We prove a family of sharp multilinear integral inequalities on real spheres involving functions that possess some symmetries that can be described by annihilation by certain sets of vector fields. The Lebesgue exponents involved are seen to be related to the combinatorics of such sets of vector fields. Moreover, we derive some Euclidean Brascamp–Lieb inequalities localized to a ball of radius R, with

Recently, Lessig et al. [Appl Comput Harmon Anal. 2019;46:384–399] introduced the notion of bendlets; a shearlet-like system that is based on anisotropic scaling, translation, shearing and bending of a compactly supported generator. The theoretical framework of the bendlet transform is yet to be explored exclusively. Taking this opportunity, our aim is to investigate the mathematical properties of

In this work, we study the existence and multiplicity of positive solutions for a nonlocal problem with changing sign data.

Let (X,d,μ) be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this setting, the author proves that the θ-type Calderón-Zygmund operator Tθ is bounded on the weighted Lebesgue space and the weighted Morrey space. Furthermore, the boundedness of the commutator [b,Tθ] generated by Tθ and b∈RBMO(μ) on weighted weak Lebesgue space

Based on limit operator techniques, we introduce and study an increasing family of closed ideals of the algebra BDON of the band-dominated operators on l2(ZN), also known as the uniform Roe algebra of the coarse space ZN. We describe the centers of the quotients of BDON by these ideals and identify them with higher-order slowly oscillating functions.

Let G be a compact Hausdorff group and H be a closed subgroup of G. In this paper, we show that every bounded linear operator T on Lp(G/H) is a pseudo-differential operator with the symbol σ for 1≤p<∞. We present necessary and sufficient conditions on symbols to ensure that a bounded pseudo-differential operator on L2(G/H) is self-adjoint, normal and present explicit formula for their symbols. A necessary

In the classical complex analysis of several variables Lars Hoermander had been proven for Hartog's extension theorem. His proof is based on solving the inhomogeneous Cauchy–Riemann system. However, it is difficult when generalizing this diagram to prove this problem in Clifford analysis because of the non-commutative property. In this paper we will propose a method to overcome this difficultly and

In this paper, the two-sided fractional Clifford–Fourier transformation (FrCFT) is defined and some of its properties are given. First, we give the definition of the two-sided FrCFT and discuss the relationship between the two-sided Clifford–Fourier transformation (CFT) and the two-sided FrCFT. Then, the inverse transformation of the two-sided FrCFT is given and the time–frequency properties are discussed

The quaternionic offset linear canonical transform (QOLCT) can be defined as a generalization of the quaternionic linear canonical transform (QLCT). In this paper, we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact, we prove the Rayleigh formula and some properties related to the QOLCT. Then, we generalize some different

In this paper, we study the nonlinear Choquard equation {−Δu+λu=(I2∗|u|p)|u|p−2uinΩ,u∈H01(Ω), where Ω⊂RN is an unbounded domain, ∂Ω≠∅ is bounded, λ∈R+, p = 2 if N = 3, 4, 5 or 2

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal p1&…&pm fractional Laplacian with critical Sobolev-Hardy exponent M1∫R2N|u(x)−u(y)|p1|x−y|N+p1s1dxdy(−Δ)p1s1u+⋯+Mm∫R2N|u(x)−u(y)|pm|x−y|N+pmsmdxdy(−Δ)pmsmu=f(x,u)+γ|u|ps∗(α)−2u|x|α+β|u|q−2u|x|αin Ω,u=0in RN∖Ω, where 0

In this paper, we study the inverse problem of the generalized X-ray transform in n-dimensional Schwartz space. We extend Natterer's inversion formula to the generalized X-ray transform. Moreover, we extend results of Rigaud and Lakhal to the n-dimensional generalized X-ray transform. Furthermore, we gain some Sobolev estimate formulas of the generalized X-ray transform.

In this paper, we discuss the existence of multiple solutions of the generalized quasilinear Schrödinger equation of Kirchhoff-type (1+b∫R3g2(u)|∇u|2dx)[−div(g2(u)∇u)+g(u)g′(u)|∇u|2]+V(x)u=f(x,u),x∈R3 where b>0 is a parameter, g∈C1(R,R+). In preceding approaches, the order of the nonlinear term f(x,t) about t at infinity is usually supposed to be higher than |t|3. But in this paper, this condition

ABSTRACT If g∈H∞, the integral operator Sg on Hp, BMOA and Bp(Besov) spaces, is defined as Sgf(z)=∫0zf′(w)g(w)dw. In this paper, we prove three necessary and sufficient conditions for the operator Sg to have closed range on Hp(1≤p<∞), BMOA and Bp(1

The aim of this paper is to study the following fractional critical problem with Hardy potential and concave–convex nonlinearities (−Δ)su−μu|x|2s=λuq+u2s∗−1,u>0 in Ω,u=0in RN∖Ω, where (−Δ)s is the fractional Laplace operator, s∈(0,1), Ω⊂RN is a smooth bounded domain containing the origin, N>2s, μ∈[0,ΛN,s), λ>0, 00. Moreover, in the convex power case  (1

We deal with Gevrey solvability of a class of complex vector fields, defined on Ω=R×T1, where T1≃R/2πZ is the unit circle, given by L=∂/∂t+(a(x)+ib(x))∂/∂x, b≢0, near the characteristic set Σ={0}×T1. We show that under certain condition involving the order of vanishing of the functions a and b at x = 0 the equation Lu = f does not have Gevrey solutions.

Suppose 0<α<∞, Bα is an integral operator of Lp(ID,dA) which is defined as follows: Bα[f](z)=∫ID1(1−w¯z)αf(w)dA(w), where D is the unit disk and dA(w) is the normalized area measure. For 0<α<2, we obtain the norm estimates of ∥Bα∥p, where 1≤p≤∞. Our results are sharp for p=1,2,∞. Moreover, we show that Bα is a compact operator and of Schatten p-class, where p>12−α. For 2<α<3, we show that Bα(Lp(ID))⊆Lq(ID)

ABSTRACT This paper considers hyperbolic wave equations with non-local in time conditions involving integrals with respect to time. It is shown that regularity of the solution can be achieved for complexified problem with integral conditions involving harmonic complex exponential weights. The paper establishes existence, uniqueness, and regularity of the solutions.

Slice quaternionic analysis in two variables is a generalization of the theory of several complex variables to quaternions. This study relies on the theory of stem functions and the theory of holomorphic functions in two complex variables. Our approach is to introduce holomorphicity for stem functions in terms of two commutative complex structures. It turns out that, locally, a function which is slice

We show that derivatives of some classes of analytic functions, univalent or locally univalent in the unit disk, are cyclic vectors for the Bergman space B2(D).

We deal with a class of one-parameter family of integral transforms of Bargmann type arising as dual transform of fractional Hankel transform. Their ranges are identified to be special subspaces of the weighted hyperholomorphic left Hilbert spaces, generalizing the slice Bergman space of the second kind. Their reproducing kernel is given by closed expression involving the ⋆-regularization of Gauss

In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff–Schrödinger–Poisson system involving a weak singularity, i.e. when 0<γ<1. Further, we obtain the existence of a solution with a strong singularity, i.e. when γ>1. We employ variational techniques to prove the existence and multiplicity results. Moreover, an L∞ estimate is obtained by using the Moser iteration

In this paper the Green's functions for three boundary value problems for the biharmonic equation are investigated. First, an integral representation of solutions to the inhomogeneous biharmonic equation is given. Then the Green's function of the Dirichlet problem is found and an integral representation of the solution to the Dirichlet problem in terms of the Green's function is given. After that,

ABSTRACT In this paper, we study the existence of compactly supported solutions for the Schrödinger equations with indefinite potentials −Δpu+V(x)u=a(x)|u|q−1u,x∈RN, where N≥ 2, 1

In this paper we study the following fractional nonlinear Schrödinger equation ε2s(−Δ)sv+V(x)v=K(x)f(v),x∈RN where ε is a positive parameter, s∈(0,1), N>2s, V(x) and K(x) are positive continuous functions. Using the variational techniques, we construct a family of positive solutions under general conditions on f. Moreover, we show that the solutions concentrate around the isolated components of positive

In this paper, we study the nonlinear Kirchhoff type equation {−(a+b∫R3|∇u|2dx)△u+λV(x)u=|u|4u+f(u),x∈R3,u∈H1(R3), where a, b are positive constants and λ>0. Suppose that the nonnegative continuous potential V represents a potential well with the bottom V−1(0) and f∈C(R,R) satisfies certain assumptions. We obtain the existence of ground state solutions by using the variational methods. Moreover, the

We show that a global holomorphic section of O(d) restricted to a closed complex subspace X⊂Pn has an interpolant if and only if it satisfies a set of moment conditions that involves a residue current associated with a locally free resolution of OX. When X is a finite set of points in Cn⊂Pn this can be interpreted as a set of linear conditions that a function on X has to satisfy in order to have a

A hypersurface X (resp., a pair of hypersurfaces {Y,Z}) is called a hypersurface (resp., a pair of hypersurfaces) of uniqueness for holomorphic mappings from C to PN(C) if for two non-degenerate holomorphic mappings f,g from C to PN(C), the condition νf(X)=νg(X) (resp., νf(Y)=νg(Y), νf(Z)=νg(Z)) implies f≡g, where for a mapping φ, νφ(V) denotes the pull-back of a divisor V in PN(C) by φ. In this paper

In this paper, we study weighted composition operators and differences of composition operators. We use the Carleson measure to estimate the operator norm and the essential norm of those operators which act between two Bergman spaces with Békollé weights.

ABSTRACT We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results

ABSTRACT The solutions of a number of well-known boundary value problems of complex analysis (for example, the Riemann boundary value problem) can be found in the form of curvilinear integrals over the boundaries of domains under consideration. In this connection, the classical results on that problems concern domains with rectifiable boundaries only. On the other hand, the boundary value problems

ABSTRACT We continue the study of open, discrete mappings satisfying generalized modular inequalities, namely the distortion and the equicontinuity of such mappings. We apply these results to certain Sobolev mappings. In the last section we study the local behaviour of open, light mappings on general metric measure spaces.

ABSTRACT In this paper, we investigate the existence of ground state solutions for fractional Schrödinger–Choquard–Kirchhoff type equations with critical growth a+b∫RN|(−Δ)s2u|2dx(−Δ)su+V(x)u=λf(x,u)+[|x|−μ∗|u|2μ,s∗]|u|2μ,s∗−2u,x∈RN,u∈Hs(RN), where a, b>0 are constants, λ>0 is a parameter, 02s, 0<μ<2s and 2μ,s∗=2N−μN−2s. When V and f are asymptotically periodic in x, we prove that the equation has

We consider the numerical range of a bounded weighted composition operator Cψ,φ on the Fock space F2, where the entire function φ must have the form az + b with a and b in C and |a|≤1. We obtain necessary and sufficient conditions for {ψ(p)an:nis a non-negative integer} to be a subset of the interior of the numerical range of Cψ,φ, where p is the fixed point of φ. We characterize when 0 belongs to

In this study, we proved the simplicity of eigenvalues and obtained asymptotic formulas for eigenvalues and eigenfunctions of second order non-local boundary-value problems with retarded argument.

In this article, we deal with the existence of solutions for a class of nonlinear elliptic system with ( p ( x ) , q ( x ) ) -Laplacian operator. The work is based on the Leray–Schauder principle with the use of the topological degree of Berkovits applied to an abstract Hammerstein equation associated to our system and presenting a certain lack of compactness. In this study, we establish existence

We are concerned with discussing the ground state solutions of the Choquard equation with the Hardy potentials and critical Sobolev exponent: − Δ u + a − μ | x | 2 u = ( I α ∗ F ( u ) ) f ( u ) + | u | 2 ∗ − 2 u , x ∈ R N ∖ { 0 } , u ∈ H 1 ( R N ) , where N ≥ 3 , α ∈ ( 0 , N ) , 0 ≤ μ < μ ¯ := ( N − 2 ) 2 4 , I α is the Riesz potential, 2 ∗ := 2 N / ( N − 2 ) is the critical Sobolev exponent, and f

ABSTRACT This paper is concerned with the following sublinear fractional Schrödinger equation: ( − Δ ) s u + V ( x ) u = a ( x ) | u | q − 1 u + f ( x ) , x ∈ R N , where s ∈ ( 0 , 1 ) , q ∈ ( ( 1 / 2 s ∗ − 1 ) , 1 ) , N>2s, 2 s ∗ = 2 N / ( N − 2 s ) , ( − Δ ) s is the fractional Laplacian operator, and V, a both change sign in R N , f ≠ 0 is a nonnegative perturbation such that f ∈ L ( q + 1 ) / q

ABSTRACT Following the ideas of recent research in Karapetyants and Samko [Mixed norm variable exponent Bergman space on the unit disc. Complex Var Elliptic Equ. 2016;61:1090–1106], we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present a general approach for the

This paper is a short essay about Sergei L'vovich Sobolev (1908–1989), an outstanding mathematician of the 20th century, who made fundamental contributions to the development of the modern theory of partial differential equations, equations of mathematical physics, functional analysis, function theory, and computational mathematics.

In this paper, we study the localization of the discrete spectrum of certain perturbations of a two-dimensional harmonic oscillator. The convergence of the expansion of the source function in terms of the eigenfunctions of a two-dimensional harmonic oscillator is investigated. A representation of Green's function of a two-dimensional harmonic oscillator is obtained. The singularities of Green's function

We prove the uniqueness of positive radial solutions to a class of singular p-Laplacian equations in a ball with Dirichlet boundary condition when a parameter is large. The reaction term exhibits infinite semipositone structure at zero and is not necessarily increasing or concave on ( 0 , ∞ ) .

ABSTRACT In this paper, we prove the existence of (global) solutions of the Poincaré-Lelong equation − 1 ∂ ∂ ¯ u = f , where f is a d-closed ( 1 , 1 ) form and is in the weighted Hilbert space with Gaussian measure, i.e. L ( 1 , 1 ) 2 ( C n , e − | z | 2 ) . The novelty of this paper is to apply a weighted L 2 version of the Poincaré Lemma for 2-forms and then apply Hörmander's L 2 solutions for Cauchy–Riemann

In this paper, we are interested in giving sufficient conditions of a quaternionic plurisubharmonic function defined on a bounded quaternionic hyperconvex domain such that it can be approximated by a decreasing sequence of smooth functions. As an application, we study the geometric property of quaternionic B-regular domains.

ABSTRACT In this paper, we investigate the existence theorem of entropy solutions for nonlinear elliptic problem of the type − div ⁡ ( a ( x , u , ∇ u ) + Φ ( x , u ) ) + g ( x , u , ∇ u ) = μ in Ω, in the setting of Musielak-Orlicz spaces. The lower order term Φ verifies the natural growth condition, the nonlinearity g has a natural growth with respect to its third argument and with sign condition

ABSTRACT In this article, we study a certain type of boundary behaviour of positive solutions of the heat equation on the Euclidean upper half-space of R n + 1 . We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover

ABSTRACT In this paper, by using Uhlenbeck–Yau's continuity method, we prove the existence of approximate Hermite–Einstein structure on the semi-stable generalized holomorphic bundles over closed generalized Kähler manifolds of symplectic type.

In this paper, we study a class of double-phase problems with sub-critical growth. Some new criteria to guarantee that the existence and multiplicity results for the considered problem is established by using variational methods and critical point theory.

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the weighted Bergman space A α 2 ( D ) . We first give properties of complex symmetric Toeplitz operators T φ on A α 2 ( D ) . Next, we prove that if T φ is complex symmetric with finite symbol, then T φ is hyponormal on A α 2 ( D ) if and only if it is hyponormal on the Hardy space H 2 ( T ) . Finally, we consider

We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.

ABSTRACT In Koo and Wang (Joint Carleson measure and the difference of composition operators on A α p ( B n ) . J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint

ABSTRACT This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices

We consider the swallowtail integral Ψ ( x , y , z ) := ∫ − ∞ ∞ e i ( t 5 + x t 3 + y t 2 + z t ) d t for large values of | z | and bounded values of | x | and | y | . The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al

In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains: ∑ i = 1 k [ u i ] s , p p θ − 1 ( − Δ ) p s u j ( x ) = λ j f j ( x ) | u j | q − 2 u j + ∑ i ≠ j β i , j h ( x ) | u i | m | u j | m − 2 u j in   Ω , u j = 0 i n   R n ∖ Ω . By using the Nehari manifold method, together with Ekeland's variational principle, we show that there

ABSTRACT There have been many extensions of the Nevanlinna's five-values theorem for meromorphic functions to the case of meromorphic maps into P n ( C ) . We improve these results by considering the degenerate case and using the weaker condition ‘ f − 1 ( H ) ⊆ g − 1 ( H ) ’ instead of the usual one ‘ f − 1 ( H ) = g − 1 ( H ) ’ for some hyperplanes H among the given hyperplanes. Our main theorems

ABSTRACT The triangular ratio metric is studied in a domain G ⊊ R n , n ≥ 2 . Several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.

This paper deals with the existence and nonexistence of positive radial solutions of the elliptic equation − Δ u = f ( | x | ,   u ) , u ∈ Ω , u | ∂ Ω = 0 ,   where Ω = { x ∈ R N : r 1 < | x | < r 2 } ,   N ≥ 3 and f ∈ C ( [ r 1 , r 2 ] × R + ) , whose sign may change. We present new eigenvalue criteria for the existence and nonexistence to this problem. The discussion is based on the fixed point index

In this paper, we consider the Choquard-type equation with the fractional Laplacian ( − Δ ) s u ( x ) + a u ( x ) = 1 | x | n − α ∗ u p u p − 1 , x ∈ R n , u ( x ) > 0 , x ∈ R n , where 0

We study the existence and asymptotic behavior of least energy sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth − Δ u + ω u + λ h 2 ( | x | ) | x | 2 + ∫ | x | ∞ h ( s ) s u 2 ( s ) d s u = f ( u ) i n     R 2 , u ∈ H r 1 ( R 2 ) , where ω , λ > 0 are constants and h ( s ) = ∫ 0 s r 2 u 2 ( r ) d r . Under some suitable assumptions on f ∈ C ( R )

ABSTRACT We give the boundedness of the Hardy-Littlewood maximal operator M λ , λ ≥ 1 , on central Herz-Morrey-Musielak-Orlicz spaces H Φ , q , ω ( X ) over bounded non-doubling metric measure spaces and to establish a generalization of Sobolev's inequality for Riesz potentials I α , τ f , τ ≥ 1 , α > 0 , of functions in such spaces. As an application and example, we obtain the boundedness of M λ and