We study the problem of best local polynomial approximation in variable Lebesgue spaces L^p(· ), where the error is measured by an averaged Luxemburg seminorm over measurable sets shrinking to a finite collection of prescribed nodes. In this context, we establish a Pólya-type inequality for algebraic polynomials and develop uniform boundedness and error estimates for nets of best approximants. These results provide the basis for analyzing the asymptotic behavior of best local approximations and their connection to Hermite interpolation. When the number of prescribed nodes does not exceed the dimension of the polynomial space, we identify the limiting approximation explicitly. In the overdetermined case, we characterize the cluster points as solutions of finite-dimensional minimization problems. As an application, we obtain sufficient conditions for the existence of best local approximations of piecewise smooth functions in variable Lebesgue spaces.
The study of linear combinations of weighted composition operators on various analytic function spaces is a natural, yet challenging topic in operator theory. Motivated by the work of Choe, Koo and Wang, we establish estimates for the essential norm of linear combinations of weighted composition operators acting on weighted Besov–Sobolev spaces. Our method integrates reproducing kernel techniques, the Fréchet–Montel principle, and approximation by dilations with a Kolmogorov–Riesz type compactness criterion. We also present several applications of our main results. Consequently, we recover and sharpen the results of Choe et al. [5] and Sharma and Sharma [22].
In [2] we considered a quadratically constrained quadratic programming problem in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Unfortunately, the proof of Theorem 3.3 in the above mentioned paper is incomplete, because [1, Thm 3.5] was mistakenly cited there. The statement of the theorem yet holds, but its proof needs to be completed. In the following we present an addendum to the proof of [2, Thm. 3.3].
In this paper, we first define the class 𝒮_g,γ^k+1(Ω ) of g-starlike mappings of real order γ ( γ∈ (0,1] ) on the bounded starlike circular domain Ω in ℂ^n and give the growth theorem for functions f∈𝒮_g,γ^k+1(Ω ) , where z=0 is a zero of order k+1 of f(z)-z and g is a Carathéodory function in 𝔻 . Next, as applications of the growth theorem, we will establish the distortion theorems of the Fréchet-derivative type for a subclass of 𝒮_g,γ^k+1(Ω ) with a weak restrictive condition. We also establish the distortion theorems of the Jacobi-determinant type for some subclasses of 𝒮_g,γ^k+1(Ω ) . Several important results can be obtained indirectly. Our works improve the classic distortion theorem of holomorphic functions from the case in ℂ to the case on the bounded starlike circular domain in ℂ^n .
We give an exact description of Calderón’s construction on a pair of cones of monotonically decreasing functions in weighted Lebesgue spaces. We also establish the interpolation and reiteration theorems for these cones.
We classify additive surjections that preserve the q -numerical radius/range of products of operators, where q∈ [0,1] .
Let ℬ_s(ℋ) be the real linear space of all self-adjoint operators on a complex Hilbert space ℋ with ℋ≥ 3 . We characterize all continuous bijections on ℬ_s(ℋ) preserving operator pairs whose pencils are nonzero projection multipliers in both directions.
For the Fredholm integral equations of the second kind, this paper introduces the mixed norm condition for kernel functions and establishes the Fredholm theory on L^p spaces ( 1 ⩽ p ⩽∞ ) by means of the degenerate kernel approximation method. The research method in this paper frees the study of Fredholm integral equations of the second kind from the limitation of Hilbert spaces and is widely applicable to other function spaces represented by L^p spaces (usually Banach spaces), which provides important theoretical support for the research on operator theory and related mathematical physics problems. Since the obtained results are independent of the specific properties of the real line, all conclusions can be directly generalized to high-dimensional Euclidean spaces, offering a reliable theoretical basis for the analysis of high-dimensional integral equations.
We investigate the existence of commuting self-adjoint extensions of the partial differential operators D_j=1/2π i∂/∂ x_j, j=1,… ,d, defined initially on C_0^∞ (Ω ) , where Ω⊂ℝ^d is an arbitrary open set, possibly disconnected and unbounded. This problem originates in a question of Segal and is closely related to Fuglede’s characterization of spectral sets through commuting self-adjoint extensions and orthogonal Fourier bases. While previous results of Fuglede and Pedersen established a complete correspondence for connected domains, the disconnected case presents new phenomena arising from interactions among distinct connected components. We extend the spectral-theoretic framework to arbitrary open sets and obtain a necessary and sufficient condition for the existence of commuting self-adjoint extensions of the operators {D_j} . The characterization is expressed in terms of a Radon measure, a multiplicity function, and a measurable matrix field that encode the spectral decomposition of the resulting commuting family. In contrast with the connected case, multiplicities greater than one may occur, and generalized eigenfunctions involve component-dependent coefficients attached to the exponential functions e_λ (x)=e^2π iλ· x . We further establish a characterization of spectral sets in terms of strongly continuous unitary representations acting as groups of local translations. For arbitrary open sets, we show that the existence of a group of local translations is equivalent to the spectrality of the domain. This extends the classical connection between spectral pairs, Fourier analysis, and translation symmetries beyond the connected setting. For open sets of finite measure with finitely many connected components, we prove that the associated spectral measure is atomic and that the joint spectrum is separated. We obtain an orthonormal basis consisting of piecewise exponential functions whose amplitudes may vary from one connected component to another. In the one-dimensional setting, where Ω is a countable union of intervals, we describe the corresponding unitary groups through a boundary transition matrix governing the propagation of translations across interval endpoints. Finally, for sets that tile ℝ^d by a discrete subgroup, we characterize the associated pair measures and derive an explicit formula for the corresponding group of local translations, thereby extending a classical result of Fuglede from lattices to arbitrary discrete subgroups.
The metric geometric and spectral geometric means are two essential concepts in the theory of matrix mean. In [15], Lemos and Soares asked whether there exists the following log-majorization relation for A, B≥ 0 , t∈ [0,1] , s(A^t(A♯ _tB)B^1-t) ≺ _(log ) s(AB). Recently, Gan and Kim prove alternative versions of above log-majorization for the metric geometric and spectral geometric means in [8]. In this paper, by several Furuta-type inequalities, we obtain several log-majorizations of Lemos–Soares type for the metric geometric mean and weak log-majorizations of Lemos–Soares type for the spectral geometric mean, which extend the recent results from Gan and Kim and other related results.
This manuscript investigates the spherical density of the lower Hewitt–Stromberg measure in ℝ^d . Also, we establish that if S=(S_1,… ,S_n) is an iterated function system fulfilling the strong open set condition for some open set U, then 𝒰^α (K_S ∩ U(x,r)) ≤ (2r)^α , for every open ball U(x,r)⊂ U with x∈ K_S and r>0 . Employing this estimate, we derive an exact formula for the lower Hewitt–Stromberg measure of self-similar sets. As a consequence, we show that the mapping M_SSC⟶ℝ: S ⟼𝒰^α (K_S), is continuous, where M_SSC denotes the class of all iterated function systems fulfilling the strong separation condition.
In this paper, we investigate the r-summing Carleson embeddings from generalized Fock spaces F_ϕ^p into L_ϕ^p(μ ) , where ϕ∈𝒞^2( ℂ^n) is real-valued and satisfies mω_0≤ dd^cϕ≤ Mω_0 for two positive constants m and M , ω_0 = dd^c| z| ^2 is the Euclidean Kähler form on ℂ^n , d^c = √(-1)/4( ∂̅ - ∂) . μ is a positive Borel measure on ℂ^n . We characterize the r-summability of the embeddings Id: F_ϕ^p→ L_ϕ^p( μ) for any r ≥ 1 and 1< p<∞ .
Given the real line interval I=[0,T] and a separable Banach space E, we consider the basic sweeping process defined by a moving set C:I→ 2^E and add at a second time a perturbation defined by a Carathéodory integrand f:I× E→ E . Assuming mainly usual assumptions and adding respectively (for the two cases) the weak compactness and the norm compactness of values of C, we obtain existence of solutions through an embedding of E in a Hilbert space. This allows to use a scalar product to perform projections. Then, describes well directed solutions with a convenient orthogonality notion, better than that represented by the (non symmetric) semi-scalar product. We adopt functions of bounded variation as solutions.
A new characterization of the conjugates of absolutely continuous operators is presented. We introduce and investigate the concept of positive p-absolutely continuous operators and show in particular that it is a true generalization of absolutely continuous operators. By quantifying the notion of positive p-absolutely continuous operators, we introduce the concept of positive (p,σ ) -absolutely continuous operators and establish the Pietsch domination/factorization theorem for it. By means of positive (p,σ ) -absolutely continuous operators, we introduce the concept of positive ((p,σ ),(q,ν )) -dominated operators and prove the famous Kwapień’s factorization theorem for it. Finally, we study the maximal properties of the classes of positive (p,σ ) -absolutely continuous operators and positive ((p,σ ),(q,ν )) -dominated operators and show that they are maximal.
Let (𝒳,d,μ ) be a metric measure space with μ being doubling. In this article, via Hajłasz gradients, we define the variable Besov–Triebel–Lizorkin spaces on 𝒳 and then establish an approximation property of those spaces in terms of discrete γ -median. By a new concept of the variable power of variable mixed Lebesgue-sequence (quasi-)norms, we introduce the variable Besov–Triebel–Lizorkin p(· ) -capacity on 𝒳 and also obtain an equivalent expression of the p(· ) -capacity. Moreover, the lower and upper bound estimates for these p(· ) -capacity in terms of a modified version of the generalized Netrusov–Hausdorff content are established
Our purpose in this paper is to explain how to calculate the relative homology corresponding to an operator ideal, presenting the raw Banach space facts as well as their homological translations. We will display a few extremal cases to show how different the standard derivation (relative to the operator ideal 𝔏 of all linear bounded maps) and relative derivation with respect to an operator ideal 𝔄 can be.
We introduce the notions of tauberian, cotauberian and weakly compact pair of closed subspaces of a Banach space. The theory produced by these notions is richer than that of the corresponding operators since an operator can be regarded as a suitable pair of closed subspaces. We investigate into these classes of pairs of subspaces and describe several applications to define some notions of indecomposability for Banach spaces and to extend definitions from the case of bounded operators to the case of closed operators.
We introduce the realification of the Siegel upper half-space, a domain in real space obtained by treating the real and imaginary parts of each complex coordinate as independent variables. On the tube domain over this realified space, we derive an explicit formula for the weighted Bergman kernel and establish necessary and sufficient conditions for the boundedness of two classes of Forelli–Rudin-type operators acting between weighted L^p and L^q spaces for all (p,q)∈ [1,∞ ]× [1,∞ ] .
