Analysis and Mathematical Physics最新文献

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.

The orthonormal Strichartz estimates for the Schrödinger equation associated to the Dunkl Laplacian and the Dunkl-Hermite operator are derived in Senapati et al. (J Geom Anal 34:74, 2024) and Mondal and Song (Israel J Math, 2023). In this article we construct a set of coherent states in the Dunkl setting and apply semi-classical analysis to derive a necessary condition on the Schatten exponent for the aforementioned orthonormal Strichartz estimates, which turns out to be optimal for the Schrödinger equations associated with Laplacian and Hermite operator as a particular case.

We study the dispersionless limit of the recently introduced Toda lattice hierarchy with constraint of type B (the B-Toda hierarchy) and compare it with that of the DKP and C-Toda hierarchies. The dispersionless limits of the B-Toda and C-Toda hierarchies turn out to be the same.

We consider families of polynomial lemniscates in the complex plane and determine if they bound a Jordan domain. This allows us to find examples of regions for which we can calculate the projection of (bar{z}) to the Bergman space of the bounded region. Such knowledge has applications to the calculation of torsional rigidity.

Let (mu ) be a positive finite Borel measure on [0, 1). Cesàro-type operators (C_{mu }) when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that (C_mu ) is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of (C_mu ) on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments ((mu _n)_{nin {mathbb {N}}_0}). The continuity, compactness and spectrum of (C_mu ) acting on Fréchet and (LB) Korenblum type spaces are also considered .

In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier’s law for heat propagation and Fick’s first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider (nge 3) orthogonal unit vectors (e_1,ldots ,e_nin {mathbb {R}}^n), and let (Omega subseteq {mathbb {R}}^n) be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator (T=sum _{i=1}^ne_ia_i(x)frac{partial }{partial x_i}) with nonconstant positive coefficients (a_i:{overline{Omega }}rightarrow (0,infty )). Under certain regularity and growth conditions on the (a_i), we identify bisectorial or strip-type regions that belong to the S-resolvent set of T. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the S-spectrum, designed to study the operators acting in Clifford modules V over the Clifford algebra ({mathbb {R}}_n), with vector operators being a specific crucial subclass. The spectral properties related to the S-spectrum of T are linked to the inversion of the operator (Q_s(T):=T^2-2s_0T+|s|^2), where (sin {mathbb {R}}^{n+1}) is a paravector, i.e., it is of the form (s=s_0+s_1e_1+cdots +s_ne_n). This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to (Q_s(T)), i.e., to the squared operator (T^2).

The notion of m-quasi-Einstein manifolds originates from the study of Einstein warped product metrics and they are influential in constructing for many physical models. For example, these manifolds arises for extremal isolated horizons in the theory of black holes. In a recent work by Cochran (arXiv:2404.17090v1, 2024), the author studied Killing vector fields on closed m-quasi-Einstein manifolds. In this short paper, we will give another proof of his main result involving the scalar curvature, which holds for all values of m and is based on the use of known formulae related to quasi-Einstein metrics.

Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of ({mathcal {S}}^0_H), the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of ({mathcal {S}}^0_H). Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of p for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well.

Let Q be the d-dimensional space of finite adeles over the algebraic number field K and let (P=Q^*) be its dual space. For a certain type of Vladimirov type time-dependent Hamiltonian (H_V(t):Qtimes Prightarrow {mathbb {C}}) we construct the Feynman formulas for the solution of the Cauchy problem with the Schrödinger operator where the caret operator stands for the qp- or pq-quantization.