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Abstract: Abstract We characterize the commutators of fractional integral operator and Calderón–Zygmund operator on differential forms. Also, the Hölder continuity for commutator of fractional integral operator with double-phase functional is derived. Finally, some estimates for the solutions to A-harmonic equations on differential forms are obtained. PubDate: 2021-10-11

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Abstract: Abstract We show that the group of isometries of an ultrametric normed space can be seen as a kind of a fractal. Then, we apply this description to study ultrametric counterparts of some classical problems in Archimedean analysis, such as the so called Problème des rotations de Mazur or Tingley’s problem. In particular, it turns out that, in contrast with the case of real normed spaces, isometries between ultrametric normed spaces can be very far from being linear. PubDate: 2021-09-15

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Abstract: Abstract We establish the strong \(L^p\) -inequality for the maximal function over hyper-surface based on Euclidean ball \(\sup _{t>0}\frac{1}{ B^n }\int _{B^n}f(x-tu,y-t^2 u ^2){\mathrm{d}}u\) . We prove the \(L^p\) -boundedness of the maximal function for \(p>1\) . Specially, when \(p=2\) , we prove the \(L^2\) -boundedness of the maximal function can be controlled by \(Cn^{\frac{7}{8}}\) , where C is independent of n. PubDate: 2021-09-09

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Abstract: Abstract We study the dominating sets and reverse Carleson measures on weighted Bergman spaces \(A^p_\omega\) with invariant weights in the spirit of Lueckings results from 1981 and 1985. Then, we characterize integrating derivatives of \(A^p_\omega\) functions, and combine this with reverse Carleson measures to characterize the reverse Carleson inequality for the derivatives of \(A^p_\omega\) functions. PubDate: 2021-09-01

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Abstract: We study generalized Drazin–Riesz invertible operators. Among other things, we show that if a bounded linear operator T is generalized Drazin–Riesz invertible then its generalized Drazin–Riesz inverse may be not unique. We also show that generalized Drazin–Riesz invertible operators are stable under commuting perturbation by Riesz operators. Applications to the abstract singular differential equations are given. PubDate: 2021-08-17

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Abstract: Abstract Let X, Y be two \(w^*\) -almost smooth Banach spaces, \(C(B(X^*),w^*)\) be the Banach space of all continuous real-valued functions on \(B(X^*)\) endowed with the supremum norm and \(C_+(B(X^*),w^*)\) be the positive cone of \(C(B(X^*),w^*)\) . In this paper, we show that if \(F: C_+(B(X^*),w^*)\rightarrow C_+(B(Y^*),w^*)\) is a standard almost surjective \(\varepsilon\) -isometry, then there exists a homeomorphism \(\tau : B(X^*)\rightarrow B(Y^*)\) in the \(w^*\) -topology such that for any \(x^*\in B(X^*)\) , we have $$\begin{aligned} \langle \delta _{x^*}, f\rangle -\langle \delta _{\tau (x^*)}, F(f)\rangle \le 2\varepsilon ,\quad \text{for all } f\in C_+(B(X^*),w^*). \end{aligned}$$ As its application, we show that if \(U:C(B(X^*),w^*)\rightarrow C(B(Y^*),w^*)\) is the canonical linear surjective isometry induced by the homeomorphism \(\gamma =\tau ^{-1}:B(Y^*)\rightarrow B(X^*)\) in the \(w^*\) -topology, then $$\begin{aligned} \Vert F(f)-U(f)\Vert \le 2\varepsilon , \quad \text{for all }f\in C_+(B(X^*),w^*). \end{aligned}$$ PubDate: 2021-08-12

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Abstract: Abstract We show that the half-line m functions associated with the vector-valued discrete Schrödinger operators are the elements in the Siegel upper half space. We introduce a metric on the space of m functions associated to these operators. Then, we show that the action of transfer matrices on these m functions is distance decreasing. PubDate: 2021-08-09

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Abstract: Abstract The main goal of this article is to establish several new \({\mathbb {A}}\) -numerical radius equalities for \(n\times n\) circulant, skew circulant, imaginary circulant, imaginary skew circulant, tridiagonal, and anti-tridiagonal operator matrices, where \({\mathbb {A}}\) is the \(n\times n\) diagonal operator matrix whose diagonal entries are positive bounded operator A. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Further, some pinching type \({\mathbb {A}}\) -numerical radius inequalities for \(n\times n\) block operator matrices are given. Some equality conditions are also given. We also provide a concluding section, which may lead to several new problems in this area. PubDate: 2021-07-21

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Abstract: Abstract We introduce a new norm on the space of all bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis–Wielandt radius norm. We study basic properties of this norm, including the upper and the lower bounds for it. As an application of the present study, we estimate bounds for the numerical radius of bounded linear operators. We illustrate that our results improve on some of the important existing numerical radius inequalities. Other application of this new norm have also studied. PubDate: 2021-07-20

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Abstract: Abstract A more accurate half-discrete Hilbert-type inequality in the whole plane with multi-parameters is established by the use of Hermite–Hadamard’s inequality and weight functions. Furthermore, some equivalent forms and some special types of inequalities and operator representations as well as reverses are considered. PubDate: 2021-06-29

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Abstract: Abstract We study the norm derivatives in the context of Birkhoff–James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff–James orthogonality in \(\ell _1^n\) and \(\ell _\infty ^n\) spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff–James orthogonality. We further study Birkhoff–James orthogonality, approximate Birkhoff–James orthogonality, smoothness and norm attainment of bounded bilinear operators between Banach spaces. PubDate: 2021-06-28

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Abstract: Abstract We obtain a Shimorin Wold-type decomposition for a doubly commuting covariant representation of a product system of \(C^*\) -correspondences over \({\mathbb {N}}_0^k\) . This result gives Shimorin-type decompositions of recent Wold-type decompositions by Jeu and Pinto (Adv Math 368:107–149, 2020) for the q-doubly commuting isometries and by Popescu (J Funct Anal 279:108798, 2020) for Doubly \(\Lambda \) -commuting row isometries. Application to the wandering subspaces of the induced representations is explored, and a version of the Beurling–Lax-type characterization is obtained to study doubly commuting invariant subspaces. PubDate: 2021-06-14

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Abstract: Abstract Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, and commutators of singular integrals in grand variable exponent Lebesgue spaces are derived. The spaces and operators are defined on quasi-metric measure spaces with doubling condition (spaces of homogeneous type). The proof of the result regarding the Hardy–Littlewood maximal operator is based on the appropriate sharp weighted norm estimates with power-type weights. To obtain the results for singular integrals and commutators we prove appropriate weighted extrapolation statement in grand variable exponent Lebesgue spaces. The extrapolation theorem deals with a family of pairs of functions (f, g). One of the consequences of the latter result is the weighted extrapolation for sublinear operators S acting in these spaces. As one of the applications of the main results we present weighted norm estimates for the Hardy–Littlewood maximal function, Cauchy singular integral operator, and its commutators in grand variable exponent Lebesgue spaces defined on rectifiable regular curves. PubDate: 2021-06-14

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Abstract: Abstract In this paper, we study a Bishop–Phelps–Bollobás type property called the property \({\mathbf{L}}_{o,o}\) of a pair of Banach spaces. Getting motivated by this, we introduce the notion of Approximate minimizing property (AMp) of a pair of Banach spaces and characterize finite dimensionality of Banach spaces with respect to this property. We further introduce the notion of approximate minimum norm attainment set of a bounded linear operator and characterize the AMp with the help of Hausdorff convergence of the sequence of approximate minimum norm attainment sets of bounded linear operators. We also investigate sufficient conditions for the holding of some weaker forms of the AMp for a pair of Banach spaces. Finally, we define and study uniform \(\varepsilon\) -approximation of a bounded linear operator in terms of its minimum norm. PubDate: 2021-06-10

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Abstract: Abstract We prove sectorial extension theorems for ultraholomorphic function classes of Beurling type defined by weight functions with a controlled loss of regularity. The proofs are based on a reduction lemma, due to the second author, which allows to extract the Beurling from the Roumieu case, which was treated recently by Jiménez-Garrido, Sanz, and the third author. To have control on the opening of the sectors, where the extensions exist, we use the (mixed) growth index and the order of quasianalyticity of weight functions. As a consequence, we obtain corresponding extension results for classes defined by weight sequences. Additionally, we give information on the existence of continuous linear extension operators. PubDate: 2021-06-04

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Abstract: Abstract The invariant subspaces of the Hardy space \(H^2(\mathbb {D})\) of the unit disc are very well known; however, in several variables, the structure of the invariant subspaces of the classical Hardy spaces is not yet fully understood. In this study, we examine the structure of invariant subspaces of Poletsky–Stessin–Hardy spaces which are the generalization of the classical Hardy spaces to hyperconvex domains in \(\mathbb {C}^n\) . We showed that not all invariant subspaces of \(H^{2}_{\tilde{u}}(\mathbb {D}^2)\) are of Beurling-type. To characterize the Beurling-type invariant subspaces of this space, we first generalized the Lax–Halmos Theorem to the vector-valued Poletsky–Stessin–Hardy spaces and then we gave a necessary and sufficient condition for the invariant subspaces of \(H^{2}_{\tilde{u}}(\mathbb {D}^2)\) to be of Beurling-type. PubDate: 2021-06-01

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Abstract: Abstract This paper investigates two class of Schrödinger equations with mixed source terms, containing, respectively, a local/non-local non-linearity and an inhomogeneous perturbation. In the energy sub-critical regime, one obtains some sharp thresholds of global/non-global existence dichotomy. PubDate: 2021-06-01

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Abstract: Abstract In this paper, We introduce a new class of multivariable operators known as joint m-quasihyponormal tuple of operators. It is a naturel extension of joint normal and joint hyponormal tuples of operators. An m-tuple of operators \(\mathbf{S}=(S_1, \ldots ,S_m)\in {{\mathcal {B}}}({{\mathcal {H}}})^m\) is said to be joint m-quasihyponormal tuple if \(\mathbf{S}\) satisfying $$\begin{aligned} \displaystyle \sum _{1\le l,\;k\;\le m}\big \langle S_k^*\big [S_k^*,\;\; S_l\big ]S_lu_k\; \;u_l\big \rangle \ge 0, \end{aligned}$$ for each finite collections \((u_l)_{1\le l\le m}\in {{\mathcal {H}}}.\) Some properties of this class of multivariable operators are studied. PubDate: 2021-05-26

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Abstract: Abstract In this paper, we study an analytic Yeh–Feynman integral and an analytic Yeh–Fourier–Feynman transform associated with Gaussian processes. Fubini theorems involving the generalized analytic Yeh–Feynman integrals are established. The Fubini theorems investigated in this paper are to express the iterated generalized Yeh–Feynman integrals associated with Gaussian processes as a single generalized Yeh–Feynman integral. Using our Fubini theorems, we next examined fundamental relationships (with extended versions) between generalized Yeh–Fourier–Feynman transforms and convolution products (with respect to Gaussian processes) of functionals on Yeh–Wiener space. PubDate: 2021-05-19

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Abstract: This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that \(\xi :H\rightarrow \mathbb {T}\) is a character, \(1\le p<\infty\) and \(L_\xi ^p(G,H)\) is the set of all covariant functions of \(\xi\) in \(L^p(G)\) . It is shown that \(L^p_\xi (G,H)\) is isometrically isomorphic to a quotient space of \(L^p(G)\) . It is also proven that \(L^q_\xi (G,H)\) is isometrically isomorphic to the dual space \(L^p_\xi (G,H)^*\) , where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact. PubDate: 2021-05-17