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

By Mitsuru Sugimoto, Naohito Tomita. 2006

170 citations recorded by the source corpus.

Abstract We consider the dilation property of the modulation spaces M p , q . Let D λ : f ( t ) ↦ f ( λ t ) be the dilation operator, and we consider the behavior of the operator norm ‖ D λ ‖ M p , q → 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 spaces, which was proved by Toft [J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal. 207 (2004) 399–429].

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f9dcae561c73b3bfb54d887f4fe2460b07f4be4f

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In Applied Mathematics

Citations in the corpus

The inclusion relation between Sobolev and modulation spaces (2010, unknown): “ere 1/p+1/p′ = 1 = 1/q+1/q′.We remark ν 2(p,q) = −ν 1(p′,q′). 0 1/2 1 1/2 1 1/q 1/p 1/q 1 1/2 0 1/2 1 1/p I 1 I 2 I 3 I ∗ 1 I∗ I 2 ∗ 3 Then the following result is known: Theorem 1.1 (Sugimoto-Tomita [15], Toft [18]). Let 1 ≤ p,q≤ ∞ and s∈ R. Then we have (1) Bp,q s (R n) → Mp,q(Rn) if and only if s”
WIENER RANDOMIZATION ON UNBOUNDED DOMAINS AND AN APPLICATION TO ALMOST SURE WELL-POSEDNESS OF NLS (2014, background): “There are several known embeddings between the Besov, Sobolev, and modulation spaces; see, for example, Okoudjou [40], Toft [51], Sugimoto-Tomita [48], and Kobayashi-Sugimoto [35].”
On the global boundedness of Fourier integral operators (2008, background): “These dilation estimates are sharp, as discussed in [43], see also [5].”
Micro-Local Analysis with Fourier Lebesgue Spaces. Part I (2011, background): “From the construction of these spaces, it turns out that modulation spaces and Besov spaces in some sense are rather similar, and sharp embeddings between these spaces can be found in [1] [28] [32] [33].”
Micro-local analysis in Fourier Lebesgue and modulation spaces: part II (2008, background): “ distributions. From the construction of these spaces, it turns out that modulation spaces and Besov spaces in some sense are rather similar, and sharp embeddings between these spaces can be found in [1,28,32,33]. (See also [9,21] for other embeddings.) Parallel to this development, modulation space”
Local well‐posedness of nonlinear dispersive equations on modulation spaces (2007, background): “There exist several embedding results between Lebesgue, Sobolev, or Besov spaces and modulation spaces, see for example [11], [12], [14]; also [1], [2].”
Weyl product algebras and modulation spaces (2007, background): “In this context we remark that the dilation results in [27] might be useful to obtain related necessary conditions.”
Functions operating on modulation spaces and nonlinear dispersive equations (2014, result): “There are several embedding results between Lebesgue, Sobolev, or Besov spaces and modulation spaces, see for example, [17, 23].”
Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices (2010, background): “The details are studied in the result by Sugimoto and Tomita [17], who have studied the dilation property of modulation spaces.”
On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, unknown): “See, for example, Okoudjou [46], Toft [58], Sugimoto-Tomita [54], and Kobayashi-Sugimoto [40].”
ESTIMATES FOR UNIMODULAR FOURIER MULTIPLIERS ON MODULATION SPACES (2009, background): “2 based on estimates in [24] for the dilation operator.”
Changes of variables in modulation and Wiener amalgam spaces (2008, background): “From the construction of these spaces, it turns out that modulation spaces and Besov spaces in some sense are rather similar (see [32]–[34], [37] for sharp embeddings).”
Sharpness of some properties of Wiener amalgam and modulation spaces (2008, unknown): “This study was carried out in depth in [17] (see also [5, 19]) in the case of modulation spaces M p , q .”