Pseudodifferential operators with rough symbols

Intrinsic pseudodifferential calculi on any compact Lie group

In this paper, we define in an intrinsic way operators on a compact Lie group by means of symbols using the representations of the group. The main purpose is to show that these operators form a symbolic pseudo-differential calculus which coincides or generalises the (local) Hörmander pseudo-differential calculus on the group viewed as a compact manifold.

Smoothing of commutators for a Hörmander class of bilinear pseudodifferential operators

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class BS_{1, 0}^1 and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.

更多关于粗糙和光滑伪微分算子的稀疏界

Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

本文扩展了Beltran和Cladek的方法，为仅在空间变量上可测的粗糙伪微分算子获得点wise稀疏界，并提供避免几何衰减稀疏界的替代证明，同时给出了稀疏形式界的充分条件。

On the Hörmander classes of bilinear pseudodifferential operators

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue…

On the Hörmander classes of bilinear pseudodifferential operators II

Boundedness properties for pseudodifferential operators with symbols in the bilinear Hörmander classes of sufficiently negative order are proved. The results are obtained in the scale of Lebesgue spaces and, in some cases, end-point estimates involving weak-type spaces and BMO are provided as well. From the Lebesgue space estimates, Sobolev ones are then easily obtained using functional calculus and interpolation. In addition, it is shown that, in contrast with the linear case, operators associated with symbols of order zero may fail to be bounded on products of Lebesgue spaces.

Sparse bounds for pseudodifferential operators

We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.

Pseudodifferential operators associated with a semigroup of operators

Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifold, fractals, graphs ...). Boundedness on $L^p$ for pseudodifferential operators of order 0 are proved. Mainly, we focus on symbols belonging to the class $S^0_{1,δ}$ for $δ\in[0,1)$. For the limit class $S^0_{1,1}$, we describe some results by restricting our attention to the case of a sub-Laplacian operator on a Riemannian manifold.

On the Fredholm property of bisingular pseudodifferential operators

For operators belonging either to a class of global bisingular pseudodifferential operators on $R^m \times R^n$ or to a class of bisingular pseudodifferential operators on a product $M \times N$ of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain associated homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to the even larger classes of Toeplitz type operators.

Fourier 积分算子 Hardy 空间上的粗糙伪微分算子

Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

本文研究 Fourier 积分算子 Hardy 空间上具有有限 x 正则性的粗糙符号伪微分算子的映射性质，证明其在相应 Sobolev 空间之间的有界性，并给出 m=0、δ=1/2、r>n-1 时对所有 p∈(1,∞) 有界作用的结果。

A bilinear pseudodifferential calculus

In this paper, we are interested in the construction of a bilinear pseudodifferential calculus. We define some symbolic classes which contains those of Coifman-Meyer. These new classes allow us to consider operators closely related to the bilinear Hilbert transform. We give a description of the action of our bilinear operators on Sobolev spaces. These classes also have a ``nice'' behavior through the transposition and the composition operations that we will present.

伪微分算子的界与 Schatten 类的 Fourier 限制

Bounds on pseudodifferential operators and Fourier restriction for Schatten classes

本文证明了具有紧 Fourier 支撑的 Weyl 伪微分算子属于 Schatten 类当且仅当其符号属于相空间的 $L^p$ 空间，并以此提供了 Luef 和 Samuelsen 关于经典与量子限制估计等价性的清晰证明。

Maximally Modulated Singular Integral Operators and their Applications to Pseudodifferential Operators on Banach Function Spaces

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calderón-Zygmund singular integral operator $T^Φ$ is of weak type $(r,r)$ for all $r\in(1,\infty)$, then $T^Φ$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above r…

Strongly singular bilinear Calderón-Zygmund operators and a class of bilinear pseudodifferential operators

Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón--Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón--Zygmund o…

An FIO calculus for marine seismic imaging, II: Sobolev estimates

We establish sharp $L^2$-Sobolev estimates for classes of pseudodifferential operators with singular symbols whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in $R^2$ with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics.