Composition of some singular Fourier integral operators and estimates for restricted $X$-ray transforms

Allan Greenleaf; Gunther Uhlmann

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Displaying similar documents to “Composition of some singular Fourier integral operators and estimates for restricted $X$ -ray transforms”

Propagation of singularities for operators with multiple involutive characteristics

Johannes Sjöstrand (1976)

Annales de l'institut Fourier

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Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty}$ manifold $X$ . Let $p_{m}$ be the principal symbol and assume that $Σ = p_{m}^{- 1} (0)$ is an involutive $C^{\infty}$ sub-manifold of $T^{*} X ∖ 0$ , satisfying a certain transversality condition. We assume that $p_{m}$ vanishes exactly to order $M$ on $Σ$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M = 2$ ). Suppose that a Levi condition is valid for the lower order symbols. If $u \in 𝒟^{'} (X)$ , $P u \in C^{\infty} (X)$ , then...

Modulation space estimates for multilinear pseudodifferential operators

Árpád Bényi, Kasso A. Okoudjou (2006)

Studia Mathematica

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We prove that for symbols in the modulation spaces $ℳ^{p, q}$ , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

On oscillatory integral operators with folding canonical relations

Allan Greenleaf, Andreas Seeger (1999)

Studia Mathematica

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Sharp $L^{p}$ estimates are proven for oscillatory integrals with phase functions Φ(x,y), (x,y) ∈ X × Y, under the assumption that the canonical relation $C_{Φ}$ projects to T*X and T*Y with fold singularities.

Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

Michael Hitrik, Karel Pravda-Starov (2013)

Annales de l’institut Fourier

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For a class of non-selfadjoint $h$ –pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description...

Unique continuation for the solutions of the laplacian plus a drift

Alberto Ruiz, Luis Vega (1991)

Annales de l'institut Fourier

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We prove unique continuation for solutions of the inequality $| Δ u (x) | \leq V (x) | \nabla u (x) |$ , $x \in Ω$ a connected set contained in $R^{n}$ and $V$ is in the Morrey spaces $F^{α, p}$ , with $p \geq (n - 2) / 2 (1 - α)$ and $α < 1$ . These spaces include $L^{q}$ for $q \geq (3 n - 2) / 2$ (see [H], [BKRS]). If $p = (n - 2) / 2 (1 - α)$ , the extra assumption of $V$ being small enough is needed.

Recovering the total singularity of a conormal potential from backscattering data

Mark S. Joshi (1998)

Annales de l'institut Fourier

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The problem of recovering the singularities of a potential from backscattering data is studied. Let $Ω$ be a smooth precompact domain in $ℝ^{n}$ which is convex (or normally accessible). Suppose $V_{i} = v + w_{i}$ with $v \in C_{c}^{\infty} (ℝ^{n})$ and $w_{i}$ conormal to the boundary of $Ω$ and supported inside $\overline{Ω}$ then if the backscattering data of $V_{1}$ and $V_{2}$ are equal up to smoothing, we show that $w_{1} - w_{2}$ is smooth.

$L^{p} (ℝ ⁿ)$ bounds for commutators of convolution operators

Guoen Hu, Qiyu Sun, Xin Wang (2002)

Colloquium Mathematicae

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The $L^{p} (ℝ ⁿ)$ boundedness is established for commutators generated by BMO(ℝⁿ) functions and convolution operators whose kernels satisfy certain Fourier transform estimates. As an application, a new result about the $L^{p} (ℝ ⁿ)$ boundedness is obtained for commutators of homogeneous singular integral operators whose kernels satisfy the Grafakos-Stefanov condition.

Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)

Maurice De Gosson (1982)

Annales de l'institut Fourier

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This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted $\partial W F_{ω} (u)$ for distributions $u \in D^{'} (R_{+}^{n})$ , regular in the normal variable $x_{n}$ (thus, $\partial W F_{ω} (u) = \emptyset$ means that $u \in \cap_{s + t = 1 / 2} H^{s + t}$ near the boundary), and it is shown that $\partial W F_{ω - m} [P (u^{0})_{x_{n} > 0}]$ is a subset of $\partial W F (u)$ if $P$ has degree $m$ and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions...

Uncertainty principles for integral operators

Saifallah Ghobber, Philippe Jaming (2014)

Studia Mathematica

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The aim of this paper is to prove new uncertainty principles for integral operators with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function $f \in L ² (ℝ^{d}, μ)$ is highly localized near a single point then (f) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function $f \in L ² (ℝ^{d}, μ)$ and...

Generalized Hörmander conditions and weighted endpoint estimates

María Lorente, José María Martell, Carlos Pérez, María Silvina Riveros (2009)

Studia Mathematica

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We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded...

Displaying similar documents to “Composition of some singular Fourier integral operators and estimates for restricted X -ray transforms”

Displaying similar documents to “Composition of some singular Fourier integral operators and estimates for restricted $X$ -ray transforms”