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Journal articles on the topic 'H-Pseudodifferential operators'

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Published: 25 May 2024

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1

Yang, Jie. "On L 2 -Boundedness of h -Pseudodifferential Operators." Journal of Function Spaces 2021 (February 20, 2021): 1–5. http://dx.doi.org/10.1155/2021/6690963.

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Let T a h be the h -pseudodifferential operators with symbol a . When a ∈ S ρ , 1 m and m = n ρ − 1 / 2 , it is well known that T a h is not always bounded in L 2 ℝ n . In this paper, under the condition a x , ξ ∈ L ∞ S ρ n ρ − 1 / 2 ω , we show that T a h is bounded on L 2 .

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2

Taylor, Michael. "The Technique of Pseudodifferential Operators (H. O. Cordes)." SIAM Review 38, no. 3 (September 1996): 540–42. http://dx.doi.org/10.1137/1038101.

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3

Deng, Yu-long. "Commutators of Pseudodifferential Operators on Weighted Hardy Spaces." Journal of Mathematics 2022 (January 20, 2022): 1–6. http://dx.doi.org/10.1155/2022/8851959.

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In this paper, we establish an endpoint estimate for the commutator, b , T , of a class of pseudodifferential operators T with symbols in Hörmander class S ρ , δ m R n . In particular, there exists a nontrivial subspace of B M O R n such that, when b belongs to this subspace, the commutators b , T is bounded from H ω 1 R n into L ω 1 R n , which we extend the well-known result of Calderón-Zygmund operators.

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4

Hitrik, Michael, and Johannes Sjöstrand. "Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point." Canadian Journal of Mathematics 60, no. 3 (June 1, 2008): 572–657. http://dx.doi.org/10.4153/cjm-2008-028-3.

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AbstractThis is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength ॉ of the perturbation is in the range h2 ≪ ॉ ≪ h1/2 (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles [−1/C, 1/C] + iॉ[F0 − 1/C, F0 + 1/C], C ≫ 1, when ॉF0 is a saddle point value of the flow average of the leading perturbation.

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5

Rabinovich, V. S. "Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators." Russian Journal of Mathematical Physics 16, no. 2 (June 2009): 300–308. http://dx.doi.org/10.1134/s1061920809020149.

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6

Rabinovich, V. "Exponential estimates of solutions of pseudodifferential equations on the lattice $${(h \mathbb{Z})^{n}}$$ : applications to the lattice Schrödinger and Dirac operators." Journal of Pseudo-Differential Operators and Applications 1, no. 2 (March 10, 2010): 233–53. http://dx.doi.org/10.1007/s11868-010-0005-2.

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7

Elong, Ouissam. "On the LP boundedness of h-Fourier integral operators with rough symbols." Mathematica Montisnigri 54 (2022): 25–39. http://dx.doi.org/10.20948/mathmontis-2022-54-3.

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We prove LP boundedness of a class of semiclassical Fourier integral operators defined by smooth phase function and semiclassical rough symbols on the spatial variable 𝑥. We also consider a spacial case of ℎ -pseudodifferential operators.

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8

Orlov, A. Yu, and P. Winternitz. "P∞ Algebra of KP, Free Fermions and 2-Cocycle in the Lie Algebra of Pseudodifferential Operators." International Journal of Modern Physics B 11, no. 26n27 (October 30, 1997): 3159–93. http://dx.doi.org/10.1142/s0217979297001532.

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The symmetry algebra P∞=W∞⊕ H ⊕ I∞ of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one ("positive") half of the point symmetries belongs to the W∞ symmetries while the other ("negative") part belongs to the I∞ ones. The corresponding action on the τ-function is obtained. A new embedding of the Virasoro algebra into gl(∞) describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.

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9

Rozendaal, Jan. "Rough Pseudodifferential Operators on Hardy Spaces for Fourier Integral Operators II." Journal of Fourier Analysis and Applications 28, no. 4 (July 13, 2022). http://dx.doi.org/10.1007/s00041-022-09959-x.

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AbstractWe obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $$a(x,\eta )$$ a ( x , η ) are elements of $$C^{r}_{*}S^{m}_{1,\delta }$$ C ∗ r S 1 , δ m classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces $${\mathcal {H}}^{s,p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO s , p ( R n ) and $${\mathcal {H}}^{t,p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO t , p ( R n ) over the Hardy space for Fourier integral operators $${\mathcal {H}}^{p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO p ( R n ) . Our main result is that for all $$r>0$$ r > 0 , $$m=0$$ m = 0 and $$\delta =1/2$$ δ = 1 / 2 , there exists an interval of p around 2 such that a(x, D) acts boundedly on $${\mathcal {H}}^{p}_{FIO}({{\mathbb {R}}^{n}})$$ H FIO p ( R n ) .

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10

"Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer." V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, no. 89 (2019). http://dx.doi.org/10.26565//2221-5646-2019-89-03.

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A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

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11

"Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer." V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, no. 89 (2019). http://dx.doi.org/10.26565/2221-5646-2019-89-03.

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A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

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12

Gorokhovsky, Alexander, Niek de Kleijn, and Ryszard Nest. "EQUIVARIANT ALGEBRAIC INDEX THEOREM." Journal of the Institute of Mathematics of Jussieu, August 27, 2019, 1–27. http://dx.doi.org/10.1017/s1474748019000380.

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We prove a $\unicode[STIX]{x1D6E4}$ -equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.

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13

Cordero, Elena. "On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces." Journal of Pseudo-Differential Operators and Applications 12, no. 1 (February 11, 2021). http://dx.doi.org/10.1007/s11868-021-00394-y.

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AbstractIn this note we consider the nonlinear heat equation associated to the fractional Hermite operator $$H^\beta =(-\Delta +|x|^2)^\beta $$ H β = ( - Δ + | x | 2 ) β , $$0<\beta \le 1$$ 0 < β ≤ 1 . We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class $$S^m_{0,0}$$ S 0 , 0 m , $$m\in \mathbb {R}$$ m ∈ R .

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14

Ding, Wei, and Guozhen Lu. "Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators." Forum Mathematicum, October 26, 2022. http://dx.doi.org/10.1515/forum-2022-0192.

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Abstract To study the boundedness of bi-parameter singular integral operators of non-convolution type in the Journé class, Fefferman discovered a boundedness criterion on bi-parameter Hardy spaces H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by considering the action of the operators on rectangle atoms. More recently, the theory of multiparameter local Hardy spaces has been developed by the authors. In this paper, we establish this type of boundedness criterion on weighted bi-parameter local Hardy spaces h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} . In comparison with the unweighted case, the uniform boundedness of rectangle atoms on weighted local bi-parameter Hardy spaces, which is crucial to establish the atomic decomposition on bi-parameter weighted local Hardy spaces, is considerably more involved. As an application, we establish the boundedness of bi-parameter pseudodifferential operators, including h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to L ω p ⁢ ( ℝ n 1 + n 2 ) {L_{\omega}^{p}(\mathbb{R}^{n_{1}+n_{2}})} and h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for all 0 < p ≤ 1 {0<p\leq 1} , which sharpens our earlier result even in the unweighted case requiring max ⁡ { n 1 n 1 + 1 , n 2 n 2 + 1 } < p ≤ 1 . \max\Bigl{\{}\frac{n_{1}}{n_{1}+1},\frac{n_{2}}{n_{2}+1}\Bigr{\}}<p\leq 1.

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15

Spence, E. A. "A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation." Advances in Computational Mathematics 49, no. 2 (April 2023). http://dx.doi.org/10.1007/s10444-023-10025-3.

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AbstractIn d dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ ≲ k requires $\sim k^{d}$ ∼ k d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k) suffers from the pollution effect if, as $k\rightarrow \infty $ k → ∞ , the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, the hp-FEM (where accuracy is increased by decreasing the meshwidth h and increasing the polynomial degree p) does not suffer from the pollution effect. The heart of the proof of this result is a PDE result splitting the solution of the Helmholtz equation into “high” and “low” frequency components. This result for the constant-coefficient Helmholtz equation in full space (i.e. in $\mathbb {R}^{d}$ ℝ d ) was originally proved in Melenk and Sauter (Math. Comp79(272), 1871–1914, 2010). In this paper, we prove this result using only integration by parts and elementary properties of the Fourier transform. The proof in this paper is motivated by the recent proof in Lafontaine et al. (Comp. Math. Appl.113, 59–69, 2022) of this splitting for the variable-coefficient Helmholtz equation in full space use the more-sophisticated tools of semiclassical pseudodifferential operators.

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