Pointwise and weighted Hessian estimates for Kolmogorov–Fokker–Planck-type operators

Abstract

In this article, we obtain Hessian estimates for Kolmogorov–Fokker–Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in harmonic analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative Hessian estimates for the Kolmogorov–Fokker–Planck operators.

Appendix A. Boundedness of maximal function on generalized Orlicz spaces

We sketch below the proof of boundedness of maximal functions in the case of $\varphi \in \Phi ({\mathbb {R}}^{N+1})$. The case of $E\subset {\mathbb {R}}^{N+1}$ is similar. We require the unit ball property which asserts that

Lemma A.1

Let $\varphi \in \Phi ({\mathbb {R}}^{N+1})$. Then,

$$\begin{aligned} \Vert f\Vert _{L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})}<1 \implies \rho _{\varphi (\cdot )}(f)\le 1 \implies \Vert f\Vert _{L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})}\le 1. \end{aligned}$$

Lemma A.2

Let $\varphi \in \Phi ({\mathbb {R}}^{N+1})$. If $\varphi$ satisfies (A0), (A1), (A2), (aInc$_p$), and (aDec$_q$) with $1<p\le q<\infty$, then we have

$$\begin{aligned}&\Vert {\mathcal {M}}f\Vert _{L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})}\le C \Vert f\Vert _{L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})},\,f\in L^{\varphi (\cdot )}({\mathbb {R}}^{N+1}) \end{aligned}$$

(5.32)

$$\begin{aligned}&\Vert {\mathcal {M}}f\Vert _{L^{\varphi ^*(\cdot )}({\mathbb {R}}^{N+1})}\le C \Vert f\Vert _{L^{\varphi ^*(\cdot )}({\mathbb {R}}^{N+1})},\,f\in L^{\varphi ^*(\cdot )}({\mathbb {R}}^{N+1}). \end{aligned}$$

(5.33)

Proof

Step 1 (Boundedness of maximal function on ultraparabolic $L^p$ spaces):

As described in the beginning of Sect. 4.1, ${\mathcal {M}}$ maps $L^p({\mathbb {R}}^{N+1},d,|\cdot |)$ to itself for $1<p\le \infty$, and $L^1({\mathbb {R}}^{N+1},d,|\cdot |)$ to $L^{1, \infty }({\mathbb {R}}^{N+1},d,|\cdot |)$.

Step 2 (Key estimate):

The following lemma from [50] goes through as it is for the homogeneous space $({\mathbb {R}}^{N+1},d,|\cdot |)$.

Theorem A.3

([50, Theorem 4.3.2]) Let $\varphi \in \Phi ({\mathbb {R}}^{N+1})$. If $\varphi$ satisfies (A0), (A1), (A2), (aInc$_p$) with $1<p<\infty$, then there exist $\beta >0$ and $h\in L^1({\mathbb {R}}^{N+1})\cap L^\infty ({\mathbb {R}}^{N+1})$ such that

(5.34)

for every ball $B\subset {\mathbb {R}}^{N+1}$, $x\in B$ and $f\in L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})$ such that $\rho _{\varphi (\cdot )}(f)\le 1$.

Step 3: Taking supremum over all balls in inequality (5.34), we obtain

$$\begin{aligned} \varphi \left( x, \beta {\mathcal {M}}f(x)\right) ^{\frac{1}{p}}\lesssim {\mathcal {M}}(\varphi (\cdot ,f)^{\frac{1}{p}})(x) + {\mathcal {M}}(h^{\frac{1}{p}})(x), \end{aligned}$$

(5.35)

for every $x\in {\mathbb {R}}^{N+1}$ and $f\in L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})$ such that $\rho _{\varphi (\cdot )}(f)\le 1$. Here, we use the fact that $h(x)^{\frac{1}{p}}\le {\mathcal {M}}(h^{\frac{1}{p}})(x)$.

Step 4: Let $f\in L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})$ such that $\Vert f\Vert _{L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})}>0$. Choose $\varepsilon :=\frac{1}{2\Vert f\Vert _{L^{\varphi (\cdot )}({\mathbb {R}}^{N+1})}}$, then by Lemma A.1, it holds that $\rho _{\varphi (\cdot )}(\varepsilon f)\le 1$ so that we may apply (5.35) to obtain

$$\begin{aligned} \varphi \left( x, \beta \varepsilon {\mathcal {M}}f(x)\right) ^{\frac{1}{p}}\lesssim {\mathcal {M}}(\varphi (\cdot ,\varepsilon f)^{\frac{1}{p}})(x) + {\mathcal {M}}(h^{\frac{1}{p}})(x). \end{aligned}$$

(5.36)

Raising both sides of (5.36) to power p and integrating over ${\mathbb {R}}^{N+1}$, we get

$$\begin{aligned} \int \limits _{{\mathbb {R}}^{N+1}}\varphi \left( x, \beta \varepsilon {\mathcal {M}}f(x)\right) \,\textrm{d}x&\lesssim \int \limits _{{\mathbb {R}}^{N+1}}\left( {\mathcal {M}}(\varphi (\cdot ,\varepsilon f)^{\frac{1}{p}})(x)\right) ^p\,\textrm{d}x + \int \limits _{{\mathbb {R}}^{N+1}}\left( {\mathcal {M}}(h^{\frac{1}{p}})(x)\right) ^p\,\textrm{d}x\nonumber \\&\le \int \limits _{{\mathbb {R}}^{N+1}}\varphi (\cdot ,\varepsilon f)(x)\,\textrm{d}x + \int \limits _{{\mathbb {R}}^{N+1}}h(x)\,\textrm{d}x\nonumber \\&=\rho _{\varphi (\cdot )}(f) + \Vert h\Vert _{L^1({\mathbb {R}}^{N+1})}\nonumber \\&\le 1 + \Vert h\Vert _{L^1({\mathbb {R}}^{N+1})} = C_1, \end{aligned}$$

(5.37)

where in the second inequality, we invoke boundedness of maximal function on $L^p({\mathbb {R}}^{N+1})$ from Step 1. Inequality (5.37) is the same as $\rho _{\varphi (\cdot )}(\beta \varepsilon {\mathcal {M}}f)\le C_1$ so that by textbfaInc$_{1}$(which is a consequence of convexity and $\phi (\cdot ,0)=0$), we have $\rho _{\varphi (\cdot )}\left( \frac{\beta }{L C_1}\varepsilon {\mathcal {M}}f\right) \le 1$. Once again, applying Lemma A.1, we get $\Vert \frac{\beta }{L C_1}\varepsilon {\mathcal {M}}f\Vert _{L^{\varphi (\cdot )}}\le 1$ so that $\Vert {\mathcal {M}}f\Vert _{L^{\varphi (\cdot )}}\le \frac{C_1 L}{\varepsilon \beta }=\frac{2C_1 L}{\beta }\Vert f\Vert _{L^{\varphi (\cdot )}}$, which is the required result.

Step 5: The result for $\varphi ^*(\cdot )$ follows by the fact that it satisfies (A0), (A1), (A2), and aInc$_{q'}$

$\square$