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In Gérard et al. (Ann Sci Ecole Norm Sup 56:127–196, 2023), the Unruh state for massless fermions on a Kerr spacetime was constructed and the authors showed its Hadmard property in the case of very slowly rotating black holes a ≪ M . In this note, we extend this result to the full non extreme case a < M . [ABSTRACT FROM AUTHOR]

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) : Q × P → 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. [ABSTRACT FROM AUTHOR]

This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel–Lizorkin-type spaces B ˙ p , q s , τ (W) and F ˙ p , q s , τ (W) . In this article, the authors establish the molecular and the wavelet characterizations of these spaces. Furthermore, as applications, the authors obtain the optimal boundedness of trace operators, pseudo-differential operators, and Calderón–Zygmund operators on these spaces. Due to the sharp boundedness of almost diagonal operators on their related sequence spaces obtained in the second article of this series, all results presented in this article improve their counterparts on matrix-weighted Besov and Triebel–Lizorkin spaces B ˙ p , q s (W) and F ˙ p , q s (W) . In particular, even when reverting to the boundedness of Calderón–Zygmund operators on unweighted Triebel–Lizorkin spaces F ˙ p , q s , these results are still better. [ABSTRACT FROM AUTHOR]

We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand's results. [ABSTRACT FROM AUTHOR]

In the setting of the lattice Z n we consider a pseudo-differential operator A whose symbol belongs to a class defined on Z n × T n , where T n is the n-torus. We realize A as an operator acting between the discrete Sobolev spaces H s j (Z n) , s j ∈ R , j = 1 , 2 , with the discrete Schwartz space serving as the domain of A. We provide a sufficient condition for the essential adjointness of the pair (A , A †) , where A † is the formal adjoint of A. [ABSTRACT FROM AUTHOR]

In this paper, we prove Hopf's lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf's lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf's lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf's lemma can be applied to the study of the symmetry of the positive solution of the nonlinear Logarithmic Laplacian problem by the moving plane method. [ABSTRACT FROM AUTHOR]

This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators: S k (D 2 w (x)) - A (x) (- Δ) α / 2 w (x) = f (z (x)) , S k (D 2 z (x)) - B (x) (- Δ) β / 2 z (x) = g (w (x)). Firstly, we discuss decay at infinity principle and narrow region principle for the k-Hessian system involving fractional order Laplacian operators. Then, by exploiting the direct method of moving planes, the radial symmetry and monotonicity of the nonnegative solutions to the coupled k-Hessian system are proved in a unit ball and the whole space, respectively. We believe that the present work will lead to a deep understanding of the coupled k-Hessian system involving different order fractional Laplacian operators. [ABSTRACT FROM AUTHOR]

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus. [ABSTRACT FROM AUTHOR]

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R 3 subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L 2 ln (L) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region L Λ ⊂ R 3 as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln (L) factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂ Λ , is not solely on its area but on the "surface perpendicular to the direction of the magnetic field". We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L 2 of the relevant trace for piecewise C 1 , α smooth surfaces ∂ Λ . [ABSTRACT FROM AUTHOR]

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space ħ Z n . The current work is an extension of the previous work (Botchway et al. in J Funct Anal 278(11):108473, 33, 2020) and agrees with it in the limit of the parameter ħ → 1 . The various representations of the operators will be studied as well as the composition, transpose, adjoint and the link between ellipticity and parametrix of operators. We also give the conditions for the ℓ p , weighted ℓ 2 boundedness and ℓ p compactness of operators. We investigate the relation between the classical and semi-classical quantization in the spirit of Ruzhansky and Turunen (Pseudo-differential operators and symmetries. Pseudo-differential operators, vol 2. Theory and Applications, Birkhäuser, Basel, 2010; J Fourier Anal Appl 16(6):943–982, 2010) RTspsJFAA and employ its applications to Schatten–von Neumann classes on ℓ 2 (ħ Z n) . We establish Gårding and sharp Gårding inequalities, with an application to the well-posedness of parabolic equations on the lattice ħ Z n . Finally we verify that in the limiting case where ħ → 0 the semi-classical calculus of pseudo-differential operators recovers the classical Euclidean calculus, but with a twist. [ABSTRACT FROM AUTHOR]

In this note, we prove that the toroidal pseudo-differential operator is bounded from L∞(Tn){L}^{\infty }\left({{\mathbb{T}}}^{n}) to BMO(Tn){\rm{BMO}}\left({{\mathbb{T}}}^{n}) if the symbol belongs to the toroidal Hörmander class Sρ,δn(ρ−1)∕2(Tn×Zn){S}_{\rho ,\delta }^{n\left(\rho -1)/2}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with 0

In this paper, we prove a potentially useful L p (d ν) -boundedness result for the pseudo-differential operators associated with the spherical mean operator. Also, boundedness result for symmetrically global pseudo-differential operator on L p (d ν) -type Sobolev space H u , v , p of order (u, v) are discussed. An application in solving a generalized heat equation is given. [ABSTRACT FROM AUTHOR]

This paper is devoted in investigations concerning the study of the coupled potential operator J s α , β and corresponding L p -Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space S α , β is introduced. Moreover, pseudo-differential operator is defined and derived one more integral representation. Further, it is shown that pseudo-differential operator associated with CFrFT is more generalization as of two dimensional fractional Fourier transform. The L p norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator J s α , β is defined as a pseudo-differential operator related with a precise symbol. The operator J s α , β is extended to a space of distributions. An L p -Sobolev boundedness result for the operator J s α , β is shown. The spaces H p m , α , β and H p m , α , β introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces. [ABSTRACT FROM AUTHOR]

We study ground states of a relativistic Fermi system involved with the pseudo-differential operator - c 2 Δ + c 4 m 2 - c 2 m in the L 2 -subcritical case, where m > 0 denotes the rest mass of fermions, and c ≥ 1 represents the speed of light. By employing Green's function and the variational principle of many-fermion systems, we prove the existence of ground states for the system. The asymptotic behavior of ground states for the system is also analyzed in the non-relativistic limit where c → ∞ . [ABSTRACT FROM AUTHOR]

We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes Γ m (R 2 d) , with negative order m. The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander’s class S 0 , 0 0 (R 2 d) . Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order m. [ABSTRACT FROM AUTHOR]

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander's class on matrix weighted Besov and Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

Pseudo-differential operators (PDO) Q (x , ℒ a , x) and Q (x , ℒ a , x) involving the index Whittaker transform are defined. Estimates for these operators in Hilbert space L2a(ℝ+;ma(x)dx) are obtained. A symbol class Ω is introduced. Later product and commutators for the PDO are investigated and their boundedness results are discussed. [ABSTRACT FROM AUTHOR]

In this paper, pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are introduced. The boundedness of pseudo-differential operators and commutator between two pseudo-differential operators on ℌα,2r are proven with the help of the Weinstein transform technique. [ABSTRACT FROM AUTHOR]

In this work we present necessary cancellation conditions for the continuity of linear operators in h p (R n) , 0 < p ≤ 1 , that map atoms into pseudo-molecules. The necessary conditions, expressed in terms of the T ∗ condition, are the same as the ones proved sufficient in our recent work, thus providing a necessary and sufficient cancellation conditions for the boundedness of inhomogeneous Calderón–Zygmund type operators. [ABSTRACT FROM AUTHOR]

We study first-order symmetrizable hyperbolic N × N systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at x = 0 , these systems take the form ∂ t u + A (t , x , y , x D x , D y) u = f (t , x , y) , (t , x , y) ∈ (0 , T) × R + × R d , where A (t , x , y , x D x , D y) is a first-order differential operator with coefficients smooth up to x = 0 and the derivative with respect to x appears in the combination x D x . No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator ∂ t + A (t , x , y , x D x , D y) is well-posed in that scale. More specifically, solutions u exhibit formal asymptotic expansions of the form u (t , x , y) ∼ ∑ (p , k) (- 1) k k ! x - p log k x u pk (t , y) as x → + 0 where (p , k) ∈ C × N 0 and ℜ p → - ∞ as | p | → ∞ , provided that the right-hand side f and the initial data u | t = 0 admit asymptotic expansions as x → + 0 of a similar form, with the singular exponents p and their multiplicities unchanged. In fact, the coefficients u pk are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients u pk solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator ∂ t + A (t , x , y , x D x , D y) is well-posed in the scale of standard Sobolev spaces H s ((0 , T) × R + 1 + d) . [ABSTRACT FROM AUTHOR]

In analogy with the definition of “extended Sobolev scale" on R n by Mikhailets and Murach, working in the setting of the lattice Z n , we define the “extended Sobolev scale" H φ (Z n) , where φ is a function which is RO-varying at infinity. Using the scale H φ (Z n) , we describe all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of discrete Sobolev spaces [ H (s 0) (Z n) , H (s 1) (Z n) ] , with s 0 < s 1 . We use this interpolation result to obtain the mapping property and the Fredholmness property of (discrete) pseudo-differential operators (PDOs) in the context of the scale H φ (Z n) . Furthermore, starting from a first-order positive-definite (discrete) PDO A of elliptic type, we define the “extended discrete A-scale" H A φ (Z n) and show that it coincides, up to norm equivalence, with the scale H φ (Z n) . Additionally, we establish the Z n -analogues of several other properties of the scale H φ (R n) . [ABSTRACT FROM AUTHOR]

This paper is devoted to a simpler derivation of energy estimates and a proof of the well-posedness, compared to previously existing ones, for effectively hyperbolic Cauchy problem. One difference is that instead of using the general Fourier integral operator, we only use a change of local coordinates x (of the configuration space) leaving the time variable invariant. Another difference is an efficient application of the Weyl-Hörmander calculus of pseudodifferential operators associated with several different metrics. [ABSTRACT FROM AUTHOR]

We consider wave equations with a nonlocal polynomial type of damping depending on a small parameter θ ∈ (0 , 1) . This research is a trial to consider a new type of dissipation mechanisms produced by a bounded linear operator for wave equations. These researches were initiated in a series of our previous works with various dissipations modeled by a logarithmic function published in (Charão et al. in Math Methods Appl Sci 44:14003-14024, 2021; Charão and Ikehata in Angew Math Phys 71:26, 2020; Piske et al. in J Diff Eqns 311:188-228, 2022). The model of dissipation considered in this work is probably the first defined by more than one sentence and it opens field to consider other more general. We obtain an asymptotic profile and optimal estimates in time of solutions as t → ∞ in L 2 -sense, particularly, to the case 0 < θ < 1 / 2 . [ABSTRACT FROM AUTHOR]

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity a (λ x , λ σ ξ) = λ 1 + σ a (x , ξ) for λ > 0 where σ > 0 is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol. [ABSTRACT FROM AUTHOR]

We establish decay estimates for Fourier transform on Hardy–Morrey spaces and its localizable version. Our work includes some aspects to these spaces linked up with pointwise Fourier estimates, in particular a natural approach on cancellation moment conditions. As application, we discuss the optimality for continuity of Fourier multipliers and pseudodifferential operators in Hardy–Morrey spaces. [ABSTRACT FROM AUTHOR]

In this work, an extended Euler–Bernoulli beam equation is addressed, where numerous phenomena are covered including damping, time-delay, and nonlinear source effects. A generalized fractional derivative is used to model dissipation of order less than one, which offers more flexibility for modeling tasks. Through a diffusive representation, the problem well-posedness is tackled and the exponential decay of the energy associated to global solutions is proved under some conditions. In order to validate our theoretical findings, we implement a finite difference scheme and we elucidate that the boundedness of the local propagation matrix may be inaccurate for the convergence evaluation in some situations. Furthermore, we show that deep neural networks are efficient alternatives to deal with computational and stability burdens resulting from the mesh refinement in standard numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper we investigate the global L p boundedness of rough h-Fourier integral operators defined by generalized rough Hörmander class amplitudes on Banach and quasi-Banach spaces. As consequences we obtain some results of L p boundedness of rough bilinear h-Fourier integral operators. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

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]

We prove M -regularity for a class of pseudodifferential operators in ultradifferentiable classes defined on the torus T m + n which are globally C ∞ hypoelliptic. The same property is also valid for certain perturbations of these operators by lower order terms. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded smooth domain in ℝN (N ≥ 3). Assuming that 0 < s < 1, 1 < p , q ≤ N + 2 s N − 2 s with (p , q) ≠ (N + 2 s N − 2 s , N + 2 s N − 2 s) , and a, b > 0 are constants, we consider the existence results for positive solutions of a class of fractional elliptic system below, { (a + b [ u ] s 2) (− Δ) s u = v p + h 1 (x , u , v , ∇ u , ∇ v) , x ∈ Ω , (− Δ) s v = u q + h 2 (x , u , v , ∇ u , ∇ v) , x ∈ Ω , u , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ℝ N \ Ω. Under some assumptions of hi(x, u, v, ∇u, ∇v)(i = 1, 2), we get a priori bounds of the positive solutions to the problem (1.1) by the blow-up methods and rescaling argument. Based on these estimates and degree theory, we establish the existence of positive solutions to problem (1.1). [ABSTRACT FROM AUTHOR]

In this paper, boundedness and compactness results for symmetrically global pseudo-differential operator on L α p (R + n + 1) -type Sobolev space H α r , s , p of order (r, s) are investigated by exploiting the theory of the Weinstein transform. Using symmetrically global symbol σ (x , ξ) in S m 1 , m 2 , m 1 , m 2 ∈ R , we have discussed various properties of minimal–maximal pseudo-differential operators involving the Weinstein transform. The weak solution of the symmetrically global pseudo-differential equation is obtained by using aforesaid theory. [ABSTRACT FROM AUTHOR]

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem 0.1 ∂ t u (t , x) = ψ (t , - i ∇) u (t , x) + f (t , x) , (t , x) ∈ (0 , T) × R d , u (0 , x) = 0 , x ∈ R d , in (Muckenhoupt) weighted L p -spaces with time-measurable pseudo-differential operators 0.2 ψ (t , - i ∇) u (t , x) : = F - 1 ψ (t , ·) F [ u ] (t , ·) (x). More precisely, we find sufficient conditions of the symbol ψ (t , ξ) (especially depending on the smoothness of the symbol with respect to ξ ) to guarantee that equation (0.1) is well-posed in (Muckenhoupt) weighted L p -spaces. Here the symbol ψ (t , ξ) is merely measurable with respect to t, and the sufficient smoothness of ψ (t , ξ) with respect to ξ is characterized by a property of each weight. In particular, we prove the existence of a positive constant N such that for any solution u to equation (0.1), 0.3 ∫ 0 T ∫ R d | (- Δ) γ / 2 u (t , x) | p (t 2 + | x | 2 ) α / 2 d x d t ≤ N ∫ 0 T ∫ R d | f (t , x) | p (t 2 + | x | 2 ) α / 2 d x d t and 0.4 ∫ 0 T ∫ R d | (- Δ) γ / 2 u (t , x) | p | x | α 2 d x q / p t α 1 d t ≤ N ∫ 0 T ∫ R d | f (t , x) | p | x | α 2 d x q / p t α 1 d t , where p , q ∈ (1 , ∞) , - d - 1 < α < (d + 1) (p - 1) , - 1 < α 1 < q - 1 , - d < α 2 < d (p - 1) , and γ is the order of the operator ψ (t , - i ∇) . [ABSTRACT FROM AUTHOR]

It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further equivalent definition that is based on the Mellin transform and it can be used when the fractional Laplacian is applied to radial functions. The main finding is tested in the case of the space-fractional diffusion equation. The one-dimensional case is also considered, such that the Mellin transform of the Riesz (namely the symmetric Riesz–Feller) fractional derivative is established. This one-dimensional result corrects an existing formula in literature. Further results for the Riesz fractional derivative are obtained when it is applied to symmetric functions, in particular its relation with the Caputo and the Riemann–Liouville fractional derivatives. [ABSTRACT FROM AUTHOR]

H-measures and semiclassical (Wigner) measures were introduced in early 1990 s and since then they have found numerous applications in problems involving L 2 weakly converging sequences. Although they are similar objects, neither of them is a generalisation of the other, the fundamental difference between them being the fact that semiclassical measures have a characteristic length, while H-measures have none. Recently introduced objects, the one-scale H-measures, generalise both of them, thus encompassing properties of both. The main aim of this paper is to fully develop this theory to the L p setting, p ∈ (1 , ∞) , by constructing one-scale H-distributions, a generalisation of one-scale H-measures and, at the same time, of H-distributions, a generalisation of H-measures to the L p setting. We also address an alternative approach to L p extension of semiclassical measures via the Wigner transform, introducing new type of objects (semiclassical distributions). Furthermore, we derive a localisation principle in a rather general form, suitable for problems with a characteristic length, as well as those not involving a specific characteristic length, providing some applications. [ABSTRACT FROM AUTHOR]

Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, offering estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivatives. We prove such estimates for Coifman–Meyer multiplier operators in the setting of Triebel–Lizorkin and Besov spaces based on quasi-Banach function spaces. In particular, these include rearrangement invariant quasi-Banach function spaces such as weighted Lebesgue spaces, weighted Lorentz spaces and generalizations, and Orlicz spaces. The method used also yields results in weighted mixed Lebesgue spaces and Morrey spaces, where we present applications to the specific case of power weights, as well as in variable Lebesgue spaces. [ABSTRACT FROM AUTHOR]

We establish the Fredholmness of a pseudo-differential operator whose symbol is of class C0,σ, 0 < σ < 1, in the spatial variable. Our work here refines the work of H. Abels, C. Pfeuffer (2020). [ABSTRACT FROM AUTHOR]

We study an anisotropic version of the Shubin calculus of pseudodifferential operators on R d . Anisotropic symbols and Gabor wave front sets are defined in terms of decay or growth along curves in phase space of power type parametrized by one positive parameter that distinguishes space and frequency variables. We show that this gives subcalculi of Shubin's isotropic calculus, and we show a microlocal as well as a microelliptic inclusion in the framework. Finally we prove an inclusion for the anisotropic Gabor wave front set of chirp type oscillatory functions with a real polynomial phase function. [ABSTRACT FROM AUTHOR]

We introduce an anisotropic global wave front set of Gelfand–Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand–Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions. [ABSTRACT FROM AUTHOR]

We prove the existence of time quasi-periodic vortex patch solutions of the 2 d -Euler equations in R 2 , close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -as well as the degeneracy of a normal frequency- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper. [ABSTRACT FROM AUTHOR]

Motivated by the recent paper of Boggiatto–Garello (J Pseudo-Differ Oper Appl 11:93–117, 2020) where a Gabor operator is regarded as pseudodifferential operator with symbol p (x , ω) periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general τ quantization. [ABSTRACT FROM AUTHOR]

Nonlocal modeling has drawn more and more attention and becomes steadily more powerful in scientific computing. In this paper, we demonstrate the superiority of a first-principle nonlocal model—Wigner function—in treating singular potentials which are often used to model the interaction between point charges in quantum science. The nonlocal nature of the Wigner equation is fully exploited to convert the singular potential into the Wigner kernel with weak or even no singularity, and thus highly accurate numerical approximations are achievable, which are hardly designed when the singular potential is taken into account in the local Schrödinger equation. The Dirac delta function, the logarithmic, and the inverse power potentials are considered. Numerically converged Wigner functions under all these singular potentials are obtained with a fourth-order accurate operator splitting spectral method, and display many interesting quantum behaviors as well. [ABSTRACT FROM AUTHOR]

We consider the boundedness of the multilinear pseudo-differential operators with symbols in the multilinear Hörmander class S 0 , 0 . The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness between local Hardy spaces. [ABSTRACT FROM AUTHOR]