Space Of Generalized Functions Research Articles

Two‐weight extrapolation on function spaces and applications

AbstractThis paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including , , and extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman–Fefferman inequalities that can be used to show some known sharp inequalities, Muckenhoupt–Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calderón–Zygmund operators. Finally, we obtain two‐weight inequalities for Littlewood–Paley operators and Fourier integral operators on weighted Banach function spaces.

The Gleason’s problem on normal weight general function spaces in the unit ball of ℂn

The Gleason’s problem on normal weight general function spaces in the unit ball of ℂn

A biorthogonal approach to the infinite dimensional fractional Poisson measure

In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure [Formula: see text], [Formula: see text], on the dual of Schwartz test function space [Formula: see text]. The Hilbert space [Formula: see text] of complex-valued functions is described in terms of a system of generalized Appell polynomials [Formula: see text] associated to the measure [Formula: see text]. The kernels [Formula: see text], [Formula: see text], of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system [Formula: see text], there is a generalized dual Appell system [Formula: see text] that is biorthogonal to [Formula: see text]. The test and generalized function spaces associated to the measure [Formula: see text] are completely characterized using an integral transform as entire functions.

Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions

It was recently shown that functions in L4([−B,B]) can be uniquely recovered up to a global phase factor from the absolute values of their Gabor transforms sampled on a rectangular lattice. We prove that this remains true if one replaces L4([−B,B]) by Lp([−B,B]) with p∈[1,∞]. To do so, we adapt the original proof by Grohs and Liehr and use a classical sampling result due to Beurling. Furthermore, we present a minor modification of a result of Müntz–Szász type by Zalik. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to Lp([−B,B]) and for more general nonuniform sampling sets.

Extrapolation in general quasi-Banach function spaces

In this work we prove off-diagonal, limited range, multilinear, vector-valued, and two-weight versions of the Rubio de Francia extrapolation theorem in general quasi-Banach function spaces. We prove mapping properties of the generalization of the Hardy-Littlewood maximal operator to very general bases that includes a method to obtain self-improvement results that are sharp with respect to its operator norm. Furthermore, we prove bounds for the Hardy-Littlewood maximal operator in weighted Lorentz, variable Lebesgue, and Morrey spaces, and recover and extend several extrapolation theorems in the literature. Finally, we provide an application of our results to the Riesz potential and the Bilinear Hilbert transform.

Integral Estimates and Multiplier Operators from Normal Weight General Function Spaces to Bloch Type Spaces

Integral Estimates and Multiplier Operators from Normal Weight General Function Spaces to Bloch Type Spaces

Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.

Summation-by-Parts Operators for General Function Spaces

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.

The maximal regularity property of abstract integrodifferential equations

We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces varPhi and/or the scales of Besov and Triebel–Lizorkin spaces defined by varPhi . We apply this result to the study of the well-posedness and maximal regularity property of abstract second-order integrodifferential equations, which model various types of elliptic and parabolic problems arising in different areas of applied mathematics.

Generalized integration operators from weighted Bergman spaces into general function spaces

Generalized integration operators from weighted Bergman spaces into general function spaces

Domination and Korenblum constants for some function spaces

The paper shows the domination in M? which is associated with the analytic Morrey space L2,?. We get the explicit expressions for upper bounds of Korenblum constants for the analytic Morrey space L2,?, the weighted Dirichlet space Dp ?, and the general function space F(p, q, s), etc. Furthermore, the domination in the Dirichlet space D? is investigated.

Regularity estimates for stationary Stokes problem in some generalized function spaces

Regularity estimates for stationary Stokes problem in some generalized function spaces

Toeplitz Operators with Radial Symbols on General Analytic Function Spaces

Toeplitz Operators with Radial Symbols on General Analytic Function Spaces

Besov regularity in non-linear generalized functions

We introduce and study new modules and spaces of generalized functions that are related to the classical Besov spaces. Various Schwartz distribution spaces are naturally embedded into our new generalized function spaces. We obtain precise criteria for detecting Besov regularity of distributions.

An Improved Regularity Criterion for the 3D Magnetic Bénard System in Besov Spaces

This article notably targets the more general (extended) function spaces by investigating the regularity of the weak solutions or turbulent solutions to the Cauchy problem of the 3D magnetic Bénard system by converting it into mathematical symmetric form, in the absence of thermal diffusion, in terms of pressure. In that regard, we successfully improved the results by obtaining sufficient integrable regularity conditions for the pressure and gradient pressure in the homogeneous Besov spaces.

The reconstruction theorem in quasinormed spaces

We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti \[EMS Surv. Math. Sci. 7.2 (2020)] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces.

On Stević-Sharma Operators from General Class of Analytic Function Spaces into Zygmund-Type Spaces

A Stevi c ′ -Sharma operator denoted by T ψ 1 , ψ 2 , φ is a generalization product of multiplication, differentiation, and composition operators. In this paper, we characterize the bounded and compact Stevi c ′ -Sharma operator T ψ 1 , ψ 2 , φ from a general class X of Banach function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. Using several restrictive terms, we show that all bounded operators T ψ 1 , ψ 2 , φ from X into the little Zygmund-type spaces are compact. As an application, we show that our results hold up for some other domain spaces of T ψ 1 , ψ 2 , φ , such as the Hardy space and the weighted Bergman space.

Essential Norms of Stević–Sharma Operators from General Banach Spaces into Zygmund-Type Spaces

A Stević–Sharma operator denoted by T ψ 1 , ψ 2 , φ is a generalization product of multiplication, differentiation, and composition operators. Using several restrictive terms, we characterize an approximation of the essential norm of the Stević–Sharma operator T ψ 1 , ψ 2 , φ from a general class X of holomorphic function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. As an application, we show that our results hold up for several other domain spaces of T ψ 1 , ψ 2 , φ , such as the Hardy space and the weighted Bergman space.

ISOMETRIES ON SOME GENERAL FAMILY FUNCTION SPACES AMONG COMPOSITION OPERATORS

In this paper, we discuss the isometries of composition operators on the holomorphic general family function spaces F(p, q, s). First, we classify the isometric composition operators acting on a general Banach spaces. For 1 < p < 2, we display that an isometry of Cφ is caused only by a rotation of the disk. We scrutinize the previous work on the case for p ≥ 2. Also, we characterize many of the foregoing results about all α-Besov-type spaces F(p, αp − 2, s), α > 0. We exhibit that in every classes F(p, αp − 2, s) except for the Dirichlet space D = F(2, 0, 0), rotations are the only that produce isometries.

Volterra integral operator from weighted Bergman spaces to general function spaces

Volterra integral operator from weighted Bergman spaces to general function spaces