Bulk universality holds in measure for compactly supported measures

Let μ be a measure with compact support, with orthonormal polynomials {p _n } and associated reproducing kernels {K _n }. We show that bulk universality holds in measure in {ξ: μ′(ξ) > 0}. More precisely, given ɛ, r > 0, the linear Lebesgue measure of the set {ξ: μ′(ξ) > 0} and for which

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Some Remarks on a Class of Nonuniformly Elliptic Equations of p-Laplacian Type

This paper deals with the existence of weak solutions in W ₀¹(Ω) to a class of elliptic problems of the form

Sharp two-sided estimates for eigenvalues of some class of sturm-liouville problems

Sharp two-sided estimates are obtained for the eigenvalues\Gl _n of the problem

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      6. On the Corona problem      E. Amar 《Journal of Geometric Analysis》1991,1(4):291-305 We show that if f1, f2 are bounded holomorphic functions in the unit ball of ℂn such that , |f1(z)|2 + |f2(z)2|2 ≥ δ2 >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f1h1 + f2h2 with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.      7. Positive Solutions for Semipositone <Emphasis Type="Italic">m</Emphasis>-point Boundary-value Problems   总被引：7，自引：0，他引：7   Ru?Yun?MaEmail author  Qiao?Zhen?Ma 《数学学报(英文版)》2004,20(2):273-282 Abstract   Let ξ i ∈ (0, 1) with 0 < ξ1 < ξ2 < ··· < ξ m−2 < 1, a i , b i ∈ [0,∞) with and . We consider the m-point boundary-value problem where f(x, y) ≥ −M, and M is a positive constant. We show the existence and multiplicity of positive solutions by applying the fixed point theorem in cones. *Supported by the NSFC (10271095). GG-110-10736-1003, NWNU-KJCXGC-212 and the Foundation of Major Project of Science and Technology of Chinese Education Ministry      8. Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems      Guo-wei?ZhangEmail author  Jing-xian?Sun 《应用数学学报(英文版)》2004,20(4):655-664 Abstract   The singular second-order m-point boundary value problem , is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ i ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξ m−2 < 1, a i ∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions. Supported by the National Natural Science Foundation of China (No.10371066, No.10371013)      9. A mixed boundary-value problem for a hyperbolic-parabolic equation      I. E. Egorov 《Mathematical Notes》1978,23(3):211-217 Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W 2 1 (QT) and W 2 2 (QT).      10. The Balian-Low theorem and regularity of Gabor systems      John?J.?BenedettoEmail author  Wojciech?Czaja  Przemystaw?Gadziński  Alexander?M.?Powell 《Journal of Geometric Analysis》2003,13(2):239-254 For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫ℝ|g(x)|2(1+|x| A )/log d (2+|x|)dx < ∞ and .      11. Multipliers of Möbius invariant <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript> spaces      Jordi Pau  José Ángel Peláez 《Mathematische Zeitschrift》2009,261(3):545-555       12. Regularity for non-autonomous functionals with almost linear growth      Dominic Breit  Bruno De Maria  Antonia Passarelli di Napoli 《manuscripta mathematica》2011,136(1-2):83-114 We consider non-autonomous functionals ${\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}$ , where the density ${f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}$ has almost linear growth, i.e., $$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$ We prove partial C 1,?? -regularity for minimizers ${u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}$ under the assumption that D ?? f (x, ??) is H?lder continuous with respect to the x-variable. If the x-dependence is C 1 we can improve this to full regularity provided additional structure conditions are satisfied.      13. On the regularity of the minima of quasiconvex integrals      Xu Chaojiang 《数学学报(英文版)》1992,8(4):362-374 We prove in this paper theC ∞ regularity for a “very strict” local minimum of classC loc ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC 1 (x)>0,C 2 (x)>0,1≥ε(x)>0, such that for all real ϕ∈c 0 ∞ (K). This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation.      14. Existence of solutions of certain quasilinear elliptic equations in ℝN without conditions at infinity      G. I. Laptev 《Journal of Mathematical Sciences》2008,150(5):2384-2394 This paper deals with conditions for the existence of solutions of the equations considered in the whole space ℝn, n ≥ 2. The functions A i (x, u, ξ), i = 1,…, n, A 0(x, u), and f(x) can arbitrarily grow as |x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝn. We prove the existence theorem for a solution u ∈ W loc 1,p (ℝn) under the condition p > n. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 133–147, 2006.      15. Bergman type operators on a class of weakly pseudoconvex domains      Shi Jihuai 《中国科学A辑(英文版)》1998,41(1):22-32 A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω a . As an appli cation, it is proved thatf?L H p (ω a ,dv λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α1,?,α n and ß = (ß1, —ß). An interesting question is whether the converse holds.      16. Inversion of exponentialk-plane transforms      Boris Rubin 《Journal of Fourier Analysis and Applications》2000,6(2):185-205 Approximate and explicit inversion formulas are obtained for a new class of exponential k-plane transforms defined by where x∈ℝn, Θ is a k-frame in ℝn, 1≤k≤n−1, μ∈ℂk is an arbitrary complex vector. The case k=1, μ∈ℝ corresponds to the exponential X-ray transform arising in single photon emission tomography. Similar inversion formulas are established for the accompanying transform where V is a real (n×k)-matrix. Two alternative methods, leading to the relevant continuous wavelet transforms, are presented. The first one is based on the use of the generalized Calderón reproducing formula and multidimensional fractional integrals with a Bessel function in the kernel. The second method employs interrelation between Pμ and the associated oscillatory potentials.      17. Subharmonic Functions in the Unit Ball      Raphaële Supper 《Positivity》2005,9(4):645-665 For functions u subharmonic in the unit ball BN of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of BN. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫BN u+(x)[-ω′(|x|2)]dx < +∞ for some positive decreasing C1 function ω, it is proved that .      18. Immersions of spheres and algebraically constructible functions      Iwona Karolkiewicz  Aleksandra Nowel  Zbigniew Szafraniec 《manuscripta mathematica》2009,128(1):77-87 Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g λ  =  g(· , λ) restricted to the sphere S n (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g λ|S n (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.      19. On the rate of convergence to a Poisson process      V. A. Egorov 《Vestnik St. Petersburg University: Mathematics》2011,44(2):103-109 Let ξ = {ξ1, ξ2, ...} be a sequence of independent, spherically symmetric, identically distributed multivariate random vectors such that the condition $\left( {\left| \xi \right| > x} \right) \to x^{ - \alpha } L^\alpha$\left( {\left| \xi \right| > x} \right) \to x^{ - \alpha } L^\alpha      20. Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces      Huang Jianfeng Wang Yuanheng 《高校应用数学学报(英文版)》2007,22(3):311-315 This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.

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On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

Positive Solutions for Semipositone <Emphasis Type="Italic">m</Emphasis>-point Boundary-value Problems

Abstract   Let ξ _i ∈ (0, 1) with 0 < ξ₁ < ξ₂ < ··· < ξ _m−2 < 1, a _i , b _i ∈ [0,∞) with and . We consider the m-point boundary-value problem

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

Regularity for non-autonomous functionals with almost linear growth

We consider non-autonomous functionals ${\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}$ , where the density ${f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}$ has almost linear growth, i.e., $$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$ We prove partial C ^1,?? -regularity for minimizers ${u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}$ under the assumption that D _?? f (x, ??) is H?lder continuous with respect to the x-variable. If the x-dependence is C ¹ we can improve this to full regularity provided additional structure conditions are satisfied.     

On the regularity of the minima of quasiconvex integrals

We prove in this paper theC ^∞ regularity for a “very strict” local minimum of classC _loc ^ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC ₁ (x)>0,C ₂ (x)>0,1≥ε(x)>0, such that for all real ϕ∈c ₀ ^∞ (K). This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation.     

Existence of solutions of certain quasilinear elliptic equations in ℝN without conditions at infinity

This paper deals with conditions for the existence of solutions of the equations

Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK _x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω _a . As an appli cation, it is proved thatf?L _H ^p (ω _a ,dv _λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α₁,?,α _n and ß = (ß₁, —ß). An interesting question is whether the converse holds.     

Inversion of exponentialk-plane transforms

Approximate and explicit inversion formulas are obtained for a new class of exponential k-plane transforms defined by where x∈ℝⁿ, Θ is a k-frame in ℝⁿ, 1≤k≤n−1, μ∈ℂ^k is an arbitrary complex vector. The case k=1, μ∈ℝ corresponds to the exponential X-ray transform arising in single photon emission tomography. Similar inversion formulas are established for the accompanying transform where V is a real (n×k)-matrix. Two alternative methods, leading to the relevant continuous wavelet transforms, are presented. The first one is based on the use of the generalized Calderón reproducing formula and multidimensional fractional integrals with a Bessel function in the kernel. The second method employs interrelation between P_μ and the associated oscillatory potentials.     

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

On the rate of convergence to a Poisson process

Let ξ = {ξ₁, ξ₂, ...} be a sequence of independent, spherically symmetric, identically distributed multivariate random vectors such that the condition

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.