Bessel Potential Research Articles

A Note on the Fourier Magnitude Data and Sobolev Embeddings

We study Sobolev Hs(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s(\\mathbb {R}^n)$$\\end{document}, s∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\in \\mathbb {R}$$\\end{document}, stability of the Fourier phase problem to recover f from the knowledge of |f^|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$| \\hat{f} |$$\\end{document} with an additional Bessel potential Ht,p(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{t,p}(\\mathbb {R}^n)$$\\end{document} a priori estimate when t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} and p∈[1,2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [1,2]$$\\end{document}. These estimates are related to the ones studied recently by Steinerberger in ”On the stability of Fourier phase retrieval” J. Fourier Anal. Appl., 28(2):29, 2022. While our estimates in general are different, they share some comparable special cases and the main improvement given here is that we can remove an additional imaginary term and obtain sharper constants. We also consider these estimates for the quotient distances related to the non-uniqueness of the Fourier phase problem. Our arguments closely follow the Fourier analysis proof of the Sobolev embeddings for Bessel potential spaces with minor modifications.

Vortex clusters and their active control in a cold Rydberg atomic system with $\mathcal{PT}$-symmetric Bessel potential

Vortex clusters and their active control in a cold Rydberg atomic system with $\mathcal{PT}$-symmetric Bessel potential

Boundedness of the Pseudo-differential Bessel Operators on Sobolev and Bessel Potential Spaces

Boundedness of the Pseudo-differential Bessel Operators on Sobolev and Bessel Potential Spaces

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

Inversion of Bessel potentials associated with the Dunkl operators on I R d

The inversion of Bessel potentials is given by using a special-type weighted wavelet transform in the context of Dunkl harmonic analysis on I R d . It has also proved the Calderón-type reproducing formula.

An alternative proof of Tataru’s dispersive estimates

Abstract The aim of this article is to give an alternative proof of Tataru’s dispersive estimates for wave equations posed on the hyperbolic space. Based on the formula for the wave kernel on ℍ n {\mathbb{H}^{n}} , we give the proof from the perspective of Bessel potentials, by exploiting various facts about Gamma functions, modified Bessel functions, and Bessel potentials. This leads to our proof being more self-contained than that in [D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 2001, 2, 795–807].

Local wellposedness of the relativistic Vlasov–Poisson equation in Besov space

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential.

Global existence and spatial analyticity for a nonlocal flux with fractional diffusion

In this paper, we study a one dimensional nonlinear equation with diffusion −ν(−∂xx)α2 for 0 ≤ α ≤ 2 and ν > 0. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space L1(R)∩H1/2(R) when 0 ≤ α ≤ 2. For the subcritical case 1 < α ≤ 2 and critical case α = 1, we obtain the global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrapping method to improve the regularity of mild solutions in the Bessel potential spaces for the subcritical case 1 < α ≤ 2. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case α = 1, if the initial data ρ0 satisfies −ν < inf ρ0 < 0, we use the method of characteristics for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If ρ0 ≥ 0, the solution exists globally and converges to steady state.

Representations and Regularity of Vector-Valued Right-Shift Invariant Operators Between Half-Line Bessel Potential Spaces

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples.

On the Theory of Spaces of Generalized Bessel Potentials

On the Theory of Spaces of Generalized Bessel Potentials

Ground state solutions for Choquard equations with Bessel potential

Firstly, we study the existence, regularity and decay of positive solutions for the elliptic system { − Δ u + u = vg ( u ) in R N , − Δv + v = G ( u ) in R N , where N ≥ 3 , g satisfies Berestycki-Lions type conditions. In Sobolev space, the system can be transformed into the Choquard equations with Bessel potential − Δ u + u = ( g 2 ⋆ G ( u ) ) g ( u ) in R N , where g 2 is the fundamental solution of the linear operator − Δ + I on R N . Moreover, by assuming that g is nondecreasing, we obtain the radial symmetry and monotonic property of positive solutions by moving plane method, and the existence of ground state solutions by non-Nehari manifold method.

On Generalized Bessel Potentials and Perfect Functional Completions

On Generalized Bessel Potentials and Perfect Functional Completions

Theory of generalized Bessel potential space and functional completion

Theory of generalized Bessel potential space and functional completion

Multidimensional optical solitons and their manipulation in a cold atomic gas with a parity-time-symmetric optical Bessel potential

We propose a scheme to realize an optical Bessel potential with parity-time ($\mathcal{P}T$) symmetry and investigate the existence, propagation, and manipulation of multidimensional optical solitons through the interplay among diffraction, Kerr nonlinearity, and potential confinement in a cold atomic gas under the condition of electromagnetically induced transparency (EIT). We show that the system supports not only two-dimensional stationary optical solitons but also rotary ones; the stability of such solitons can be actively controlled by the gain-loss component (imaginary part), while the rotary motions can be tuned by the refractive-index component (real part) of the $\mathcal{P}T$-symmetric potential. Moreover, we demonstrate that the system allows the existence of stable three-dimensional spatiotemporal optical solitons, i.e., optical bullets, which have ultraslow propagation velocity and display helicoidal motions with controllable propagation trajectories. Due to the Kerr nonlinearity enhanced by the EIT effect, extremely low power is needed to create these multidimensional optical solitons. The results reported here are useful not only for the generation and manipulation of high-dimensional solitons via $\mathcal{P}T$-symmetric potentials, but also for promising applications in optical information processing and transmission.

The nontrivial solutions for nonlinear fractional Schrödinger-Poisson system involving new fractional operator

In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem in combination with the perturbation method, we prove the existence of solutions.

The fractional p,-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (-Delta )^2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces H^{t,p} for any tin {mathbb R}, 1 le p < infty and s in {mathbb R}_+ {setminus } {mathbb N}: If uin H^{t,p}({mathbb R}^n) satisfies (-Delta )^su=u=0 in a nonempty open set V, then uequiv 0 in {mathbb R}^n. This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension.

Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

A maximal $$L_p$$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Let $$Z=(Z_t)_{t\ge 0}$$ be an additive process with a bounded triplet $$(0,0,\varLambda _t)_{t\ge 0}$$ . Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: $$\begin{aligned} {\mathcal {A}}_Z(t)u(t,x)&=\lim _{h \downarrow 0}\frac{{\mathbb {E}}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h} \\&=\int _{{\mathbb {R}}^d}(u(t,x+y)-u(t,x)-y\cdot \nabla _x u(t,x)1_{|y|\le 1})\varLambda _t(dy). \end{aligned}$$ Suppose that for any Schwartz function $$\varphi $$ on $${\mathbb {R}}^d$$ whose Fourier transform is in $$C_c^{\infty }(B_{c_s} {\setminus } B_{c_s^{-1}} )$$ , there exist positive constants $$N_0$$ , $$N_1$$ , and $$N_2$$ such that $$\begin{aligned} \int _{{\mathbb {R}}^d}|{\mathbb {E}}[\varphi (x+r^{-1}Z_t)]|dx\le N_0 e^{- \frac{ N_1 t}{s(r)}},\quad \forall (r,t)\in (0,1)\times [0,T], \end{aligned}$$ and $$\begin{aligned} \Vert \psi ^{\mu }(r^{-1}D)\varphi \Vert _{L_1({\mathbb {R}}^d)}\le \frac{N_2}{s(r)},\quad \forall r\in (0,1). \end{aligned}$$ where s is a scaling function (Definition 2.4), $$c_s$$ is a positive constant related to s, $$\mu $$ is a symmetric Lévy measure on $${\mathbb {R}}^d$$ , $$\psi ^{\mu }(r^{-1}D)\varphi (x)= {\mathcal {F}}^{-1} \left[ \psi ^{\mu }(r^{-1}\xi ) {\mathcal {F}}[\varphi ]\right] (x)$$ and $$\psi ^{\mu }(\xi ){:=}\int _{{\mathbb {R}}^d}(e^{iy\cdot \xi }-1-iy\cdot \xi 1_{|y|\le 1})\mu (dy)$$ . In particular, above assumptions hold for Lévy measures $$\varLambda _t$$ having a nice lower bound and $$\mu $$ satisfying a weak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Lévy measures $$\varLambda _t$$ and they do not have to be symmetric. In this paper, we establish the $$L_p$$ -solvability to the initial value problem 0.2 $$\begin{aligned} \frac{\partial u}{\partial t}(t,x)={\mathcal {A}}_Z(t)u(t,x),\quad u(0,\cdot )=u_0,\quad (t,x)\in (0,T)\times {\mathbb {R}}^d, \end{aligned}$$ where $$u_0$$ is contained in a scaled Besov space $$B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb {R}}^d)$$ (see Definition 2.8) with a scaling function s, exponent $$p \in (1,\infty )$$ , $$q\in [1,\infty )$$ , and order $$\gamma \in [0,\infty )$$ . We show that equation (0.2) is uniquely solvable and the solution u obtains full-regularity gain from the diffusion generated by a stochastic process Z. In other words, there exists a unique solution u to equation (0.2) in $$L_q((0,T);H_p^{\mu ;\gamma }({\mathbb {R}}^d))$$ , where $$H_p^{\mu ;\gamma }({\mathbb {R}}^d)$$ is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies $$\begin{aligned} \Vert u\Vert _{L_q((0,T);H_p^{\mu ;\gamma }({\mathbb {R}}^d))}\le N\Vert u_0\Vert _{B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb {R}}^d)}, \end{aligned}$$ where N is independent of u and $$u_0$$ . We finally remark that our operators $${\mathcal {A}}_{Z}(t)$$ include logarithmic operators such as $$-a(t)\log (1-\varDelta )$$ (Corollary 3.2) and operators whose symbols are non-smooth such as $$-\sum _{j=1}^dc_j(t)(-\varDelta )^{\alpha /2}_{x^j}$$ (Corollary 3.9).

Derivatives of Sub-Riemannian Geodesics are Lp-Hölder Continuous

This article is devoted to the long-standing problem on the smoothness of sub-Riemannian geodesics. We prove that the derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous. Additionally, this result has several interesting implications. These include (i) the decay of Fourier coefficients on abnormal controls, (ii) the rate at which they can be approximated by smooth functions, (iii) a generalization of the Poincaré inequality, and (iv) a compact embedding of the set of shortest paths into the space of Bessel potentials.

Two-dimensional spatial optical solitons in Rydberg cold atomic system under the action of optical lattice

Realizing stable high-dimensional light solitons is a long-standing goal in the study of nonlinear optical physics. However, in high-dimensional space, the light field will inevitably be distorted due to diffraction. In order to solve the diffraction effect in nonlinear Kerr media and achieve the spatial localization of light fields, we propose a scheme to generate stable two-dimensional (2D) solitons in a cold Rydberg atomic system with a Bessel optical lattice, where a three-level atomic structure, a weak probe laser field, and a strong control field constitute the Rydberg-dressed atomic system. When the local nonlinearity, Bessel potential, and nonlocal nonlinearity which is caused by the long-range Rydberg-Rydberg interaction (RRI) between Rydberg atoms are balanced, the probe field can be localized. Under the approximation of electric dipole and rotating wave, the stable solution of probe field is obtained by solving Maxwell-Bloch equations numerically. A cluster of 2D spatial solitons, including fundamental, two-pole, quadrupole and vortex solitons, is found in this system. Among them, the fundamental, dipole and quadrupole have, one, two, and four intensity centers, respectively. Vortex solitons, on the other hand, exhibit vertical characters in profiles and phase structures. The formation and transmission of these solitons can be controlled by system parameters, such as the propagation coefficient, the degree of nonlocal nonlinearity, and Bessel lattice strength. The stable regions of these solitons are determined by anti Vakhitov Kolokolov (anti-VK) criterion and linear stability analysis method. It is found that four kinds of solitons can be generated and stably propagate in space with proper parameters. Owing to the different structures of the poles, the fundamental state and vortex state remain stable, while the quadrupole ones are unstable. In the modulation of solitons, there is a cutoff value of propagation constant &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}${b_{{\text{co}}}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20230096_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, only below which value, the solitons can propagate stably. The light intensity of soliton shows a periodic behavior by tuning Bessel lattice strength. The period of the intensity decreases with the order of the solitons as a result of the interaction between the poles. It is also found that the solitons are more stable with weak nonlocal nonlinearity coefficient. This study provides a new idea for the generation and regulation of optical solitons in high dimensional space.