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Naohito Tomita

Publications and source records attributed to Naohito Tomita.

At least 19 recordsLinked to original sources
arXiv2026

Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators

We study optimal multiple weight assumptions in the weighted theory of multilinear Fourier multipliers and multilinear pseudo-differential operators. For multilinear Fourier multipliers, we revisit the weighted Hörmander-type theorem of Li and Sun, as a multilinear version of Kurtz and Wheeden, and show that their multiple weight condition is sharp. This provides the sharp necessary condition in the multilinear setting and simultaneously improves the classical linear necessity established by Kurtz and Wheeden. In the pseudo-differential setting, we consider recent weighted estimates of the authors for symbols in the multilinear Hörmander class and prove that their multiple weight hypothesis is also best possible. As a corollary, we can obtain the optimality of sharp maximal function estimates for multilinear pseudo-differential operators in the papers of the authors which originated from the results of Chanillo and Torchinsky.

math.CA↗
arXiv2024

On some bilinear Fourier multipliers with oscillating factors, II

For $s > 0$, $s \neq 1$, bilinear Fourier multipliers of the form $e^{i (|ξ|^s + |η|^s+ |ξ+ η|^s)} σ(ξ, η)$ are considered, where $σ(ξ, η)$ belongs to the Hörmander 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 $e^{i (|ξ| + |η|+ |ξ+ η|)} σ(ξ, η)$ are considered. It is proved that if $σ(ξ, η)$ is in the Hörmander 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 Rodríguez-López, Rule and Staubach.

math.CA↗
arXiv2024

Sharp maximal function estimates for linear and multilinear pseudo-differential operators

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González, which provides more elaborate and natural weighted estimates in the multilinear setting.

math.AP↗
arXiv2024

Sharp Maximal function estimates for Multilinear pseudo-differential operators of type (0,0)

In this paper, we study sharp maximal function estimates for multilinear pseudo-differential operators. Our target is operators of type (0, 0) for which a differentiation does not make any decay of the associated symbol. Analogous results for operators of type (ρ, ρ), 0 < ρ< 1, appeared in an earlier work of the authors, but a different approach is given for ρ= 0

math.AP↗
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.

math.CA↗
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.

math.CA↗
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.

math.CA↗
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.

math.CA↗
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.

math.CA↗
arXiv2019

Limited smoothness conditions with mixed norms for bilinear Fourier multipliers

In this paper, the $L^2 \times L^{\infty} \to L^2$ and $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier multiplier operators is discussed under weak smoothness conditions on multipliers. As an application, we prove the $L^2 \times BMO \to L^2$ and $L^2 \times L^2 \to H^1$ boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.

math.CA↗
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 $0 < p,q \leq \infty$.

math.FA↗
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.

math.CA↗
arXiv2018

Bilinear pseudo-differential operators with exotic symbols

The boundedness from $L^p \times L^q$ to $L^r$, $1<p,q \le \infty$, $0<1/p+1/q=1/r \le 1$, of bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $BS^m_{ρ,ρ}$, $0 \le ρ<1$, is proved for the critical order $m$. Related results for the cases $p=1$, $q=1$ or $r=\infty$ are also obtained.

math.CA↗