K, there is a δ _K >0 such that for 0<δ<δ _K , there exists a standing-wave solution (u _δ ,ν _δ ) to the FitzHugh-Nagumo system with the property that u _δ has K spikes Q ^δ ₁,. . .,Q ^δ _K and approaches an optimal configuration in (*), where .     

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

Converse and Smoothness Theorems for Erdo&#x030B;s Weights inL_(0<p?∞)

We prove converse and smoothness theorems of polynomial approximation in weightedL_pspaces with norm ‖fW‖_Lp()(0<p?∞) for Erdo&#x030B;s weights on the real line. In particular we prove characterization theorems involving realization functionals and thereby establish some interesting properties of our weighted modulus of continuity.     

On complemented subspaces of (Σl_2 )_{l_p }

It is shown that ifX is a complemented subspace of (Σ (1<p<∞), thenX is isomorphic to eitherl ₂,l _p,l ₂⊗l _p or (Σ . IfX is a complemented subspace ofC _p(1<p<∞) which does not contain an isomorph of (Σ which does not contain an isomorph of thenX is isomorphic to a complemented subspace of (Σ ⊗l ₂. This research was partially supported by NSF MPS 72-04634-A03.     

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

Bipartite Subgraphs and Quasi-Randomness

We say that a family of graphs is p-quasi-random, 0<p<1, if it shares typical properties of the random graph G(n,p); for a definition, see below. We denote by the class of all graphs H for which and the number of not necessarily induced labeled copies of H in G_n is at most (1+o(1))p^e(H)n^v(H) imply that is p-quasi-random. In this note, we show that all complete bipartite graphs K_a,b, a,b2, belong to for all 0<p<1.Acknowledgments We would like to thank Andrew Thomason for fruitful discussions and Yoshi Kohayakawa for organizing Extended Workshop on Combinatorics in eq5 Paulo, Ubatuba, and Rio de Janeiro, where a part of this work was done. We also thank the referees for their careful work.The first author was partially supported by NSF grant INT-0072064The second author was partially supported by NSF grants DMS-9970622, DMS-0301228 and INT-0072064Final version received: October 24, 2003     

Multipliers in Besov Spaces and in LpΩ-Spaces (The Cases 0<p≦1 and p=∞)

The paper contains sufficient conditions for multipliers in two types of quasi-BANACH spaces: (i) weighted L_p-spaces of entire analytic functions of exponential type; 0<p≦∞, (ii) Besov spaces B, where ?∞<s<∞; 0<p≦∞; 0<q≦∞.     

Several Dimensional θ-Summability and Hardy Spaces

The d-dimensional Hardy spaces H_p ( T × … × T ) (d = d₁ + … + d_kand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θ_j having integrable Fourier transforms. Under some conditions on θ_j we show that the maximal operator of the θ-means of a distribution is bounded from H_p ( T × … × T ) to L_p ( T ^d) where p₀ < p < ∞ and p₀ < 1 is depending only on the functions θ_j. By an interpolation theorem we get that the maximal operator is also of weak type ( L₁) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)^k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L₁. Moreover, we prove that the θ-means are uniformly bounded on the spaces H_p ( T × … × T ) whenever p₀ <p < ∞, thus the θ-means converge to f in ( T × … × T ) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.     

On a diophantine inequality involving prime numbers (III)

Let 1<c<11/10. In the present paper it is proved that there exists a numberN(c)>0 such that for each real numberN>N(c) the inequality is solvable in prime numbersp ₁,p ₂,p ₃, wherec ₁ is some absolute positive constant. Project supported by the National Natural Science Foundation of China (grant: 19801021) and by MCSEC     

Some characterizations of weighted Herz-type Hardy spaces and their applications

In this paper, the authors first establish some new real-variable characterizations of Herz-type Hardy spaces and , where ω₁,ω₃ ∈ A₁-weight, 1<q>∞,n(1−1/q)≤α<∞ and 0<p<∞. Then, using these new characterizations, they investigate the convergence of a bounded set in these spaces, and study the boundedness of some potential operators on these spaces. Supported by the NNSF of China     

On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements

Let {Y _i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a _i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V _i =∑ _k=−∞^∞ a _i+k Y _i ,i≥1. In this paper, we derive that if and E|X| ^μ log  ^ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).     

An Integration by Parts Formula in Sobolev Spaces

We prove the following formula

Multiple blowup of solutions for a semilinear heat equation

The present paper is concerned with a Cauchy problem for a semilinear heat equation with u₀L(R^N). A solution u of (P) is said to blow up at t=T<+ if lim sup_tT|u(t)|=+ with the supremum norm |·| in R^N. We show that if and N11, then there exists a proper solution u of (P) which blows up at t=T₁, becomes a regular solution for t(T₁,T₂) and blows up again at t=T₂ for some T₁,T₂ with 0<T₁<T₂<+.Mathematics Subject Classification (2000): 35K20, 35K55, 58K57Revised version: 20 July 2004Acknowledgment The author expresses her gratitude to Professor Marek Fila for useful discussion.     

Nodal solutions for a second-order m-point boundary value problem

We study the existence of nodal solutions of the m-point boundary value problem

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality is solvable, where p ₁,···,p ₅ are primes.