Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.     

On the sharpness of Bruen’s bound for intersection sets in Desarguesian affine spaces

In this note we investigate the sharpness of Bruen’s bound on the size of a t-fold blocking set in \(AG(n,q)\) with respect to the hyperplanes. We give a construction for t-fold blocking sets meeting Bruen’s bound with \(t=q-n+2\) . This construction is used further to find the minimal size of a t-fold affine blocking set with \(t=q-n+1\) . We prove that for blocking sets in the geometries \(AG(n,2)\) the difference between the size of an optimal t-fold blocking set and tn exceeds any given number. In particular, we deviate infinitely from Bruen’s bound as n goes to infinity. We conclude with a construction that gives t-fold blocking sets with \(t=q-n+3\) whose size is close to the lower bounds known so far.     

Jensen’s inequality on convex spaces

We prove a Jensen’s inequality on $p$ -uniformly convex space in terms of $p$ -barycenters of probability measures with $(p-1)$ -th moment with $p\in ]1,\infty [$ under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT $(\kappa )$ -space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into $2$ -uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.     

Jackson’s theorem in Q_p spaces

A Jackson type inequality in Q p spaces is established, i.e., for any f (z) = Σ∞ j=0 ajzj ∈ Qp , 0≤p ∞, a 1, and k-1 ∈ N,where ω(1/k, f, Q p ) is the modulus of continuity in Q p spaces and C(a) is an absolute constant depending only on the parameter a.     

Trigonometric Cesàro bases in the spaces of functions integrable with power weight

(2), k1, >0, L _p (0,), 1p L =C. , , p, k, (C, )- L _p (0,), , , {sinnx} _n ^=k/ (C, )- L _p (0,) |x|^p . , 1p, {x sinnx} _n=k , k2 2k–2–1/p<2k–1/p, (C, )- L ^p [0,] , >–(p–1)/.     

On the proof of Erdős’ inequality

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality \({\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}\) for a constrained polynomial p of degree at most n, initially claimed by P. Erd?s, which is different from the one in the paper of T.Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval (?1, 1) and establish a new asymptotically sharp inequality.     

Wolff’s inequality in multi-parameter Morrey spaces

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L ^p (R ^d ) and multi-parameter Morrey spaces ${L^p_\lambda (R^d)}$ , where ${R^d=R^{n_1} \times R^{n_2} \times \cdots \times R^{n_k},\, \lambda = (\lambda _1,\ldots ,\lambda _k})$ and 0?<?λ _i ≤ n _i , 1?≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.     

The sharp Jackson inequality in the space L2 on the segment [?1, 1] with the power weight

In the space L ₂ on the segment [−1, 1] with the power weight |x|^2λ+1, λ ≥ −1/2, we define a complete orthogonal system, the value of the best approximation with respect to this system, the operator of generalized shift, and the modulus of continuity and prove the sharp Jackson inequality.     

Sharp constant in Jackson’s inequality with modulus of smoothness for uniform approximations of periodic functions

It is proved that, in the space C_2π, for all k, n ∈ ?,n > 1, the following inequalities hold: where e _n?1(f) is the value of the best approximation of f by trigonometric polynomials and ω ₂(f, h) is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.     

A sharp Markov brothers-type inequality in the spaces L ∞ and L 1 on the segment

The inequality between the uniform norm of the derivative of order ? of an algebraic polynomial of degree n and the L ₁-norm of the polynomial itself on a segment are studied. For all ? ≥ (n ? 1)/3, the exact constant and the extremal polynomial are written out.     

On furstenberg’s characterization of harmonic functions on symmetric spaces

We provide a necessary and sufficient condition on a radial probability measureμ on a symmetric space for whichf =f *μ, f bounded, implies thatf is harmonic. In particular, we obtain a short and elementary proof of a theorem of Furstenberg which says that iff is a bounded function on a symmetric space which satisfiesf =f *μ for some radialabsolutely continuous probability measureμ, thenf is harmonic.     

On probability analogs of Rosenthal’s inequality

We obtain probability combinatorial inequalities for independent random variables, strengthening the well-known Rosenthal inequality. As a corollary, we prove that the generalized Rosenthal inequality for identically distributed independent functions remains valid in the case of quasinormed symmetric spaces.     

On the spaces C k (ℝ) ∩ L p (ℝ)

We show that the Fréchet-Sobolev spaces C(ℝ) ∩ L ^p (ℝ) and C ^k (ℝ) ∩ L ^p (ℝ) are not isomorphic for p ≠ 2 and k ∈ ℕ. Research supported by the Italian MURST.     

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R ^m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A ^?1) and Tr(A ^?2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A ^?1) and Tr(A ^?2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A ^?1)}²/Tr(A ^?2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.     

Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,κ(Rd) by a subspace E2κ(σ)(SE2κ(σ)), which is a subspace of entire functions of exponential type(spherical exponential type) at most σ. Here L2,κ(Rd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight v2κ(x) =ξ∈R, which is defined by a positive+|(ξ, x)|κ(ξ)subsystem R+ of a finite root system RRdand a function κ(ξ) : R → R+ invariant under the reflection group G(R) generated by R. In the case G(R) = Zd2, we get some exact results. Moreover,the deviation of best approximation by the subspace E2κ(σ)(SE2κ(σ)) of some class of the smooth functions in the space L2,κ(Rd) is obtained.     

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.