Convolution Algebraic Structures Defined by Hardy-Type Operators

This work establishes the existence of weak solutions for p-Laplacian-like equations in weighted Sobolev spaces under Dirichlet boundary conditions, assuming the data is in the weighted Lebesgue space.

On constructing multiplications on Banach spaces

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, and they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also uncovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works.

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

On the stability of the quadratic mapping in Banach modules

Modular inequalities and inequalities for the norms of Hardy-type operators on the cone Ω of positive functions and on the cone of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Ω is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces.

In this work, we present results on the embeddings of fractional Riemann-Liouville Sobolev spaces, using an important relationship between Riemann-Liouville Sobolev spaces and ordinary Sobolev spaces.This relationship allows us to prove compact embeddings after establishing continuous embeddings based on the continuity of the Riemann-Liouville fractional integral operators between Lebesgue spaces under certain conditions.We provide an example of a boundary problem where existence and uniqueness are addressed using two methods: the fixed point method and the Faedo-Galerkin method.Both methods require specific fractional type embeddings.

On the constants for multiplication in Sobolev spaces

Applying the theory of pseudodifferential and Calderón-Zygmund operators, we study the compactness of commutators of multiplication operators aI and convolution operators \( W^{0}(b) \) on weighted Lebesgue spaces \( L^{p}(\mathbb{R},w)\;{\rm with}\; p \in (1,\infty)\) and Muckenhoupt weights ?? for some classes of piecewise slowly oscillating functions \( a \in PSO^{\diamond} \; and b \in PSO^{\diamond}_{p,w}\) on the real line ℝ. Then we study the Banach algebra \( Z_{p,w} \) generated by the operators \( aW^{0}(b)\) with functions \( a \in SO^{\diamond} \; and b \in SO^{\diamond}_{p,w}\) admitting slowly oscillating discontinuities at every point \( lambda \in \mathbb{R} \cup \left\{\infty \right\} \). Applying the method of limit operators under some condition on Muckenhoupt weights w, we describe the maximal ideal space of the commutative quotient Banach algebra \( Z^{\pi}_{p,w}\;=\;Z_{p,w}/K_{p,w}{\rm where}\;K_{p,w}\) is the ideal of compact operators on \( L^{p}(\mathbb{R,w}) \) define the Gelfand transform for \( Z^{\pi}_{p,w}) \) and establish the Fredholmness for the operators \( A\in Z_{p,w}\)

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

The Cauchy problem for the elliptic–hyperbolic Davey–Stewartson system in Sobolev space

Let X, Y be Banach modules over a -algebra and let be given. We prove the generalized Hyers–Ulam stability of the following functional equation in Banach modules over a unital -algebra: We show that if and an odd mapping satisfies the functional equation (0.1) then the odd mapping is Cauchy additive. As an application, we show that every almost linear bijection of a unital -algebra A onto a unital -algebra B is a -algebra isomorphism when for all unitaries , all , and all . The concept of generalized Hyers–Ulam stability originated from Th.M. Rassias' stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type neces- sary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.