We consider the initial value problems (IVPs) associated to the extended nonlinearSchr & ouml;dinger (e-NLS) equation partial derivative(t)v + i alpha partial derivative(2)(x)v - partial derivative(3)(x)v + i beta...
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We consider the initial value problems (IVPs) associated to the extended nonlinearSchr & ouml;dinger (e-NLS) equation partial derivative(t)v + i alpha partial derivative(2)(x)v - partial derivative(3)(x)v + i beta|v|(2)v = 0, x, t is an element of R and the higher order nonlinear Schr & ouml;dinger (h-NLS) equation partial derivative(t)u - i alpha partial derivative(2)(x)u + partial derivative(3)(x)u - i beta|u|(2)u + gamma|u|(2)partial derivative(x)u + delta partial derivative(x)(|u|(2))u = 0, x, t is an element of R, for given data in the modulation space M-s(2,p)(R). We derive a trilinear estimate for functions with negative Sobolev regularity and use it in the contraction mapping principle to prove that the IVPs associated to the e-NLS equation and the h-NLS equation are locally well-posed for s > -1/4 and s >= 14 respectively.
We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations...
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We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the submanifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces.
We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Ba...
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We study the mapping properties of metaplectic operators S is an element of Mp(2,Double-struck capital R) on modulation spaces of the type M (Double-struck capital R). Our main result is a full characterization of the...
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We study the mapping properties of metaplectic operators S is an element of Mp(2,Double-struck capital R) on modulation spaces of the type M (Double-struck capital R). Our main result is a full characterization of the pairs (,M,(Double-struck capital R)) for which the operator : M,(Double-struck capital R) -> M,(Double-struck capital R) is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of : M,(Double-struck capital R)-> M,(Double-struck capital R) transfers to : M, (Double-struck capital R) -> M, (Double-struck capital R).
The spherical average A(1)(f) and its iteration (A(1))(N) are important operators in harmonic analysis and probability theory. Also Delta(A(1))(N) is used to study the K functional in approximation theory, where Delta...
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The spherical average A(1)(f) and its iteration (A(1))(N) are important operators in harmonic analysis and probability theory. Also Delta(A(1))(N) is used to study the K functional in approximation theory, where Delta is the Laplace operator. In this paper, we obtain the sufficient and necessary conditions to ensure the boundedness of Delta(A(1))(N) from the modulation space M-p1,q1(s1) to the modulation space M-p2,q2(s2) for 1 <= p(1), p(2), q(1), q(2) <= infinity and s(1), s(2) is an element of R. (C) 2020 Elsevier Ltd. All rights reserved.
We give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates for such operators on modulation spaces, and a...
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We give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates for such operators on modulation spaces, and apply the results to prove Strichartz estimates for such propagators when acting on Pilipovic and modulation spaces. Especially we extend some results by Balhara, Cordero, Nicola, Rodino and Thangavelu. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.& COPY;2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
In this paper, we consider the operator e i t ϕ ( h ( D ) ) where h defined on R n is a C ∞ ( R n ∖ { 0 } ) positive homogeneous function with degree λ > 0 and ϕ : R → R is a smooth function satisfying the follo...
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In this paper, we consider the operator e i t ϕ ( h ( D ) ) where h defined on R n is a C ∞ ( R n ∖ { 0 } ) positive homogeneous function with degree λ > 0 and ϕ : R → R is a smooth function satisfying the following. (A 1 ) There exists a constant m 1 > 0 such that for all μ ∈ N 0 : = N ∪ { 0 } | ϕ ( μ ) ( r ) | ≲ r m 1 − μ , r ≥ 1 . (A 2 ) There exists a constant m 2 > 0 such that for all μ ∈ N 0 | ϕ ( μ ) ( r ) | ≲ r m 2 − μ , 0 < r < 1 . We prove the boundedness of the operator e i t ϕ ( h ( D ) ) on the modulation spaces and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solutions for the Cauchy problems for the generalized wave, Klein–Gordon and Schrödinger equations.
In the present paper we define weighted modulation spaces on a LCA group with respect to a window function drawn from a suitable Banach space of test functions and prove a theorem to establish uncertainty principle fo...
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In the present paper we define weighted modulation spaces on a LCA group with respect to a window function drawn from a suitable Banach space of test functions and prove a theorem to establish uncertainty principle for these modulation spaces. Also, using the concept of Zak transform, we generalize an earlier result of Heil (1990) on the Balian–Low theorem for the Wiener amalgam space . Our theorems include the corresponding results on Euclidean spaces as particular cases.
We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Ba...
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We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.
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