Search

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, (F , f) , on ℝ n in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley g -functions, Lusin area functions, Littlewood–Paley g λ ∗ -functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley g -functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ℝ n in the aforementioned function spaces. [ABSTRACT FROM AUTHOR]

In the paper the Lorentz space L p , τ (T m) , 2 π of periodic functions of many variables and spaces with mixed logarithmic smoothness is considered. Equivalent norms of a space with mixed logarithmic smoothness are found and embedding theorems are proved. [ABSTRACT FROM AUTHOR]

We study the almost everywhere pointwise convergence of the Boussinesq operator along sequences {tn}∞n=1 with lim n→∞ tn = 0 in one dimension. We obtain a characterization of convergence almost everywhere when {tn} ∈ lr;∞(N) for all f ∈ Hs(R) provided 0 < s < 1/2. [ABSTRACT FROM AUTHOR]

We study the almost everywhere pointwise convergence of the Boussinesq operator along sequences $ \{t_n\}_{n = 1}^\infty $ with $ \lim\limits_{n\rightarrow \infty} t_n = 0 $ in one dimension. We obtain a characterization of convergence almost everywhere when $ \{t_n\}\in l^{r, \infty}(\mathbb{N}) $ for all $ f\in H^s(\mathbb{R}) $ provided $ 0 < s < \frac12 $.

In this paper, we show that Fourier restriction theorem implies the maximal and variational Fourier restriction in Lorentz space frame. The key ingredient is that we work out the induction argument in Lorentz space, based on some good properties of Lorentz space. This extends Kovač's result [J. Funct. Anal., 2019, 277(10): 3355–3372] to Lorentz space frame. [ABSTRACT FROM AUTHOR]

In this note, we investigate steady compressible nematic liquid crystal flows in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$. We establish Liouville‐type theorems for smooth solutions (ρ,u,d)$$ \left(\rho, u,d\right) $$ that fulfill specific conditions within Lorentz spaces. [ABSTRACT FROM AUTHOR]

The equivalence of the Haar system in a rearrangement invariant space on and a sequence of pairwise disjoint functions in some Lorentz space is known to imply that up to the equivalence of norms. We show that the same holds for the class of uniform disjointly homogeneous rearrangement invariant spaces and obtain a few consequences for the properties of isomorphic embeddings of such spaces. In particular, the space with is the only uniform -disjointly homogeneous rearrangement invariant space on with nontrivial Boyd indices which has two rearrangement invariant representations on the half-axis . [ABSTRACT FROM AUTHOR]

For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set of all p ∈ [ 1 , ∞ ] such that ℓ p is finitely represented in X in such a way that the unit basis vectors of ℓ p ( c 0 if p = ∞ ) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a follow up of a paper by the first-named author related to separable rearrangement invariant spaces on (0 , ∞) . [ABSTRACT FROM AUTHOR]

Our main purpose in this paper is to obtain the anisotropic Moser-Trudinger type inequality in Lorentz space $ L(n, q) $, $ 1 \leq q \leq \infty $. It can be seen as a generation result of the Moser-Trudinger type inequality in Lorentz space.

In the paper, the Lorentz space Lq,τ (Tm) of periodic functions of several variables, the Nikol'skii--Besov class Sq,τ,θr B and the associated class Wq,τ, a,b,r 1q,τ, q, τ1 and Sq,τ,θr B in the norm of the space Lp,τ2(Tm) for different relations between the parameters q,τ1,p,τ1,a,θ.The proofs of the theorems are based on the constructive method developed by V.N. Temlyakov. [ABSTRACT FROM AUTHOR]

In this paper, with the help of some new atomic decomposition theorems, several Φ-moment Banach space valued martingale inequalities associated with concave functions in the context of Lorentz spaces are deduced. Our results are closely related with the geometrical properties of the underlying Banach spaces. [ABSTRACT FROM AUTHOR]

We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and lower bounds on the Hausdorff dimension of the residual set of a generalized Apollonian circle packing. This dimension (which also coincides with a critical exponent) is strictly greater than that of the Apollonian packing. [ABSTRACT FROM AUTHOR]

In this research, spinor descriptions of the curves in surfaces have been given according to the B-Darboux frame in Lorentzian 3-space E. The relations between B- Darboux and Darboux frames have been specified via their spinor conceptions which expressed in both timelike and spacelike surfaces, separately. Additionally, all these spinor representations have been portrayed in the view of the B-Darboux frame (via the curvatures) in Lorentzian 3-space. The findings are supported by some theorems and corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some Liouville-type properties for the stationary incompressible MHD equations in three dimensions. Liouville-type properties are concerned with the uniqueness of trivial solutions to the stationary MHD equations. We show that the velocity field u and the magnetic field b are vanishing provided that u ∈ BMO - 1 (R 3) and b ∈ L p , ∞ (R 3) with 2 < p ≤ 6 . Moreover, we also prove that if then u and b are vanishing for the stationary 3D MHD equations with partial viscosity and partial diffusivity. [ABSTRACT FROM AUTHOR]

In the paper spaces of periodic functions of several variables were considered, namely the Lorentz space L2,τ(Tm), the class of functions with bounded mixed fractional derivative Wr2,τ, 1 ≤ τ < ∞, and the order of the best M-term approximation of a function f ∈ Lp,τ(Tm) by trigonometric polynomials was studied. The article consists of an introduction, a main part, and a conclusion. In the introduction, basic concepts, definitions and necessary statements for the proof of the main results were considered. One can be found information about previous results on the mentioned topic. In the main part, exact-order estimates are established for the best M-term approximations of functions of the Sobolev class Wr2,τ1 in the norm of the space Lp,τ2(Tm) for various relations between the parameters p,τ1,τ2.

The paper is dedicated to the study of embeddings of the anisotropic Besov spaces B p , θ 1 , ... , θ n β 1 , ... , β n (ℝn) into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of the exponents βk tend to 1 (βk < 1). In particular, these results give an extension of the estimate proved by Bourgain, Brezis, and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link with some known embeddings of anisotropic Lipschitz spaces. One of the key results of the paper is an anisotropic type estimate of rearrangements in terms of partial moduli of continuity. [ABSTRACT FROM AUTHOR]

Let φ : [ 0 , 1) × [ 0 , ∞) → [ 0 , ∞) be a Musielak–Orlicz function and q ∈ (0 , ∞ ] . In this article, the authors characterize the martingale Musielak–Orlicz Hardy space H φ [ 0 , 1) and the martingale Musielak–Orlicz–Lorentz Hardy space H φ , q [ 0 , 1) via dyadic maximal operators. As applications, the authors prove that the maximal Fejér operator is bounded from the space H φ [ 0 , 1) to the Musielak–Orlicz space L φ [ 0 , 1) and from H φ , q [ 0 , 1) to the Musielak–Orlicz–Lorentz space L φ , q [ 0 , 1) , which further implies some convergence results of the Fejér means. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, explores the representability of lr-spaces in Orlicz function spaces. The author introduces the concepts of lattice finitely representable and symmetrically finitely representable spaces and defines the sets L(X), S(X), and J(X) for a symmetric space X. The article focuses on separable Orlicz spaces and presents a theorem stating that for certain Orlicz functions, the sets S(LN), L(LN), and J(LN) coincide. The article concludes with implications and suggestions for further research. The text includes a mathematical proof of Theorem 1 by proving Proposition 1, which involves the use of Matuszewska-Orlicz indices and various inequalities. Funding support for the work is acknowledged, and references to related research are provided. [Extracted from the article]

The aim of this study is to investigate the associated curves of the normal indicatrix of a non-null curve in Lorentz 3-space. To achieve this goal, firstly, the vectors used in the Frenet elements of the normal indicatrix is utilized to create version of the Bishop frame by taking advantage of the rotations around them. The relationships between the elements of the obtained frames and the associated curves of the normal indicatrix is examined. Additionally, the similarities of these associated curves to the evolute, Mannheim, and Bertrand curve pairs, which are commonly encountered in differential geometry, are studied. [ABSTRACT FROM AUTHOR]

We explore boundedness properties of kernel integral operators acting on rearrangement‐invariant (r.i.) spaces. In particular, for a given r.i. space X we characterize its optimal range partner, that is, the smallest r.i. space Y such that the operator is bounded from X to Y. We apply the general results to Lorentz spaces to illustrate their strength. [ABSTRACT FROM AUTHOR]

We study mapping properties of the centered Hardy--Littlewood maximal operator M acting on Lorentz spaces. Given p ∈ (1,∞) and a metric measure space X = (X, ρ, µ) we let ΩHLp (X) ⊂ [0, 1]² be the set of all pairs (1/q, 1/r) such that M is bounded from Lp,q(X) to Lp,r(X). Under mild assumptions on µ, for each fixed p all possible shapes of ΩHLp(X) are characterized. Namely, we show that the boundary of ΩHLp (X) either is empty or takes the form {δ} × [0, ... F(u)] ∪ {(u, F(u)): u ∈(δ, 1]}, where δ ∈ [0, 1] and F : [δ, 1] → [0, 1] is concave, nondecreasing, and satisfies F(u) ≤ u. Conversely, for each such F we find X such that M is bounded from Lp,q(X) to LP,R(X) if and only if the point (1/q, 1/r) lies on or under the graph of F, that is, 1/q ≥ δ and 1/r ≤ F(1/q). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous Navier–Stokes flows with vacuum, and obtain regularity criteria and Liouville type theorems in the Lorentz space if a smooth solution (ρ , u) satisfies suitable conditions. [ABSTRACT FROM AUTHOR]

In this study, Rodrigues parameters have been first calculated for a homothetic rotation around spacelike and timelike axis using one parameter homothetic motions in Lorentz 3-space E 1 3 . The behavior of the vectors during the homothetic rotation, which is the increase or decrease (about size of one of the objects in motion) notion, has been investigated as a three dimensional shape with the help of Cinema 4D program. The behavioral differences have been observed on figures. Then, Lorentz motions that corresponding to the homothetic rotation matrices in terms of spacelike and timelike axes have been examined and expressed in E 1 3 . Some definitions, theorems, corollaries and three dimensional figures have been given in Lorentz 3-space. [ABSTRACT FROM AUTHOR]

In this paper, it is investigated the optimal range of the classical Hilbert transform in Lorentz spaces of functions and its non-commutative counterparts including a triangular truncation operator in Schatten-Lorentz ideals. Some applications of obtained results to operator Lipschitz functions and commutator estimates in Schatten-Lorentz ideals of compact operators are presented. [ABSTRACT FROM AUTHOR]

The classical inequalities of Bochkarev play a very important role in harmonic analysis. The meaning of these inequalities lies in the connection between the metric characteristics of functions and the summability of their Fourier coefficients. One of the most important directions of harmonic analysis is the theory of Fourier series. His interest in this direction is explained by his applications in various departments of modern mathematics and applied sciences, as well as the availability of many unsolved problems. One of these problems is the study of the interrelationships of the integral properties of functions and the properties of the sum of its coefficients. The solution of these problems was dedicated to the efforts of many mathematicians. And further research in this area are important and interesting problems and can give new, unexpected effects. In the article we receive a two-dimensional analog of the Bochkarev type theorem for the Fourier transform.

It is shown that the space is crudely finitely representable in the Lorentz space , , if and only if or . To the best of the author's knowledge, this is the first example of a "natural" rearrangement-invariant space on such that the set of all numbers for which is crudely finitely representable in is not an interval of the real line. [ABSTRACT FROM AUTHOR]

We use a fixed point theorem and an approximation scheme to obtain the existence of at least one positive solution of a p-Laplacian equation with an exponential term that may have subcritical, critical or supercritical growth. [ABSTRACT FROM AUTHOR]

Given m ∈ N ∖ { 0 } and a compact Riemannian manifold N , we construct for every map u in the critical Sobolev space W m / (m + 1) , m + 1 (S m , N) , a map U : B 1 m + 1 → N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity. [ABSTRACT FROM AUTHOR]

Copyright of Erzincan University Journal of Science & Technology is the property of Erzincan Binali Yildirim Universitesi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the present paper, we obtain sharper analogues of the Hardy-Littlewood-Stein inequalities for double trigonometric series. We also establish a new unified version of the Hardy-Littlewood-Stein inequalities for Fourier series in regular systems, which covers the whole range 1 < p < ∞ including the critical case p = 2. [ABSTRACT FROM AUTHOR]

We provide a constructive proof of H1 (ℝd) (the classical Hardy space) factorization in terms of fractional commutators in Lorentz spaces. As a direct application, we obtain a characterization of functions in BMO space. Furthermore, we also obtain a Lorentz compactness characterization of fractional commutators. [ABSTRACT FROM AUTHOR]

We derive three generalized Gagliardo-Nirenberg inequalities in Lorentz, BMO and homogeneous Lipschitz spaces. They have the forms... whose parameters satisfy specific conditions. We use the so-called Muramatu's integral formula as the main approach throughout the paper. [ABSTRACT FROM AUTHOR]

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space L q , τ (T m) , the class of functions with bounded mixed fractional derivative W q , τ r ¯ , 1 < q , τ < ∞ , and studies the order of the best M-term approximation of a function f ∈ L p , τ (T m) by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class W q , τ 1 r ¯ in the norm of the space L p , τ 2 (T m) for various relations between the parameters p , q , τ 1 , τ 2 . [ABSTRACT FROM AUTHOR]

For a separable rearrangement invariant space X on (0 , ∞) of fundamental type we identify the set of all p ∈ [ 1 , ∞ ] such that ℓ p is finitely represented in X in such a way that the unit basis vectors of ℓ p ( c 0 if p = ∞ ) correspond to pairwise disjoint and equimeasurable functions. This characterization hinges upon a description of the set of approximate eigenvalues of the doubling operator x (t) ↦ x (t / 2) in X. We prove that this set is surprisingly simple: depending on the values of some dilation indices of such a space, it is either an interval or a union of two intervals. We apply these results to the Lorentz and Orlicz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range 1 < p < 2 to 2 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

The logarithm space Lp log L can be defined by two equivalent, but very differently looking norms. In this note we give estimates for the best constants for the ratio of these norms. [ABSTRACT FROM AUTHOR]

In this work we study the non-commutative Hardy-Littlewoodmaximal operator on Lorentz spacesofτ-measurable operators. Non-commutative maximal inequalities were studied, in particular,in [1–3]. Another version of the (non-commutative) Hardy-Littlewood maximal operator was in-troduced by T. Bekjan [4]. Later J. Shao investigated the Hardy-Littlewood maximal operator onnon-commutative Lorentz spaces associated with finite atomless von Neumann algebra (see [5]).

Let (Ω , F , P) be a probability space, φ : Ω × [ 0 , ∞) → [ 0 , ∞) a Musielak–Orlicz function, and q ∈ (0 , ∞ ] . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of σ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space L φ , q (Ω) . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejér operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space H φ , q [ 0 , 1) to L φ , q [ 0 , 1) , which further implies some convergence results of the Fejér means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis. [ABSTRACT FROM AUTHOR]

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N (N ≥ 4). Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C (R , L s , ∞ (R N)) when S (n) = n − 1 2 . Next for the case S (n) = 0 , we study the mild solution and point out it can become strong solution in B C (R , L s , ∞ (R N)) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations. [ABSTRACT FROM AUTHOR]

We study mapping properties of the centered Hardy–Littlewood maximal operator M acting on Lorentz spaces L p , q (X) in the context of certain non-doubling metric measure spaces X . The special class of spaces for which these properties are very peculiar is introduced and many examples are given. In particular, for each p 0 , q 0 , r 0 ∈ (1 , ∞) with r 0 ≥ q 0 we construct a space X for which the associated operator M is bounded from L p 0 , q 0 (X) to L p 0 , r (X) if and only if r ≥ r 0 . [ABSTRACT FROM AUTHOR]

Let (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H*p,q(𝒳)H_*^{p,q}\left( \mathcal{X} \right) with the optimal range p∈(ωω+η,∞)p \in \left( {{\omega \over {\omega + \eta }},\infty } \right) and q ∈ (0, ∞]. When and p∈(ωω+η,1]p \in ({\omega \over {\omega + \eta }},1] q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

As we all know for a surfaceMand a point P ∈ M in Euclidean 3-space, the Weingarten map of M on P is always diagonalizable. Also if H²(P) -- K(P) = 0 then we say that P is an umbilical point of M: Here H is the mean curvature and K is the Gauss curvature of the surface. If all P ∈ M is umbilical then M is an umbilical surface. In Lorentz 3-space the situation is different. The equation of H²(P) -- K(P) = 0 for a point P ∈ M doesn't mean that P is an umbilical point and theWeingarten map of M on P can be diagonalizable. In this paper we find the surfaces with the equation H² -- K = 0, whose generating curve is a graph of a polynomial under homothetic motion groups in Lorentz 3-space. We show that, this equation is possible when the axis of the homothetic motion is timelike or spacelike and the degree of the polynomial is 0 or 1, so the surfaces with H² -- K = 0 are ruled surfaces. Also, despite having the equation H²(P) -- K(P) = 0 for some P ∈ M; these points are not umbilical on M. [ABSTRACT FROM AUTHOR]

A complete description of multipliers of Fourier–Haar series in Lorentz spaces is given. Necessary and sufficient conditions for sequences of complex numbers to belong to the class of multipliers of Fourier–Haar series are obtained. [ABSTRACT FROM AUTHOR]

Let X (R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X (R) and on its associate space X ′ (R) . The algebra C X (R ˙) of continuous Fourier multipliers on X (R) is defined as the closure of the set of continuous functions of bounded variation on R ˙ = R ∪ { ∞ } with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X (R) is reflexive, then the ideal of compact operators is contained in the Banach algebra A X (R) generated by all multiplication operators aI by continuous functions a ∈ C (R ˙) and by all Fourier convolution operators W 0 (b) with symbols b ∈ C X (R ˙) . We show that there are separable and non-reflexive Banach function spaces X (R) such that the algebra A X (R) does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces L p , 1 (R) with 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

In this paper we give a simple proof of an estimate for Bessel potentials acting on Lorentz spaces in the supercritical exponent: let 1 < p = d α - 1 and 1 ≤ q ≤ + ∞ . If f ∈ L p , q (R d) , then there exists a constant C = C (α , d , p , q) such that | g α ∗ f (x) - g α ∗ f (z) | ≤ C | x - z | | ln (| x - z |) | + 1 1 q ′ ‖ f ‖ L p , q . [ABSTRACT FROM AUTHOR]

We refine some earlier results by Flores, Hernández, Semenov, and Tradacete on compactness of the square of strictly singular endomorphisms and identifying general Banach lattices with the Kato property in the setting of rearrangement invariant spaces on [0, 1]. A Banach space X is said to have the Kato property if every strictly singular operator acting in X is compact. We show that each strictly singular operator bounded in a disjointly homogeneous rearrangement invariant space with the non-trivial Boyd indices has compact square, and that the Kato property is shared by a 2-disjointly homogeneous rearrangement invariant space X whenever X ⊃ G , where G is the closure of L ∞ in the Orlicz space, generated by the function e u 2 - 1 . Moreover, a partial converse to the latter result is given under the assumption that X ⊂ L log 1 / 2 L . As an application we find rather sharp conditions, under which a Lorentz space Λ (2 , ψ) possesses the Kato property. In particular, Λ (2 , log - α (e / u)) , with 0 < α ≤ 1 , is a 2-DH and 2-convex rearrangement invariant space, which does not have the Kato property. This gives a negative answer to the question posed by Hernández, Semenov, and Tradacete. [ABSTRACT FROM AUTHOR]

In this paper, by using the idea of linearizing maximal operators originated by Charles Fefferman and the TT* method of Stein-Wainger, we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by Andersen and John on the weighted Lorentz spaces with vector-valued functions. [ABSTRACT FROM AUTHOR]