Showing papers on "Lp space published in 2017"
TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.
372 citations
01 Jan 2017
TL;DR: In this article, a motor stator is mounted on a motor housing and the motor housing is flexed by the surface of the stator face so that the terminal housing is firmly retained on the surface.
Abstract: A terminal housing which is intended to be mounted on a motor stator comprises a housing member having a terminal-receiving face and a base surface which is directed in the opposite direction from the terminal-receiving face. The housing has an integral open rectangular mounting frame which comprises arms extending in opposite directions from the endwalls of the housing, parallel legs which extend from the ends of the arms, and a strut which extends between the ends of the legs. The housing and the integral mounting frame are dimensioned such that the housing can be mounted on one face of the stator with the legs extending across the circumferential surface of the stator and with the strut extending across the other face of the stator. When the stator is assembled to the motor housing, the frame is captured by the motor housing and flexed by the surface of the motor housing so that the terminal housing is firmly retained on the surface of the stator face.
82 citations
TL;DR: In this paper, a broad family of test function spaces and their dual (distribution) spaces is considered, including Gelfand-Shilov spaces and a family of Test Function Spaces introduced by Pilipovic.
Abstract: We consider a broad family of test function spaces and their dual (distribution) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.
74 citations
TL;DR: In this paper, Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations are studied and the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, and Poisson maximal operators coincide.
Abstract: We introduce and explore Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations. We prove that the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, and Poisson maximal operators coincide. We also study the relation between anisotropic mixed-norm Hardy spaces and mixed-norm Lebesgue spaces.
72 citations
TL;DR: In this paper, it was shown that bilinear fractional integral integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed lebesgue spaces.
Abstract: We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.
60 citations
TL;DR: In this article, new classes of functions are defined, which generalize Morrey spaces and give a refinement of Lebesgue spaces, and some embeddings between these new classes are also proved.
Abstract: In this paper, new classes of functions are defined. These spaces generalize Morrey spaces and give a refinement of Lebesgue spaces. Some embeddings between these new classes are also proved. Finally, the authors apply these classes of functions to obtain regularity results for solutions of partial differential equations of parabolic type.
50 citations
Book•
10 Jan 2017
TL;DR: In this paper, the authors give a coherent account of the theory of Banach spaces and Banach lattices, using the spaces C_0(K) of continuous functions on a locally compact space K as the main example.
Abstract: This book gives a coherent account of the theory of Banach spaces and Banach lattices, using the spaces C_0(K) of continuous functions on a locally compact space K as the main example. The study of C_0(K) has been an important area of functional analysis for many years. It gives several new constructions, some involving Boolean rings, of this space as well as many results on the Stonean space of Boolean rings. The book also discusses when Banach spaces of continuous functions are dual spaces and when they are bidual spaces.
47 citations
Posted Content•
TL;DR: In this article, it was shown that any Lorentz-Zygmund space is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that
Abstract: In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally $G\Gamma$-spaces. As a direct consequence of our results any Lorentz-Zygmund space $L^{a,r}({\rm Log}\, L)^\beta$, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that $ 1
44 citations
TL;DR: In this paper, a new differential variational inequality (DVI) was proposed, which is composed of an evolution equation and a variational inequalities in infinite Banach spaces, and based on the Browder's theorem and the optimal control theory, the existence of solutions to the mentioned problem was shown.
40 citations
Posted Content•
TL;DR: In this paper, the nonasymptotical pairwise bilateral exact up to multiplicative constants interrelations between exponential decreasing tail behavior, moments (Grand Lebesgue Spaces) norm and moment generating functions norm for random variables and vectors (r.v.).
Abstract: We offer in this paper the non-asymptotical pairwise bilateral exact up to multiplicative constants interrelations between exponential decreasing tail behavior, moments (Grand Lebesgue Spaces) norm and moment generating functions norm for random variables and vectors (r.v.).
39 citations
TL;DR: In this article, the Troisi inequality and anisotropic Lebesgue spaces were used to study the regularity criteria for the incompressible Navier-Stokes equations in the R 3 space.
Abstract: We study the regularity criteria for the incompressible Navier–Stokes equations in the whole space R 3 based on one velocity component, namely u 3 , ∇ u 3 and ∇ 2 u 3 . We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition ∇ u 3 ∈ L β ( 0 , T ; L p ) , where 2 / β + 3 / p = 7 / 4 + 1 / ( 2 p ) and p ∈ ( 2 , ∞ ] , yields the regularity of u on ( 0 , T ] .
TL;DR: In this article, weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Ho¨ rmander's condition on Lebesgue spaces are obtained.
Abstract: In this paper, weighted inequalities for a certain general commutator associated to a singular integral operator satisfying a variant of Ho¨ rmander’s condition on Lebesgue spaces are obtained. To do this, some weighted sharp maximal function inequalities for the commutator are proved.
TL;DR: In this paper, the authors obtained weighted norm inequalities for different conical square functions associated with the Heat and the Poisson semigroups generated by a second order divergence form elliptic operator with bounded complex coefficients.
Abstract: This is the first part of a series of three articles. In this paper, we obtain weighted norm inequalities for different conical square functions associated with the Heat and the Poisson semigroups generated by a second order divergence form elliptic operator with bounded complex coefficients. We find classes of Muckenhoupt weights where the square functions are comparable and/or bounded. These classes are natural from the point of view of the ranges where the unweighted estimates hold. In doing that, we obtain sharp weighted change of angle formulas which allow us to compare conical square functions with different cone apertures in weighted Lebesgue spaces. A key ingredient in our proofs is a generalization of the Carleson measure condition which is more natural when estimating the square functions below $p=2$.
TL;DR: In this paper, the authors investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups and establish new Hormander-Mikhlin criteria for spectral and non-spectral multipliers.
Abstract: We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hormander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative Lp spaces. This general principle —exploited in Euclidean harmonic analysis during the last 40 years— is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimension by the Sobolev dimension. This is based on a noncommutative form of the Sobolev embedding theory for Markov semigroups initiated by Varopoulos, and yields more flexibility to measure the smoothness of the symbol. Third, we introduce a dual notion of polynomial growth to further exploit our maximal principle for non-spectral Fourier multipliers. The combination of these ingredients yields new Lp estimates for smooth Fourier multipliers in group algebras.
TL;DR: In this paper, the authors studied the non-uniform sampling and reconstruction problem in shift-invariant subspaces of mixed Lebesgue spaces and proposed a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X = { ( x j, y k ) : k, j ∈ J } is sufficiently dense.
TL;DR: In this paper, it was shown that any Lorentz-Zygmund space L a, r ( Log L ) β, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that β ≠ 0.
Abstract: In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz–Zygmund spaces or more generally G Γ -spaces. As a direct consequence of our results any Lorentz–Zygmund space L a , r ( Log L ) β , is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1 a ∞ , β ≠ 0 . The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.
01 Mar 2017
TL;DR: Mitsuo Izuki was partially supported by Grand-in-Aid for Scientific Research (C), No. 15K04928, for Japan Society for the Promotion of Science as mentioned in this paper.
Abstract: Mitsuo Izuki was partially supported by Grand-in-Aid for Scientific Research (C), No. 15K04928, for
Japan Society for the Promotion of Science.
Yoshihiro Sawano was partially supported by Grand-in-Aid for Scientific Research (C), No. 16K05209,
for Japan Society for the Promotion of Science.
TL;DR: In this article, the Hausdorff operator on weighted Herz spaces was studied in the setting of the Heisenberg group and its sharp boundedness on power-weighted Herz spaces and powerweighted Lebesgue spaces was established.
Abstract: In this article, we study the Hausdorff operator, defined via a general linear mapping A, on weighted Herz spaces in the setting of the Heisenberg group. Under some assumptions on the mapping A, we establish its sharp boundedness on power-weighted Herz spaces and power-weighted Lebesgue spaces in the Heisenberg group. Our proof is heavily based on the block decomposition of the Herz space, which is quite different from any other function spaces. Our results extend and improve some existing theorems.
TL;DR: In this article, the boundedness of the Hardy-Littlewood maximal function on Lebesgue spaces and Morrey spaces was studied. But the bounded properties of the Lipschitz spaces were not considered.
TL;DR: The main purpose of as discussed by the authors is to carry on with this process by introducing and investigating the spaces Lp(T) with p∈[1, ∞] and LpT with p ∈ [1,∞] for conditional expectation T with natural domain a Riesz space L1(T).
TL;DR: In this paper, the existence and uniqueness of l p -solutions for discrete time fractional models in the form Δ α u (n, x ) = A u ( n, x) + ∑ j = 1 k β j u(n − τ j, x ) + f ( n, u( n, x) ), n ∈ Z, x ∈ R N, β j ∈ N, τ j ∆ R N, and τ n ∆ Z, where A is a closed linear operator defined on a Banach space X
TL;DR: In this paper, the authors extend the extrapolation theory to Morrey spaces associated with Banach function spaces and obtain the John-Nirenberg inequalities on these spaces, as well as characterizations of the function spaces of bounded mean oscillation in terms of these spaces.
Abstract: We extend the extrapolation theory to Morrey spaces associated with Banach function spaces. Some applications by this theory such as the Fefferman–Stein vector-valued maximal inequalities are obtained. By using this extrapolation theory, we obtain the John-Nirenberg inequalities on Morrey spaces associated with Banach function spaces. In addition, by using the John-Nirenberg inequalities, we have the characterizations of the function spaces of bounded mean oscillation in terms of Morrey spaces associated with Banach function spaces.
TL;DR: In this paper, the authors provided characterizations for boundedness of multilinear Fourier multiplier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces.
Abstract: We provide characterizations for boundedness of multilinear Fourier multiplier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0 \lt q \le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1 \lt q \le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderon and Torchinsky [1] and of the bilinear results of Miyachi and Tomita [17]. The extension to general $m$ is significantly more complicated both technically and combinatorially; the optimal Sobolev space smoothness required of the symbol depends on the Hardy–Lebesgue exponents and is constant on various convex simplices formed by configurations of $m2^{m-1}+1$ points in $[0,\infty)^m$.
TL;DR: In this paper, the authors proved that solitary wave solutions are asymptotically stable in the even subspace of perturbations (to ignore translations and eigenvalues ± 2 ω i ).
TL;DR: In this article, the authors prove maximum and comparison principles for the discrete fractional derivatives in the integers and prove the convergence to the Grunwald-Letnikov derivative for Holder continuous functions.
TL;DR: In this paper, the hypercyclicity of forward and backward shifts on weighted L p spaces of a directed tree was studied and necessary and sufficient conditions for the backward shift were obtained.
TL;DR: In this article, the authors take up the study of more general Lq(ℝ3) → Lr(S) Fourier restriction estimates, by studying a prototypical model class of two-dimensional surfaces for which the Gaussian curvature degenerates in one-dimensional subsets.
Abstract: The problem of Lq(ℝ3) → L2(S) Fourier restriction estimates for smooth hypersurfaces S of finite type in ℝ3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general Lq(ℝ3) → Lr(S) Fourier restriction estimates, by studying a prototypical model class of two-dimensional surfaces for which the Gaussian curvature degenerates in one-dimensional subsets. We obtain sharp restriction theorems in the range given by Tao in 2003 in his work on paraboloids. For high-order degeneracies this covers the full range, closing the restriction problem in Lebesgue spaces for those surfaces. A surprising new feature appears, in contrast with the nonvanishing curvature case: there is an extra necessary condition. Our approach is based on an adaptation of the bilinear method. A careful study of the dependence of the bilinear estimates on the curvature and size of the support is required.
TL;DR: In this paper, the authors define a Hardy type space, by taking in the maximal characterization of Hardy spaces, the Wiener amalgam norms of the maximal functions, instead of the Lebesgue norms.
TL;DR: In this paper, the authors studied multivariable spectral multipliers of the Cartesian product of the Riesz-transform-like operators and double Bochner-Riesz means and showed that if a function satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space.
Abstract: Let $$X_1$$
and $$X_2$$
be metric spaces equipped with doubling measures and let $$L_1$$
and $$L_2$$
be nonnegative self-adjoint operators acting on $$L^2(X_1)$$
and $$L^2(X_2)$$
respectively. We study multivariable spectral multipliers $$F(L_1, L_2)$$
acting on the Cartesian product of $$X_1$$
and $$X_2$$
. Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators $$L_1$$
and
$$L_2$$
, we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $$F(L_1, L_2)$$
is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space $$X_1\times X_2$$
. We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner–Riesz means.
Posted Content•
TL;DR: In this paper, a new weak norm, iterated weak norm in Lebesgue spaces with mixed norms, was proposed and the convergence of the mixed weak norm and the truncated norm was studied.
Abstract: In this paper, we consider a new weak norm, iterated weak norm in Lebesgue spaces with mixed norms. We study properties of the mixed weak norm and the iterated weak norm and present the relationship between the two weak norms. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for both weak norms. We study the convergence in truncated norm, which is a substitution of the convergence in measure for mixed Lebesgue spaces. And we give a characterization of the convergence in truncated norm. We show that Holder's inequality is not always true on mixed weak spaces and we give a complete characterization of indices which admit Holder's inequality. As applications, we establish some geometric inequalities related to fractional integration in mixed weak spaces and in iterated weak spaces which essentially generalize the Hardy-Littlewood-Sobolev inequality.