<span class="mathjax-formula">$ {L}^{p}$</span>-maximal regularity of nonlocal parabolic equations and applications

Ann. I. H. Poincaré – AN 30 (2013) 573–614

www.elsevier.com/locate/anihpc

L ^p -maximal regularity of nonlocal parabolic equations and applications

Xicheng Zhang

School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, PR China Received 18 March 2012; received in revised form 9 September 2012; accepted 8 October 2012

Available online 15 November 2012

Abstract

By using Fourier’s transform and Fefferman–Stein’s theorem, we investigate theL^p-maximal regularity of nonlocal parabolic and elliptic equations with singular and non-symmetric Lévy operators, and obtain the unique strong solvability of the correspond- ing nonlocal parabolic and elliptic equations, where the probabilistic representation plays an important role. As a consequence, a characterization for the domain of pseudo-differential operators of Lévy type with singular kernels is given in terms of the Bessel potential spaces. As a byproduct, we also show that a large class of non-symmetric Lévy operators generates an analytic semigroup inL^p-spaces. Moreover, as applications, we prove Krylov’s estimate for stochastic differential equations driven by Cauchy pro- cesses (i.e. critical diffusion processes), and also obtain the global well-posedness for a class of quasi-linear first order parabolic systems with critical diffusions. In particular, critical Hamilton–Jacobi equations and multidimensional critical Burger’s equations are uniquely solvable and the smooth solutions are obtained.

©2012 Elsevier Masson SAS. All rights reserved.

Keywords:L^p-regularity; Lévy process; Krylov’s estimate; Sharp function; Critical Burger’s equation

1. Introduction

Consider the following Cauchy problem of fractional Laplacian heat equation in the domain[0,∞)×R^d with α∈(0,2)andλ0:

∂_tu+(−)^α2u+b· ∇u+λu=f, u(0)=ϕ, (1.1)

whereb: [0,∞)×R^d→R^dis a measurable vector field,f : [0,∞)×R^d→Randϕ:R^d→Rare two measurable functions, and(−)^α2 is the fractional Laplacian (also called Lévy operator) defined by

(−)^α2u=F⁻¹

| · |^αF(u)

, u∈S R^d

, (1.2)

whereF (resp.F⁻¹) denotes the Fourier (resp. inverse) transform, S(R^d)is the Schwartz class of smooth real or complex-valued rapidly decreasing functions.

E-mail address:XichengZhang@gmail.com.

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.10.006

Let(L_t)_t0be a symmetric and rotationally invariantα-stable process. Letb, f∈C_b^∞([0,∞)×R^d)andX_t,s(x) solve the following stochastic differential equation (SDE):

X_t,s(x)=x+ s t

b

−r, X_t,r(x) dr+

s t

dLr, ts0, x∈R^d.

It is well known that forϕ∈C_b^∞(R^d), the unique solution of Eq. (1.1) can be represented by the Feyman–Kac formula as (see Theorem5.2below):

u(t, x)=Eϕ

X_−t,0(x) +E

0

−t

e^{−λ(s+t )}f

−s, X_−t,s(x) ds

, t0. (1.3)

In connection with this representation, the first order termb· ∇u is also called the drift term, and the fractional Laplacian term(−)^α2uis also called the diffusion term.

Let nowu(t, x)satisfy (1.1). Forr >0 and(t, x)∈ [0,∞)×R^d, define u^r(t, x):=r^−αu

r^αt, rx

, b^r(t, x):=b r^αt, rx

, f^r(t, x):=f r^αt, rx

, then it is easy to see thatu^r satisfies

∂_tu^r+(−)^α2u^r+r^α−1

b^r· ∇u^r

+λr^αu^r=f^r. (1.4)

If one letsr→0, this scaling property leads to the following classification:

• (Subcritical case:α∈(1,2).) The drift term is controlled by the diffusion term at small scales.

• (Critical case:α=1.) The fractional Laplacian has the same order as the first order term.

• (Supercritical case:α∈(0,1).) The effect of the drift term is stronger than the diffusion term at small scales.

In recent years there is great interest to study the above nonlocal equation, since it has appeared in numerous disciplines, such as quasi-geostrophic fluid dynamics (cf.[10,9]), stochastic control problems (cf. [34]), non-linear filtering with jump (cf.[28]), mathematical finance (cf.[5]), anomalous diffusion in semiconductor growth (cf.[38]), etc. In[12], Droniou and Imbert studied the first order Hamilton–Jacobi equation with the fractional diffusion(−)^α2 basing upon a “reverse maximal principle”. Therein, when α∈(1,2), the classical solution was obtained; when α∈(0,2), the existence and uniqueness of viscosity solutions in the class of Lipschitz functions was also established.

In[9], Caffarelli and Vasseur established the global well-posedness of critical dissipative quasi-geostrophic equations (see also [21]for a simple proof in the periodic and two-dimensional case). On the other hand, Hölder regularity theory for the viscosity solutions of fully non-linear and nonlocal elliptic equations was also developed by Caffarelli and Silvestre[8], and Barles, Chasseigne and Imbert[4](see also[3] and the series of works of Silverstre[30,31, 33,32], etc.). We emphasize that the arguments in[8]and[4]are different: the former is based on the Alexandorff–

Backelman–Pucci’s (ABP) estimate, and the latter is based on the Ishii–Lions’ simple method. Moreover, in the subcritical case, Kurenok[25]established Krylov’s type estimate for one-dimensional stable processes with drifts (see[39]for multidimensional extension).

The purpose of this paper is to develop anL^p-regularity theory for nonlocal equations with general Lévy operators.

We describe it as follows. Letνbe a Lévy measure inR^d, i.e., aσ-finite measure satisfyingν({0})=0 and

R^d

min 1,|y|²

ν(dy) <+∞.

Forα∈(0,2), we write

y^(α):=1α∈(1,2)y+1α=1y1_|y|1.

In this article we are mainly concerned with the following pseudo-differential operator of Lévy type:

L^νf (x):=

R^d

f (x+y)−f (x)−y^(α)· ∇f (x)

ν(dy), f ∈S R^d

, (1.5)

whereνsatisfies

ν₁^(α)(B)ν(B)ν₂^(α)(B), B∈B R^d

, (1.6)

and 1α=1

r|y|R

yν(dy)=0, 0< r < R <+∞. (1.7)

Here,ν_i^(α),i=1,2 are the Lévy measures of twoα-stable processes taking the form

ν_i^(α)(B):=

S^d−1

∞ 0

1B(rθ )dr r^1+α

Σ_i(dθ ), (1.8)

whereS^d−1= {θ∈R^d: |θ| =1}is the unit sphere inR^d, andΣ_i called the spherical part ofν_i^(α)is a finite measure on S^d−1. We remark that condition (1.7) is a common assumption in the critical case (see[27,11]), and is clearly satisfied whenνis symmetric. Moreover, in the case ofα∈(1,2), for the convenience of proof, we useyrather thany1_|y|1

in (1.5). This is not essential since one can always minus a first order term( _|y|>1yν(dy))· ∇f (x)in (1.5).

One of the aims of the present paper is to determineD^p(L^ν), the domain of the Lévy operatorL^ν inL^p-space. We shall prove that under (1.6) and (1.7), ifν₁^(α)is nondegenerate (see Definition2.6below), then for anyp∈(1,∞),

D^p L^ν

=H^α,p,

whereH^α,p is theα-order Bessel potential space. Whenν(dy)=a(y)dy/|y|^d+α withc1|a(y)|c2, this charac- terization was obtained recently by Dong and Kim[11]. It is remarked that the technique in[4]was used by Dong and Kim to derive some local Hölder estimate for nonlocal elliptic equation in order to prove their characterization.

However, the following sum of nonlocal operators is not covered by[11]:

Lf (x)= d i=1

R

f (x₁, . . . , x_i−1, x_i+y_i, x_i+1, . . . x_d)−f (x)−y_i^(α)·∂_if (x)

|y_i|^1+α dyi,

since in this case, the Lévy measure (or the Lévy symbol) is very singular (or non-smooth) (see Remark2.7). Notice that if the Lévy symbol is smooth and its derivatives satisfy suitable conditions, the above characterization falls into the classical multiplier theorems about pseudo-differential operators (cf.[36,17]). We also mention that Farkas, Jacob and Schilling [13, Theorem 2.1.15]gave another characterization for D^p(L^ν) in terms of the so calledψ-Bessel potential space, whereψis the symbol ofL^ν.

The strategy for proving the above characterization is to prove the following Littlewood–Paley type inequality: for anyp∈(1,∞), there exists aC >0 such that for anyλ0,f ∈L^p(R⁺×R^d),

∞ 0

L^ν2

t 0

e^−λ(t−s)Pt^ν−¹sf (s,·)ds

p

p

dtC ∞ 0

f (t,·)^p

pdt,

whereν₁, ν₂are two Lévy measures satisfying (1.6) and (1.7), and(Pt^ν1)_t0is the semigroup associated withL^ν1. Indeed, this estimate is the key ingredient inL^p-theory of PDE (see[26,24]), and corresponds to the optimal regu- larity of nonlocal parabolic equation. Likewise[11], whenν(dy)=a(y)dy/|y|^d+αwith smooth and 0-homogeneous a(y)andc₁|a(y)|c₂, Mikulevicius and Pragarauskas[27]proved this type of estimate by showing some weak (1,1)-type estimate. In a different way, the proof given here is based on Fourier’s transform and Fefferman–Stein’s theorem about sharp functions (cf. [22,24]). We stress that probabilistic representation (1.3) will play an impor- tant role in reducing the general non-homogeneous operator to homogeneous operator (see Step 1 in the proof of Theorem4.2).

Another aim of this paper is to solve the linear and quasi-linear first order nonlocal parabolic equation with critical diffusions in the L^p-sense rather than the viscosity sense [12]. The critical case is specially interesting not only

because it appears naturally in quasi-geostrophic flows, but also it is an attractive object in mathematics. In particular, we care about the following multidimensional critical Burger’s equation:

∂_tu+(−)¹²u+u· ∇u=0, u(0)=ϕ. (1.9)

In one-dimensional case, this equation has a natural variational formulation and admits a unique global smooth so- lution (see [7,20]) under some regularity assumption on ϕ. In multidimensional case, the local well-posedness of Burger’s equation is relatively easy (cf. [18,40]). However, the global well-posedness of Eq. (1.9) seems to be un- known. The reason lies in two aspects: on one hand, there is no energy inequality and thus, the variational method seems not to be applicable; on the other hand, the first order term has the same order as the diffusion term. In fact, Kiselev, Nazarov and Schterenberg[20]have showed the existence of blow up solutions for 1-D supercritical Burger’s equation. The idea here is to establish some a priori Hölder estimate for Eq. (1.1) and then use the classical method of freezing coefficients. In[32], Silvestre proved an a priori Hölder estimate for Eq. (1.1) with only bounded measur- ableb. This is the key point for us. However, the assumption of scale invariance on Lévy operators seems to be crucial in[32]since the proof is by the iteration of the diminish of oscillation at all scales. As above, we shall use probabilistic representation (1.3) like a perturbation argument to extend Silvestre’s estimate to the more general non-homogeneous Lévy operator (see Corollary6.2).

This paper is organized as follows. In Section 2, we prepare some lemmas and recall some facts for later use. In Section 3, the basic maximum principles for nonlocal parabolic and elliptic equation are proved. In Section 4, we prove the main Theorem4.2, and give a comparison result between two Lévy operators. In particular, we show that(P_t^ν)_t0 forms an analytic semigroup inL^p-space. In Section 5, we prove the existence of a unique strong solution for the first order nonlocal parabolic equation with critical diffusion and variable coefficients. Here we assume that the first order coefficient is uniformly continuous with respect to the spatial variable since we are working in the critical case, and the non-homogeneous term is in someL^p-space. As an application, we also prove Krylov’s estimate for critical diffusion processes. We mention that in the subcritical case, Krylov’s estimate was proved in[25]and[39]. In Section 6, we investigate the quasi-linear first order nonlocal parabolic system, and get the existence of smooth solutions and strong solutions. In particular, the global solvability of Eq. (1.9) is obtained. In this section, the coefficients are assumed to be locally Lipschitz continuous, the zero order term is also required to be linear growth, and the initial value is in some fractional Sobolev spaces. In the whole proofs, basing upon the a priori estimates, we use the mollifying technique in many places.

Notations. We collect some frequently used notations below for the reader’s convenience.

• R⁺:=(0,∞),R⁺₀ := [0,∞). For a complex numberz, Re(z) (Im(z)): real (image) part ofz.

• S(R^d): the Schwartz class of smooth real or complex-valued rapidly decreasing functions. C_b^∞(R^d) (resp.

C_b^k(R^d),C₀^∞(R^d)): the space of all bounded smooth functions with bounded derivatives of all orders (resp. up to k-order, with compact support).

• FandF⁻¹: Fourier’s transform and Fourier’s inverse transform.

• ν: Lévy measure;ν^(α): the Lévy measure ofα-stable process;Σ: a finite measure onS^d−1, called the spherical part ofν^(α).

• L^ν_t: the Lévy process associated with Lévy measureν;Pt^μ: the semigroup associated withL^μ_t.L^ν: the generator ofL^μ_t,L^ν∗: the adjoint operator ofL^ν;p_t^ν: the heat kernel ofL^ν∗.

• B_r(x₀):= {x:∈R^d: |x−x₀|r},B_r:=B_r(0),B_r^c: the complement ofB_r.

• H^α,p: Bessel potential space;W^α,p: Sobolev–Slobodeckij space;W^∞:=

k,pW^k,p.

• ωb: the continuous modulus function ofb, i.e.,ωb(s):=sup_|x−y|s|b(x)−b(y)|.

• H^β: the space of Hölder continuous functions with the norm_[β]

k=0∇^kf∞+ ∇^[β]f_H^β, where[β]denotes the integer part ofβ, and∇^[β]f_H^β:=sup_|x−y|1|∇^[β]f (x)− ∇^[β]f (y)|/|x−y|^β.

• (ρ_ε)_ε∈(0,1): a family of mollifiers inR^d withρ_ε(x)=ε^−dρ(ε⁻¹x), whereρ is a nonnegative smooth function with support inB₁and satisfies _Rdρ(x)dx=1.

Convention. The letterC with or without subscripts will denote an unimportant constant. The inner product in Eu- clidean space is denoted by “·”.

2. Preliminaries

Forα∈(0,2), letν be a Lévy measure in R^d and satisfy (1.6) and (1.7). Let (L^ν_t)t0 be the d-dimensional Lévy process, a stationary and independent increment process defined on some probability space(Ω,F, P ), with characteristic function

Ee^iξ·Lνt =e^{−t ψν(ξ )}, ξ∈R^d, (2.1)

whereψvis the Lévy exponent with the following form by Lévy–Khintchine’s formula (cf.[2,29]), ψ_ν(ξ ):=

R^d

1+iξ·y^(α)−e^iξ·y

ν(dy). (2.2)

Letν^(α)take the form (1.8) and satisfy (1.7). It is well known that(L^ν_t^(α))_t0is ad-dimensionalα-stable process and has the following self-similarity (cf.[29, Proposition 13.5 and Theorem 14.7]):

L^ν_rt^(α)

t0 (d)=

r^1/αL^ν_t^(α)

t0, ∀r >0, (2.3)

where^(d)=means that the two processes have the same laws. Moreover, from expression (1.8), it is easy to see that for anyβ∈(0, α),

R^d

min

|y|^β,|y|²

ν^(α)(dy) <+∞, (2.4)

and Re

ψ_ν(α)(ξ )

= ∞

0

(1−cosr)dr r^1+α

S^d−1

|ξ·θ|^αΣ (dθ ). (2.5)

The Feller semigroup associated with(L^ν_t)_t0is defined by Pt^νf (x):=Ef

L^ν_t +x

, f ∈S R^d

.

The generator of(Pt^ν)t0is then given by (cf.[2, Theorem 3.3.3]) L^νf (x)=

R^d

f (x+y)−f (x)−y^(α)· ∇f (x)

ν(dy), (2.6)

i.e.,

∂_tPt^νf (x)=L^νPt^νf (x)=Pt^νL^νf (x), t >0. (2.7)

Moreover, F

L^νf

(ξ )= −ψ_ν(ξ )·F(f )(ξ ),

andψ_ν is also called the Lévy symbol of the operatorL^ν. From (2.5), one sees that if the spherical partΣ ofν^(α) is the uniform distribution (equivalently, rotationally invariant) onS^d−1, thenψ_ν(α)(ξ )=c_d,α|ξ|^α for some constant c_d,α>0, and thus, by (1.2),

−L^ν(α)f (x)=c_d,α(−)^α2f (x). (2.8)

On the other hand, from expression (2.6) and assumption (1.7), it is easy to see thatL^νhas the following invariance:

• Forz∈R^d, definefz(x):=f (z+x), then L^νf_z(x)=L^νf_x(z), L^νf_z

p=L^νf

p, (2.9)

wherep1 and · pdenotes the usualL^p-norm inR^d.

• Forr >0, definef_r(x):=f (rx), then

L^νf (rx)=L^ν(r·)f_r(x)=r^−αL^rαν(r·)f_r(x). (2.10) We remark thatr^αν^(α)(r·)=ν^(α) by (1.8).

• L^ν(C_b^∞(R^d))⊂C_b^∞(R^d), and for anyk2,L^ν :C_b^k(R^d)→C_b^k−2(R^d)is a continuous linear operator, where C_b^∞(R^d)(resp.C_b^k(R^d)) is the space of all bounded smooth functions with bounded derivatives of all orders (resp.

up tok-order).

The adjoint operator ofL^ν is given by L^ν∗f (x)=

R^d

f (x−y)−f (x)+y^(α)· ∇f (x)

ν(dy), (2.11)

i.e.,

R^d

L^νf (x)·g(x)dx=

R^d

f (x)·L^ν∗g(x)dx, f, g∈S R^d

.

Clearly,L^ν∗=L^ν(−), whereν(−)denotes the Lévy measureν(−dy).

Definition 2.1.Letν1andν2be two Borel measures. We say thatν1is less thanν2if ν₁(B)ν₂(B), B∈B

R^d , and we simply writeν₁ν₂in this case.

Lemma 2.2. Letν be a Lévy measure less thanν^(α) for someα∈(0,2), whereν^(α) takes the form(1.8). We also assume(1.7)forν. Then for someκ0>0,

ψ_ν(ξ )κ₀|ξ|^α, ξ∈R^d. (2.12)

Proof. Writeξˆ:=ξ /|ξ|. Forα∈(1,2), by the definitions ofψ_ν andν^(α), we have Im

ψ_ν(ξ )^(2.2)

R^d

ξ·y−sin(ξ·y)ν(dy)

R^d

ξ·y−sin(ξ·y)ν^(α)(dy)

(1.8)

=

S^d−1

∞ 0

|ξ·(rθ )−sin(ξ·rθ )|

r^1+α drΣ (dθ )

= |ξ|^α

S^d−1

∞ 0

|ˆξ·rθ−sin(ˆξ·rθ )|

r^1+α drΣ (dθ )C|ξ|^α. Forα=1, by (1.7), we have

Im

ψ_ν(ξ )=

R^d

ξ·y1_|y||ξ|−1−sin(ξ·y) ν(dy)

R^d

ξ·y1_|y||ξ|−1−sin(ξ·y)ν⁽¹⁾(dy)

=

S^d−1

∞ 0

|ξ·(rθ )1_r|ξ|−1−sin(ξ·rθ )|

r² drΣ (dθ )

= |ξ|

S^d−1

∞ 0

|ˆξ·rθ1r1−sin(ˆξ·rθ )|

r² drΣ (dθ )C|ξ|.

Forα∈(0,1), we have Im

ψ_ν(ξ )

R^d

sin(ξ·y)ν(dy)

R^d

sin(ξ·y)ν^(α)(dy)

= |ξ|^α

S^d−1

∞ 0

|sin(ˆξ·rθ )|

r^1+α drΣ (dθ )C|ξ|^α. Thus, combining with (2.5), we obtain (2.12). 2

Fork∈Nandp∈ [1,∞], letW^k,p be the usual Sobolev space with the norm fk,p:=

k j=0

∇^jf

p,

where∇^j denotes thej-order gradient.

We need the following simple interpolation result.

Lemma 2.3.Letp∈ [1,∞]andβ∈ [0,1]. For anyf∈W^1,pandy∈R^d, we have f (· +y)−f (·)

p

2fp

1−β

∇fp|y|β

. (2.13)

Proof. Observing that forf ∈S(R^d), f (x+y)−f (x)|y|

1 0

|∇f|(x+sy)ds,

by a density argument, we have for anyf∈W^1,p, f (· +y)−f (·)

p∇fp|y|. Thus, for anyβ∈ [0,1],

f (· +y)−f (·)

p

2fp

∧

∇fp|y|

2fp

1−β

∇fp|y|β

. The result follows. 2

The following lemma will be used to derive some asymptotic estimate of large time for the heat kernel of Lévy operator (see Corollary2.9below).

Lemma 2.4.Assume that Lévy measureνis less thanν^(α) for someα∈(0,2), whereν^(α)takes the form(1.8). Then for anyp∈ [1,∞]andf∈W^2,p, we have

L^νf

pC

⎧⎪

⎪⎨

⎪⎪

⎩

∇f¹p^−γ∇²f^γp+ ∇f¹p^−β∇²f^βp, α∈(1,2), γ∈(α−1,1], β∈ [0, α−1),

∇f¹p^−γ∇²f^γp+ f¹p^−β∇f^βp, α=1, γ ∈(0,1], β∈ [0,1), f¹p^−γ∇f^γp+ f¹p^−β∇f^βp, α∈(0,1), γ∈(α,1], β∈ [0, α), where the constantCdepends only onα, β, γ and the Lévy measureν^(α).

Proof. Let us first look at the case ofα∈(1,2). In this case, we have L^νf (x)=

R^d

y· 1

0

∇f (x+sy)− ∇f (x) ds

ν(dy).

Sinceνis bounded byν^(α), by Minkowski’s inequality and Lemma2.3, we have forγ∈(α−1,1]andβ∈ [0, α−1), L^νf

p

2∇fp

1−γ∇²f^γ

p

|y|1

|y|^1+γν^(α)(dy)+

2∇fp

1−β∇²f^β

p

|y|>1

|y|^1+βν^(α)(dy).

In the case ofα=1, we similarly have forγ∈(0,1]andβ∈ [0,1), L^νf

p

2∇fp

1−γ∇²f^γ

p

|y|1

|y|^1+γν⁽¹⁾(dy)+ 2fp

1−β

∇f^βp

|y|>1

|y|^βν⁽¹⁾(dy).

In the case ofα∈(0,1), we have forγ∈(α,1]andβ∈ [0, α), L^νf

p

2fp

1−γ

∇f^γp

|y|1

|y|^γν^(α)(dy)+ 2fp

1−β

∇f^βp

|y|>1

|y|^βν^(α)(dy).

The proof is complete by (2.4). 2

We also need the following estimate, which will be used frequently in localizing the nonlocal equation.

Lemma 2.5.Assume that Lévy measureν is less thanν^(α) for someα∈(0,2), whereν^(α)takes the form(1.8). Let ζ ∈S(R^d)and setζz(x):=ζ (x−z)forz∈R^d.

1. For anyβ∈(0∨(α−1),1)andp∈ [1,∞), there exists a constantC=C(ν^(α), β, p, d) >0 such that for all f ∈W^1,p,

R^d

L^ν(f ζ_z)− L^νf

ζ_z^p

pdz 1/p

Cζ2,pf¹_p^−βf^β_1,p. (2.14)

2. For anyβ∈(0∨(α−1),1)andγ ∈ [0, α), there exists a constantC=C(ν^(α), β, γ , d) >0 such that for any p∈ [1,∞]andf ∈H^β,

L^ν(f ζ )− L^νf

ζ

pCL^νζ

p+ ζ¹p^−γ∇ζ^γp

f∞+ ∇ζpf_H^β

, (2.15)

wheref_H^β :=sup_x=y,|x−y|1|f (x)−f (y)|/|x−y|^β, and for anyp∈ [1,∞]andf∈W^1,p, L^ν(f ζ )−

L^νf ζ

pCL^νζ

∞+ ζ¹_∞^−γ∇ζ^γ_∞

fp+ ∇ζ∞f¹p^−β∇f^βp

. (2.16)

Proof. (i) By formula (2.6), we have

L^ν(f ζ_z)(x)−L^νf (x)·ζ_z(x)−f (x)·L^νζ_z(x)

=

R^d

f (x+y)−f (x)

ζ_z(x+y)−ζ_z(x) ν(dy)

=

|y|1

f (x+y)−f (x)

ζ_z(x+y)−ζ_z(x) ν(dy)+

|y|>1

f (x+y)−f (x)

ζ_z(x+y)−ζ_z(x) ν(dy)

=:I_z⁽¹⁾(x)+I_z⁽²⁾(x). (2.17)

ForIz⁽¹⁾(x), by Fubini’s theorem, Minkowski’s inequality and Lemma2.3, we have

R^d

I_z^(1)p

pdz

R^d

|y|1

f (· +y)−f (·)1 0

|∇ζz|(· +sy)ds

|y|ν(dy)

p

p

dz

∇ζ^pp

R^d |y|1

f (x+y)−f (x)· |y|ν(dy) p

dx

∇ζ^pp

|y|1

f (· +y)−f (·)

p· |y|ν(dy) p

∇ζ^pp

2fp

p(1−β)

∇f^pβp

|y|1

|y|^1+βν^(α)(dy) p

.

ForIz⁽²⁾(x), we similarly have

R^d

I_z^(2)p

pdz4^p ν^(α)

B₁^cp

ζ^ppf^pp. Moreover, by (2.9) and Lemma2.4, we also have

R^d

fL^νζ_z^p

pdz=L^νζ^p

pf^ppCζ^p_2,pf^pp. Combining the above calculations, we obtain (2.14).

(ii) We have I₀⁽¹⁾

pf_H^β∇ζp

|y|1

|y|^1+βν(dy)f_H^β∇ζp

|y|1

|y|^1+βν^(α)(dy), and by Lemma2.3,

I₀⁽²⁾

pf_∞ 2ζp

1−γ

∇ζ^γp

|y|>1

|y|^γν(dy)f_∞ 2ζp

1−γ

∇ζ^γp

|y|>1

|y|^γν^(α)(dy).

Estimate (2.15) follows by (2.17) andfL^νζpf_∞L^νζp. As for (2.16), it is similar. 2 We introduce the following notion about the nondegeneracy ofν^(α).

Definition 2.6.Letν^(α)be a Lévy measure with the form (1.8). We say thatν^(α)is nondegenerate if the spherical part Σofν^(α) satisfies

S^d−1

|θ0·θ|^αΣ (dθ )=0, ∀θ0∈S^d−1. (2.18)

By the compactness ofS^d−1and (2.5), the above condition is equivalent to saying that for some constantκ1>0, Re

ψ_ν(α)(ξ )

κ₁|ξ|^α, ξ∈R^d. (2.19)

Remark 2.7.LetL¹_t, . . . , Lⁿ_t ben-independent copies of Lévy processL^ν_t. Write Lt=

L¹_t, . . . , Lⁿ_t .

ThenLt is annd-dimensional Lévy process and the characteristic function ofL1is given byψ(ξ )=ψ_ν(ξ¹)+ · · · + ψ_ν(ξⁿ), whereξ=(ξ¹, . . . , ξⁿ)∈R^ndwithξⁱ∈R^d. Clearly, if

Re ψ_ν(ξ )

κ₁|ξ|^α, ξ∈R^d, then

Reψ(ξ )

κ₁|ξ|^α, ξ∈R^nd.

It should be noticed that the Lévy measureνofLt is very singular and has the expression ν(dx) =ν

dx¹ ₀

dx², . . . ,dxⁿ

+ · · · +₀

dx¹, . . . ,dxⁿ⁻¹ ν

dxⁿ ,

wherex=(x¹, . . . , xⁿ)∈R^ndwithxⁱ∈R^d,0denotes the Dirac measure inR^(n−1)d, and the generator ofLtis given by

Lf (x) = n i=1

R^d

f

x¹, . . . , xⁱ+y, . . . , xⁿ

−f (x) −y^(α)· ∇_xⁱf (x)

ν(dy). (2.20)

We need the following simple result about the smoothness of the distribution density of Lévy process (see[16, Lemma 3.1]for the symmetric case).

Proposition 2.8.Letψ_νbe defined by(2.2)and satisfy Re

ψν(ξ )

κ1|ξ|^α, ξ∈R^d. (2.21)

Then for each t >0, the law ofL^ν_t inR^d has a C^∞-density p_t^ν with respect to the Lebesgue measure, andp_t^ν∈

k∈NW^k,1. In particular, by(2.7),

∂_tp_t^ν(x)=L^ν∗p_t^ν(x), (t, x)∈R⁺×R^d, (2.22)

whereL^ν∗is defined by(2.11), andp^ν_t(x)is also called the heat kernel ofL^ν∗.

Proof. By (2.21) and[29, p. 190, Proposition 28.1],L^ν_t has a smooth densityp^ν_t. Let us now prove that for each n∈N,∇ⁿp_t^ν∈L¹(R^d). By Fourier’s transform (2.1), one sees that

p^ν_t(x)= 1 (2π )^d

R^d

e^−iξ·xe^{−t ψν(ξ )}dξ.

Set

φ(ξ ):=

|y|1

1+iξ·y−e^iξ·y ν(dy).

It is easy to see thatφis a smooth complex-valued function, and by (2.21), for anyn∈Nandj1, . . . , jn∈ {1, . . . , d}, ξ→ξ_j1· · ·ξ_jne^{−t φ (ξ )}∈S

R^d ,

whereξ=(ξ₁, . . . , ξ_d). Since Fourier’s transformF is a bijective and continuous linear operator fromS(R^d)onto itself, there is a functionf ∈S(R^d)such that

f (ξ )ˆ :=F(f )(ξ )=ξ_j1· · ·ξ_jne^{−t φ (ξ )}.

On the other hand, by Lévy–Khintchine’s representation theorem (cf. [2, Theorem 1.2.14]), there is a probability measureμonR^dsuch that

ˆ μ(ξ ):=

R^d

e^iξ·yμ(dy)=e^{−t (ψν−φ)(ξ )}.

Thus, by the property of Fourier’s transform, we have

∂_xj

1· · ·∂_xjnp^ν_t(x)= (−i)ⁿ (2π )^d

R^d

e^−iξ·x

ξ_j1· · ·ξ_jne^{−t φ (ξ )}

e^{−t (ψν−φ)(ξ )}dξ

= (−i)ⁿ (2π )^d

R^d

e^−iξ·xf (ξ )ˆ μ(ξ )ˆ dξ=(−i)ⁿ

R^d

f (x−y)μ(dy).

From this, we immediately deduce that∇ⁿp_t^ν∈L¹(R^d). 2

Using Proposition2.8and Lemma2.4, we have the following useful estimates about the heat kernel.

Corollary 2.9.Letν_i^(α),i=1,2be two Lévy measures with the form (1.8), whereν₁^(α) is nondegenerate. Letν be another Lévy measure less than ν₂^(α). Then, there are two indexesδ₁, δ₂>1 (depending only onα)and constants C₁, C₂>0 (depending only ond, α,ν^(α)_i and not onν)such that for allt1,

∇L^νp^ν

(α) 1

t

1C1t^−δ1, (2.23)

∂_tL^νp^ν

(α) 1

t

1C₂t^−δ2. (2.24)

Proof. First of all, by the scaling property (2.3) and Proposition2.8, we have p^ν

(α) 1

t (x)=t^−d/αp^ν

(α) 1

1

t^−1/αx , and for eachn∈N,

R^d

∇ⁿp^ν

(α) 1

t (x)dx=t^−n/α

R^d

∇ⁿp^ν

(α) 1

1 (x)dxCt^−n/α. (2.25)

Estimate (2.23) follows from Lemma2.4by suitable choices ofβ andγ. Notice that by (2.22),

∂_tL^νp^ν

(α) 1

t (x)=L^νL^ν1(α)∗p^ν

(α) 1

t (x).

Estimate (2.24) follows by using Lemma2.4twice. 2

Now we turn to recall the classical Fefferman–Stein’s theorem. Fixα∈(0,2). LetQ^(α) be the collection of all parabolic cylinders

Q_r:=

t₀, t₀+r^α

×

x∈R^d: |x−x₀|r .

Forf ∈L¹_loc(R^d+1), define the Hardy–Littlewood maximal function by Mf (t, x):= sup

Q∈Q^(α),(t,x)∈Q

−

Q

f (s, y)dyds,

and the sharp function by f(t, x):= sup

Q∈Q^(α),(t,x)∈Q

−

Q

f (s, y)−fQdyds,

where f_Q:= −_Qf (s, y)dyds= _|Q¹_{| Q}f (s, y)dyds and|Q| is the Lebesgue measure of Q. One says that f ∈ BMO(R^d+1)iff∈L^∞(R^d+1). Clearly,f ∈BMO(R^d+1)if and only if there exists a constantC >0 such that for anyQ∈Q^(α), and for somea_Q∈R,

−

Q

f (s, y)−a_QdydsC.

The following theorem is taken from[24, Chapter 3](see also[36, p. 148, Theorem 2]).

Theorem 2.10(Fefferman–Stein’s theorem). Forp∈(1,∞), there exists a constantC=C(p, d, α)such that for all f ∈L^p(R^d+1),

fpCf

p. (2.26)