Akihiko Miyachi

Estimates for unimodular Fourier multipliers on modulation spaces

We study the action on modulation spaces of Fourier multipliers with symbols e^iμ(ξ), for real-valued functions μ having unbounded second derivatives. In a simplified form our result reads as follows: if μ satisfies the usual symbol estimates of order α ≥ 2, or if μ is a positively homogeneous function of degree α, then the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces M^{p,q}_s and M^{p,q}, for all 1 ≤ p, q ≤ ∞ and s ≥ (α − 2)n|1/p − 1/2|. Here s represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function μ whose Hessian matrix is non-degenerate at some point

Minimal smoothness conditions for bilinear Fourier multipliers.

The problem of finding the differentiability conditions for bilinear Fourier multipliers that are as small as possible to ensure the boundedness of the corresponding operators from products of Hardy spaces H1 ×Hp2 to L, 1/p1 + 1/p2 = 1/p, is considered. The minimal conditions in terms of the product type Sobolev norms are given for the whole range 0 < p1, p2 ≤ ∞.

On Multilinear Fourier Multipliers of Limited Smoothness

In this paper, we prove the L2-boundedness of multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we can extend Calderón and Torchinsky’s result in the linear theory to the multilinear case. The sharpness of our results is also discussed.

A transplantation theorem for Jacobi series in weighted Hardy spaces

Muckenhoupts transplantation theorem for Jacobi series in weighted Lp spaces is extended to weighted Hardy spaces.

Weighted Hardy Spaces on a Domain

Weighted Hardy spaces were first studied by Gacia-Cuerva [GC] for the case of weight functions in the A ∞ class. Generalization to the case of weight functions satisfying only the doubling condition was given by Stromberg and Torchinsky [ST]. Both of these works are concerned with the weighted Hardy spaces on ℝ n . If we try to generalize the arguments of [GC] and [ST] so that we can deal with weighted Hardy spaces on a domain of ℝ n , we meet with considerable technical difficulties since in those papers the convolution and the Fourier transform on ℝ n are used as basic tools and the use of testing functions in the Schwartz class S seems to be unavoidable.

Hardy Space Estimate for the Product of Singular Integrals

H p estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given.

Weighted Hardy Spaces on a Domain and its Application

We give basic properties of the Hardy spaces on an open subset \(\Omega \subset \mathbb{R}^n \)with respect to a doubling measure on Ω, and apply it to give the transplantation theorem for Jacobi series in Hardy spaces on a finite interval of \(\mathbb{R}\).

Extension theorems for real variable Hardy and Hardy-Sobolev spaces

For 0 < p ≤1, let Hp(Rn) denote the real variable Hardy space as given in the paper of C. Fefferrnan and E. M. Stein [2; Section 11]. For 0< p≤ 1 and k ∈ N (= the set of positive integers), we define \( H_{p}^{k}({R^{n}}) \) as the set of the distributions f on Rn for which the derivatives ∂αf belong to Hp(Rn) for |α| = k. (Contrary to the custom, we do not require ∂αf ∈ Hp (Rn) for |α| k.) We shall consider \( H_{p}^{k}({R^{n}}) \) only for k n/p-n.

A Factorization Theorem for the Real Hardy Spaces

Abstract The purpose of this article is to give a generalization of the factorization theorem for the real Hardy spaces and its application to the majorant property of the real Hardy spaces.

Bilinear pseudo-differential operators with exotic symbols, II

The boundedness from