Lorentz Spaces Research Articles

A study on convolutions in weighted lorentz spaces using steklov means

In this study, some estimations of convolution type operators defined with the help of Steklov operator in weighted Lorentz space L_ω^(p,q) (T) are obtained. Also, some basic properties of convolution type operators in these spaces are investigated.

The dual spaces of variable anisotropic Hardy-Lorentz spaces and continuity of a class of linear operators

In this paper, the authors obtain the continuity of a class of linear operators on variable anisotropic Hardy--Lorentz spaces. In addition, the authors also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type spaces with variable exponents. This result is still new even when the exponent function $p(\cdot)$ is $p$.

Some Liouville-type theorems for the stationary density-dependent Navier–Stokes equations

This paper is devoted to the Liouville-type theorems for the 3D stationary density-dependent Navier–Stokes equations, without any integrability assumption on ∇u. More precisely, we prove that the velocity field u, being in some Lorentz spaces, must be zero, provided the density ρ is essentially bounded.

Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ H: = -\Delta+V $\end{document}</tex-math></inline-formula> be a nonnegative Schrödinger operator on <inline-formula><tex-math id="M2">\begin{document}$ L^2({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a radially symmetric inverse square potential. Let <inline-formula><tex-math id="M5">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> be the operator norm of <inline-formula><tex-math id="M6">\begin{document}$ \nabla^\alpha e^{-tH} $\end{document}</tex-math></inline-formula> from the Lorentz space <inline-formula><tex-math id="M7">\begin{document}$ L^{p, \sigma}({\bf R}^N) $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M8">\begin{document}$ L^{q, \theta}({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">\begin{document}$ \alpha\in\{0, 1, 2, \dots\} $\end{document}</tex-math></inline-formula>. We establish both of upper and lower decay estimates of <inline-formula><tex-math id="M10">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> and study sharp decay estimates of <inline-formula><tex-math id="M11">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>. Furthermore, we characterize the Laplace operator <inline-formula><tex-math id="M12">\begin{document}$ -\Delta $\end{document}</tex-math></inline-formula> from the view point of the decay of <inline-formula><tex-math id="M13">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>.</p>

A Köthe-Toeplitz dual of a generalized Cesaro difference sequence space, a degenerate Lorentz space, their corresponding function spaces and fpp

In 1970, Cesaro sequence spaces was introduced by Shiue. In 1981, Kızmaz defined difference sequence spaces for ${\ell }^{\infty }$, ${\mathrm{c}}_0$ and $\mathrm{c}$. Then, in 1983, Orhan introduced Cesaro difference sequence spaces. Both works used difference operator and investigated the Köthe-Toeplitz dual for the new Banach spaces they introduced. Later, various authors generalized these new spaces, especially the one introduced by Orhan. In this study, first we discuss the fixed point property for these spaces and for the corresponding function space of the Köthe-Toeplitz dual. Moreover, we consider another generalized space which is a degenerate Lorentz space because the spaces we consider are somehow related in terms of their construction. In fact, the Köthe-Toeplitz dual is contained in $\ell^1$, the corresponding function space contains Lebesgue space, the Banach space of Lebesgue integrable functions on $[0,1]$, $L_1[0,1]$ but for the degenerate Lorentz space and its corresponding function space, this is reversed completely. Then, as an important result, for both function spaces we show that they fail the weak fixed point property for isometries and even for contractions. As the second main result, by passing to counting measure, we take the corresponding sequence spaces, and for both spaces we get large classes of closed, bounded and convex subsets satisfying the fixed point property for nonexpansive mappings as a Goebel and Kuczumow analogy after noting that in $\ell^1$ Goebel and Kuczumow found a large class with the fixed point property for nonexpansive mappings.

Periodic solutions and their stability to the Navier-Stokes equations on a half space

Consider the Navier-Stokes equations in a half space. It is shown that this system admits a unique, $ T $-periodic mild solution in some suitable Lorentz space provided the outer force is $ T $-periodic and small enough. This result is then extended to the case of almost periodic functions. Moreover, it is shown that these solutions are polynomially stable.

A note on the nonexistence of spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form

We obtain nonexistence results concerning complete noncompact spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form, under the assumption that the support functions with respect to a fixed nonzero vector are linearly related. Our approach is based on a suitable maximum principle recently established by Alías, Caminha and do Nascimento [3].

On differential analysis of spacelike flows on normal congruence of surfaces

<abstract><p>The present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the $ \operatorname{curl} $ of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both $ s-lines $ and $ b-lines. $ Also, we examine a normal congruence of surfaces containing the $ s-lines $ and $ b-lines $. By compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this normal congruence of surface. Intrinsic geometric properties of this normal congruence of surfaces are given.</p></abstract>

Classification of f-biharmonic submanifolds in Lorentz space forms

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

The embedding theorems for anisotropic Nikol’skii-Besov spaces with generalized mixed smoothness

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables). This work also extends some abstract results proved in Nguyen and Tran (2021) [43] by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimate of the gradient of weak solutions via fractional maximal operators and Riesz potential of data.

Оценки норм Шаттена–Неймана одного класса интегральных операторов

The article considers an integral operator acting from Lebesque spaces to Lorentz spaces. The conditions are found under which the compact operator belongs to the Shatten-Neumann classes.

Local and global existence in Lp for the inhomogeneous nonlinear Schrödinger equation

This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|−b|u|βu. It is show that a global solution exists in the mass‐subcritical for large data in the spaces Lp, p < 2 under some suitable conditions on b, β, and p. The solution is established using a data‐decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.

Functions with general monotone Fourier coefficients

This paper is a study of trigonometric series with general monotone coefficients in the class with . Sharp estimates are proved for the Fourier coefficients of integrable and continuous functions. Also obtained are optimal results in terms of coefficients for various types of convergence of Fourier series. For two-sided estimates are obtained for the -moduli of smoothness of sums of series with -coefficients, as well as for the (quasi-)norms of such sums in Lebesgue, Lorentz, Besov, and Sobolev spaces in terms of Fourier coefficients. Bibliography: 99 titles.

Complete spacelike hypersurfaces with constant scalar curvature: Descriptions and gaps

We provide sharp lower and upper bounds for the supremum of the norm of the total umbilicity tensor of complete spacelike hypersurfaces with constant scalar curvature immersed in a Lorentzian space form and satisfying a suitable Okumura-type inequality, which corresponds to a weaker hypothesis when compared with the geometric condition of the hypersurface having two distinct principal curvatures. Furthermore, we give a complete description and the gaps of the spacelike hypersurfaces which realize our estimates, obtaining as a consequence new characterizations of totally umbilical spacelike hypersurfaces and hyperbolic cylinders of Lorentzian space forms. Our approach is based on a version of Omori–Yau’s maximum principle for trace-type differential operators defined on a complete Riemannian manifold.

Electromagnetic curves and Rytov curves based on the hyperbolic split quaternion algebra

This paper aims to investigate the hyperbolically the motion of the polarization plane traveling along the linearly polarized light wave in the optical fiber via the hyperbolic split quaternion algebra. The motion of the electric field (polarization vector) about an axis on the general hyperboloid is described by the Lorentzian scalar product space Ra1,a2,a32,1. The hyperbolic geometric (Berry) phase models are generated through the pseudo-spheres of Ra1,a2,a32,1. Then, the parametric representations of the Rytov curves are obtained via the hyperbolic split quaternion product and one-parameter homothetic motion. Then, the hyperbolically motion of the electric field is expressed by the Fermi–Walker parallel transportation law. Moreover, the electromagnetic curves (EM-curves) associated with the electric field E are determined via the hyperbolic geometric phase models in the optical fiber. Furthermore, some motivating examples are presented by using the MAPLE program.

Extrapolation to mixed norm spaces and applications

This paper establishes extrapolation theory to mixed norm spaces. By applying this extrapolation theory, we obtain the mapping properties of the Rubio de Francia Littlewood-Paley functions and the geometrical maximal functions on mixed norm spaces. As special cases of these results, we have the mapping properties on the mixed norm Lebesgue spaces with variable exponents and the mixed norm Lorentz spaces.

A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Given a bounded measurable function σ on Rn, we let Tσ be the operator obtained by multiplication on the Fourier transform by σ. Let 0<s1≤s2≤⋯≤sn<1 and ψ be a Schwartz function on the real line whose Fourier transform ψˆ is supported in [−2,−1/2]∪[1/2,2] and which satisfies ∑j∈Zψˆ(2−jξ)=1 for all ξ≠0. In this work we provide a sharp form of the Marcinkiewicz multiplier theorem on Lp by finding an almost optimal function space with the property that, if the function(ξ1,…,ξn)↦∏i=1n(I−∂i2)si2[∏i=1nψˆ(ξi)σ(2j1ξ1,…,2jnξn)] belongs to it uniformly in j1,…,jn∈Z, then Tσ is bounded on Lp(Rn) when |1p−12|<s1 and 1<p<∞. In the case where si≠si+1 for all i, it was proved in [12] that the Lorentz space L1s1,1(Rn) is the function space sought. Here we address the significantly more difficult general case when for certain indices i we might have si=si+1. We obtain a version of the Marcinkiewicz multiplier theorem in which the space L1s1,1 is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among s2,…,sn that equal s1. Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space.

Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials

We establish the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey–Lorentz spaces. As a by-product we derive estimates on the restriction operator in block spaces. Direct application to solvability of a fourth order nonlinear equation related to higher order boundary conformally invariant problem is considered. By employing a nonvariational approach, we obtain a unique solution under suitable smallness conditions on the boundary data and potentials prescribed in Morrey–Lorentz spaces which allow for singular functions. Moreover, these solutions are shown to be C∞ in the interior, Cloc2,μ(R+n+1¯), μ∈(0,1) in some special case and satisfy interesting qualitative properties including positivity. The results are extended to a larger class of problems involving the polyharmonic operator. In particular, the higher order nonlocal equation (−Δ)2m−12v=K(x)|v|σ−1v+H(x)v+ginRn,m∈Nfor 2≤2m<n+1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ>n/(n−2m+1). We also prove that no positive solution in the latter framework exists if σ≤n/(n−2m+1).

Modified general relativity and quantum theory in curved spacetime

With appropriate modifications, the multi-spin Klein–Gordon (KG) equation of quantum field theory can be adapted to curved space–time for spins 0, 1, 1/2. The associated particles in the microworld then move as a wave at all space–time coordinates. From the existence in a Lorentzian space–time of a line element field [Formula: see text], the spin-1 KG equation [Formula: see text] is derived from an action functional involving [Formula: see text] and its covariant derivative. The spin-0 KG equation and the KG equation of the outer product of a spin-1/2 Dirac spinor and its Hermitian conjugate are then constructed. Thus, [Formula: see text] acts as a fundamental quantum vector field. The symmetric part of the spin-1 KG equation, [Formula: see text], is the Lie derivative of the metric. That links the multi-spin KG equation to Modified General Relativity (MGR) through its energy–momentum tensor of the gravitational field. From the invariance of the action functionals under the diffeomorphism group Diff(M), which is not restricted to the Lorentz group, [Formula: see text] can instantaneously transmit information along [Formula: see text]. That establishes the concept of entanglement within a Lorentzian formalism. The respective local/nonlocal characteristics of MGR and quantum theory no longer present an insurmountable problem to unify the theories.