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Abstract: Abstract The theory of slice regular functions of a quaternionic variable on the unit ball of the quaternions was introduced by Gentili and Struppa in 2006 and nowadays it is a well established function theory, especially in view of its applications to operator theory. In this paper, we introduce the notion of fractional slice regular functions of a quaternionic variable defined as null-solutions of a fractional Cauchy–Riemann operators. We present a fractional Cauchy–Riemann operator in the sense of Riemann–Liouville and then in the sense of Caputo, with orders associated to an element of \((0,1)\times {\mathbb {R}} \times (0,1)\times {\mathbb {R}}\) for some axially symmetric slice domains which are new in the literature. We prove a version of the representation theorem, of the splitting lemma and we discuss a series expansion. PubDate: 2023-12-02

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Abstract: Abstract In the paper we consider a general inequality \( p_{n-1}p_{n+1}-p_n{}^2 \le 4- p_1 ^2\) involving coefficients of functions with a positive real part. We prove this inequality for \(n=2\) and \(n=3\) . Consequently, the relative inequalities involving coefficients of Schwarz functions are obtained. As an application, the two sharp estimates of the Hankel determinants \(H_{3,1}\) and \(H_{2,3}\) are proved for functions in \({\mathcal S}^*(1/2)\) and \({\mathcal {M}}\) , respectively. PubDate: 2023-12-02

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Abstract: Abstract In this paper, we obtain Jackson type theorem of approximation by truncated max-product sampling Kantorovich operators in \(L^p\) spaces. Our results generalize those of Coroianu et al. (Anal Appl 19:219–244, 2021) and Coroianu and Gal (J Integral Equat Appl 29:349–364, 2017). We use the equivalence between the \(K-\) functional and the modulus of continuity of f, and the Hardy–Littlewood maximal function as the main tools in the proofs, and also give some examples to apply the main result. PubDate: 2023-12-02

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Abstract: Abstract By employing Riesz decomposition theorem about monotone functions, we prove a Flett type theorem for the Riemann–Stieltjes integral and related integral mean value results. Then, since monotone functions are inner functions for gauge integrable functions, we revisit classical mean value integral theorems in terms of the gauge integral. PubDate: 2023-12-02

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Abstract: Abstract The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range \(1<p\le q<\infty \) . We also calculate the precise value of sharp constants in respective inequalities, improving the result of Ruzhansky and Verma (Proc R Soc A 475:20180310, 2019) in the case of homogeneous groups. PubDate: 2023-12-02

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Abstract: Abstract In this paper, we deduce several multiple expansion formulas over root systems. These formulas give some multiple extensions of an expansion formula of Liu. From these formulas, we establish multiple expansion formulas for infinite products, a \(C_{n}\) extension of Rogers’ non-terminating \(\text {}_{6}\phi _{5}\) summation formula and multiple expansion formulas for \((q)_{\infty }^{m}.\) As applications, we deduce several \(C_{n}\) and \(D_{n}\) extensions of finite/semi-finite forms of the quintuple product identity, a multiple extension of an expansion formula of Liu and a \(C_{n}\) extension of Liu’s extension of Rogers’ \(\text {}_{6}\phi _{5}\) summation formula. PubDate: 2023-12-02

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Abstract: Abstract Several authors studied the so called exponential polynomials, characterized as solutions to the equation $$\begin{aligned} f(x+y) = \sum _{k=1}^nu_i(x)v_i(y). \end{aligned}$$ In the present paper we deal with a more general equation $$\begin{aligned} \sum _{j=1}^M P_j(x,y) f_j(a_jx + c_j y) = \sum _{k=1}^n u_k(x)v_k(y). \end{aligned}$$ Here all \(f_j, \ u_k, \ v_k\) are assumed to be unknown scalar functions on \({\mathbb {R}}^d,\) while \(P_j\) are polynomials. We prove that \(f_j\) are ratios of exponential polynomials and polynomials, or sums of exponential functions multiplied by rational functions. PubDate: 2023-12-02

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Abstract: Abstract In this paper, we study the rough fractional Hausdorff operator on variable exponent Morrey–Herz spaces in the setting of the Heisenberg group. We define Morrey–Herz spaces with three variable exponents and then give sufficient and necessary conditions for the boundedness of the rough fractional Hausdorff operator. PubDate: 2023-11-29

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Abstract: Abstract Let \(\omega \) and \(\nu \) be radial weights on the unit disc of the complex plane, and denote \(\sigma =\omega ^{p'} \nu ^{-\frac{p'}{p}}\) and \(\omega _x =\int _0^1\,s^x \omega (s)\,ds\) for all \(1\le x<\infty \) . Consider the one-weight inequality $$\begin{aligned} \Vert P_\omega (f)\Vert _{L^p_\nu }\le C\Vert f\Vert _{L^p_\nu },\quad 1<p<\infty , \qquad \qquad (\dagger ) \end{aligned}$$ for the Bergman projection \(P_\omega \) induced by \(\omega \) . It is shown that the moment condition $$\begin{aligned} D_p(\omega ,\nu )=\sup _{n\in {\mathbb {N}}\cup \{0\}} \frac{\left( \nu _{np+1}\right) ^\frac{1}{p}\left( \sigma _{np'+1} \right) ^\frac{1}{p'}}{\omega _{2n+1}}<\infty \end{aligned}$$ is necessary for ( \(\dagger \) ) to hold. Further, \(D_p(\omega ,\nu )<\infty \) is also sufficient for ( \(\dagger \) ) if \(\nu \) admits the doubling properties \(\sup _{0\le r<1}\frac{\int _r^1\nu (s) s\,ds}{\int _{\frac{1+r}{2}}^1\nu (s)s\,ds}<\infty \) and \(\sup _{0\le r<1}\frac{\int _r^1\nu (s)s\,ds}{\int _r^{1-\frac{1-r}{K}} \nu (s)s\,ds}<\infty \) for some \(K>1\) . In addition, an analogous result for the one weight inequality \(\Vert P_\omega (f)\Vert _{D^p_{\nu ,k}} \le C\Vert f\Vert _{L^p_\nu }\) , where $$\begin{aligned} \Vert f \Vert _{D^p_{\nu , k}}^p =\sum \limits _{j=0}^{k-1}\vert f^{(j)}(0)\vert ^p +\int _{{\mathbb {D}}} \vert f^{(k)}(z)\vert ^p (1-\vert z \vert )^{kp} \nu (z)\,dA(z)<\infty , \quad k\in {\mathbb {N}}, \end{aligned}$$ is established. The inequality ( \(\dagger \) ) is further studied by using the necessary condition \(D_p(\omega ,\nu )<\infty \) in the case of the exponential type weights \(\nu (r)=\exp \left( -\frac{\alpha }{(1-r^l)^{\beta }} \right) \) and \(\omega (r)= \exp \left( -\frac{ \widetilde{\alpha }}{(1-r^{\widetilde{l}})^{\widetilde{\beta }}} \right) \) , where \(0<\alpha , \, \widetilde{\alpha }, \, l, \, \widetilde{l}<\infty \) and \(0<\beta , \, \widetilde{\beta }\le 1\) . PubDate: 2023-11-28

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Abstract: Abstract For a non-negative integer p, the p-Frobenius number, which is one of the generalized Frobenius numbers in terms of the number of representations, is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers \(a_1,a_2,\ldots ,a_k\) with \(\gcd (a_1,a_2,\ldots ,a_k)=1\) . When \(p=0\) , it reduces to the classical Frobenius number. One of the most natural questions is to find a closed explicit form of the Frobenius number. When \(k=2\) , its explicit formula was discovered in the nineteenth century. When \(k\ge 3\) , explicit formulas are very difficult to obtain even for \(p=0\) . The case of \(p>0\) is even more difficult, and until recently there was no explicit formula for the Frobenius number even in a single case. However, we finally found explicit formulas for the p-Frobenius number, such as in the case of repunit. In this paper, we give closed formulas for the p-Frobenius number for the generalized repunit. The method is to analyze the structure of the p-Apéry set, which is a more general Apéry set. In the generalized repunit, the structure of the Apéry set is similar, but the position of the element that takes the maximum value is different, making it more difficult to identify. PubDate: 2023-11-27

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Abstract: Abstract Let \(n \ge 2\) be a natural number, and denote by \({\mathcal {M}}_{n}\) the algebra of all \(n\times n\) matrices over the complex field \({\mathbb {C}}\) . Let \(C\in {\mathcal {M}}_{n}\) be a fixed matrix, and let \(x_{0},y_{0}\in {\mathbb {C}} ^{n}\) be fixed vectors with \(x_{0}\ne 0\) . In this paper, we characterize linear maps \(\varphi ,\psi :{\mathcal {M}}_{n}\rightarrow {\mathcal {M}}_{n} \) , with \(\varphi \) bijective, having the property that \(\varphi \left( A\right) \psi \left( B\right) x_{0}=y_{0}\) in \({\mathbb {C}}^{n}\) whenever \(AB=C\) in \( {\mathcal {M}}_{n} \) . PubDate: 2023-11-25

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Abstract: Abstract This paper is concerned with the mass-energy threshold dynamics for the elliptic–elliptic Davey–Stewartson system in dimension three. By proving the compactness of minimizing sequence for the Weinstein functional, we show the long time behavior of solutions with data being at the mass-energy threshold. Our proof is based on a Gagliardo–Nirenberg type inequality, localized virial estimates and the concentration-compactness lemma. PubDate: 2023-11-25

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Abstract: Abstract We introduce a novel family of two-dimensional surfaces in \({\mathbb {E}}^4\) which generalize the classical Dini surfaces in \({\mathbb {E}}^3\) by inheriting their geometric features concerning the degeneration of the Bianchi-Bäcklund transformation. PubDate: 2023-11-22

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Abstract: Abstract We prove that the lower density operator associated with intensity points on the real line has Borel values. We also prove that the simple density operator and the complete density operator have Borel values. PubDate: 2023-11-22

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Abstract: Abstract A third order self-adjoint differential operator with periodic boundary conditions and an one-dimensional perturbation has been considered. For this operator, we first show that the spectrum consists of simple eigenvalues and finitely many eigenvalues of multiplicity two. Then the expressions of eigenfunctions and resolvent are described. Finally, the inverse problems for recovering all the components of the one-dimensional perturbation are solved. In particular, we prove the Ambarzumyan-type theorem and show that the even or odd potential can be reconstructed by three spectra. PubDate: 2023-11-22

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Abstract: Abstract We establish some Lie–Trotter formulae for unital complex Jordan–Banach algebras, showing that for any elements a, b, c in a unital complex Jordan–Banach algebra \(\mathfrak {A}\) the identities $$\begin{aligned}{} & {} \lim _{n\rightarrow \infty } \left( e^{\frac{a}{n}}\circ e^{\frac{b}{n}} \right) ^{n} = e^{a+b},\ \lim _{n\rightarrow \infty } \left( U_{e^{\frac{a}{n}}} \left( e^{\frac{b}{n}}\right) \right) ^{n} = e^{2 a+b}, \hbox { and }\\{} & {} \lim _{n\rightarrow \infty } \left( U_{e^{\frac{a}{n}},e^{\frac{c}{n}}} \left( e^{\frac{b}{n}}\right) \right) ^{n} = e^{a+b + c} \end{aligned}$$ hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in \(\mathfrak {A}\) . These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals \(f:\mathfrak {A}\rightarrow \mathbb {C}\) satisfying \(f(U_x (y))=U_{f(x)}f(y),\) for all \(x,y\in \mathfrak {A}\) . We prove that for any such a functional f, there exists a unique continuous (Jordan-) multiplicative linear functional \(\psi :\mathfrak {A}\rightarrow \mathbb {C}\) such that \( f(x)=\psi (x),\) for every x in the connected component of the set of all invertible elements of \(\mathfrak {A}\) containing the unit element. If we additionally assume that \(\mathfrak {A}\) is a JB \(^*\) -algebra and f is continuous, then f is a linear multiplicative functional on \(\mathfrak {A}\) . The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Touré. PubDate: 2023-11-22

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Abstract: Abstract In this paper, we first give a notion of \({\mathcal {M}}\) -convexity, and then under suitable settings related to this superconvexity, we can obtain the existence of solutions to prescribed shifted Gaussian curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of \({\mathcal {M}}\) -convex, closed hypersurface (which is graphic with respect to the base manifold and whose shifted Gaussian curvature satisfies some constraint) in a given warped product manifold of special type can be assured. Besides, different from prescribed Weingarten curvature problems in space forms, due to the \({\mathcal {M}}\) -convexity of hypersurfaces in the warped product manifold considered, we do not need to impose a sign condition for the radial derivative of the prescribed function in the shifted Gaussian curvature equation to prove the existence of solutions. PubDate: 2023-11-22

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Abstract: Abstract In set theory without the full power of the axiom of choice ( \(\textbf{AC}\) ), we resolve open problems from Keremedis, Olfati and Wajch “On P-spaces and \(G_{\delta }\) -sets in the absence of the Axiom of Choice” on the deductive strength of statements concerning P-spaces and strongly quasi Baire spaces via positive and independence results. For some of the independence results, we construct three new permutation models of \(\textbf{ZFA}+\lnot \textbf{AC}\) , where \(\textbf{ZFA}\) denotes the Zermelo–Fraenkel set theory with atoms. Part of our \(\textbf{ZFA}\) -independence proofs are transferable to \(\textbf{ZF}\) (i.e. Zermelo–Fraenkel set theory without \(\textbf{AC}\) ). PubDate: 2023-11-22

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Abstract: Abstract We derive asymptotic estimates for the growth of the norm of the deformed Hankel transform on the deformed Hankel–Lipschitz space defined via a generalised modulus of continuity. The established results are similar in nature to the well-known Titchmarsh theorem, which provide a characterization of the square integrable functions satisfying certain Cauchy–Lipschitz condition in terms of an asymptotic estimate for the growth of the norm of their Fourier transform. We also give some necessary conditions in terms of the generalised modulus of continuity for the boundedness of the Dunkl transform of functions in Dunkl–Lipschitz spaces, improving the Hausdorff–Young inequality for the Dunkl transform in this special scenario. PubDate: 2023-11-22

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Abstract: Abstract We develop an invariant approach to SU(2)–structures on spin 5–manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to SU(2). Moreover, we show how to induce quaternionic structure on the contact distribution of the considered SU(2)–structure. We show the invariance of certain components of the covariant derivative \(\nabla \varphi \) , where \(\varphi \) is any spinor field defining SU(2)–structure. This shows, as expected, that (at least some of) the intrinsic torsion modules can be derived invariantly with the spinorial approach. We conclude with the explicit description of the intrinsic torsion and the characteristic connection. PubDate: 2023-11-18