Weighted Hardy and potential operators in the generalized Morrey spaces

We study the weighted p→q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces Lp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, for such a p→q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p→q-boundedness of the Riesz potential operator.     

Weighted modular inequalities for Hardy-Steklov operators

We characterize weighted modular inequalities of weak and strong type for the Hardy-Steklov operators T defined by , where g is a positive function and s, h are increasing and continuous functions such that s(x)?h(x) for all x.     

Weighted approximation by modified Kantorovich–Bernstein operators

We introduce a new type of Kantorovich–Bernstein operators. Direct and converse theorems and a Voronovskaya-type relation are given for the weighted approximation with Jacobi weights w(x)=x ^α (1?x) ^β by the new operator. None of the results involved have the restriction ${\alpha,\beta<1-\frac{1}{p}}$ .     

The Christoffel function for the Hermite weight is bell-shaped

We show that the Christoffel function λ_n associated with the Hermite weight function w_H(x)=exp(−x²) is bell-shaped. As a consequence, we describe completely how the weights in a Gauss-type quadrature formula associated with w_H(x) are arranged in magnitude.     

A Fan Type Condition For Heavy Cycles in Weighted Graphs

 A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree d ^w (v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. max{d ^w (x),d ^w (y)∣d(x,y)=2}≥c/2; 2. w(x z)=w(y z) for every vertex z∈N(x)∩N(y) with d(x,y)=2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least c. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result. Received: February 7, 2000 Final version received: June 5, 2001     

Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type

We determine the exact asymptotic order of the entropy numbers of compact embeddings of weighted Besov spaces in the case where the ratio of the weights w(x) = w ₁(x)/w ₂(x) is of logarithmic type. This complements the known results for weights of polynomial type. The estimates are given in terms of the number 1/p = 1/p ₁ − 1/p ₂ and the function w(x). We find an interesting new effect: if the growth rate at infinity of w(x) is below a certain critical bound, then the entropy numbers depend only on w(x) and no longer on the parameters of the two Besov spaces. All results remain valid for Triebel–Lizorkin spaces as well.     

Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in Carro et al. (2005) [7] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator Mα,B associated to a Young function B and the multilinear maximal operators Mψ=M0,ψ, ψ(t)=B(t1−α/(nm))^nm/(nm−α). As an application of these estimate we obtain a direct proof of the Lp−Lq boundedness results of Mα,B for the case B(t)=t and Bk(t)=tk(1+log⁺t) when 1/q=1/p−α/n. We also give sufficient conditions on the weights involved in the boundedness results of Mα,B that generalizes those given in Moen (2009) [22] for B(t)=t. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.     

Small alliances in a weighted graph

We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑_{x∈NG(v)∩S}w(x)+w(v)≥∑_{x∈NG(v)−S}w(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3].     

Integral inequalities for the Hilbert transform applied to a nonlocal transport equation

We prove several weighted inequalities involving the Hilbert transform of a function f(x) and its derivative. One of those inequalities,

is used to show finite time blow-up for a transport equation with nonlocal velocity.     

Bounds for ratios of modified Bessel functions and associated Turán-type inequalities

New sharp inequalities for the ratios of Bessel functions of consecutive orders are obtained using as main tool the first order difference-differential equations satisfied by these functions; many already known inequalities are also obtainable, and most of them can be either improved or the range of validity extended. It is shown how to generate iteratively upper and lower bounds, which are converging sequences in the case of the I-functions. Few iterations provide simple and effective upper and lower bounds for approximating the ratios Iν(x)/Iν−1(x) and the condition numbers for any x,ν?0; for the ratios Kν(x)/Kν+1(x) the same is possible, but with some restrictions on ν. Using these bounds Turán-type inequalities are established, extending the range of validity of some known inequalities and obtaining new inequalities as well; for instance, it is shown that Kν+1(x)Kν−1(x)/(Kν²(x))<|ν|/(|ν|−1), x>0, ν∉[−1,1] and that the inequality is the best possible; this proves and improves an existing conjecture.     

Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds

In this paper we obtain essentially sharp generalized Keller-Osserman conditions for wide classes of differential inequalities of the form Lu?b(x)f(u)?(|∇u|) and Lu?b(x)f(u)?(|∇u|)−g(u)h(|∇u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and ?. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry-Emery Ricci tensor, are presented.     

On the total vertex irregularity strength of trees

A vertex irregular total k-labelling λ:V(G)∪E(G)?{1,2,…,k} of a graph G is a labelling of vertices and edges of G done in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex irregular total k-labelling is called the total vertex irregularity strength of G, denoted by . In this paper, we determine the total vertex irregularity strength of trees.     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .     

Claw Conditions for Heavy Cycles in Weighted Graphs

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d^w(x),d^w(y)} c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.     

On the relation between two Minkowski functions

The definition of Minkowski's “Fragefunktion”? (x) is recapitulated. This function is compared to a function L(x) introduced by the author in 1926. It is shown that the inverse function P(w) = x to the function L(x) is related to the Fragefunktion through ?(w) = P(w)?1.     

Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation

We establish local and global norm inequalities for solutions of the nonhomogeneous A-harmonic equation A(x,g+du)=h+d^?v for differential forms. As applications of these inequalities, we prove the Sobolev-Poincaré type imbedding theorems and obtain Lp-estimates for the gradient operator ∇ and the homotopy operator T from the Banach space Ls(D,Λl) to the Sobolev space W1,s(D,Λl−1), l=1,2,…,n. These results can be used to study both qualitative and quantitative properties of solutions of the A-harmonic equations and the related differential systems.     

A general class of inequalities with mixed means

Suppose (T, Σ, μ) is a space with positive measure,f: ? → ? is a strictly monotone continuous function, and &;(T) is the set of real μ-measurable functions onT. Letx(·) ∈ &;(T) andf ○x)(·) ∈L ₁(T,μ). Comparison theorems are proved for the means $\mathfrak{M}_{(T,{\mathbf{ }}\mu ,{\mathbf{ }}f)} (x( \cdot ))$ and the mixed means $\mathfrak{M}_{(T_1 ,{\mathbf{ }}\mu _1 ,{\mathbf{ }}f_1 )} (\mathfrak{M}_{(T_2 ,{\mathbf{ }}\mu _2 ,{\mathbf{ }}f_2 )} (x( \cdot )))$ these inequalities imply analogs and generalizations of some classical inequalities, namely those of Hölder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies.     

Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials

Let u be the weak solution to the degenerate Schrödinger equation with singular coefficients in Lipschitz domain as following

−div(w(x)A(x)∇u(x))+V(x)u(x)w(x)=0,     

The failure of the Hardy inequality and interpolation of intersections

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=?1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L _p(w₀), N∩L_p(w₁)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L _p(w₀),N∩L _p (w₁)), which turns out to be essentially different from theK-functional for (L _p(w₀), L_p(w₁)), even for the case whenN∩L _p(w_i) is dense inL _p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.