Vector valued bilinear maximal operator and method of rotations

In this paper we obtain a bilinear analogue of Fefferman-Stein?s vector valued inequality for classical Hardy-Littlewood maximal function. Also, we prove the boundedness of bilinear Hardy-Littlewood maximal operator from Lp₁(Rn)×Lp₂(Rn)→L¹(Rn), where , by applying the method of rotations.     

Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type

Let 0<γ<1, b be a BMO function and the commutator of order m for the fractional integral. We prove two type of weighted Lp inequalities for in the context of the spaces of homogeneous type. The first one establishes that, for A_∞ weights, the operator is bounded in the weighted Lp norm by the maximal operator Mγ(Mm), where Mγ is the fractional maximal operator and Mm is the Hardy-Littlewood maximal operator iterated m times. The second inequality is a consequence of the first one and shows that the operator is bounded from to , where [(m+1)p] is the integer part of (m+1)p and no condition on the weight w is required. From the first inequality we also obtain weighted Lp-Lq estimates for generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.     

Parabolic mean values and maximal estimates for gradients of temperatures

We aim to prove inequalities of the form for solutions of on a domain Ω=D×R⁺, where δ(x,t) is the parabolic distance of (x,t) to parabolic boundary of Ω, is the one-sided Hardy-Littlewood maximal operator in the time variable on R⁺, is a Calderón-Scott type d-dimensional elliptic maximal operator in the space variable on the domain D in Rd, and 0<λ<k<λ+d. As a consequence, when D is a bounded Lipschitz domain, we obtain estimates for the Lp(Ω) norm of δ2n−λn(∇2,1)u in terms of some mixed norm for the space with denotes the Besov norm in the space variable x and where .     

Kuznietsov trace formula and weighted distribution of Hecke eigenvalues

Let (X,μ) be a measurable topological space. Let S₁,S₂,… be a family of finite subsets of X. Suppose each x∈S_i has a weight w_ix∈R⁺ assigned to it. We say {S_i} is {w_i}-distributed with respect to the measure μ if for any continuous function f on X, we have .Let S(N,k) be the space of modular cusp forms over Γ₀(N) of weight k and let be a basis which consists of Hecke eigenforms. Let a_r(h) be the rth Fourier coefficient of h. Let x_p^h be the eigenvalue of h relative to the normalized Hecke operator T′_p. Let ||·|| be the Petersson norm on S(N,k). In this paper we will show that for any even integer k?3, is -distributed with respect to a polynomial times the Sato-Tate measure when N→∞.     

Stable extendibility of normal bundles over lens spaces

We study the stable extendibility of R-vector bundles over the (2n+1)-dimensional standard lens space Ln(p) with odd prime p, focusing on the normal bundle to an immersion of Ln(p) in the Euclidean space R2n+1+t. We show several concrete cases in which is stably extendible to Lk(p) for any k with k?n, and in several cases we determine the exact value m for which is stably extendible to Lm(p) but not stably extendible to Lm+1(p).     

Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L²(R₊,S) by the differential expression

     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

The number of reachable pairs in a digraph

For a digraph G, let R(G) (respectively, R^(k)(G)) be the number of ordered pairs (u,v) of vertices of G such that u≠v and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range S_n of R(G) and the range of R^(k)(G) as G varies over all possible digraphs on n vertices. We give a sufficient condition and a necessary condition for an integer to belong to S_n. These determine the set S_n for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)^0.57+2, for arbitrary k.     

Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that over a cone-like set a.e. for all f∈L₁(Rd). Moreover, converges to f(x) over a cone-like set at each Lebesgue point of f∈L₁(Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.     

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T(a)∈L(?p), 1<p<∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections Tn(a)=PnT(a)Pn depends heavily on the Fredholm properties of the operators T(a) and . In particular, if the operators T(a) and are Fredholm on ?p, then the approximation numbers of Tn(a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than .     

On weak type inequalities for dyadic maximal functions

We obtain sharp estimates for the localized distribution function of the dyadic maximal function , given the local L¹ norms of ? and of G○? where G is a convex increasing function such that G(x)/x→+∞ as x→+∞. Using this we obtain sharp refined weak type estimates for the dyadic maximal operator.     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).