[Paper Review] Vector valued $q$-variation for differential operators and semigroups I

Guixiang Hong, Tao Ma|arXiv (Cornell University)|Nov 5, 2014

Advanced Harmonic Analysis Research33 references3 citations

TL;DR

This paper establishes vector-valued $q$-variational inequalities for differential operators and symmetric diffusion semigroups in Banach spaces with martingale cotype $q_0$, generalizing Pisier and Xu's martingale results and extending classical variational inequalities in harmonic analysis and ergodic theory. The key contribution is showing that Rademacher cotype $q$ is necessary for such inequalities, and the results provide quantitative rates of pointwise convergence dependent on the Banach space's geometric properties.

ABSTRACT

In this paper, we establish $\mathcal B$-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space $\mathcal B$ has martingale cotype property. These results generalize, on the one hand Pisier and Xu's result on the variational inequalities for $\mathcal B$-valued martingales, on the other hand many classical variational inequalities in harmonic analysis and ergodic theory. Moreover, we show that Rademacher cotype $q$ is necessary for the $\mathcal B$-valued $q$-variational inequalities. As applications of the variational inequalities, we deduce the jump estimates and obtain quantitative information on the rate of convergence. It turns out the rate of convergence depends on the geometric property of the Banach space under consideration, which considerably improve Cowling and Leinert's result where it is shown that the convergence always holds for all Banach spaces.

Motivation & Objective

• To extend scalar-valued variational inequalities in harmonic analysis and ergodic theory to the vector-valued setting using Banach space geometry.
• To identify the necessary geometric condition—martingale cotype $q_0$—for $\mathcal{B}$-valued $q$-variational inequalities to hold.
• To provide quantitative estimates on the rate of pointwise convergence for semigroups and differential operators, improving upon Cowling and Leinert’s general convergence result.
• To derive jump estimates and link them to the $q$-variational norm, showing sharpness of the exponent $q$.

Proposed method

• Introduces the $V_q(\mathcal{B})$-norm for families $(a_t)_{t>0}$ in a Banach space $\mathcal{B}$, defined via suprema over increasing sequences of times.
• Adapts Pisier and Xu’s martingale $q$-variational inequality to differential operators and symmetric diffusion semigroups using the martingale cotype property of $\mathcal{B}$.
• Applies the theory of contractively regular semigroups and density arguments to extend $L^p$-boundedness to $L^p(\Omega; \mathcal{B})$-valued operators.
• Uses the $\lambda$-jump function $N(a,\lambda)$ to derive $L^p$-estimates for the number of jumps exceeding $\lambda$, linking it to the $V_q(\mathcal{B})$-norm.
• Employs interpolation theory, assuming $\mathcal{B}$ is an interpolation space between a Hilbert space and another Banach space of martingale cotype $q_1 > q_0$.
• Applies the mean ergodic theorem to decompose $L^p(\Omega)$ and deduce individual ergodic convergence in $\mathcal{B}$-norm.

Experimental results

Research questions

• RQ1 Under what geometric conditions on a Banach space $\mathcal{B}$ do $\mathcal{B}$-valued $q$-variational inequalities hold for differential operators and semigroups?
• RQ2 Is the Rademacher cotype $q$ condition necessary for such inequalities, and can it be weakened?
• RQ3 Can variational inequalities provide quantitative information on the rate of pointwise convergence for $T_t f$ as $t \to 0^+$ or $t \to \infty$?
• RQ4 How do jump estimates relate to the $V_q(\mathcal{B})$-norm, and what is the sharpness of the exponent $q$?
• RQ5 To what extent do the convergence rates depend on the Banach space’s geometry, as opposed to being uniform across all spaces?

Key findings

• The $\mathcal{B}$-valued $q$-variational inequality holds for differential operators and symmetric diffusion semigroups if $\mathcal{B}$ has martingale cotype $q_0$ with $2 \leq q_0 < \infty$, and $q > q_0$.
• Rademacher cotype $q$ is necessary for the $\mathcal{B}$-valued $q$-variational inequality to hold, as shown by sharpness in the Hilbert space case.
• The jump estimate $\left\| \omega \mapsto N((T_t f)(\omega), \lambda)^{1/q} \right\|_p \lesssim \|f\|_{L^p(\mathcal{B})}/\lambda$ holds for $q > q_0$, with $\mu\{\omega : N(\cdot, \lambda) > K\} \lesssim \|f\|_{L^p(\mathcal{B})}^{p} / (\lambda^p K^{p/q})$.
• The rate of pointwise convergence of $T_t f(\omega)$ to $f(\omega)$ as $t \to 0^+$ and to $P_A f(\omega)$ as $t \to \infty$ is quantitatively controlled by the geometric property of $\mathcal{B}$, specifically its martingale cotype $q_0$.
• The results improve Cowling and Leinert’s general convergence result by showing that convergence speed depends on the Banach space’s geometry, not just its existence.
• The $q$-variational inequality fails for $q < q_0$, as demonstrated by the failure of the 2-variational inequality in the Hilbert space case, establishing sharpness of the exponent $q_0$.