Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with Jacobi weight ?^(α,β)(x) = (1 ? x)^α(1 + x)^β α ≥ β > ?1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L _q ^(α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.     

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator M _s is bounded from the weighted Lebesgue spaces L ^p(l ^r, vdx) into L ^q(l ^r, udx) whenever 0 s < n, 1 < p, r < , and 1 q < .     

Schrödinger operators with singular potentials

Recently B. Simon proved a remarkable theorem to the effect that the Schrödinger operatorT=?Δ+q(x) is essentially selfadjoint onC ₀ ^∞ (R ^m if 0≦q ∈L ²(R ^m). Here we extend the theorem to a more general case,T=?Σ _j ^=1/m (?/?x _j ?ib _j(x))² +q ₁(x) +q ₂(x), whereb _j, q₁,q ₂ are real-valued,b _j ∈C(R ^m),q ₁ ∈L _loc ² (R ^m),q ₁(x)≧?q*(|x|) withq*(r) monotone nondecreasing inr ando(r ²) asr → ∞, andq ₂ satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq ₁ is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.     

Hardy-Littlewood Maximal Operator,Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces

Any continuous linear operator T: L^p → L^q has a natural vector-valued extension T: L^p(l) → L^q(l) which is automatically continuous. Relations between the norms of these operators in the cases of p = q and r = 2 were considered by Marcinkiewicz -Zygmund [28], Herz [14] and Krivine [19] - [21]. In this paper we study systematically these relations and given some applications. It turns out that some known results can be proved in a simple way as a consequence of these developments.     

Weighted restriction theorems for space curves

Consider a nondegenerate Cn curve γ(t) in Rn, n?2, such as the curve γ₀(t)=(t,t²,…,tn), t∈I, where I is an interval in R. We first prove a weighted Fourier restriction theorem for such curves, with a weight in a Wiener amalgam space, for the full range of exponents p, q, when I is a finite interval. Next, we obtain a generalization of this result to some related oscillatory integral operators. In particular, our results suggest that this is a quite general phenomenon which occurs, for instance, when the associated oscillatory integral operator acts on functions f with a fixed compact support. Finally, we prove an analogue, for the Fourier extension operator (i.e. the adjoint of the Fourier restriction operator), of the two-weight norm inequality of B. Muckenhoupt for the Fourier transform. Here I may be either finite or infinite. These results extend two results of J. Lakey on the plane to higher dimensions.     

An approximation technique for nonlinear integral operations

Approximation results for J. S. Mac Nerney's theory of nonlinear integral operations are established. For the nonlinear product integral _xΠ^y (1 + V)P, approximations of the form Π_{i = 1}ⁿ [1 + L_q(x_i?1, x_i)]P are considered, where L₁(u, v)P = ∝_u^vVP and L_q(u, v)P = ∝_u^vV(r, s)[1 + L_q?1(s, v)]P for q = 2, 3,…. Error bounds are obtained for the difference between the product integral and the preceding product.     

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from L^p(v) to L^q,∞(u), 1<p?q<∞, provided the pair of weights (u,v) verifies a Muckenhoupt condition with a “power-bump” on the weight u.     

Quadratic residues and the distribution of prime numbers

D. Shanks [11] has given a heuristical argument for the fact that there are “more” primes in the non-quadratic residue classes modq than in the quadratic ones. In this paper we confirmShanks' conjecture in all casesq<25 in the following sense. Ifl ₁ is a quadratic residue,l ₂ a non-residue modq, ε(n, q, l ₁,l ₂) takes the values +1 or ?1 according ton?l ₁ orl ₂ modq, then $$\mathop {\lim }\limits_{x \to \infty } \sum\limits_p {\varepsilon (p,q,l_1 ,l_2 )} \log pp^{ - \alpha } \exp ( - (\log p)^2 /x) = - \infty$$ for 0≤α<1/2. In the general case the same holds, if all zeros ?=β+yγ of allL(s, χ modq),q fix, satisfy the inequality β²?γ²<1/4.     

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II₁ factors and Mn(C)) and symmetric gauge norms on L^∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II₁ factor coincides with the class of symmetric gauge norms on L^∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286-300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M₂(C)) and some extreme points of N(Mn(C)) (n?3). For a type II₁ factor M, we prove that if t (0?t?1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).     

非齐型空间中一类满足Hörmander条件的Marcinkiewicz交换子估计

记μ为上的非负Radon测度,且仅满足对固定的C₀>0和n∈(0,d],及所有的和r>0, μ(B(x,r))≤C₀ rⁿ.作者建立了一类核函数满足Hörmander条件的Marcinkiewicz积分与Lip_β(μ)(0<β)函数生成的交换子由L^p(μ)到L^q(μ),由L^p(μ) 到Lip_β-n/p(μ)及L^n/β(μ)到RBMO(μ)有界.部分结论对经典 Marcinkiewicz积分也是新的.       

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Essential self-adjointness of Schrödinger-type operators

The essential self-adjointness of the strongly elliptic operator L = ∑_j,k=1ⁿ (?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)) + q(x) acting on C₀^∞(Rⁿ) is considered, where the matrix (a_jk) is real and symmetric, b_j and q are real, a_jk and b_j are sufficiently smooth, and q?L_loc². It has been shown by Ural'ceva and also Laptev that if q is bounded below and n ? 3 the minimal operator may not be self-adjoint if the principal coefficients rise too rapidly. Thus a theorem of Weyl for ordinary differential operators does not extend to partial differential operators. In this paper it is shown that if q is bounded below and if the principal coefficients are “well behaved” within a sequence of closed shells which go to infinity, then the minimal operator is self-adjoint. It is also shown that a number of results which were known to be true when q is sufficiently smooth may be extended to include the case where q?L_loc². The principal tools used are a distribution inequality due to Tosio Kato and a general maximum principle due to Walter Littman.     

On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

Generalized comparison theorems for disfocality types of the equation Lny + py = 0

Comparison theorems for disfocality types on [a, ∞) of a pair of equations L_n(r_n, r_{n ? 1},…, r₀) y + py = 0 and L_v(?_v, ?_{v ? 1},…, ?₀) y + qy = 0 are given, where L_n and L_v are disconjugate linear differential operators, not necessarily of the same order, and p and q are continuous and of constant sign.     

Delta-subharmonic functions of completely regular gamma-growth in a half-plane

We consider entire, analytic, and delta-subharmonic functions of completely regular growth relative to the proximate order in a half-plane and to an arbitrary function of growth ??(r), which satisfies to the inequality ??(2r)????K??(r) at some K?>?0 and all r?>?0. The concept of indicator h(??, v) of a delta-subharmonic function v in a half-plane relative to ??(r) is introduced. It is proved that the indicator h(??, v) belongs to the space L ₂[0, ??].     

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? belongs to L ^p,∞ where $\mathcal{M}$ is the dyadic maximal operator. We obtain these estimates given the L ¹ and L ^q norm, q<p and certain weak-L ^p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L ¹ and the L ^q norm for a fixed q such that 1<q<p, where ∥?∥ _p,∞ is the usual quasi norm on L ^p,∞.     

Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces

Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hyto¨nen.We prove that the L p(μ)-boundedness with p∈(1,∞)of the Marcinkiewicz integral is equivalent to either of its boundedness from L1(μ)into L1,∞(μ)or from the atomic Hardy space H1(μ)into L1(μ).Moreover,we show that,if the Marcinkiewicz integral is bounded from H1(μ)into L1(μ),then it is also bounded from L∞(μ)into the space RBLO(μ)(the regularized BLO),which is a proper subset of RBMO(μ)(the regularized BMO)and,conversely,if the Marcinkiewicz integral is bounded from L∞b(μ)(the set of all L∞(μ)functions with bounded support)into the space RBMO(μ),then it is also bounded from the finite atomic Hardy space H1,∞fin(μ)into L1(μ).These results essentially improve the known results even for non-doubling measures.     

Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ ^k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ ^k for almost all (in induced measure) points x onM according as the sum Σ _r ^{= 1/∞} Ψ(r)r ^k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r ^−t . τ >k − 1.     

Polynomial approximation on the sphere using scattered data

We consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space ℝ^{q +1} by using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi‐projection in the de la Vallée Poussin style, i.e. it reproduces spherical polynomials up to a certain degree and has uniformly bounded L^p operator norm for 1 ≤ p ≤ ∞. Using certain positive quadrature rules for scattered sites due to Mhaskar, Narcowich and Ward, we discretize this operator obtaining a polynomial approximation of the target function which can be computed from scattered data and provides the same approximation degree of the best polynomial approximation. To establish the error estimates we use Marcinkiewicz–Zygmund inequalities, which we derive from our continuous approximating operator. We give concrete bounds for all constants in the Marcinkiewicz–Zygmund inequalities as well as in the error estimates. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)