[Paper Review] On the R-boundedness of stochastic convolution operators

Jan van Neerven, Mark Veraar+1 more|arXiv (Cornell University)|Apr 13, 2014

Advanced Harmonic Analysis Research24 references4 citations

TL;DR

This paper establishes a characterization of R-boundedness for stochastic convolution operators with scalar-valued square-integrable kernels in Banach lattices, showing it is equivalent to ℓ¹-boundedness of associated deterministic convolution operators with squared kernels. The key contribution is a counterexample demonstrating that R-boundedness fails in certain UMD Banach lattices with type 2, such as $\ell^2(\ell^4)$, despite their favorable geometric properties.

ABSTRACT

The $R$-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal $L^p$-regularity, $2

Motivation & Objective

• To systematically analyze the R-boundedness of vector-valued stochastic convolution operators with scalar-valued square-integrable kernels in general Banach lattices.
• To establish a precise equivalence between R-boundedness of stochastic convolution families and ℓ¹-boundedness of associated deterministic convolution operators with kernels equal to the square of the original.
• To investigate the connection between ℓ¹-boundedness and the boundedness of the X-valued Hardy-Littlewood maximal function.
• To determine whether stochastic maximal L^p-regularity holds in UMD Banach lattices with type 2, particularly beyond L^q spaces.
• To construct a counterexample showing that R-boundedness can fail in UMD Banach lattices with type 2, challenging the expectation that such spaces support maximal regularity.

Proposed method

• The authors introduce a class $\mathscr{S}$ of kernels $k$ satisfying $\int_{\mathbb{R}_+} \sqrt{t} |k'(t)| \, dt \leq 1$, ensuring $k \in L^2(\mathbb{R}_+)$ and $k^2 \in \widetilde{\mathscr{K}}$, a class of kernels with integrable derivative in a weighted sense.
• They establish an equivalence: the family $\{S_k^H\}$ of stochastic convolution operators is R-bounded if and only if the associated family $\{T_{k^2}\}$ of deterministic convolution operators is ℓ¹-bounded.
• The analysis connects ℓ¹-boundedness of $T_{k^2}$ to the boundedness of the X-valued Hardy-Littlewood maximal function, particularly through the Hardy-Littlewood property of the dual of the 2-concavification of $X$.
• The authors use a duality argument to reduce the problem of ℓ¹-boundedness to ℓ^∞-boundedness on the dual space, leveraging the fact that ℓ¹-boundedness on $L^p(\mathbb{R}_+; X^2)$ is equivalent to ℓ^∞-boundedness on $L^p(\mathbb{R}_+; (X^2)^*)$.
• A counterexample is constructed using $X = \ell^2(\ell^4)$, whose 2-concavification is $\ell^1(\ell^2)$ and dual $\ell^\infty(\ell^2)$, which is shown to fail the Hardy-Littlewood property.
• The failure of the Hardy-Littlewood property in $\ell^\infty(\ell^2)$ is proven via a construction of a function in $L^2(\mathbb{R}; \ell^\infty(\ell^2))$ with unbounded maximal function, implying ℓ¹-boundedness fails.

Experimental results

Research questions

• RQ1 Is R-boundedness of stochastic convolution operators with scalar-valued kernels equivalent to ℓ¹-boundedness of the associated deterministic convolution operators with squared kernels in general Banach lattices?
• RQ2 Does the Hardy-Littlewood property of the dual of the 2-concavification of a Banach lattice $X$ ensure R-boundedness of stochastic convolution operators on $X$?
• RQ3 Can stochastic maximal $L^p$-regularity hold in UMD Banach lattices with type 2 beyond $L^q$ spaces?
• RQ4 Does the failure of the Hardy-Littlewood property in $\ell^\infty(\ell^2)$ imply that $\ell^2(\ell^4)$ is a counterexample to R-boundedness of stochastic convolutions?
• RQ5 Is there a UMD Banach lattice with type 2 where R-boundedness of stochastic convolution operators fails?

Key findings

• The family of stochastic convolution operators $\{S_k^H\}$ is R-bounded on $L^p(\mathbb{R}_+ \times \Omega; X)$ for $2 < p < ∞$ if and only if the associated family of deterministic convolution operators $\{T_{k^2}\}$ is ℓ¹-bounded on $L^{p/2}(\mathbb{R}_+; X^2)$.
• For Banach lattices $X$ with type 2 such that the dual of $X^2$ has the Hardy-Littlewood property, the family $\{S_k^H\}$ is R-bounded, generalizing previous results on $L^q$ spaces.
• The space $\ell^\infty(\ell^2)$ fails the Hardy-Littlewood property, as demonstrated by constructing a function in $L^2(\mathbb{R}; \ell^\infty(\ell^2))$ whose maximal function has norm growing like $\sqrt{N}/4$ as $N \to \infty$.
• The family $\{T_k\}$ with $k^2 \in \widetilde{\mathscr{K}}$ fails to be ℓ¹-bounded on $L^p(\mathbb{R}_+; \ell^1(\ell^2))$ for any $1 < p < \infty$, implying failure of ℓ¹-boundedness.
• Consequently, the family $\{S_k^H\}$ fails to be R-bounded from $L^p_{\mathscr{F}}(\mathbb{R}_+ \times \Omega; \gamma(H, \ell^2(\ell^4)))$ to $L^p(\mathbb{R}_+ \times \Omega; \ell^2(\ell^4))$ for any $2 < p < \infty$, providing a counterexample in a UMD Banach lattice with type 2.
• This counterexample shows that stochastic maximal $L^p$-regularity does not hold in general for UMD Banach lattices with type 2, even when the space is well-behaved in other geometric senses.