Operator-valued Fourier Multiplier Theorems Research Articles

Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = -\Delta - |x|^2\partial_t^2 are bounded on L^p(R^n+1). We also establish an analogue of H\"ormander-Mihlin multiplier theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator valued Fourier multiplier theorem due to L. Weis.

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations (P2): (Mu)″(t) + α(Mu)′(t) = Au(t) + Σ−∞ta(t − s)Au(s)ds + f(t), 0 ⩽ t ⩽ 2π, with periodic boundary conditions Mu(0) = Mu(2π), (Mu)′(0) = (Mu)′(2π), in periodic Lebesgue-Bochner spaces Lp(\(\mathbb{T}\),X), periodic Besov spaces Bp,qs(\(\mathbb{T}\),X) and periodic Triebel-Lizorkin spaces Fp,qs (\(\mathbb{T}\),X), where A and M are closed linear operators on a Banach space X satisfying D(A) ⊂ D(M), a ∈ L1(ℝ+) and α is a scalar number. Using known operatorvalued Fourier multiplier theorems, we completely characterize the well-posedness of (P2) in the above three function spaces.

On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems

In this paper, we proved the generalized resolvent estimate and the maximal $L_p$-$L_q$ regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the $\mathcal{R}$ boundedness of solution operators defined in a sector $\Sigma_{\epsilon,\gamma_0} = \{\lambda \in C \: \backslash \: \{0\} \: | \: \left|\mathrm{arg}\lambda\right| \leq \pi - \epsilon, \left|\lambda\right| \geq \gamma_0 \}$ with $0 < \epsilon < \pi / 2$ and $\gamma_0 \geq 0$. This $\mathcal{R}$ boundedness implies the resolvent estimate of the Stokes operator and the combination of this $\mathcal{R}$ boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal $L_p$-$L_q$ regularity of the non-stationary Stokes. For a densely defined closed operator $A$, we know that what $A$ has maximal $L_p$ regularity implies that the resolvent estimate of $A$ in $\lambda \in \Sigma_{\epsilon,\gamma_0}$, but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the $\mathcal{R}$ boundedness of the operator family in the sector $\Sigma_{\epsilon,\gamma_0}$, we derive a systematic way to prove the resolvent estimate and the maximal $L_p$ regularity at the same time.

On strong solutions of a Robin problem modelling heat conduction in materials with corroded boundary

Motivated by a practical problem on a corrosion process, we shall study a third kind of BVP for a large class of elliptic equations in vector-valued L p spaces. Particularly we will determine optimal spaces for boundary data and get maximal regularity for inhomogeneous equations. Then based on these results we shall treat some nonlinear problems. Our approach will be based on the semigroup theory, the interpolation theory of Banach spaces, fractional powers of positive operators, operator-valued Fourier multiplier theorems and the Banach fixed point theorem.

Maximal [formula omitted]– [formula omitted] regularity for the two-phase Stokes equations; Model problems

In this paper we prove the generalized resolvent estimate and maximal L p – L q regularity of the Stokes equation with and without surface tension and gravity in the whole space with flat interface. We prove R boundedness of solution operators defined in a sector Σ ϵ , γ 0 = { λ ∈ C ∖ { 0 } | | arg λ | ⩽ π − ϵ , | λ | ⩾ γ 0 } with 0 < ϵ < π / 2 and γ 0 ⩾ 0 , which combined with the Fourier multiplier theorem of S.G. Mihlin and the operator valued Fourier multiplier theorem of L. Weis yields the required generalized resolvent estimate and maximal L p – L q regularity at the same time. One of the character of the paper is to introduce special function spaces E q ( R ˙ n , Σ ϵ , γ 0 ) and E p , q , γ 0 ( R ˙ n × R ) (cf. (1.7) and (1.8)), which is necessary to treat the situation that the normal component of velocity fields jumps across the interface. Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al., 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003 [22], Shibata and Shimada, 2007 [23], Shibata and Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary.

A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications

We study the fractional differential equation (*) Dαu(t) + BDβu(t) + Au(t) = f(t), 0 ⩽ t ⩽ 2π (0 ⩽ β < α ⩽ 2) in periodic Lebesgue spaces Lp(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R-boundedness of the sets {(ik)α((ik)α + (ik)βB + A)−1}k∈Z and {(ik)βB((ik)α + (ik)βB + A)−1}k∈Z. Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset-Boussinesq-Oseen equation are treated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Maximal [formula omitted]- [formula omitted] regularity of the linearized initial–boundary value problem for motion of compressible viscous fluids

We discuss the initial–boundary value problem for the linearized system of equations which describe motion of compressible viscous barotropic fluids in a bounded domain with the Navier boundary condition. This problem has uniquely a solution in the anisotropic Sobolev space W q 1 ( W ˙ p 1 ) × ( W p , q 2 , 1 ) n for any 1 < p < ∞ , 1 < q < ∞ globally in time. Moreover, exponentially weighted L p - L q estimates for solutions globally in time can be established. We prove the above properties by resolvent estimates for the linearized operator of the above system, the theory of analytic semigroups on Banach spaces and the operator-valued Fourier multiplier theorem on UMD spaces.

The Stokes operator in weighted L q -spaces II: weighted resolvent estimates and maximal L p -regularity

In this paper we establish a general weighted L q -theory of the Stokes operator \({\mathcal{A}}_{q,\omega}\) in the whole space, the half space and a bounded domain for general Muckenhoupt weights \(\omega \in A_q\). We show weighted L q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator \({\mathcal{A}}_{q,\omega}\) generates a bounded analytic semigroup but even yield the maximal L p -regularity of \({\mathcal{A}}_{q,\omega}\) in the respective weighted L q -spaces for arbitrary Muckenhoupt weights \(\omega \in A_q\). This conclusion is archived by combining a recent characterisation of maximal L p -regularity by \({\mathcal{R}}\)-bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L p -regularity. Preprint (1999)] with the fact that for L q -spaces \({\mathcal{R}}\) -boundedness is implied by weighted estimates.

On operator-valued Fourier multiplier theorems

The classical Fourier multiplier theorems of Marcinkiewicz and Mikhlin are extended to vector-valued functions and operator-valued multiplier functions on Z d or R d which satisfy certain R-boundedness conditions.

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Let $${\Omega = \Sigma \times \mathbb{R}}$$ be an infinite cylinder of $${\mathbb{R}^n}$$ , n ≥ 3, with a bounded cross-section $${\Sigma \subset \mathbb{R}^{n-1}}$$ of C 1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in $${L^{q}(\mathbb{R}; L^{r}_{\omega}(\Sigma))}$$ for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A r with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the $${\mathcal{R}}$$ -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional partial Fourier transform.

Singular convolution integrals with operator-valued kernel

We study operators \(f\mapsto Kf\) of the form \((Kf)(t)=\int_{{\bf R}^{n}} k(t-s)f(s) {\rm d}s\), where f is a vector-valued function and k an operator-valued singular kernel. Sufficient conditions for boundedness on Lp-spaces of UMD-valued functions are given in terms of a Hormander-type condition involving R-boundedness. The results cover large classes of kernels and also provide new proofs of recent operator-valued Fourier multiplier theorems. Moreover, they give new information about families of singular integral operators.

Fourier Multipliers and Integro‐Differential Equations in Banach Spaces

Operator-valued Fourier multiplier theorems are used to establish maximal regularity results for an integro-differential equation with infinite delay in Banach spaces. Results are obtained under general conditions for periodic solutions in the vector-valued Lebesgue and Besov spaces. The latter scale includes in particular the Holder spaces $C^{\alpha},\,0\,{

Fourier multipliers for Hölder continuous functions and maximal regularity

Two operator-valued Fourier multiplier theorems for Holder spaces are proved, one periodic, the other on the line. In contrast to the L p -situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Holder can be characterized by simple resolvent estimates of the underlying operator.

Operator-valued Fourier multiplier theorems on Lp( X) and geometry of Banach spaces

This paper gives the optimal order l of smoothness in the Mihlin and Hörmander conditions for operator-valued Fourier multiplier theorems. This optimal order l is determined by the geometry of the underlying Banach spaces (e.g. Fourier type). This requires a new approach to such multiplier theorems, which in turn leads to rather weak assumptions formulated in terms of Besov norms.