Volume 33, Issue 1

In this paper, we introduce a property of the inverse Semicircle and the free Gamma laws based on the dilation of measures in the context of Cauchy–Stieltjes Kernel (CSK) families. Assume that the CSK family produced by a non-degenerate probability measure λ on ℝ {\mathbb{R}} with support limited from above is ℱ + ⁢ ( λ ) = { 𝒬 m λ ⁢ ( d ⁢ y ) : m ∈ ( m 1 λ , m + λ ) } {\mathcal{F}_{+}}(\lambda)=\{\mathcal{Q}_{m}^{\lambda}(dy):m\in(m_{1}^{\lambda% },m_{+}^{\lambda})\} . For α ∈ ℝ ∖ 0 {\alpha\in\mathbb{R}\setminus 0} , consider 𝐇 α {\mathbf{H}_{\alpha}} , x ↦ α ⁢ x {x\mapsto\alpha x} and provide the set of measures 𝐇 α ⁢ ( ℱ + ⁢ ( λ ) ) = { 𝐇 α ⁢ ( 𝒬 m λ ⁢ ( d ⁢ y ) ) : m ∈ ( m 1 λ , m + λ ) } . \mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))=\{\mathbf{H}_{\alpha}(\mathcal% {Q}_{m}^{\lambda}(dy)):m\in(m_{1}^{\lambda},m_{+}^{\lambda})\}. Let us say that α > 0 {\alpha>0} . We demonstrate that if 𝐇 α ⁢ ( ℱ + ⁢ ( λ ) ) {\mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))} is a re-parametrization of ℱ + ⁢ ( λ ) {{\mathcal{F}_{+}}(\lambda)} (i.e., 𝐇 α ⁢ ( ℱ + ⁢ ( λ ) ) = ℱ + ⁢ ( λ ) {\mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))={\mathcal{F}_{+}}(\lambda)} ), then λ is either the free Gamma type law or the inverse Semicircle type law up to scaling.

Let 𝔸 {\mathbb{A}} be the affine group and let Φ 1 , Φ 2 {\Phi_{1},\Phi_{2}} be Young functions. We study the Orlicz amalgam spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) {W(L^{\Phi_{1}}(\mathbb{A}),L^{\Phi_{2}}(\mathbb{A}))} defined on 𝔸 {\mathbb{A}} , where the local and global component spaces are the Orlicz spaces L Φ 1 ⁢ ( 𝔸 ) {L^{\Phi_{1}}(\mathbb{A})} and L Φ 2 ⁢ ( 𝔸 ) {L^{\Phi_{2}}(\mathbb{A})} , respectively. In this paper, we obtain norm estimates of the dilation operator on the amalgam space W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) {W(L^{\Phi_{1}}(\mathbb{A}),L^{\Phi_{2}}(\mathbb{A}))} using constructions related to the affine group 𝔸 {\mathbb{A}} .

In present paper, the second-order accurate stable compact difference schemes (DSs) for the delay Schrödinger-type partial differential equation (DSPDE) in a Hilbert space are constructed. The stability of these DSs is established. As applications, stability estimates (SEs) for the solutions of DSs for two types of DSPDEs are derived. A numerical method is proposed for solving one and two-dimensional DSPDEs.

In this paper, we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these operators indeed are bounded from the martingale Hardy space H p ⁢ ( G ) {H_{p}(G)} to the Lebesgue space L p ⁢ ( G ) {L_{p}(G)} for 0 < p < 1 {0<p<1} . Moreover, we also prove the sharpness of this result. As a consequence, we obtain some new and well-known results.

In this article, we show that the approximation order of Kantorovich-type max-product Bernstein operators can be improved. We are able to obtain a better approximation order for q -Bernstein operators. In addition, we present a Kantorovich-type generalization of max-product operators based on q -integers. In particular, we find the approximation order (via the classical modulus of continuity) of the Kantorovich-type max-product q -Bernstein operator.

In this paper, A class of overlooked trigonometric-like functions is explored, along with the relevant applications. We indeed show that Taylor series, resembling that of an ordinary cosine, are representative of wider classes of functions that are naturally suited to problems ranging from molecular to Laser Physics. The article goes through the original motivations of the proposal and studies the relevant properties within the context of an Umbral interpretation. Their use in applications is discussed within the framework of Free Electron Laser theory, Lennard–Jones potentials and Kramers–Kronig causality identities.

We introduce q -analog of Schröder matrix and discuss its domains in the spaces c and c 0 {c_{0}} . Also, the Schauder basis and α-, β-, γ-duals are given and certain classes of matrix mappings are characterized on these spaces. Moreover, the characterizations of certain compact operators are established via Hausdorff measure of noncompactness.

In this article, we present new contributions to the study of endomorphisms within algebraic structures, particularly focusing on prime Banach algebras. We establish that any prime Banach algebra must be commutative if it admits endomorphisms and multipliers satisfying specific algebraic identities. Our main results generalize several known theorems. Additionally, we provide an illustrative example demonstrating that the primeness assumption in several theorems is indeed essential.

We are concerned with weak-type estimates for variable Riesz potentials I α ⁢ ( ⋅ ) , τ ⁢ f {I_{\alpha(\,\cdot\,),\tau}f} with respect to variable dimensional Hausdorff content over bounded metric measure spaces X equipped with lower Ahlfors Q ⁢ ( ⋅ ) {Q(\,\cdot\,)} -regular measures. As an application, we give an exponential integrability for I α ⁢ ( ⋅ ) , τ ⁢ f {I_{\alpha(\,\cdot\,),\tau}f} with respect to variable dimensional Hausdorff content.

In this paper, we study oscillatory singular integrals with radial rough kernels: T Q , h ⁢ f ⁢ ( x ) =  p.v.  ⁢ ∫ ℝ n f ⁢ ( x - y ) ⁢ Ω ⁢ ( y ) ⁢ h ⁢ ( | y | ) | y | n ⁢ e i ⁢ Q ⁢ ( | y | ) ⁢ d y , T_{Q,h}f(x)=\text{ p.v. }\int_{\mathbb{R}^{n}}f(x-y)\frac{\Omega(y)h(|y|)}{|y|% ^{n}}e^{iQ(|y|)}\,\mathrm{d}y, where Q ⁢ ( t ) {Q(t)} is a real polynomial on ℝ {\mathbb{R}} , Ω ∈ H ⁢ ( S n - 1 ) {\Omega\in H(S^{n-1})} is a homogeneous function of degree zero on the unit sphere S n - 1 {S^{n-1}} with vanishing mean value, and h satisfies the following conditions: 1 R ⁢ ∫ 0 R | h ⁢ ( t ) | q ⁢ d t ≤ C , | 1 p - 1 2 | < min ⁡ { 1 2 , 1 q ′ } , \frac{1}{R}\int_{0}^{R}|h(t)|^{q}\,\mathrm{d}t\leq C,\quad\bigg{|}\frac{1}{p}-% \frac{1}{2}\bigg{|}<\min\biggl{\{}\frac{1}{2},\frac{1}{q^{\prime}}\bigg{\}}, where C is a constant independent of R > 0 {R>0} , and h is the radial rough kernel. Under the assumption Ω ∈ H 1 ⁢ ( S n - 1 ) {\Omega\in H^{1}(S^{n-1})} , we prove that T Q , h {T_{Q,h}} is bounded on the Lebesgue space L p {L^{p}} for 1 < p < ∞ {1<p<\infty} , provided that h satisfies aforementioned condition. The methods we mainly use are created by S. Guo.

Let C be a semidualizing module over a commutative ring R and let R ⋉ C {R\ltimes C} be the trivial extension of R by C . We prove that (1) ( M , f ) {(M,f)} is a Gorenstein projective R ⋉ C {R\ltimes C} -module if and only if M is a C -Gorenstein projective R -module; (2) [ M , g ] {[M,g]} is a Gorenstein injective R ⋉ C {R\ltimes C} -module if and only if M is a C -Gorenstein injective R -module; (3) if R is a coherent ring, then ( M , f ) {(M,f)} is a Gorenstein flat R ⋉ C {R\ltimes C} -module if and only if M is a C -Gorenstein flat R -module. Some applications of these results are given.

Let ℬ {\mathcal{B}} be a Banach algebra. The interest of this article lies in the study of the commutativity of ℬ {\mathcal{B}} if certain specific algebraic identities hold over a non-empty open subset of ℬ {\mathcal{B}} . The limitations imposed in the hypothesis of our results are justified by some examples.

We construct a convergent recurrence scheme for a solution of the three-dimensional Dirichlet boundary value problem of the elasticity theory. By the potential method, the Dirichlet problem is reduced to the uniquely solvable Fredholm integral equation of the first kind with a weakly singular boundary integral operator generated by the single layer potential. First, we construct a sequence of successive approximations which converges to the solution of the boundary integral equation in appropriate Bessel-potential spaces of functions defined on the boundary. Afterwards, using these approximations as densities of the single layer potential, we formulate another iteration which converges to the solution of the Dirichlet boundary value problem in the appropriate Sobolev–Slobodetskii spaces of functions defined in the region occupied by the elastic body.

In this paper, we establish a new characterization of weights in dynamic inequality of Wirtinger’s type involving higher order delta derivatives on time scales. More precisely, conditions on nonnegative rd-continuous weighted functions u and υ are given which ensure that a mixed norm dynamic inequality of the form ∥ ψ σ ∥ 𝕃 u q q ⁢ ( [ a , ∞ ) 𝕋 ) ≤ C ⁢ ∥ ψ Δ n ∥ 𝕃 υ p p ⁢ ( [ a , ∞ ) 𝕋 ) , \|\psi^{\sigma}\|_{\mathbb{L}_{u^{q}}^{q}([a,\infty)_{\mathbb{T}})}\leq C\big{% \|}\psi^{\Delta^{n}}\big{\|}_{\mathbb{L}_{\upsilon^{p}}^{p}([a,\infty)_{% \mathbb{T}})}, holds for 1 < p ≤ q ≤ p ′ {1<p\leq q\leq p^{\prime}} , p ′ = p p - 1 {p^{\prime}=\frac{p}{p-1}} and ψ ∈ C r ⁢ d ( n ) ⁢ ( [ a , ∞ ) 𝕋 , ℝ ) {\psi\in C_{rd}^{(n)}([a,\infty)_{\mathbb{T}},\mathbb{R})} . As a special case, these results encompass the results obtained for integral inequalities when 𝕋 = ℝ {\mathbb{T=R}} , for the discrete inequalities when 𝕋 = ℕ {\mathbb{T=N}} , and for the quantum inequalities of quantum series when 𝕋 = q ℕ {\mathbb{T=}q^{\mathbb{N}}} .

In this paper we prove a Cauchy–Kowalevski theorem for the solutions of the system ∂ x ψ ∂ x ψ f = 0 {{}^{\psi}\partial_{x}{}^{\psi}\partial_{x}f=0} , where ∂ x ψ {{}^{\psi}\partial_{x}} stands for the Cauchy–Riemann operator, associated to the orthonormal basis ψ of ℝ m + 1 {{\mathbb{R}}^{m+1}} .

In this paper, we focus on studying the existence of positive weak solutions for a class of fractional Laplacian systems with respect to another function ψ ⁢ ( ⋅ ) {\psi(\,\cdot\,)} with multiple parameters. These systems arise in various applications, such as Lévy flights. We establish new conditions for the existence of a positive solution for the considered system using the method of sub-super solutions, which extends and complements previously known results in the literature.

The paper presents an algebraic characterization of a twist-free weakly locally compact group whose factor group is linearly compact with respect to the zero component. It is shown that the local direct product of weakly locally compact groups is weakly locally compact. Additionally, a necessary and sufficient condition is established for determining when a weakly locally compact group G , containing a given subgroup H , can be decomposed into a local direct product of subgroups of rank one.

In this paper, we study multilinear Dunkl–Calderón–Zygmund operators, focusing on both the establishment of endpoint estimates for the operators and the boundedness of two types of commutators generated by these operators and BMO d {\mathrm{BMO}_{d}} functions, as demonstrated through sharp maximal function estimates.