Journal of Pseudo-Differential Operators and Applications最新文献

A geodetically convex set in the Heisenberg group ({mathbb {H}}^n), (nge 1) is defined to be a set with the property that a geodesic joining any two points in the set lies completely in it. Here we classify the geodetically convex sets to be either an empty set, a singleton set, an arc of a geodesic or the whole space ({mathbb {H}}^n). We also show that a geodetically convex function on ({mathbb {H}}^n)is a constant function. These results generalize the known results of ({mathbb {H}}^1) to higher dimensional Heisenberg group.

We review at first the role of localization operators as a meeting point of three different areas of research, namely: signal analysis, quantization and pseudo-differential operators. We extend then the correspondence between symbol and operator which characterizes localization operators to a more general situation, introducing the class of bilocalization operators. We show that this enlargement yields a quantization rule that is closed under composition. Some boundedness results are deduced both for localization and bilocalization operators. In particular for bilocalization operators we prove that square integrable symbols yield bounded operators on (L^2) and that the class of bilocalization operators with integrable symbols is a subalgebra of bounded operators on every fixed modulation space.

In this paper we consider the Laguerre operator (L=-frac{d^2}{dx^2}-frac{alpha }{x}frac{d}{dx}+x^2) on the Euclidean space (mathbb R_{+}). The main aim of this article is to develop a theory of homogeneous Besov type spaces associated to the Laguerre operator. To achieve our expected goals, Schwartz type spaces on (mathbb R_{+}) are introduced and then tempered type distributions are constructed. Using a suitable distribution of the Laguerre operator, the Calderón reproducing formula and the Harnack type inequality for subharmonic functions are established. With these tools in hand, we define the Besov type spaces (dot{B}_{p,q}^{s,L,m}) and obtain the molecular decompositions of (dot{B}_{p,q}^{s,L,m}). As applications, the embedding theorem and square functions characterization of Besov type spaces (dot{B}_{p,q}^{s,L,m}) are also investigated.

In this paper, we initially concentrate on the concept of complex convolutional neural networks, constructing the essential frameworks required for managing complex-valued inputs. We subsequently introduce a novel neural network architecture that replaces the standard convolution operator with a more general operator known as pseudo-differential operators. This unique modification ensures the effective handling of an input’s frequency information through the application of appropriate filters. To validate this approach, we conducted empirical testing on one-dimensional and two-dimensional datasets. The results affirm the convergence and efficacy of this novel architecture, indicating a potential significant advancement in the field of complex neural network development.

We consider multilinear pseudo-differential operators with symbols in the multilinear Hörmander class (S_{0, 0}). The aim of this paper is to discuss the boundedness of these operators in the settings of Besov spaces.

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two-term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and to obtain optimal spectral asymptotics again under certain regularity conditions. For the Weyl law, we assume that the coefficients are differentiable with Hölder continuous derivatives, while for the Riesz means we assume that the coefficients are twice differentiable with Hölder continuous derivatives.

Abstract

This paper tackles a class of nonlinear parabolic equations driven by the fractional p-Laplacian operator with a vanishing initial datum. Our primary aim is to investigate the well-posedness (existence and uniqueness) of solutions to the proposed model. Notably, we will establish two interesting results concerning the existence and uniqueness of weak solutions. The first result pertains to the scenario where the source term is independent of the solution. In this case, we demonstrate the existence and uniqueness of the solution via the classical monotone operator theory modulus vanishing initial datum. The second result deals with the case where the source term is nonlinear and strongly dependent on the solution. To establish the existence of a weak solution in this scenario, we will rely essentially on the use of Schaefer’s fixed point theorem and supplement our approach with some new technical estimates.

The purpose of this work is to develop a new model of fractional operators called Hilfer-fractional random nonlinear integro-differential equations. In this paradigm, a further discussion is encouraged under almost sectorial operators. The results are supported by fractional calculus, stochastic analysis theory, and Bohnenblust–Karlin’s fixed point theorem for multi-valued mappings. In addition, a mild solution to the model under consideration is presented. Ultimately, an example is provided to support our results.

The frame theory is a dynamic and exciting field with various applications in pure and applied mathematics. In this paper, we introduce and study the concept of controlled continuous (*)-g-frames in Hilbert (C^{*})-modules, which is a generalization of discrete controlled (*)-g-frames in Hilbert (C^{*})-modules. Additionally, we present some properties.

In this paper, let (T_{a,varphi }) be a Fourier integral operator with rough amplitude (a in {L^infty }S_rho ^m) and rough phase (varphi in {L^infty }{Phi ^2}) which satisfies a new class of rough non-degeneracy condition. When (0 leqslant rho leqslant 1), if (m < frac{{n(rho - 1)}}{2} - frac{{rho (n - 1)}}{4}), we obtain that (T_{a,varphi }) is bounded on ({L^2}). Our main result extends and improves some known results about ({L^2}) boundedness of Fourier integral operators.