Analysis and Mathematical Physics最新文献

In general a Schrödinger operator with a sparse potential has singular continuous spectrum, and some open interval is purely singular continuous spectrum. We give a sufficient condition so that the endpoint of the open interval is not an eigenvalue. An example of a Schrödinger operator with a negative sparse potential on the half-line which has no nonnegative embedded eigenvalue for any boundary conditions is given.

For a function defined on ({mathbb {R}}_q) we define two new variants of a modulus of smoothness and give a Boas type result about connection between the smoothness of this function and the behavior of its q-Dunkle Fourier transform near zero and at infinity.

We show that the coupling operator between distinct modes of a second-order hyperbolic system is smoothing of degree one, where we assume that the eigenvalues of the symbol are of constant rank, and that the coefficients of the system have bounded derivatives of second order. An important example is the wave equation for linear isotropic elasticity, where our assumption states that the Lamé parameters and mass density have bounded derivatives of second order. This extends a result for the elastic wave equation established by Brytik, et.al.

This work studies the Hardy number of hyperbolic planar domains satisfying Abel’s inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that the Hardy number of a Koenings domains whose complement is non-polar is greater than or equal to 1/2, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.

In this research, we investigate the error bounds associated with Milne’s formula, a well-known open Newton–Cotes approach, initially focused on differentiable convex functions within the frameworks of fractional calculus. Building on this work, we investigate fractional Milne-type inequalities, focusing on their application to the more refined class of twice-differentiable convex functions. This study effectively presents an identity involving twice differentiable functions and Riemann–Liouville fractional integrals. Using this newly established identity, we established error bounds for Milne’s formula in fractional and classical calculus. This study emphasizes the significance of convexity principles and incorporates the use of the Hölder inequality in formulating novel inequalities. In addition, we present precise mathematical illustrations to showcase the accuracy of the recently established bounds for Milne’s formula.

We establish two key results regarding pseudo symmetric and pseudo Ricci symmetric space-times. Firstly, we demonstrate that in pseudo symmetric generalized Robertson-Walker space-times either the scalar curvature remains constant or the associated vector field (B_{i}) is irrotational. Secondly, in pseudo Ricci symmetric generalized Robertson-Walker space-times, we establish that either the scalar curvature is zero or the associated vector field (A_{i}) is irrotational. We identify the conditions to ensure both (B_{i}) and (A_{i}) of these manifolds are acceleration-free and vorticity-free. We provide evidence that a pseudo symmetric and pseudo Ricci symmetric GRW space-time can be described as a perfect fluid. In a pseudo symmetric space-time, the state equation is given by (p=frac{4-n}{ 2n-2}mu ), whereas in a pseudo Ricci symmetric space-time, the state equation takes the form (p=frac{3-n}{n-1}mu ), where p and (mu ) are the isotropic pressure and the energy density. It is noteworthy that if (n=4) , a pseudo symmetric space-time corresponds to the dust matter era, while a pseudo Ricci symmetric space-time corresponds to the phantom era.

In this paper, we introduce the weighted variable anisotropic Hardy spaces (H_{omega ,A}^{p(cdot )}left( mathbb {R}^nright) ) via the nontangential grand maximal function. We also establish the atomic decompositions for the weighted variable anisotropic Hardy spaces (H_{omega ,A}^{p(cdot )}left( mathbb {R}^nright) ). In addition, we obtain the duality between (H_{omega ,A}^{p(cdot )}left( mathbb {R}^nright) ) and the weighted anisotropic Campanato spaces with variable exponents. We also obtain equivalent characterizations of the weighted variable anisotropic Hardy spaces by means of the anisotropic Lusin area function, the Littlewood–Paley g-function and the Littlewood–Paley (g_lambda ^*)-function. As applications, we study the boundedness of Calderón–Zygmund singular integral operators on the weighted variable anisotropic Hardy spaces.

Let X be a compact Kähler manifold of complex dimension (kge 2) and (f: X rightarrow X) a holomorphic correspondence with simple action on cohomology such that (f^{-1}) is also a holomorphic correspondence. We prove that the sequence of normalized pull-backs of a non-pluripolar current under iterates of f converges to the Green current associated with f.

In this paper, we study the necessary and sufficient conditions for the quantitative weighted bounds of the Calderón type commutator for the Littlewood–Paley operator. Let (g_{Omega ,1;b}) be the Calderón type commutator for the Littlewood–Paley operator where (Omega ) is homogeneous of degree zero and satisfies the cancellation condition on the unit sphere, and (bin Lip(mathbb {R}^n)). More precisely, for the sufficiency, we use a new operator (widetilde{G}_{Omega ,m;b}^j). Through the Calderón–Zygmund decomposition and the grand maximal operator (mathcal {M}_{widetilde{G}_{Omega ,m;b}^j}) of weak type (1,1), we establish a sparse domination of (widetilde{G}_{Omega ,m;b}^j). And then applying the interpolation theorem with change of measures and the relationship between the operators (g_{Omega ,1;b}) and (widetilde{G}_{Omega ,m;b}^j), we get the weighted bounds of the Calderón type commutators for the Littlewood–Paley operator (g_{Omega ,1;b}). In addition, for the necessity, through the local mean oscillation, we obtain Lip-type characterizations of (Lip(mathbb {R}^n)) via the weighted bounds of the Calderón type commutators for the Littlewood–Paley operator.

A normalized analytic function defined on the open unit disk is a bounded turning function if its derivative has positive real part. Such functions are univalent, and therefore, we find sufficient conditions for a function to be a bounded turning function. In this paper, we prove a general differential subordination theorem in terms of the derivative, the pre-Schwarzian derivative, and the Schwarzian derivative, providing sufficient conditions for a function to be a bounded turning function. We then apply the result to obtain several simple sufficient conditions.