Spherical square functions of Marcinkiewicz type with Riesz potentials

We prove a pointwise equivalence between a spherical square function composed with the Riesz potential and a Littlewood–Paley function arising from the Bochner–Riesz operators. Also, its application to the theory of Sobolev spaces will be given.     

Sublinear operators in generalized weighted Morrey spaces

Generalized weighted Morrey spaces defined on spaces of homogeneous type are introduced by using weight functions in the Muckenhoupt class. Theorems on the boundedness of a large class of sublinear operators on these spaces are presented. The classes of sublinear operators under consideration contain a whole series of important operators of harmonic analysis, such as, e.g., maximal functions, singular and fractional integrals, Bochner–Riesz means, and so on.     

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner–Riesz summability results with sharp exponent.     

An alternative proof of the Aubin–Lions lemma

Using a generalized version of the Weyl–Riesz criterion for compactness of subsets of Lebesgue–Bochner spaces, we present in this short note an alternative proof of a result by J. Simon [4] that extends the classical result by J.P. Aubin and J.L. Lions on compact embeddings in Lebesgue–Bochner spaces to the non-reflexive Banach space case.     

Weighted norm inequalities of the maximal commutator of quasiradial Bochner-Riesz operators

In this paper, we obtain certain Lpw(Rn)-mapping properties for the maximal operator associated with the commutators of quasiradial Bochner-Riesz means with index δ under certain surface condition on Σed , provided that δ (n-1)/2, b ∈ BMO(Rn), 1 p ∞ and w ∈ A1 . Moreover, if δ (n-1)/2, then we prove that the above maximal operator admits weak type (H1w(Rn), L1w(Rn))-mapping properties for b ∈ BMO(Rn) and w ∈ A1 under the surface condition on Σed .     

Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let \(p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]\) be a variable exponent function satisfying the globally log-Hölder continuous condition, \(q\in (0,\infty ]\) and A be a general expansive matrix on \({\mathbb {R}}^n\). Let \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) by means of the Littlewood–Paley g-function or the Littlewood–Paley \(g_\lambda ^*\)-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space \(L^{p(\cdot ),q}({\mathbb {R}}^n)\). Moreover, the finite atomic characterization of \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) to \(L^{p(\cdot ),q}({\mathbb {R}}^n)\) and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

     

Intrinsic characterizations for complex hypercylinders and complex hyperspheres

In terms of conditions on the curvature tensors of Riemann-Christoffel, Ricci, Weyl and Bochner we obtain several new characterizations of complex hyperspheres in complex projective spaces, of complex hypercylinders in complex Euclidean spaces and of complex hyperlanes in complex space forms.Aspirant N.F.W.O. (België).     

Hobson's formula for Dunkl operators and its applications

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.     

On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space

For the Riesz potential of variable order over bounded domains in Euclidean space, we prove the boundedness result from variable exponent Morrey spaces to variable exponent Campanato spaces. A special attention is paid to weaken assumptions on variability of the Riesz potential.     

Riesz transforms for the Dunkl harmonic oscillator

We propose an approach to the theory of Riesz transforms in a framework emerging from certain reflection symmetries in Euclidean spaces. Relying on Rösler’s construction of multivariable generalized Hermite functions associated with a finite reflection group on \({\mathbb R^d}\), we define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator. In the case isomorphic with the group \({\mathbb{Z}^d_2}\) it is proved that the Riesz transforms are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, thus their mapping properties follow from the general theory.     

On differential inclusions with additive generalized functions

We study a space of potentials on the n-dimensional Euclidean space that are constructed on the basis of rearrangement-invariant spaces (RISs) by means of convolutions with kernels of general form. These spaces include the classical spaces of Bessel and Riesz potentials as particular cases. We examine the integral properties of the potentials and find necessary and sufficient conditions for their embedding in an RIS. Optimal RISs for such embeddings are also described.     

Approximate Radon-Nikodým representations on Riesz algebras

For the finitely additive case approximate Radon-Nikodym representations are obtained in the setting of Riesz algebras, complementing and generalizing results of Bochner [4], Fefferman [10], de Amo, Chitescu and Díaz Carrillo [1]. Furthermore the possible relations between the various spaces which appear here are given, answering a question of [1]. Various examples show that our results are sharp, there also the class of approximately representable functionals is explicitly characterized, a partial answer to another question of [1]. Finally some open questions are listed.     

Characterization of balls by generalized Riesz energy

We show that balls, circles and 2‐spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution.     

Riesz Potentials and Liouville Operators on Fractals

An analogue to the theory of Riesz potentials and Liouville operators in R ⁿ for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of Euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved.     

求解非线性算子方程的梯形牛顿法

在Banach空间中研究非线性算子方程F（x）=0的近似求解问题．首先,把实函数数值积分的梯形公式推广到非线性泛函的Bochner积分中来,得到Bochner积分的梯形公式;然后,利用这一公式来构造牛顿迭代法的变形格式,从而得到梯形牛顿法,并在弱条件的α-判据下借助于优函数技巧证明了它的收敛性．     

Riesz Means on Compact Riemannian Symmetric Spaces

We study approximation properties of the Riesz means on compact symmetric spaces of rank one. To do so we establish equivalences between the Riesz means and Peetre K-moduli and estimate the weak type and the uniform approximation of the Riesz means at the critical index. The relations between the Riesz means and the best approximation as well as the Cesàro means are also considered.     

球面Hardy空间上Riesz平均的逼近

引进了球面Hrady空间上Riesz平均算子及Peetre K模。讨论了Riesz平均算子在Hardy空间上的逼近性质。证明了Riesz平均算子与Peetre K模的强渐近等价关系。所得结果表明Peetre K模完全刻划了Riesz平均的逼近。     

On the order of approximation by Riesz means in multiplicative systems in the classes E X [ɛ]

We establish the order of approximation by Riesz means of the Fourier series in a multiplicative system of a class of functions with given majorant of the sequence of best approximations. In some cases, approximations by Riesz means and best approximations are considered in a specific space, but, in other cases, approximations by Riesz means are considered in spaces with a stronger norm.     

Vector valued Riesz distributions on Euclidian Jordan algebras

Let V be a Euclidean Jordan algebra, Гthe associated symmetric cone and G be the identity component of the linear automorphism group of Г.In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain Г +iV.     

Growth properties for generalized Riesz potentials of functions satisfying Orlicz conditions

Riesz decomposition theorem says that superharmonic functions on the punctured unit ball are represented as the sum of generalized (Newtonian) potentials and harmonic functions. In this paper we study growth properties near the origin of spherical means for generalized Riesz potentials of functions satisfying Orlicz conditions in the punctured unit ball.