Analysis and Mathematical Physics最新文献

In this paper, we show that the semiclassical calculus recently developed on nilpotent Lie groups and nilmanifolds include the functional calculus of suitable subelliptic operators. Moreover, we obtain Weyl laws for these operators. Amongst these operators are sub-Laplacians in horizontal divergence form perturbed with a potential and their generalisations.

We study mappings that satisfy the inverse modulus inequality of Poletsky type in a fixed domain. It is shown that, under some additional restrictions, the image of a ball under such mappings contains a fixed ball uniformly over the class. This statement can be interpreted as the well-known analogue of Koebe’s theorem for analytic functions. As an application of the obtained result, we show that, if a sequence of mappings belonging to the specified class converges locally uniformly, then the limit mapping is open.

In this paper, we consider integral operators with non-negative kernels that satisfy conditions less restrictive than those studied earlier. We establish criteria for the boundedness of these operators in Lebesgue spaces.

Invoking the concept of (alpha )-dense curves, we develop a new fixed point approach for multi-valued mappings which works under more general conditions than Darbo multi-valued fixed point theorem and its generalizations. We prove some generalizations of Krasnosielskii’s type fixed point theorems and we establish nonlinear alternatives of Leray-Schauder’s type for multi-valued mappings. This theory is then applied to investigate general existence principles of nonlinear hybrid fractional integral inclusions in abstract Banach spaces. Our results extend and generalize a number of earlier works.

In this paper, we study the growth of the (m-th) ((mge 1)) derivatives of an arbitrary algebraic polynomial in weighted Lebesgue spaces over the whole complex plane. We first study the growth of the (m-th) derivatives of an arbitrary algebraic polynomial over unbounded regions of the complex plane, and then we obtain estimates for the growth of the (m-th) derivatives of this polynomial over the closure of the given region. Combining both estimates, we find estimates for the growth of the (m-th) derivatives of an arbitrary algebraic polynomial over the whole complex plane.

Using Malliavin’s calculus, it is proved that the generator of the one-parameter unitary semigroup of Schrödinger type on the complex Hilbert space (L^2_mathbb {C}(mathbb {R}^n,gamma )) equipped with the Gaussian measure (gamma ) on (mathbb {R}^n) takes the form (sum _j^n(mathfrak {h}_2(phi _{jmath })+1)), where (mathfrak {h}_2(phi _{jmath })) are second-order Hermite polynomials of independent random variables (phi _jmath ), generated by an orthonormal basis in (mathbb {R}^n) using the Paley-Wiener maps. The Weyl-Schrödinger unitary irreducible representation of Heisenberg matrix group (mathbb {H}_{2n+1}) and the Segal-Bargmann transform are essentially used. By applying the inverse Gauss transform, it is found that this representation of (mathbb {H}_{2n+1}) can be fully described by complex Weyl pairs, generated using the multiplication operator with a real Gaussian variable on (mathbb {R}^n).

In this paper, we present a version of the Kleinecke–Shirokov Theorem applicable to isometries on a Hilbert space ({mathcal {H}}). Specifically, we demonstrate that if ( V in {mathfrak {B}}({mathcal {H}})) is a quasinormal partial isometry and (T in {mathfrak {B}}({mathcal {H}})) satisfies ({mathcal {R}}(T) subseteq {mathcal {R}}(V)), then

We also consider the mixed commutators of two isometries, and their belonging to the Schatten-von Neumann classes. Finally, we show that the corresponding classical statement regarding normal operators can be derived from our results.

Using couples of weighted Radon measures an extension of the J-method of interpolation is described. This method allows for the representation also of non-regular interpolation spaces. In particular, Banach couples with the Calderon–Mitjagin property has all its interpolation spaces described by this extension.

In this paper, we investigate the generalized Boussinesq equation (gBq) as a model for the water wave problem with surface tension. Our study begins with the analysis of the initial value problem within Sobolev spaces, where we derive improved conditions for global existence and finite-time blow-up of solutions, extending previous results to lower Sobolev indices. Furthermore, we explore the time-decay behavior of solutions in Bessel potential and modulation spaces, establishing global well-posedness and time-decay estimates in these function spaces. Using Pohozaev-type identities, we demonstrate the non-existence of solitary waves for specific parameter regimes. A significant contribution of this work is the numerical generation of solitary wave solutions for the gBq equation using the Petviashvili iteration method. Additionally, we propose a Fourier pseudo-spectral numerical method to study the time evolution of solutions, particularly addressing the gap interval where theoretical results on global existence or blow-up are unavailable in the Sobolev spaces. Our numerical results provide new insights by confirming theoretical predictions in covered cases and filling gaps in unexplored scenarios. This comprehensive analysis not only clarifies the theoretical and numerical landscape of the gBq equation but also offers valuable tools for further investigations.

In this paper, one class of reductions of modified KP hierarchy is introduced, which is called the BC(_r)–modified KP hierarchy ((rge 0)) here, since it contains modified BKP and modified CKP hierarchies as special cases. For BC(_r)–modified KP hierarchy, we discuss Lax equations and bilinear descriptions, where the corresponding equivalence is also investigated.