AN IMPROVED HARDY'S INEQUALITY

In this paper we shall extend Hardy＇s inequality associated with Fourier trans- form to the strip n（2-p） ≤σ〈 n＋p（N＋ 1） where N = [n（1/p- 1）], the greatest integer not exceeding n（1/p - 1）.

NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESàRO TYPE

This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro oper- ators （with symbols in central BMO spaces） are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

GRADIENT ESTIMATES FOR SOLUTIONS TO QUASILINEAR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV GROWTH AND HARDY POTENTIAL

This note is a continuation of the work[17].We study the following quasilinear elliptic equations- △pu-μ/｜x｜p ｜u｜p-2 u=Q（x）｜u｜Np/N-p -2u,x∈R N,where 1 〈 p 〈 N,0 ≤ μ 〈（（N-p）/p）p and Q ∈ L∞（RN）.Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.

Experimentally testing Hardy's theorem on nonlocality with entangled mixed states

Hardy＇s theorem on nonlocality has been verified by a series of experiments with two-qubit entangled pure states.However,in this paper we demonstrate the experimental test of the theorem by using the two-photon entangled mixed states.We first investigate the generic logic in Hardy＇s proof of nonlocality,which can be applied for arbitrary two-qubit mixed polarization entangled states and can be reduced naturally to the well-known logic tested successfully by the previous pure state experiments.Then,the optimized violations of locality for various experimental parameters are delivered by the numerical method.Finally,the logic argued above for testing Hardy＇s theorem on nonlocality is demonstrated experimentally by using the mixed entangled-photon pairs generated via pumping two type-I BBO crystals.Our experimental results shows that Hardy＇s proof of nonlocality can also be verified with two-qubit polarization entangled mixed states,with a violation of about 3.4 standard deviations.

A HEAT KERNEL VERSION OF HARDY’S THEOREM FOR THE LAGUERRE HYPERGROUP

In this article, we prove a heat kernel version of Hardy＇s theorem for the Laguerre hypergroup.

HEAT KERNELS AND HARDY'S UNCERTAINTY PRINCIPLE ON H-TYPE GROUPS

This article obtains an explicit expression of the heat kernels on H-type groups and then follow the estimate of heat kernels to deduce the Hardy＇s uncertainty principle on the nilpotent Lie groups.

The relation between Hardy's non-locality and violation of Bell inequality

We give an analytic quantitative relation between Hardy＇s non-locality and Bell operator. We find that Hardy＇s non-locality is a sufficient condition for the violation of Bell inequality, the upper bound of Hardy＇s non-locality allowed by information causality just corresponds to Tsirelson bound of Bell inequality and the upper bound of Hardy＇s non- locality allowed by the principle of no-signaling just corresponds to the algebraic maximum of Bell operator. Then we study the CabeUo＇s argument of Hardy＇s non-locality （a generalization of Hardy＇s argument） and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy＇s non-locality under the constraint of information causality with the aid of the above derived relation between Hardy＇s non-locality and Bell operator.

A MATHEMATICAL PROOF OF A PROBABILISTIC MODEL OF HARDY'S INEQUALITY

In this paper using an argument from [1] , we prove one of the probabilistic version of Hardy＇s inequality.

Some integral inequalities on time scales

In this article, we study the reverse HSlder type inequality and HSlder inequality in two dimensional case on time scales. We also obtain many integral inequalities by using HSlder inequalities on time scales which give Hardy＇s inequalities as spacial cases.

DEGENERATE BOUNDARY LAYER SOLUTIONS TO THE GENERALIZED BENJAMIN-BONAMAHONY-BURGERS EQUATION

This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers（BBM-Burgers） equation to the corresponding degenerate boundary layer solutions in the half-space.It is shown that the convergence rate is t-（α/4） as t →∞ provided that the initial perturbation lies in H α 1 for α 〈 α（q）：= 3 ＋（2/q）,where q is the degeneracy exponent of the flux function.Our analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant introduced in [1]

Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.

Atomic decompositions and Hardy's inequality on weak Hardy-Morrey spaces

We introduce the weak Hardy-Morrey spaces in this paper.We also obtain the atomic decompositions of the weak Hardy-Morrey spaces.By using these decompositions,we establish the Hardy inequalities on the weak Hardy-Morrey spaces.

Experimentally testing Bell's theorem based on Hardy's nonlocal ladder proofs

Bell＇s theorem argues the existence of quantum nonlocality which goes basically against the hidden variable theory （HVT）. Many experiments have been done via testing the violations of Bell＇s inequalities to statistically verify the Bell＇s theorem. Alternatively, by testing the Hardy＇s ladder proofs we experimentally demonstrate the deterministic violation of HVT and thus confirm the quantum nonlocality. Our tests are implemented with non-maximal entangled photon pairs generated by spontaneous parametric down conversions （SPDCs）. We show that the degree freedom of photon entanglement could be significantly enhanced by using interference filters. As a consequence, the Hardy＇s ladder proofs could be tested and Bell＇s theorem is verified robustly. The probability of violating the locality reach to 41.9%, which is close to the expectably ideal value 46.4% for the photon pairs with degree of entanglement e = 0.93. The higher violating probability is possible by further optimizing the experimental parameters.

On a New Reverse Extended Hardy’s Integral Inequality

In this paper,a new reverse extended Hardy＇s integral inequality is proved by means of weight coefficients and the technique of real analysis.Some particular results are considered.

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the HSrmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy Spaces for k≥ 3 using the multi- parameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k 3, and the method works for all the cases k ≥ 3： Tmf（x1,x2,x3） =1/（（2π）＋n1＋n2＋n3） ∫ R n1×R n2×R n3 m（ξ）f（ξ）e 2π ix.ξ dξ. where x = （x1,x2,x3） ∈ Rn1 × Rn2 × R n3 and ξ = （ξ1,ξ2,ξ3） ∈ R n1 × Rn2 ×R n3. One of our main results is the following： Assume that m（ξ） is a function on Rn1＋n2＋n3 satisfying sup j,k,l ∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3）〈∞ with si 〉 ni（1/p-1/2） for 1 ≤ i ≤ 3. Then Tm is bounded from HP（R n1 × R n2 ×R n3） to HP（R n1 ×R n2 × R n3） for all 0 〈 p ≤ 1 and ｜｜Tm｜｜ Hp→Hp≤ sup j,k,l∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3） Moreover, the smoothness assumption on sl for 1 ≤ i ≤ 3 is optimal. Here we have used the notations mj,k,l （ξ）= m（2 j ξ1,2 k ξ2, 2 l ξ3） ψ（ξ1） ψ（ξ2） ψ（ξ3） and ψ（ξi） is a suitable cut-off function on R ni for 1 ≤ i ≤ 3, and W（s1,s2,s3） is a three-parameter Sobolev space on R n × R n2 × Rn 3. Because the Fefferman criterion breaks down in three parameters or more, we consider the Lp boundedness of the Littlewood-Paley square function of T mf to establish its boundedness on the multi-parameter Hardy spaces.

Existence and regularity of solutions to semi-linear Dirichlet problem of infinitely degenerate elliptic operators with singular potential term

In this paper,we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term.By using the logarithmic Sobolev inequality and Hardy's inequality,the existence and regularity of multiple nontrivial solutions have been proved.