arXiv subjects
Publications and source records attributed to Shohei Nakamura.
In this paper, motivated by recent important works due to Frank-Lewin-Lieb-Seiringer \cite{FLLS} and Frank-Sabin \cite{frank-sabin-1}, we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates in certain sense. An application of the inequality shows the well-posedness to the periodic Hartree equation describing the infinitely many quantum particles with the power type interaction.
The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the form \[ \bigg\|\sum_jλ_j|e^{itΔ}f_j|^2\bigg\|_{L^p_tL^q_x}\lesssim\|λ\|_{\ell^α} \] for orthonormal systems $(f_j)_j$ of initial data in $L^2$, firstly in work of Frank--Lewin--Lieb--Seiringer and later by Frank--Sabin. The primary objective is identifying the largest possible $α$ as a function of $p$ and $q$, and in contrast to the classical case, for such estimates the critical case turns out to be $(p,q) = (\frac{d+1}{d},\frac{d+1}{d-1})$. We consider the case of orthonormal systems $(f_j)_j$ in the homogeneous Sobolev spaces $\dot{H}^s$ for $s \in (0,\frac{d}{2})$ and we establish the sharp value of $α$ as a function of $p$, $q$ and $s$, except possibly an endpoint in certain cases, at which we establish some weak-type estimates. Furthermore, at the critical case $(p,q) = (\frac{d+1}{d-2s},\frac{d(d+1)}{(d-1)(d-2s)})$ for general $s$, we show the veracity of the desired estimates when $α= p$ if we consider frequency localised estimates, and the failure of the (non-localised) estimates when $α= p$; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.
In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $σ(ξ_1,\dots , ξ_m)$ and their derivatives vanish on the hyperplane $ξ_1+\cdots+ξ_m=0$.
We obtain boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman-Meyer class, sums of products of linear Calderon-Zygmund operators and combinations of these two types.
A necessary condition and a sufficient condition for one weight norm inequalities on Morrey spaces to hold are given for the fractional maximal operator and the fractional integral operator. We clarify the difference between the behavior of the fractional maximal operator and the one of the fractional integral operator which is originated from the structure of Morrey spaces. Both the necessary condition and the sufficient condition are also verified for the power weights.
In this paper, we give necessary conditions and sufficient conditions respectively for the boundedness of the singular integral operator on the weighted Morrey spaces. We observe the phenomenon unique to the case of Morrey spaces; the $A_q$-theory by Muckenhoupt and Wheeden does not suffice. We also discuss the boundedness of the commutators. The difference between the maximal operator and the singular integral operators will be clarified. Further, the property of the sharp maximal operator is investigated.
The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.