New characterizations of Hajłasz-Sobolev spaces on metric spaces

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type, including metric spaces and fractals. These Sobolev spaces include the well-known Hajłasz-Sobolev spaces as special models. The author establishes various characterizations of (sharp) maximal functions for these spaces. As applications, the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces. Moreover, some embedding theorems are also given.     

A characterization of Haj?asz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions

In this paper, we establish the equivalence between the Haj?asz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p∈(n/(n+1),∞), we give a new characterization of the Haj?asz-Sobolev spaces via a grand Littlewood-Paley function.     

Haj?asz-Sobolev imbedding and extension

The author establishes some geometric criteria for a Haj?asz-Sobolev -extension (resp. -imbedding) domain of Rn with n?2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel-Lizorkin space on Ω.     

Embeddings of variable Haj?asz-Sobolev spaces into Hölder spaces of variable order

Pointwise estimates in variable exponent Sobolev spaces on quasi-metric measure spaces are investigated. Based on such estimates, Sobolev embeddings into Hölder spaces with variable order are obtained. This extends some known results to the variable exponent setting.     

A note on λ-compact spaces

We extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.     

A note on certain square functions on product spaces

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog L(Sn-1 × Sm-1) satisfying the cancellation condition.     

A note on spaces with a countable μ-base

In topological spaces every normal space with a countable base is metacompact. We show that this is not necessarily true in generalized topological spaces; more exactly we give an example of a μ-normal space with a countable μ-base which has a μ-open cover with no μ-open point-finite refinement.     

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Haj?asz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Haj?asz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.     

A note on weakly Lindelöf determined Banach spaces

We prove that weakly Lindelöf determined Banach spaces are characterized by the existence of a “full” projectional generator. Some other results pertaining to this class of Banach spaces are given.     

A note on Herglotz’s theorem for time series on function spaces

In this article, we prove Herglotz’s theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz’s theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cramér representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist.     

A note on Thomassen?s conjecture

In 1983 C. Thomassen [8] conjectured that for every k,g∈N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. A result of Pyber, Szemerédi, and the second author implies that the conjecture is true for every graph G with average .We strengthen this and show that the conjecture holds for every graph G with average for some constants α, β depending on k and g.     

A note on Stevi?'s iteration method

In the present paper, motivated by Stevi?'s iteration method, we show the stability result of the following unified iterative scheme

xn+1=tnTvn+(1−tn)xn+un     

A note on Somos? quadratic recurrence constant

We show how to calculate Somos? quadratic recurrence constant to a high degree of accuracy, and give its value to 300 decimal places.     

A note on Chudnovsky?s Fuchsian equations

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.     

A note on θ-tightness

In questo lavoro chiariamo il ruolo che la θ-tightness (introdotta dagli autori in [2]) ha nel determinare alcune estensioni di sottoinsiemi di uno spazio topologico.     

A note on Marino-Vafa formula

Hodge integrals over moduli spaces of curves appear naturally during the localization procedure in computation of Gromov-Witten invariants. A remarkable formula of Marino-Vafa expresses a generation function of Hodge integrals via some combinatorial and algebraic data seemingly unrelated to these apriori algebraic geometric objects. We prove in this paper by directly expanding the formula and estimating the involved terms carefully that except a specific type all the other Hodge integrals involving up to three Hodge classes can be calculated from this formula. This implies that amazingly rich information about moduli spaces and Gromov-Witten invariants is encoded in this complicated formula. We also give some low genus examples which agree with the previous results in literature. Proofs and calculations are elementary as long as one accepts Mumford relations on the reductions of products of Hodge classes.