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Tomoya Kato

Publications and source records attributed to Tomoya Kato.

18 recordsLinked to original sources
arXiv2024

On some bilinear Fourier multipliers with oscillating factors, II

For $s > 0$, $s \neq 1$, bilinear Fourier multipliers of the form $e^{i (|\xi|^s + |\eta|^s+ |\xi + \eta|^s)} \sigma (\xi, \eta)$ are considered, where $\sigma(\xi, \eta)$ belongs to the H\"ormander class $S^{m}_{1, 0}(\mathbb{R}^{2n})$. A criterion for $m$ to ensure the $L^{\infty}\times L^{\infty} \to L^\infty$, $L^{1} \times L^{\infty} \to L^{1}$, and $L^{\infty} \times L^{1} \to L^{1}$ boundedness of the corresponding bilinear operators is given.

math.CA↗
arXiv2024

On some bilinear Fourier multipliers with oscillating factors, I

Bilinear Fourier multipliers of the form $e^{i (|\xi| + |\eta|+ |\xi + \eta|)} \sigma (\xi, \eta)$ are considered. It is proved that if $\sigma (\xi, \eta)$ is in the H\"ormander class $S^{m}_{1,0} (\mathbb{R}^{2n})$ with $m=-(n+1)/2$ then the corresponding bilinear operator is bounded in $L^{\infty} \times L^{\infty} \to bmo$, $h^{1} \times L^{\infty} \to L^{1}$, and $L^{\infty} \times h^{1} \to L^{1}$. This improves a result given by Rodr\'iguez-L\'opez, Rule and Staubach.

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arXiv2024

Bilinear oscillatory Fourier multipliers

For bilinear Fourier multipliers that contain some oscillatory factors, boundedness of the operators between Lebesgue spaces is given including endpoint cases. Sharpness of the result is also considered.

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arXiv2023

Estimates for a certain bilinear Fourier integral operator

In this paper, we consider the boundedness from $H^{1} \times L^{\infty}$ to $L^{1}$ of bilinear Fourier integral operators with non-degenerate phase functions and amplitudes in $BS_{1,0}^{-n/2}$. Our result gives an improvement of Rodríguez-López, Rule, and Staubach's result.

math.CA↗
arXiv2023

Estimates for some bilinear wave operators

We consider some bilinear Fourier multiplier operators and give a bilinear version of Seeger, Sogge, and Stein's result for Fourier integral operators. Our results improve, for the case of Fourier multiplier operators, Rodríguez-López, Rule, and Staubach's result for bilinear Fourier integral operators. The sharpness of the results is also considered.

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arXiv2021

Notes on bilinear lattice bump Fourier multipliers

We consider the bilinear Fourier multiplier operator with the multiplier written as a linear combination of a fixed bump function. For those operators we prove two transference theorems, one in amalgam spaces and the other in Wiener amalgam spaces.

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arXiv2021

Boundedness of multilinear pseudo-differential operators with symbols in the Hörmander class $S_{0,0}$

The multilinear pseudo-differential operators with symbols in the multilinear Hörmander class $S_{0,0}$ are considered. A complete identification of the cases where those operators define bounded operators between local Hardy spaces is given. Some results for the boundedness between Wiener amalgam spaces are also given. These are extensions and improvements of the results known in the bilinear case.

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arXiv2020

Notes on lattice bump Fourier multiplier operators on $L^2 \times L^2$

Given a smooth bump function, we consider the multiplier formed by taking the linear combination of the translations of the bump function and the corresponding bilinear Fourier multiplier operator. Under certain condition on the bump function, we give a complete characterization of the coefficients of the linear combination for which the corresponding bilinear operator defines a bounded operator from $L^2\times L^2$ to $L^2$-based amalgam spaces.

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arXiv2019

Boundedness of multilinear pseudo-differential operators of $S_{0,0}$-type in $L^2$-based amalgam spaces

We consider the multilinear pseudo-differential operators with symbols in a generalized $S_{0,0}$-type class and prove the boundedness of the operators from $(L^2,\ell^{q_1}) \times \dots \times (L^2,\ell^{q_N})$ to $(L^2,\ell^{r})$, where $(L^2, \ell^{q})$ denotes the $L^2$-based amalgam space. This extends the previous result by the same authors, which treated the bilinear pseudo-differential operators and gave the $L^2 \times L^2 $ to $(L^2, \ell^{1})$ boundedness.

math.CA↗
arXiv2019

Pseudodifferential operators with symbols in the Hörmander class $S^0_{α,α}$ on $α$-modulation spaces

In this paper, we study the boundedness of pseudodifferential operators with symbols in the Hörmander class $S^0_{ρ,ρ}$ on $α$-modulation spaces $M_{p,q}^{s,α}$, and consider the relation between $α$ and $ρ$. In particular, we show that pseudodifferential operators with symbols in $S^0_{α,α}$ are bounded on all $α$-modulation spaces $M^{s,α}_{p,q}$, for arbitrary $s\in\mathbb{R}$ and for the whole range of exponents $0 < p,q \leq \infty$.

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arXiv2019

Boundedness of bilinear pseudo-differential operators of $S_{0,0}$-type on $L^2 \times L^2$

We extend the known result that the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $BS^{-n/2}_{0,0}(\mathbb{R}^n)$ are bounded from $L^2 \times L^2$ to $h^1$. We show that those operators are also bounded from $L^2 \times L^2$ to $L^r $ for every $1< r \le 2$. Moreover we give similar results for symbol classes wider than $BS^{-n/2}_{0,0}(\mathbb{R}^n)$. We also give results for symbols of limited smoothness.

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arXiv2018

Nonlinear operations on a class of modulation spaces

We discuss when the nonlinear operation $f\mapsto F(f)$ maps the modulation space $M^{p,q}_s(\mathbb{R}^n)$ ($1 \leq p,q \leq \infty$) to the same space again. It is known that $M^{p,q}_s(\mathbb{R}^n)$ is a multiplication algebra when $s > n-n/q$, hence it is true for this space if $F$ is entire. We claim that it is still true for non-analytic $F$ when $q\geq4/3$.

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