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In this paper, we are concerned with the anti-symmetric solutions to the following elliptic system involving fractional Laplacian \[ \begin{cases} (-\Delta)^{s}u(x)=u^{m_{1}}(x)v^{n_{1}}(x), & u(x)\geq0,\ x\in\mathbb{R}_{+}^{n} ,\\ \ (-\Delta)^{s}v(x)=u^{m_{2}}(x)v^{n_{2}}(x), & v(x)\geq0,\ x\in\mathbb{R}_{+}^{n} ,\\ \ u(x^{\prime},-x_{n})=-u(x^{\prime},x_{n}), & x=(x^{\prime},x_{n})\in\mathbb{R}^{n} ,\\ \ v(x^{\prime},-x_{n})=-v(x^{\prime},x_{n}), & x=(x^{\prime},x_{n})\in\mathbb{R}^{n} , \end{cases} \] { (− Δ) s u (x) = u m 1 (x) v n 1 (x) , u (x) ≥ 0 , x ∈ R + n , (− Δ) s v (x) = u m 2 (x) v n 2 (x) , v (x) ≥ 0 , x ∈ R + n , u (x ′ , − x n) = − u (x ′ , x n) , x = (x ′ , x n) ∈ R n , v (x ′ , − x n) = − v (x ′ , x n) , x = (x ′ , x n) ∈ R n , where 00\ _{(i=1,2)},n>2s,\mathbb {R}_{+}^{n} =\{(x^{\prime},x_{n})|x_{n}>0\} $ m i , n i > 0 (i = 1 , 2) , n > 2 s , R + n = { (x ′ , x n) | x n > 0 }. We first show that the solutions only depend on $ x_{n} $ x n variable by the method of moving planes. Moreover, we can obtain the monotonicity of solutions with respect to $ x_{n} $ x n variable (for the critical and subcritical cases $ m_{i}+n_{i}\leq \frac {n+2s}{n-2s}\ _{(i=1,2)} $ m i + n i ≤ n + 2 s n − 2 s (i = 1 , 2) in the $ \mathcal {L}_{2s} $ L 2 s space). Furthermore, when $ m_{1}=n_{2}=p,n_{1}=m_{2}=q $ m 1 = n 2 = p , n 1 = m 2 = q , in the cases $ p+q+2s\geq 1 $ p + q + 2 s ≥ 1 , we obtain a Liouville theorem for the cases $ p+q\leq \frac {n+2s}{n-2s} $ p + q ≤ n + 2 s n − 2 s in the $ \mathcal {L}_{2s} $ L 2 s space. Then, through the doubling lemma, we obtain the singularity estimates of the positive solutions on a bounded domain Ω. Using the anti-symmetric property of the solutions, one can extend the space from $ \mathcal {L}_{2s} $ L 2 s to $ \mathcal {L}_{2s+1} $ L 2 s + 1 , we can still prove the Liouville theorem in the extended space. With the extension, we prove the existence of nontrivial solutions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to enrich the theoretical system of the linear canonical Fourier transform (LCFT) by introducing the canonical potential J m , A and corresponding L p -Sobolev space. Moreover, the Schwartz-type space S A is introduced. Further, pseudo-differential operator (PDO) is defined and obtained its another integral representation. The L p -boundedness result for the pseudo-differential operator associated with the LCFT is discussed. Some applications of Sobolev-type spaces H A m , p and H A m , p are given. [ABSTRACT FROM AUTHOR]

In this paper, some elementary and useful terminologies associated to Lebedev–Skalskaya transform (LS-transform) are notified and then estimates for R κ 1 2 + i τ (x) , F κ 1 2 + i τ (x) , translation and convolution operators associated to LS-transform are obtained. The continuity of LS-transforms, translation and convolution operators on functions spaces S α , k , H α , k and G α , k are discussed. The continuity of pseudo-differential operators in terms of LS-transforms is discussed on functions spaces S α , k , H α , k , G α , k and Lebesgue space. At the end, an application of LS-transform is shown by solving a differential equation of convolution type. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped σ-evolution models with different power nonlinearities. By assuming additional L m regularity on the initial data, with m ∈ [ 1 , 2) , we use (L m ∩ L 2) − L 2 and L 2 − L 2 estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces for any σ ≥ 1 and δ ∈ (0 , σ). Moreover, not only the nonexistence of global solution is indicated by using test function method, but also a lifespan estimate is discussed when σ and δ ∈ (0 , σ) are integers. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of three weak solutions for a Kirchhoff-type elliptic system involving nonlocal fractional p-Laplacian with homogeneous Dirichlet boundary conditions. The approach is based on the three critical points theorem and some variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional Schrödinger equations (− Δ) s u + V (x) u = f (x , u) in R N , where s ∈ (0,1) and N>2s. Firstly, we prove that the problem admits a nontrivial solution and infinitely many nontrivial solutions under the assumptions that V is allowed to be indefinite potential and f is superlinear and subcritical. In addition, we establish an existence criterion of infinitely many nontrivial solutions for the aforementioned problem with concave and critical nonlinearities as well as indefinite potential, which improves the result of Du and Tian [Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete Contin Dyn Syst B. 2016;21(10):3407–3428 (Theorem 1.6)]. [ABSTRACT FROM AUTHOR]

In this article, we present some boundedness results of fractional Hankel transform on Gelfand–Shilov spaces of type S. Certain inequalities are obtained for pseudo-differential operators involving the fractional Hankel transform on Gelfand–Shilov spaces of type S. This article made further discussion on the continuity properties of fractional Hankel transform and pseudo-differential operator on some ultradifferentiable function spaces. [ABSTRACT FROM AUTHOR]

In this paper, some basic results and notations related to Lebedev–Skalskaya transforms (LS-transforms) are introduced and then estimates of translation and convolution operators associated to LS-transform are obtained. Continuity of LS-transforms on Lebesgue space as well as on function spaces S β , k and G α , k are discussed. Pseudo-differential operators in terms of LS-transforms are defined and their integral representations have been studied in various ways. Moreover, their estimates in Lebesgue space are obtained. At the end, some applications have been discussed. [ABSTRACT FROM AUTHOR]

If M is a compact, contact manifold, with contact line bundle Σ, L a "sublaplacian" on M, characteristic on Σ ∖ 0 , and if { φ j } is an orthonormal basis of eigenfunctions of –L, with eigenvalues λ j ↗ + ∞ , then { φ j } concentrates microlocally on Σ ∖ 0. We study a microlocal Weyl formula, giving the limit as N → ∞ of N − 1 ∑ j = 1 N (A φ j , φ j) for a class of pseudodifferential operators A on M. Such a limit for A ∈ O P S 0 (M) , the algebra of classical pseudodifferential operators, was evaluated by Colin, Hillairet, and Trelat and used to study quantum ergodic theorems on compact 3 D contact manifolds. Here we take A to belong to a different pseudodifferential operator algebra, O P Ψ ˜ 0 (M) , tuned to the contact structure of M. This produces a Weyl formula exhibiting richer behavior than one sees for A ∈ O P S 0 (M). [ABSTRACT FROM AUTHOR]

The main aim of this paper is to find out the estimates of convolution for zero-order Mehler–Fock transform with various approaches. Pseudo-differential operator in terms of zero-order Mehler–Fock transform is defined and obtained its another integral representation. Further, its estimate in Lebesgue space has been studied. At the end some applications of zero-order Mehler–Fock transform and of convolution are discussed. [ABSTRACT FROM PUBLISHER]

A method is developed for analysing asymptotic behaviour of terms involving an arbitrary integer order powers offunctions by means of H-measures. It is applied to the small amplitude homogenisation problem for a stationary diffusion equation, in which coefficients are assumed to be analytic perturbations of a constant, enabling formulæ for higher order correction terms in a general, non-periodic setting. Explicit expressions in terms of Fourier coefficients are obtained under periodicity assumption. The method allows of its generalisation and application to the corresponding non-stationary equation, as well as to some other small amplitude homogenisation problems. [ABSTRACT FROM PUBLISHER]

We seek to extend work by Christianson–Hassell–Toth [3] on restrictions of Neumann data of Laplacian eigenfunctions to interior hypersurfaces to a general semiclassical setting. In the semiclassical regime the appropriate generalization is to study the restrictions of the functionv = ν(x,hD)uwhereν(x,hD) is the operator defined by quantizing the normal velocity observable. For the Laplacianwhereνis the normal to the hypersurface. We find thatprovideduis anquasimode of the semiclassical pseudodifferential operatorp(x,hD). This statement should be interpreted as a statement of non-concentration for the quantization of normal velocity. [ABSTRACT FROM AUTHOR]

We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator. [ABSTRACT FROM PUBLISHER]

We prove that Hörmander’s global wave front set and Nakamura’s homogeneous wave front set of a tempered distribution coincide. In addition, we construct a tempered distribution with a given wave front set, and we develop a pseudodifferential calculus adapted to Nakamura’s homogeneous wave front set. [ABSTRACT FROM PUBLISHER]

We consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a compact Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral geometry question of some attenuated geodesic ray transform through a pseudolinearization argument. We show injectivity (up to natural obstructions) and stability estimates for both the linear and nonlinear problems for generic simple metrics and generic connections and Higgs fields, including the real-analytic ones. We consider the problems on simple manifolds to make the exposition of the main ideas clear and concise, many results of this paper still hold under some assumptions weaker than simplicity. [ABSTRACT FROM PUBLISHER]

A pair of pseudo-differential operators (p.d.o.) involving the Kontorovich–Lebedev transform (KL-transform) as well as its adjoint is defined. Estimates for translation and convolution function associated with the Kontorovich–Lebedev transform are obtained. Continuity of KL-transform and p.d.o. onandare discussed. [ABSTRACT FROM AUTHOR]

We obtain a global version in theN-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context. [ABSTRACT FROM AUTHOR]

The pseudo-differential operator involving the Kontorovich–Lebedev transform is defined. The boundedness of pseudo differential operator in a certain Sobolev-type space is shown. Composition of pseudo-differential operatorsis defined in terms of the two symbols, and also its boundedness in certain Sobolev-type space is given. Some special cases are also discussed. Finally, the Kontorovich–Lebedev transform is used in the solution of some partial differential equations. [ABSTRACT FROM PUBLISHER]

Given a smoothly bounded domain Ω ⊂ ℝnwithn ≥ 1 odd, we study the blowup of bounded sequencesof solutions to the nonlocal equationwhere λk → λ∞ ∈ [0, ∞), anddenotes the Lions–Magenes spaces of functionsu ∈ L2(ℝn) which are supported in Ω and with. Extending previous works of Druet, Robert-Struwe, and Martinazzi, we show that if the sequence (uk) is not bounded inL∞(Ω), a suitably rescaled subsequence ηkconverges to the function, which solves the prescribed nonlocalQ-curvature equationrecently studied by Da Lio–Martinazzi–Rivière whenn = 1, Jin–Maalaoui–Martinazzi–Xiong whenn = 3, and Hyder whenn ≥ 5 is odd. We infer that blowup can occur only if. [ABSTRACT FROM AUTHOR]

Letbdbe the Weyl symbol of the inverse to the harmonic oscillator on Rd. We prove thatbdand its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions forbd. In the even-dimensional case we characterizebdin terms of elementary functions. In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions. [ABSTRACT FROM PUBLISHER]

Fors ∈ (0, 1) we introduce (integro-differential) harmonic mapsv: Ω ⊂ ℝn → ℝN, which are defined as critical points of the Gagliardo/Slobodeckij energywith the condition thatv(Ω) ⊂ 𝕊N−1, for the (N − 1)-sphere 𝕊N−1 ⊂ ℝN. Ifp = 2 these are the classical fractional harmonic maps first considered by Da Lio and Rivière. Forp ≠ 2 this is a new energy which has degenerate, non-local Euler-Lagrange equations. They are different from then/p-harmonic maps introduced by Da Lio and the author, and have to be treated with new arguments, which might be of independent interest for further applications on geometric energies. The main result is Hölder continuity for these maps in the critical case. [ABSTRACT FROM AUTHOR]

We prove that a function or distribution onRdis radial symmetric, if and only if its Bargmann transform is a composition by an entire function onCand the canonical quadratic function fromCdtoC. [ABSTRACT FROM AUTHOR]

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in quantum chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit ℏ → 0 we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis. [ABSTRACT FROM AUTHOR]