[Paper Review] Operator-valued Fourier multipliers in Besov spaces and its applications

Veli Shakhmurov, Rishad Shahmurov|ArXiv.org|Aug 19, 2008

Advanced Harmonic Analysis Research15 references3 citations

TL;DR

This paper establishes operator-valued Fourier multiplier theorems in Besov spaces of Banach-valued functions, proving optimal embeddings and maximal regularity for differential-operator equations. It demonstrates that certain differential operators generate analytic semigroups by showing their resolvents satisfy uniform estimates, extending maximal regularity theory to vector-valued Besov spaces with applications to boundary value problems and infinite systems of quasi-elliptic equations.

ABSTRACT

The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here, we show that these differential operators generate analytic semigroup.

Motivation & Objective

• To establish operator-valued Fourier multiplier theorems from $B_{q_1,r}^s$ to $B_{q_2,r}^s$ in Banach-valued function spaces.
• To investigate optimal embeddings between Besov spaces and the separability and positivity of differential operators.
• To prove that certain differential operators generate analytic semigroups via resolvent estimates.
• To extend maximal regularity theory to operator convolution equations and boundary value problems in vector-valued Besov spaces.
• To apply the results to infinite systems of quasi-elliptic equations with positive operator coefficients.

Proposed method

• The authors use Fourier multipliers in Besov spaces of $E$-valued functions, where $E$ is a Banach space, to analyze operator families.
• They define operator-valued Fourier multipliers via the Fourier transform of operator families $A(t)$, and study their boundedness on Besov spaces.
• The analysis relies on the concept of $\varphi$-positive operators and resolvent estimates in sectorial regions $S_\varphi$, ensuring boundedness of resolvents.
• The method involves proving uniform operator-norm bounds for multiplier functions $\sigma_i(\xi)$ derived from the resolvent of the differential operator.
• The framework incorporates interpolation theory and properties of Schwartz class functions to derive estimates in $B_{p,q}^s$ spaces.
• The theory is applied to differential-operator equations and infinite systems by embedding the problem into a vector-valued Besov space framework with positive operators.

Experimental results

Research questions

• RQ1 Under what conditions is an operator-valued Fourier multiplier bounded from $B_{q_1,r}^s(R^n;E)$ to $B_{q_2,r}^s(R^n;E)$?
• RQ2 What are the optimal embedding conditions between Besov spaces of different integrability and smoothness indices?
• RQ3 When does a differential operator generate an analytic semigroup in a vector-valued Besov space?
• RQ4 How can maximal regularity be established for boundary value problems involving operator coefficients in Besov spaces?
• RQ5 What conditions ensure the existence and coercive estimates for solutions of infinite systems of quasi-elliptic equations in vector-valued Besov spaces?

Key findings

• The operator $Q + \lambda$ has a bounded inverse on $B_{q_1,r}^s(R;E)$ for $\lambda \in S_\varphi$ with $|\lambda| \geq \lambda_0 > 0$, and the resolvent satisfies uniform estimates.
• The solution $u$ to the equation $Qu = f$ exists uniquely in $B_{q_2,r}^{l,s}(R;E(A),E)$ for $f \in B_{q_1,r}^s(R;E)$, with coercive estimate $\|u\|_{B_{q_2,r}^{l,s}} \leq C\|f\|_{B_{q_1,r}^s}$.
• The operator $Q + a$ for $a > 0$ is positive and generates an analytic semigroup on $B_{q_2,r}^s(R;E)$, provided $A$ is strongly positive.
• For infinite systems of quasi-elliptic equations, the solution $u = \{u_m\}$ belongs to $B_{q_2,r}^{2l,s}(R^N;l_q(D),l_q)$ and satisfies $\|u\|_{B_{q_2,r}^{2l,s}} + \|Au\|_{B_{q_2,r}^s} \leq C\|f\|_{B_{q_1,r}^s}$.
• The resolvent $ (Q + \lambda)^{-1} $ exists and satisfies $ \|D^\alpha (Q + \lambda)^{-1}\| + \|A(Q + \lambda)^{-1}\| \leq C $ uniformly in $\lambda \in S_\varphi$, ensuring maximal regularity.
• The results are general and apply to various concrete cases by choosing specific Banach spaces $E$ and positive operators $A$, such as matrices or differential operators.