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Register a new userUnimodular multipliers on $alpha$-modulation spaces: A revisit with new method under weaker conditions
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By a new method derived from Nicola--Primo--Tabacco[24], we study the boundedness on $alpha$-modulation spaces of unimodular multipliers with symbol $e^{imu(xi)}$. Comparing with the previous results, the boundedness result is established for a larger family of unimodular multipliers under weaker assumptions.
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We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{imu(xi)}$, for some real-valued functions $mu(xi)$ whose prototype is $|xi|^{beta}$ with $betain (0,2]$. Under some suitable assumptions on $mu$, we give the characterization of $W^{p,q}_srightarrow W^{p,q}$ boundedness of $e^{imu(D)}$, for arbitrary pairs of $0< p,qleq infty$. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case $mu(xi)=|xi|^{beta}$ with $1<beta<2$.
A new characterization of CMO(R^n) is established by the local mean oscillation. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.
In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc. In addition, we consider functions on the polydisc whose restrictions to lower dimensional polydiscs lie in the corresponding Dirichet type spaces. We see that such functions need not be in the Dirichlet type space of the whole polydisc. Similar observations are made regarding multipliers.
The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:ell_{p_{1}}^{n}times cdotstimesell_{p_{m}}^{n}longrightarrowmathbb{R}$ (or $mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $left{ p_{1},...,p_{m}right} subsetlbrack2,infty]$ and when $left{ p_{1},...,p_{m}right} subsetlbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $left{ p_{1},...,p_{m}right} $ intercepts both $[2,infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
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