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Naohito Tomita

Publications and source records attributed to Naohito Tomita.

31 records · Page 2Linked to original sources
arXiv2018

Bilinear pseudo-differential operators with exotic symbols, II

The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear Hörmander class $BS^m_{ρ,ρ}$, $0 \le ρ<1$, of critical order $m$, where $H^p$ is the Hardy space. This combined with the previous results of the same authors establishes the sharp boundedness from $H^p \times H^q$ to $L^r$, $1/p+1/q=1/r$, of those operators in the full range $0< p, q \le \infty$, where $L^r$ is replaced by $BMO$ if $r=\infty$.

math.CA↗
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 $q\geq4/3$.

math.FA↗
arXiv2015

Multilinear Fourier Multipliers with Minimal Sobolev Regularity, II

We provide characterizations for boundedness of multilinear Fourier operators on Hardy-Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0<q\le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1<q\le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón and Torchinsky [http://www.sciencedirect.com/science/article/pii/S0001870877800169] and of the bilinear results of Miyachi and Tomita [http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4]. The extension to general $m$ is significantly more complicated both technically and combinatorially, the optimal Sobolev space smoothness required of the symbol depends on the Hardy-Lebesgue exponents and is constant on various convex simplices formed by configurations of $m2^{m-1} +1$ points in $[0,\infty)^m$.

math.AP↗
arXiv2008

On the Hörmander multiplier theorem and modulation spaces

It is known that the Sobolev space $L^2_s$ with $s>n/2$ appeared in the Hörmander multiplier theorem can be replaced by the Besov space $B^{2,1}_{n/2}$. On the other hand, the Besov space $B_{n/2}^{2,1}$ is continuously embedded in the modulation space $M^{2,1}_0$. In this paper, we consider the problem whether we can replace $B_{n/2}^{2,1}$ by $M^{2,1}_0$.

math.FA↗
arXiv2008

Changes of variables in modulation and Wiener amalgam spaces

In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier Lebesgue spaces. Finally, counterparts of these results are discussed for spaces on the torus as well as for weighted spaces.

math.FA↗
arXiv2007

Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on $M^{p,q}$ whose symbols are of the class $S_{1,δ}^0$ with $0<δ<1$.

math.FA↗
arXiv2006

The dilation property of modulation spaces and their inclusion relation with Besov spaces

We consider the dilation property of the modulation spaces $M^{p,q}$. Let $D_λ:f(t)\mapsto f(λt)$ be the dilation operator, and we consider the behavior of the operator norm $\|D_λ\|_{M^{p,q}\to M^{p,q}}$ with respect to $λ$. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov space, which was proved by Toft.

math.FA↗