Fourier Multiplier Theorem Research Articles

Mikhlin-Type Hp Multiplier Theorem on the Heisenberg Group

The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type Hp multiplier theorem on the Heisenberg group. If an operator-valued function M(λ) satisfies certain conditions, the right-multiplier operator TM is bounded on the Hardy space Hp(Hn), which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems.

Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

AbstractWe prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from $$H^\alpha ({\mathbb {R}},U)$$ H α ( R , U ) to $$H^\beta ({\mathbb {R}},U)$$ H β ( R , U ) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Titchmarsh and Boas-type theorems related to (κ,n)-Fourier transform

Abstract The aim of this paper is to prove a generalization of Titchmarsh’s theorems for the generalized Fourier transform called ( κ , n {\kappa,n} )-Fourier transform, where n is a positive integer and κ is a constant coming from Dunkl theory. As an application, we derive a ( κ , n ) {(\kappa,n)} -Fourier multiplier theorem for L 2 {L^{2}} Lipschitz spaces. Moreover, we give necessary conditions to ensure that f belongs to either one of the generalized Lipschitz classes of order m. This allows us to establish the analogue of the Boas-type result for ℱ κ , n {\mathcal{F}_{\kappa,n}} .

Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ( $P_4$ ): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$ , where $A$ , $B$ , $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$ , $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$ ) into $X$ , $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$ . We completely characterize the well-posedness of ( $P_4$ ) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

Well-posedness of fractional differential equations with finite delay in complex Banach spaces

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems.

Periodic solutions of second‐order degenerate differential equations with infinite delay in Banach spaces

AbstractWe consider the well‐posedness of the second‐order degenerate differential equations with infinite delay on [0, 2π] in Lebesgue–Bochner spaces and periodic Besov spaces , where , and M are closed linear operators in a Banach space X satisfying and the kernels . Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in and . These results are applied to examine some concrete examples.

Half-Space Model Problem for Navier-Lamé Equations with Surface Tension

Recently, we have seen the phenomena in use of partial differential equations (PDEs) especially in fluid dynamic area. The classical approach of the analysis of PDEs were dominated in early nineteenth century. As we know that for PDEs the fundamental theoretical question is whether the model problem consists of equation and its associated side condition is well-posed. There are many ways to investigate that the model problems are well-posed. Because of that reason, in this paper we consider the <img src=image/13426244_01.gif>-boundedness of the solution operator families for Navier-Lamé equation by taking into account the surface tension in a bounded domain of <img src=image/13426244_02.gif>-dimensional Euclidean space (<img src=image/13426244_02.gif>≥ 2) as one way to study the well-posedess. We investigate the <img src=image/13426244_01.gif>-boundedness in half-space domain case. The <img src=image/13426244_01.gif>-boundedness implies not only the generation of analytic semigroup but also the maximal <img src=image/13426244_03.gif> regularity for the initial boundary value problem by using Weis's operator valued Fourier multiplier theorem for time dependent problem. It was known that the maximal <img src=image/13426244_03.gif> regularity class is the powerful tool to prove the well-posesness of the model problem. This result can be used for further research for example to analyze the boundedness of the solution operators of the model problem in bent-half space or general domain case.

Maximal Lp-Lq regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation

This paper is concerned with the generalized resolvent estimate and the maximal Lp-Lq regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of R-bounded solution operators for the model problems and the maximal Lp-Lq regularity for the system. The key step is to prove the maximal Lp-Lq regularity theorem for the linearized heat equation with the help of the R-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.

On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2&lt;p&lt;∞ and N&lt;q&lt;∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

Regularity criterion for weak solutions to the 3D Navier–Stokes equations via two vorticity components in [formula omitted

In this paper, we provide a regularity criterion for 3D Navier–Stokes equations in terms of two vorticity components, which extends a recent result established by Guo, Kučera and Skalák (2018). More precisely, we prove that a unique local strong solution u to 3D Navier–Stokes equations does not blow up at time T provided only two components of vorticity belongs to L2(0,T;BMO−1). We also prove that u can be smoothly extended beyond T, if the horizontal gradient of horizontal components of velocity u belong to L2(0,T;BMO−1). The proof relies on the technique of the Bony decomposition and some Fourier multiplier theorems.

Solutions of second-order degenerate equations with infinite delay in Banach spaces

We consider the well-posedness of the second-order degenerate differential equations with infinite delay (P2): (Mu′)′(t)+(Lu)′(t)=Au(t)+ ∫ −∞ta(t−s)Bu(s)ds+f(t), (0≤t≤2π) with periodic boundary conditions u(0)=u(2π), (Mu′)(0)=(Mu′)(2π), in Lebesgue–Bochner spaces Lp(𝕋;X) and periodic Besov spaces Bp,qs(𝕋;X), where A, B, L and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(L), D(A)∩D(B)≠{0} and a∈L1(ℝ+). We completely characterize the well-posedness of (P2) in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.

Solutions of third order degenerate equations with infinite delay in Banach spaces

We study the well-posedness of the third order degenerate differential equations with infinite delay$$(P_3): (Mu)'''(t) + (Lu)''(t) + (Bu)'(t)= Au(t) + \int _{-\infty }^t a(t-s)Au(s)ds + f(t){\text{ on }}[0, 2\pi ]$$in Lebesgue–Bochner spaces $$L^p(\mathbb{T};\; X)$$ and periodic Besov spaces $$B_{p,\,q}^s(\mathbb{T};\; X)$$, where A, B, L and M are closed linear operators on a Banach space X satisfying $$D(A)\subset D(B)\cap D(L)\cap D(M)$$ and $$ a\in L^1(\mathbb{R}_+)$$. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for $$(P_3)$$ to be $$L^p$$-well-posed(or $$B_{p,q}^s$$-well-posed). Concrete examples are also given to support our main abstract results.

Lp-Lq multipliers on locally compact groups

In this paper we discuss the Lp-Lq boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups G for the range 1<p≤2≤q<∞. As a consequence of the established Fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the Lp-Lq norms of the heat kernels for general positive unbounded invariant operators on G. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group, as well as for higher order operators. We show that our results imply the known results for Lp-Lq multipliers such as Hörmander's Fourier multiplier theorem on Rn or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on advancing the analysis in the group von Neumann algebra and its application to the derivation of the desired multiplier theorems.

Well-Posedness of Fractional Degenerate Differential Equations in Banach Spaces

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 &lt; β &lt; α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

Strong solutions of a neutral type equation with finite delay

This paper is concerned to study the existence and uniqueness of solution of neutral type differential equations, by using the maximal regularity property of the first-order abstract Cauchy problem with finite delay on Lebesgue spaces defined at the line. The main tools that we use to achieve our goals are an operator-valued version of Miklhin’s Fourier multiplier theorem, weighted Sobolev spaces on the real line and fixed point arguments.

Titchmarsh theorems for Fourier transforms of H\xf6lder\u2013Lipschitz functions on compact homogeneous manifolds

In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder–Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for L^2-Hölder–Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini–Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

AbstractWe study the well‐posedness of the fractional differential equations with infinite delay urn:x-wiley:0025584X:media:mana201800104:mana201800104-math-0001on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying , and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.

Operator-Valued Triebel–Lizorkin Spaces

This paper is devoted to the study of operator-valued Triebel–Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel–Lizorkin spaces on $$\mathbb {R}^d$$ . As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood–Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel–Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

AbstractWe study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.