Weak approximation of kinetic SDEs: closing the criticality gap

WEAK APPROXIMA TION OF KINETIC SDES: CLOSING THE CRI TICALI T Y GAP ZIMO HA O , KHOA LÊ AND CHENGCHENG LING A b s t r a c t . W e study the weak convergence of a generic tamed Euler-Maruyama scheme for the kinetic stochastic dierential e quations (SDEs) with integrable drifts. W e show that the marginal density of the considered scheme converges at rate 1 / 2 to the corresponding marginal density of the SDE. The convergence rate is independent from the criticality gap , which is new compared to previous results. M at h e m at i c s S u b j e c t C l a s s i f i c at i o n ( 2 0 2 0 ) : Primary 60H35, 65C30; Se condary 60H10, 60H50. K e y w o r d s : Singular SDEs; degenerate noise; weak approximation; kinetic SDEs; Second order SDEs; (tamed-)Euler-Maruyama scheme; regularization by noise 1. I n t r o d u c t i o n Let 𝑑 ⩾ 1 be an integer dimension, 𝜉 , 𝜂 be vectors in ℝ 𝑑 and 𝑊 be a 𝑑 -dimensional standard Brownian motion on a ltered probability space ( Ω , ℱ , ( ℱ 𝑠 ) 𝑠 ⩾ 0 , ℙ ) . W e consider the se cond order stochastic dierential system  𝑑 𝑋 𝑡 = 𝑉 𝑡 𝑑 𝑡 , 𝑑𝑉 𝑡 = 𝑏 ( 𝑡 , 𝑋 𝑡 , 𝑉 𝑡 ) 𝑑 𝑡 + 𝑑𝑊 𝑡 , ( 𝑋 0 , 𝑉 0 ) = ( 𝜉 , 𝜂 ) ∈ ℝ 𝑑 × ℝ 𝑑 . (1.1) The drift 𝑏 : ℝ + × ℝ 𝑑 × ℝ 𝑑 → ℝ 𝑑 is a measurable function which satises the following integrability condition. Assumption 1.1 . Ther e exists a constant 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 2 , ∞] 2 such that 𝒂 · 𝑑 𝒑 : = 3 𝑑 𝑝 𝑥 + 𝑑 𝑝 𝑣 < 1 and ∥ 𝑏 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) : = sup 𝑡   ℝ 𝑑   ℝ 𝑑 | 𝑏 ( 𝑡 , 𝑥 , 𝑣 ) | 𝑝 𝑥 𝑑 𝑥  𝑝 𝑣 𝑝 𝑥 𝑑 𝑣  1 𝑝 𝑣 < ∞ . (1.2) (The Lebesgue integration norm is replaced by sup -norm when 𝑝 𝑥 = ∞ or 𝑝 𝑣 = ∞ .) System ( 1.1 ) is one of the typical models that describes the Hamiltonian mechanics in the form of Langevin equation ([ Soi94 , T al02 ]). In such context, 𝑋 𝑡 , 𝑉 𝑡 usually r epresent respe ctiv ely the position and velocity of a moving particle at a time 𝑡 . The probability density 𝜌 𝑡 ( 𝑥 , 𝑣 ) of Date : Februar y 23, 2026. 1 𝑍 𝑡 : = ( 𝑋 𝑡 , 𝑉 𝑡 ) (which e xists under Theorem 1.1 , see [ RZ24 ]) is a common subject in statistical mechanics, and it satises the following Fokker–Planck equation ([ RS76 , Vil02 ]) 𝜕 𝑡 𝜌 = Δ 𝑣 𝜌 + 𝑣 · ∇ 𝑥 𝜌 + 𝑏 · ∇ 𝑣 𝜌 . (1.3) Let 𝑛 ⩾ 1 be an integer . W e dene 𝑘 𝑛 ( 𝑡 ) : = ⌊ 𝑛𝑡 ⌋ 𝑛 for each 𝑡 ∈ ℝ + . W e let ( 𝑏 𝑛 ) 𝑛 be a sequence of bounded functions that converges to 𝑏 . Let { 𝑍 𝑛 𝑡 } : = { ( 𝑋 𝑛 𝑡 , 𝑉 𝑛 𝑡 ) } be the solution to the following tamed Euler scheme  𝑋 𝑛 𝑡 = 𝜉 +  𝑡 0 𝑉 𝑛 𝑠 𝑑 𝑠 , 𝑉 𝑛 𝑡 = 𝜂 +  𝑡 0 𝑏 𝑛 ( 𝑠 , 𝑋 𝑛 𝑘 𝑛 ( 𝑠 ) + ( 𝑠 − 𝑘 𝑛 ( 𝑠 ) ) 𝑉 𝑛 𝑘 𝑛 ( 𝑠 ) , 𝑉 𝑛 𝑘 𝑛 ( 𝑠 ) ) 𝑑 𝑠 + 𝑊 𝑡 . (1.4) The probability density of 𝑍 𝑛 𝑡 , which exists, is denoted by 𝜌 𝑛 𝑡 . The goal of the curr ent article is to obtain convergence of 𝜌 𝑛 𝑡 to 𝜌 𝑡 with a rate that is independent from 𝑑 , 𝑝 𝑥 , 𝑝 𝑣 . T o be more precise, we assume that ( 𝑏 𝑛 ) 𝑛 satises the following condition. Assumption 1.2 . Ther e are nite constants 𝜁 ∈ [ 0 , 1 2 ] , 𝜗 > 0 , 𝛿 ∈ ( 1 , 2 ) and 𝜅 𝑏 > 0 such that sup 𝑛 ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) ⩽ 𝜅 𝑏 , (1.5) sup 𝑛 𝑛 𝜗 𝛿 ∥ 𝑏 𝑛 − 𝑏 ∥ 𝐿 ∞ ( ( 0 , 1 ] , 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ) ⩽ 𝜅 𝑏 , (1.6) sup 𝑛 𝑛 − 𝜁 sup ( 𝑡 ,𝑧 ) ∈ [ 0 , 𝑛 − 1 ] × ℝ 2 𝑑 | 𝑏 𝑛 ( 𝑡 , 𝑧 ) | ⩽ 𝜅 𝑏 , (1.7) and lim ℎ ↓ 0 sup 𝑛 ( 𝑛 − 1 / 2 ∧ ℎ ) sup ( 𝑡 ,𝑧 ) ∈ [ 0 , 1 ] × ℝ 2 𝑑 | 𝑏 𝑛 ( 𝑡 , 𝑧 ) | = 0 . (1.8) In ( 1.6 ) , 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 is an anisotropic Besov space of distributions on ℝ 2 𝑑 with regularity index − 𝛿 . Precise denitions are provided later in Section 2 . Heuristically , condition ( 1.5 ) asserts that 𝑏 𝑛 belongs to the same function space as 𝑏 uniformly . The constant 𝜗 parametrizes and quanties the approximating sequence ( 𝑏 𝑛 ) 𝑛 . Condition ( 1.8 ) is a technical one which is necessary for the application of Girsanov theorem. Theorem 1.3. Assume that Theorems 1.1 and 1.2 hold and initial data ( 𝜉 , 𝜂 ) ∈ ℝ 𝑑 × ℝ 𝑑 is given. Then for any 𝒒 ∈ [ 2 , ∞) 2 with 𝒒 ⩾ 𝒑 , there is a constant 𝐶 = 𝐶 ( 𝑑 , 𝒑 , 𝒒 , 𝜁 , 𝜗 , 𝛿 , 𝜅 𝑏 , ∥ 𝑏 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) ) > 0 such that for all 𝑡 ∈ ( 0 , 1 ] , ∥ 𝜌 𝑡 − 𝜌 𝑛 𝑡 ∥ 𝒒 ′ ≲ 𝐶 𝑛 − ( 𝜗 ∧ 1 2 )  𝑡 − 𝜁 − 1 2 𝒂 · 𝑑 𝒒 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 )  , (1.9) where 𝒒 ′ = ( 𝑞 𝑥 𝑞 𝑥 − 1 , 𝑞 𝑣 𝑞 𝑣 − 1 ) is the Hölder conjugate of 𝒒 . W e note that the constant 𝐶 in ( 1.9 ) is independent from the initial position. The restriction on the time interval [ 0 , 1 ] is conventional which simplies our exposition. The results discussed herein can be extended straightfor wardly to arbitrary time inter vals. In such case, the implicit constant in Theorem 1.3 would depend on the the length of the temporal inter val. W e give two 2 typical e xamples of ( 𝑏 𝑛 ) 𝑛 which satises Theorem 1.2 . The rst one is given by the convolution 𝑏 𝑛 = 𝑏 ∗ 𝜙 𝑛 , 𝜙 𝑛 ( 𝑥 , 𝑣 ) : = 𝑛 4 𝑑 𝜗 𝜙 ( 𝑛 3 𝜗 𝑥 , 𝑛 𝜗 𝑣 ) , (1.10) where 𝜙 is any b ounded probability density function with 𝜑 ( 𝑥 , 𝑣 ) = 𝜑 ( 𝑥 , − 𝑣 ) and 𝜗 is any constant in ( 0 , 1 2 ( 𝒂 · 𝑑 𝒑 ) − 1 ] . When ( ( 𝑝 𝑥 ∧ 𝑝 𝑣 ) − 1 ) 𝒂 · 𝑑 𝒑 > 1 or when 𝒑 = ( ∞ , ∞) , one can choose the other example of taming sequence is obtained through the truncation 𝑏 𝑛 ( 𝑡 , 𝑥 , 𝑣 ) : = 1 | 𝑏 ( 𝑡 ,𝑥 ,𝑣 ) | > 0 | 𝑏 ( 𝑡 , 𝑥 , 𝑣 ) | ∧ ( 𝐶 2 𝑛 𝜅 ) | 𝑏 ( 𝑡 , 𝑥 , 𝑣 ) | 𝑏 ( 𝑡 , 𝑥 , 𝑣 ) , (1.11) where 𝜅 is any constant in ( 0 , 1 / 2 ) . T o obtain rate 1 / 2 in ( 1.9 ) , it is sucient to tune the parameters so that 𝜗 > 1 2 in case of ( 1.10 ) and 𝜅 > 1 2 ( 𝒂 · 𝑑 𝒑 ) in case of ( 1.11 ) , se e details in Section 2.2 below . Discussion. Comparing ( 1.4 ) with the standard Euler-Maruyama scheme  𝑋 𝑛 𝑡 = 𝜉 +  𝑡 0 𝑉 𝑛 𝑘 𝑛 ( 𝑠 ) 𝑑 𝑠 , 𝑉 𝑛 𝑡 = 𝜂 +  𝑡 0 𝑏 ( 𝑠 , 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) 𝑑 𝑠 + 𝑊 𝑡 , (1.12) we have replaced  𝑡 0 𝑉 𝑛 𝑘 𝑛 ( 𝑠 ) 𝑑 𝑠 and  𝑡 0 𝑏 ( 𝑠 , 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) 𝑑 𝑠 respectively by  𝑡 0 𝑉 𝑛 𝑠 𝑑 𝑠 and  𝑡 0 𝑏 𝑛 ( 𝑠 , 𝑋 𝑛 𝑘 𝑛 ( 𝑠 ) + ( 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑉 𝑛 𝑘 𝑛 ( 𝑠 ) , 𝑉 𝑛 𝑘 𝑛 ( 𝑠 ) ) 𝑑 𝑠 . This replacement is more favorable due to the se veral reasons. First, e ven though  𝑡 0 𝑉 𝑛 𝑠 𝑑 𝑠 has continuum variable , the scheme ( 1.4 ) is still recursive. Indee d, let ℎ : = 1 / 𝑛 , for any 𝑘 ∈ ℕ , we obtain from ( 1.4 ) that 𝑋 𝑛 ( 𝑘 + 1 ) ℎ = 𝑋 𝑛 𝑘 ℎ +  ( 𝑘 + 1 ) ℎ 𝑘 ℎ 𝑉 𝑛 𝑠 𝑑 𝑠 , 𝑉 𝑛 𝑠 = 𝑉 𝑛 𝑘 ℎ +  𝑠 𝑘 ℎ 𝑏 𝑛 ( 𝑟 , 𝑋 𝑛 𝑘 ℎ + ( 𝑟 − 𝑘 ℎ ) 𝑉 𝑛 𝑘 ℎ , 𝑉 𝑛 𝑘 ℎ ) 𝑑 𝑟 + 𝑊 𝑠 − 𝑊 𝑘 ℎ , 𝑠 ∈ [ 𝑘 ℎ, ( 𝑘 + 1 ) ℎ ] . This implies the recursive relation  𝑋 𝑛 ( 𝑘 + 1 ) ℎ 𝑉 𝑛 ( 𝑘 + 1 ) ℎ  = 𝐹 𝑛 ℎ ( 𝑋 𝑛 𝑘 ℎ , 𝑉 𝑛 𝑘 ℎ ) + 𝜉 𝑛 𝑘 , (1.13) where 𝐹 𝑛 ℎ ( 𝑥 , 𝑣 ) : =  𝑥 + ℎ𝑣 +  ℎ 0 ( ℎ − 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑥 + 𝑟 𝑣 , 𝑣 ) 𝑑 𝑟 𝑣 +  ℎ 0 𝑏 𝑛 ( 𝑟 , 𝑥 + 𝑟 𝑣 , 𝑣 ) 𝑑 𝑟  and 𝜉 𝑛 𝑘 =   ( 𝑘 + 1 ) ℎ 𝑘 ℎ ( 𝑊 𝑠 − 𝑊 𝑘 ℎ ) 𝑑 𝑠 𝑊 ( 𝑘 + 1 ) ℎ − 𝑊 𝑘 ℎ  . W e note that 𝜉 𝑛 𝑘 are i.i.d. Gaussian random variables with covariance matrix  ℎ 3 3 𝕀 𝑑 × 𝑑 ℎ 2 2 𝕀 𝑑 × 𝑑 ℎ 2 2 𝕀 𝑑 × 𝑑 ℎ 𝕀 𝑑 × 𝑑  . 3 Consequently , in the case of time-independent smooth drift 𝑏 , the scheme ( 1.4 ) is completely implementable through ( 1.13 ). Second, we use tamed term 𝑏 𝑛 instead of 𝑏 itself, which is known as “taming te chnique ” [ HJK12 , LL25 ], to aid the hardly controllable case when 𝑏 has singularities. The taming of the drift turns out to be technically b enecial in analyzing convergence rates. There are limited works on the study of the weak w ell-posedness of ( 1.1 ) . Sharp conditions given from [ RM22 , Theorem 1] are either 𝑏 ∈ 𝐿 ∞ 𝑇 ( 𝐶 𝛽 𝑥 ,𝑣 ) with 𝛽 ∈ ( 0 , 1 ) or 𝑏 ∈ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) or 𝐿 𝑞 𝑇 ( 𝕃 𝑝 ) with 2 𝑞 + 4 𝑑 𝑝 < 1 and 𝑝 ⩾ 2 , 𝑞 > 2 . [ RZ24 , Theorem 1.4] obtains, among other things, weak well-posedness for ( 1.1 ) with unbounded singular drifts in Kato ’s class, which includes Theorem 1.1 . Considering the fact that there are few results on w eak well-posedness, it se ems that the current result is the rst work that quanties the weak convergence rate of order 1 / 2 for ( 1.1 ) under Theorem 1.1 . The work [ LM10 ] also studies the law of the discrete Euler–Maruyama equation but does not provide any quantitative bounds. When 𝑏 = 𝑏 ( 𝑡 , 𝑣 ) ∈ 𝕃 ( 𝑝 𝑥 ,𝑝 𝑣 ) with 𝑝 𝑥 = ∞ and 𝑝 𝑣 > 𝑑 , the kinetic SDE ( 1.1 ) reduces to the non-degenerate SDE 𝑑𝑉 𝑡 = 𝑏 ( 𝑡 , 𝑉 𝑡 ) 𝑑 𝑡 + 𝑑𝑊 𝑡 . The work [ JM24 ] considered the convergence of ( 1.4 ) with the approximating drift 𝑏 𝑛 ( 𝑡 , 𝑣 ) : = 1 𝑡 > 1 / 𝑛, | 𝑏 ( 𝑡 ,𝑣 ) | > 0 | 𝑏 ( 𝑡 , 𝑣 ) | ∧ ( 𝐶 𝑛 𝜅 ) | 𝑏 ( 𝑡 , 𝑣 ) | 𝑏 ( 𝑡 , 𝑣 ) , 𝜅 = 𝑑 𝑝 𝑣 , (1.14) and established a pointwise convergence rate for 𝜌 𝑛 of order 𝑛 − 1 − 𝑑 / 𝑝 𝑣 2 , which vanishes when 𝑑 / 𝑝 𝑣 approaches 1 . Due to technical reasons, [ JM24 ] requires that 𝑏 𝑛 vanishes when 𝑡 < 1 / 𝑛 , which causes ( 1.14 ) to violate ( 1.6 ) . In contrast, our method does not treat the initial time step dierently and ther efore allows the use of ( 1.11 ) . In this case , by choosing 𝜅 > 1 2 ( 𝒂 · 𝑑 𝒑 ) , Theorem 1.3 yields a convergence rate (with respect to a weaker topology) of order 𝑛 − 1 2 , which is independent from the criticality gap 1 − 𝑑 / 𝑝 𝑣 . While [ JM24 ] also considers the case ( 1.14 ) with 𝜅 = 1 / 2 , the choice ( 1.11 ) with 𝜅 = 1 / 2 is ruled out by condition ( 1.8 ) (see Theorem 4.3 for further details). Idea of the proof. W e briey sketch the general arguments showing ( 1.9 ) . By duality , it suces to estimate | 𝔼 𝜑 ( 𝑍 𝑡 ) − 𝔼 𝜑 ( 𝑍 𝑛 𝑡 ) | , for each measurable function 𝜑 satisfying ∥ 𝜑 ∥ 𝕃 𝒒 = 1 . Applying Itô ’s formula to 𝑟 → 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) and 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) , we have | ⟨ 𝜑 , 𝜌 𝑡 − 𝜌 𝑛 𝑡 ⟩ | ⩽    𝔼  𝑡 0 ( 𝑏 − 𝑏 𝑛 ) ( 𝑟 , 𝑍 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) 𝑑 𝑟    +    𝔼  𝑡 0 𝑏 𝑛 ( 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) − 𝑏 𝑛 ( 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟    +    𝔼  𝑡 0  𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑟 ) − Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) )  · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟    = : 𝐼 𝑛 1 + 𝐼 𝑛 2 + 𝐼 𝑛 3 . 4 The rst term, 𝐼 𝑛 1 , encodes the taming error arising from approximating 𝑏 by 𝑏 𝑛 . The convergence rate of 𝐼 1 𝑛 is therefor e governed by ( 1.6 ) . T o utilize ( 1.6 ) , one must ee ctively control the regularity of the product ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 · ( 𝑏 − 𝑏 𝑛 ) ( 𝑠 ) . Standard product estimates would yield unsatisfactory rates. W e overcome this diculty by using Bony’s paraproduct decomposition, writing ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ≺ ( 𝑏 𝑛 − 𝑏 ) ( 𝑠 ) + ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ≽ ( 𝑏 𝑛 − 𝑏 ) ( 𝑠 ) . Each term in Bony’s decomposition p ossesses a dierent degree of regularity and integrability , and by treating them separately , we are able to apply condition ( 1.6 ) eectively (see The or em 4.4 ). The second term, 𝐼 𝑛 2 , has a recursive structure and can be estimated directly in terms of the map 𝑠 ↦→ ∥ 𝜌 𝑠 − 𝜌 𝑛 𝑠 ∥ 𝒑 ′ . By applying a Grönwall argument, this recursive contribution b ecomes negligible. The last term, 𝐼 𝑛 3 , is a weighted quadrature error arising from the discretizations of the time steps. T o control it, we use r ened estimates related to the transition probabilities associate d with 𝑍 and 𝑍 𝑛 , see Theorems 4.5 to 4.7 . The application of Itô’s formula leading to the de composition into 𝐼 𝑛 1 , 𝐼 𝑛 2 , 𝐼 𝑛 3 is already pre- sented in literature . For instance, it is used in [ Hol24 ] in a slightly dierent context, where the test function 𝜑 is Hölder continuous, which leads to convergence in a W asserstein metric. Our analysis of each term is nevertheless no vel, and new techniques have been introduced to handle the lack of regularity present in the coecients and the test functions. Organization of the paper . Precise denitions, notations and examples are stated and dis- cussed in Section 2 . Section 3 colle cts some analytic estimates on the free Markov semigroup and exponential functionals. The proof of Theorem 1.3 is provided in Section 4 . Section A collects several technical lemmas for calculations appeared in the main proofs. Paraproduct estimates in anisotropic Besov spaces ar e summarized in Section B . Conventions. A vector 𝑧 in ℝ 2 𝑑 is often decomposed into two components 𝑧 = ( 𝑥 , 𝑣 ) ∈ ℝ 𝑑 × ℝ 𝑑 . W e denote ∇ 𝑥 = ( 𝜕 𝑥 𝑘 ) 1 ⩽ 𝑘 ⩽ 𝑑 , ∇ 𝑣 = ( 𝜕 𝑣 𝑘 ) 1 ⩽ 𝑘 ⩽ 𝑑 and ∇ = ( ∇ 𝑥 , ∇ 𝑣 ) . W e use relations < , = , ⩽ and > , ⩾ between vectors if each corresponding component shares the same relation, e.g. 𝒒 : = ( 𝑞 𝑖 ) 𝑖 ⩾ 1 ⩽ 𝒑 : = ( 𝑝 𝑖 ) 𝑖 ⩾ 1 if 𝑞 𝑖 ⩽ 𝑝 𝑖 for each 𝑖 . For each 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 1 , ∞] 2 , we denote 𝒑 − 1 : = 1 𝒑 : = ( 1 𝑝 𝑥 , 1 𝑝 𝑣 ) , with convention that 1 /∞ = 0 . The notation 𝑎 ≲ 𝑏 means that 𝑎 ⩽ 𝐶 𝑏 for some nite non-negative constant 𝐶 which dep ends only on the parameters in the corresponding statement. If there are further dep endence on another parameter 𝑐 , we incorporate it in the notation by writing 𝑎 ≲ 𝑐 𝑏 . 2. N o tat i o n s a n d e x a m p l e s 2.1. Notations. For each 𝑝 ∈ [ 1 , ∞) , ∥ · ∥ 𝑝 denotes the classical Lebesgue norm on ℝ 𝑑 , while ∥ · ∥ ∞ denotes the supremum norm over ℝ 𝑑 , i.e. ∥ 𝑓 ∥ ∞ : = sup 𝑥 ∈ ℝ 𝑑 | 𝑓 ( 𝑥 ) | . W e note that this diers from the usual 𝐿 ∞ Lebesgue norm dened via the essential supremum. This choice is purely for convenience since w e consider only functions that are bounded every where . 5 For each vector index 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 1 , ∞] 2 , we dene the mix 𝕃 𝒑 space as the collection of measurable functions 𝑓 on ℝ 2 𝑑 such that ∥ 𝑓 ∥ 𝒑 : =   ℝ 𝑑 ∥ 𝑓 ( · , 𝑣 ) ∥ 𝑝 𝑣 𝑝 𝑥 𝑑 𝑣  1 / 𝑝 𝑣 < ∞ if 𝑝 𝑣 < ∞ , and ∥ 𝑓 ∥ 𝒑 : = sup 𝑣 ∈ ℝ 𝑑 ∥ 𝑓 ( · , 𝑣 ) ∥ 𝑝 𝑥 < ∞ if 𝑝 𝑣 = ∞ . When 𝒑 = ( ∞ , ∞) , we abbreviate ∥ · ∥ 𝒑 = ∥ · ∥ ∞ . For each 𝑞 ∈ [ 1 , ∞] and 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 1 , ∞] 2 , 𝐿 𝑞 𝑇 ( 𝕃 𝒑 ) denotes the space of measurable functions 𝑓 on [ 0 , 1 ] × ℝ 2 𝑑 such that ∥ 𝑓 ∥ 𝐿 𝑞 𝑇 ( 𝕃 𝒑 ) : =   𝑇 0 ∥ 𝑓 ( 𝑡 , · ) ∥ 𝑞 𝒑 𝑑 𝑡  1 / 𝑞 < ∞ if 𝑞 < ∞ and ∥ 𝑓 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) : = sup 𝑡 ∈ [ 0 , 1 ] ∥ 𝑓 ( 𝑡 , · ) ∥ 𝑞 𝒑 < ∞ if 𝑞 = ∞ . Anisotropic Besov spaces. For a Lebesgue integrable function 𝑓 in ℝ 2 𝑑 , let ˆ 𝑓 and ˇ 𝑓 respec- tively be the Fourier transform of 𝑓 and its inverse which are dened by ˆ 𝑓 ( 𝜉 ) : = ( 2 𝜋 ) − 2 𝑑  ℝ 2 𝑑 e − i 𝜉 · 𝑧 𝑓 ( 𝑧 ) 𝑑 𝑧 , ˇ 𝑓 ( 𝜉 ) : = ( 2 𝜋 ) − 2 𝑑  ℝ 2 𝑑 e i 𝜉 · 𝑧 𝑓 ( 𝑧 ) 𝑑 𝑧 , 𝜉 ∈ ℝ 2 𝑑 . Put 𝒂 = ( 3 , 1 ) and dene the anisotropic distance | 𝑧 − 𝑧 ′ | 𝒂 : = | 𝑥 − 𝑥 ′ | 1 / 3 + | 𝑣 − 𝑣 ′ | , ∀ 𝑧 = ( 𝑥 , 𝑣 ) , 𝑧 ′ = ( 𝑥 ′ , 𝑣 ′ ) ∈ ℝ 𝑑 × ℝ 𝑑 . For each 𝑟 > 0 and 𝑧 ∈ ℝ 2 𝑑 , we denote 𝐵 𝒂 𝑟 ( 𝑧 ) : = { 𝑧 ′ ∈ ℝ 2 𝑑 : | 𝑧 ′ − 𝑧 | 𝒂 ⩽ 𝑟 } and 𝐵 𝒂 𝑟 : = 𝐵 𝒂 𝑟 ( 0 ) . Let 𝜒 𝒂 0 be a 𝐶 ∞ 𝑐 -function on ℝ 2 𝑑 which is symmetric in the direction of 𝑥 and 𝑣 with 𝜒 𝒂 0 ( 𝜉 ) = 1 if 𝜉 ∈ 𝐵 𝒂 1 and 𝜒 𝒂 0 ( 𝜉 ) = 0 if 𝜉 ∉ 𝐵 𝒂 2 . W e dene 𝜙 𝒂 𝑗 ( 𝜉 ) : =  𝜒 𝒂 0 ( 2 − 𝑗 𝒂 𝜉 ) − 𝜒 𝒂 0 ( 2 − ( 𝑗 − 1 ) 𝒂 𝜉 ) , 𝑗 ⩾ 1 , 𝜒 𝒂 0 ( 𝜉 ) , 𝑗 = 0 , where for each 𝑠 ∈ ℝ and 𝜉 = ( 𝜉 1 , 𝜉 2 ) ∈ ℝ 𝑑 × ℝ 𝑑 , 2 𝑠 𝒂 𝜉 = ( 2 3 𝑠 𝜉 1 , 2 𝑠 𝜉 2 ) . Note that supp ( 𝜙 𝒂 𝑗 ) ⊂  𝜉 : 2 𝑗 − 1 ⩽ | 𝜉 | 𝒂 ⩽ 2 𝑗 + 1  , 𝑗 ⩾ 1 , supp ( 𝜙 𝒂 0 ) ⊂ 𝐵 𝒂 2 , and 𝑘  𝑗 = 0 𝜙 𝒂 𝑗 ( 𝜉 ) = 𝜒 𝒂 0 ( 2 − 𝑘 𝒂 𝜉 ) → 1 , as 𝑘 → ∞ , ∀ 𝜉 ∈ ℝ 2 𝑑 . (2.1) 6 Let S be the space of all Schwartz functions on ℝ 2 𝑑 and S ′ be its topological dual, i.e. the space of tempered distributions. For given 𝑗 ⩾ 0 , the dyadic anisotropic block operator R 𝒂 𝑗 is dened on S ′ by R 𝒂 𝑗 𝑓 ( 𝑧 ) : = ( 𝜙 𝒂 𝑗 ˆ 𝑓 ) ˇ ( 𝑧 ) = ˇ 𝜙 𝒂 𝑗 ∗ 𝑓 ( 𝑧 ) , where the convolution is understood in the distributional sense. W e conventionally put R 𝒂 𝑖 : = 0 for 𝑖 < 0 . W e give a denition for anisotr opic Besov spaces ( cf. [ Tri06 , Chapter 5]). Denition 2.1 . Let 𝑠 ∈ ℝ , 𝑞 ∈ [ 1 , ∞] and 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 1 , ∞] 2 . The anisotropic Besov space is dened by 𝔹 𝑠 , 𝑞 𝒑 ; 𝒂 : =        𝑓 ∈ S ′ : ∥ 𝑓 ∥ 𝔹 𝑠 , 𝑞 𝒑 ; 𝒂 : =   𝑗 ⩾ 0  2 𝑗 𝑠 ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑  𝑞  1 / 𝑞 < ∞        . When 𝑞 = ∞ , we write 𝔹 𝑠 𝒑 ; 𝒂 : = 𝔹 𝑠 , ∞ 𝒑 ; 𝒂 . For any 𝒑 ∈ [ 1 , ∞] 2 , 𝑠 ′ > 𝑠 and 𝑞 ∈ [ 1 , ∞] , it holds that ([ HRZ23 , App endix B]) 𝔹 0 , 1 𝒑 ; 𝒂 ↩ → 𝕃 𝒑 ↩ → 𝔹 0 , ∞ 𝒑 ; 𝒂 , 𝔹 𝑠 ′ , ∞ 𝒑 ; 𝒂 ↩ → 𝔹 𝑠 , 1 𝒑 ; 𝒂 ↩ → 𝔹 𝑠 , 𝑞 𝒑 ; 𝒂 . (2.2) Lemma 2.2. For any 𝒑 ∈ [ 1 , ∞] 2 and 𝑚, 𝑛 ∈ ℕ 0 , there is a constant 𝐶 = 𝐶 ( 𝒑 , 𝑑 ) > 0 such that for any 𝑓 ∈ 𝔹 𝑚 + 3 𝑛, 1 𝒑 ; 𝒂 , ∥ ∇ 𝑚 𝑣 ∇ 𝑛 𝑥 𝑓 ∥ 𝒑 ≲ ∥ 𝑓 ∥ 𝔹 𝑚 + 3 𝑛, 1 𝒑 ; 𝒂 . (2.3) Proof. W e will make use of Bernstein’s inequalities (see [ ZZ24 ]): for any 𝑗 ⩾ 0 , ∥ ∇ 𝑘 1 𝑥 ∇ 𝑘 2 𝑣 R 𝒂 𝑗 𝑓 ∥ 𝒑 ′ ≲ 2 𝑗 𝒂 · ( 𝒌 + 𝑑 𝒑 − 𝑑 𝒑 ′ ) ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑 , ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑 ≲ ∥ 𝑓 ∥ 𝒑 . (2.4) It follows that ∥ ∇ 𝑚 𝑣 ∇ 𝑛 𝑥 𝑓 ∥ 𝒑 ⩽  𝑗 ⩾ − 1 ∥ ∇ 𝑚 𝑣 ∇ 𝑛 𝑥 R 𝒂 𝑗 𝑓 ∥ 𝒑 ≲  𝑗 ⩾ − 1 2 𝑗 ( 𝑚 + 3 𝑛 ) ∥ ∇ 𝑚 𝑣 ∇ 𝑛 𝑥 R 𝒂 𝑗 𝑓 ∥ 𝒑 ≲ ∥ 𝑓 ∥ 𝔹 𝑚 + 3 𝑛, 1 𝒑 ; 𝒂 , as desired. □ 2.2. Examples. W e discuss two specic examples of ( 1.4 ) and their corresponding convergence rates. Example 2.3 (Convolution) . W e show that 𝑏 𝑛 dened in ( 1.10 ) satises Theorem 1.2 for any 𝜗 ∈ ( 0 , 1 2 ( 𝒂 · 𝑑 𝒑 ) − 1 ] . W e se e that ( 1.5 ) holds because sup 𝑛 ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) ⩽ ∥ 𝑏 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) < ∞ . For each 𝑁 > 1 , Y oung’s convolution inequality implies that ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) ≲ ∥ ( 𝑏 1 | 𝑏 | < 𝑁 ) ∗ 𝜙 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) + ∥ ( 𝑏 1 | 𝑏 | ⩾ 𝑁 ) ∗ 𝜙 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) ≲ ∥ 𝑏 1 | 𝑏 | < 𝑁 ∥ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) + 𝑛 𝜗 𝒂 · 𝑑 𝒑 ∥ 𝑏 1 | 𝑏 | ⩾ 𝑁 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) ⩽ 𝑁 + 𝑛 𝜗 𝒂 · 𝑑 𝒑 ∥ 𝑏 1 | 𝑏 | ⩾ 𝑁 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) . 7 Hence, lim ℎ ↓ 0 sup 𝑛 ( 𝑛 − 1 / 2 ∧ ℎ ) ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) ≲ lim ℎ ↓ 0 ℎ 𝑁 2 + 𝑛 − 1 / 2 𝑛 𝜗 𝒂 · 𝑑 𝒑 ∥ 𝑏 1 | 𝑏 | ⩾ 𝑁 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) ⩽ ∥ 𝑏 1 | 𝑏 | ⩾ 𝑁 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) , (2.5) which vanishes as 𝑁 → ∞ . This shows ( 1.8 ) . A similar application of Y oung’s convolution inequality also shows that ( 1.7 ) with 𝜁 = 𝜗 𝒂 · 𝑑 𝒑 . Moreover , Theorem A.4 implies that for any 𝛿 ∈ ( 1 , 2 ) , ∥ 𝑏 𝑛 − 𝑏 ∥ 𝐿 ∞ ( [ 0 , 1 ] , 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ) ⩽ 𝑐 𝛿 𝑛 − 𝛿 𝜗 , which shows ( 1.6 ). Thus, by choosing 𝜗 ∈ [ 1 2 , 1 2 ( 𝒂 · 𝑑 𝒑 ) − 1 ] , ( 1.9 ) yields ∥ 𝜌 𝑡 − 𝜌 𝑛 𝑡 ∥ 𝒒 ′ ≲ 𝐶  𝑡 − ( 𝜗 + 1 2 ) 𝒂 · 𝑑 𝒒 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 )  𝑛 − 1 2 . Example 2.4 (Cut-o ) . W e show that 𝑏 𝑛 dened by ( 1.11 ) satises Theorem 1.2 with 𝜁 = 𝜅 and 𝜗 = 𝜅 ( 𝒂 · 𝑑 𝒑 ) − 1 . Obviously , sup 𝑛 ∥ 𝑏 𝑛 ∥ 𝕃 ∞ 𝑇 ( 𝕃 𝒑 ) ⩽ ∥ 𝑏 ∥ 𝕃 ∞ 𝑇 ( 𝕃 𝒑 ) < ∞ , so that ( 1.5 ) holds. When 𝒑 ≠ ( ∞ , ∞) , applying Theorem A.5 , we have ∥ 𝑏 𝑛 − 𝑏 ∥ 𝐿 ∞ ( [ 0 , 1 ] , 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ) ≲ ∥ 𝑏 1 | 𝑏 | > 𝑛 𝜅 ∥ 𝐿 ∞ ( [ 0 , 1 ] , 𝔹 − 𝛿 𝒑 ; 𝒂 ) ≲ 𝑛 − 𝛿 𝜅 ( 𝒂 · 𝑑 / 𝒑 ) − 1 , provided that 𝛿 ∈ ( 0 , ( ( 𝑝 𝑥 ∧ 𝑝 𝑣 ) − 1 ) 𝒂 · 𝑑 𝒑 ) . T o see that ( 1.8 ) holds, we note that ∥ 𝑏 𝑛 ∥ ∞ ≲ 𝑛 𝜅 so that lim ℎ ↓ 0 sup 𝑛 ( 𝑛 − 1 / 2 ∧ ℎ ) ∥ 𝑏 𝑛 ∥ ∞ ⩽ 𝐶 lim ℎ ↓ 0 sup 𝑛 ( 𝑛 − 1 / 2 ∧ ℎ ) 𝑛 𝜅 = 0 provided that 𝜅 < 1 2 . Lastly , ( 1.7 ) holds with 𝜁 = 𝜅 be cause sup 𝑛 ∈ ℝ 𝑑 𝑛 − 𝜅 sup 𝑡 ∈ [ 0 , 𝑛 − 1 ] ∥ 𝑏 𝑛 ( 𝑡 ) ∥ ∞ ≲ 1 . Hence in this case, we obtain fr om Theorem 1.3 that ∥ 𝜌 𝑡 − 𝜌 𝑛 𝑡 ∥ 𝒒 ′ ≲ 𝐶  𝑡 − 𝜅 − 1 2 𝒂 · 𝑑 𝒒 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 )  ( 𝑛 − 1 2 + 𝑛 − 𝜅 ( 𝒂 · 𝑑 𝒑 ) − 1 ) . 3. P r e l i m i n a r i e s 3.1. Estimates on Markov semigroup. For each 𝑡 > 0 , 𝑧 = ( 𝑥 , 𝑣 ) ∈ ℝ 𝑑 × ℝ 𝑑 and any bounde d measurable function 𝑓 : ℝ 2 𝑑 → ℝ , we denote 𝐺 𝑡 : =   𝑡 0 𝑊 𝑠 𝑑 𝑠 , 𝑊 𝑡  , 𝑀 𝑡 ( 𝑧 ) : = 𝐺 𝑡 + Γ 𝑡 𝑧 , (3.1) 𝑃 𝑡 𝑓 ( 𝑧 ) : = 𝔼 𝑓  𝑥 + 𝑡 𝑣 +  𝑡 0 𝑊 𝑠 𝑑 𝑠 , 𝑣 + 𝑊 𝑡  = 𝔼 𝑓  𝑀 𝑡 ( 𝑧 )  . (3.2) 8 The process ( 𝑀 𝑡 ( 𝑧 ) ) 𝑡 is the solution starting at 𝑧 to SDE ( 1.1 ) without the drift term (i.e. the underlying free process) and ( 𝑃 𝑡 ) 𝑡 is its associated Markov semigroup. Let 𝑔 𝑡 be the probability density of 𝐺 𝑡 , which is given by 𝑔 𝑡 ( 𝑥 , 𝑣 ) : =  𝜋 2 𝑡 4 3  − 𝑑 / 2 exp  − 3 | 𝑥 | 2 + | 3 𝑥 − 2 𝑡 𝑣 | 2 2 𝑡 3  . (3.3) It is straightforward to obtain the following relations 𝑔 𝑡 ( 𝑥 , 𝑣 ) = 𝑡 − 2 𝑑 𝑔 1 ( 𝑡 − 3 2 𝑥 , 𝑡 − 1 2 𝑣 ) , Γ 𝑡 𝑔 𝑡 ( 𝑥 , 𝑣 ) = 𝑡 − 2 𝑑 𝑔 1 ( 𝑡 − 3 2 𝑥 + 𝑡 − 1 2 𝑣 , 𝑡 − 1 2 𝑣 ) = 𝑡 − 2 𝑑 Γ 1 𝑔 1 ( 𝑡 − 3 2 𝑥 , 𝑡 − 1 2 𝑣 ) , (3.4) 𝑃 𝑡 𝑓 ( 𝑧 ) = ( 𝑔 𝑡 ∗ 𝑓 ) ( Γ 𝑡 𝑧 ) . (3.5) Lemma 3.1. Let 𝒑 ∈ [ 1 , ∞] 2 ; 𝛼 , 𝛽 ⩾ 0 and 𝑚, 𝑛 ∈ ℕ 0 . Then, there exist nite constants 𝐶 1 = 𝐶 1 ( 𝑑 , 𝒑 ) and 𝐶 2 = 𝐶 2 ( 𝑑 , 𝛼 , 𝛽 , 𝒑 , 𝑚, 𝑛 ) such that for every 𝑡 ∈ ( 0 , 1 ] , ∥ 𝑔 𝑡 ∥ 𝒑 ⩽ 𝐶 1 𝑡 − 𝒂 · 𝑑 2 ( 1 − 1 𝒑 ) , (3.6) and   | 𝑥 | 𝛼 | 𝑣 | 𝛽 | ∇ 𝑚 𝑥 ∇ 𝑛 𝑣 𝜕 𝑡 ( Γ 𝑡 𝑔 𝑡 ) ( 𝑥 , 𝑣 ) |   𝒑 ⩽ 𝐶 2 𝑡 3 ( 𝛼 − 𝑚 ) + ( 𝛽 − 𝑛 ) − 2 2 − 𝒂 · 𝑑 2 ( 1 − 1 𝒑 ) . (3.7) Proof. W e only show ( 3.7 ) , ( 3.6 ) can be obtained analogously . It follows from the scaling ( 3.4 ) that 𝜕 𝑡 ( Γ 𝑡 𝑔 𝑡 ) ( 𝑥 , 𝑣 ) = ( − 2 𝑑 ) 𝑡 − 2 𝑑 − 1 Γ 1 𝑔 1 ( 𝑡 − 3 / 2 𝑥 , 𝑡 − 1 / 2 𝑣 ) + ( − 3 / 2 ) 𝑡 − 2 𝑑 − 5 / 2 𝑥 · ∇ 𝑥 Γ 1 𝑔 1 ( 𝑡 − 3 / 2 𝑥 , 𝑡 − 1 / 2 𝑣 ) + ( − 1 / 2 ) 𝑡 − 2 𝑑 − 3 / 2 𝑣 · ∇ 𝑣 Γ 1 𝑔 1 ( 𝑡 − 3 / 2 𝑥 , 𝑡 − 1 / 2 𝑣 ) . Dening 𝐺 ( 𝑥 , 𝑣 ) : = ( − 2 𝑑 ) Γ 1 𝑔 1 ( 𝑥 , 𝑣 ) + ( − 3 / 2 ) 𝑥 · ∇ 𝑥 Γ 1 𝑔 1 ( 𝑥 , 𝑣 ) + ( − 1 / 2 ) 𝑣 · ∇ 𝑣 Γ 1 𝑔 1 ( 𝑥 , 𝑣 ) , one sees that 𝜕 𝑡 ( Γ 𝑡 𝑔 𝑡 ) ( 𝑥 , 𝑣 ) = 𝑡 − 2 𝑑 − 1 𝐺 ( 𝑡 − 3 / 2 𝑥 , 𝑡 − 1 / 2 𝑣 ) and then | 𝑥 | 𝛼 | 𝑣 | 𝛽 ∇ 𝑚 𝑥 ∇ 𝑛 𝑣 𝜕 𝑡 ( Γ 𝑡 𝑔 𝑡 ) ( 𝑥 , 𝑣 ) = 𝑡 − 2 𝑑 − 1 + [ 3 ( 𝛼 − 𝑚 ) + ( 𝛽 − 𝑛 ) ] / 2 | 𝑡 − 3 / 2 𝑥 | 𝛼 | 𝑡 − 1 / 2 𝑣 | 𝛽 ∇ 𝑚 𝑥 ∇ 𝑛 𝑣 𝐺 ( 𝑡 − 3 / 2 𝑥 , 𝑡 − 1 / 2 𝑣 ) . Therefore , we have   | 𝑥 | 𝛼 | 𝑣 | 𝛽 | ∇ 𝑚 𝑥 ∇ 𝑛 𝑣 𝜕 𝑡 ( Γ 𝑡 𝑔 𝑡 ) ( 𝑥 , 𝑣 ) |   𝒑 = 𝑡 − 2 𝑑 − 1 + [ 3 ( 𝛼 − 𝑚 ) + ( 𝛽 − 𝑛 ) ] / 2 𝑡 𝒂 · 𝑑 𝒑   | 𝑥 | 𝛼 | 𝑣 | 𝛽 | ∇ 𝑚 𝑥 ∇ 𝑛 𝑣 𝐺 ( 𝑥 , 𝑣 ) |   𝒑 ≲ 𝑡 3 ( 𝛼 − 𝑚 ) + ( 𝛽 − 𝑛 ) − 2 2 − 𝒂 · 𝑑 2 ( 1 − 1 𝒑 ) , because Γ 1 𝑔 1 is a Schwartz function. This completes the proof. □ W e note that for any 𝑡 ⩾ 0 and 𝒑 ∈ [ 1 , ∞] 2 , ∥ Γ 𝑡 𝑓 ∥ 𝒑 = ∥ 𝑓 ∥ 𝒑 , which shows that ( Γ 𝑡 ) 𝑡 ⩾ 0 is invariant on 𝕃 𝒑 . 9 Lemma 3.2. Let 𝛼 ∈ ( 0 , 1 ) , 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 1 , ∞] 2 and put 𝑝 : = 𝑝 𝑥 ∧ 𝑝 𝑣 . Let ( 𝑀 𝑡 ( 𝑧 ) ) , 𝑧 ∈ ℝ 2 𝑑 be as in ( 3.1 ) and 𝑓 ∈ 𝔹 𝛼 𝒑 , 𝒂 . There exists constant 𝐶 = 𝐶 ( 𝛼 , 𝑝 , 𝑑 , 𝒑 ) such that for every 0 < 𝑠 ⩽ 𝑡 and 𝑟 ⩾ 0 , for all 𝑧 ∥ 𝑓 ( 𝑀 𝑡 ( 𝑧 ) ) ∥ 𝐿 𝑝 ( Ω ) ⩽ 𝐶 𝑡 − 𝒂 · 𝑑 2 𝒑 ∥ 𝑓 ∥ 𝒑 . (3.8) Proof. W e recall that 𝑔 𝑡 , as dened in ( 3.3 ) , is the density of 𝐺 𝑡 from ( 3.1 ) . By Hölder’s inequality for 𝒒 : = ( 𝑝 𝑥 𝑝 𝑥 − 𝑝 , 𝑝 𝑣 𝑝 𝑣 − 𝑝 ) , we have ∥ 𝑓 ( 𝑀 𝑡 ( 𝑧 ) ) ∥ 𝑝 𝐿 𝑝 ( Ω ) =  ℝ 2 𝑑 | 𝑓 ( 𝑧 ′ + Γ 𝑡 𝑧 ) | 𝑝 𝑔 𝑡 ( 𝑧 ′ ) 𝑑 𝑧 ′ ≲ ∥ 𝑓 ∥ 𝑝 𝒑 ∥ 𝑔 𝑡 ∥ 𝒒 . In view of ( 3.6 ), this yields ( 3.8 ). □ Corollary 3.3. Let 𝑓 ∈ 𝕃 𝒑 ′ with 𝒑 ′ = ( 𝑝 ′ 𝑥 , 𝑝 ′ 𝑣 ) ∈ [ 1 , ∞] 2 . For any 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ [ 𝑝 ′ 𝑥 , ∞] × [ 𝑝 ′ 𝑣 , ∞] , any 𝛿 ∈ [ 0 , 1 ] , and any 𝑘 ∈ ℕ 0 , there exists constant 𝐶 > 0 depending on 𝒑 ′ , 𝑑 , 𝒑 , 𝛿 , 𝑘 such that for any ( 𝑠 , 𝑡 ) ∈ ( 0 , 1 ] 2 ⩽ , ∥ ∇ 𝑘 𝑣 ( 𝑃 𝑡 𝑓 − 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 ) ∥ 𝒑 ⩽ 𝐶 ∥ 𝑓 ∥ 𝒑 ′ | 𝑡 − 𝑠 | 𝛿 𝑠 𝑑 ( 3 2 ( 1 𝑝 𝑥 − 1 𝑝 ′ 𝑥 ) + 1 2 ( 1 𝑝 𝑣 − 1 𝑝 ′ 𝑣 ) ) − 𝑘 2 − 𝛿 . (3.9) In particular , for 𝑓 ∈ 𝕃 ∞ , we have for any ( 𝑠 , 𝑡 ) ∈ ( 0 , 1 ] 2 ⩽ that ∥ 𝑃 𝑡 𝑓 − 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 ∥ ∞ ⩽ 𝐶  [ ( 𝑡 − 𝑠 ) 𝑠 − 1 ] ∧ 1  ∥ 𝑓 ∥ ∞ . (3.10) Proof. It suces to sho w ( 3.9 ). Putting 𝐹 𝑡 : = Γ 𝑡 𝑓 and using ( 3.5 ), one sees that 𝑃 𝑡 𝑓 − 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 = ( Γ 𝑡 𝑔 𝑡 ) ∗ ( Γ 𝑡 𝑓 ) − ( Γ 𝑠 𝑔 𝑠 ) ∗ ( Γ 𝑡 𝑓 ) =  𝑡 𝑠 𝜕 𝑟 Γ 𝑟 𝑔 𝑟 ∗ 𝐹 𝑡 𝑑 𝑟 . From here , we apply Y oung’s inequality for convolution with 1 𝒑 + ( 1 , 1 ) = 1 𝒑 ′ + 1 𝒒 and ( 3.7 ) to obtain that ∥ ∇ 𝑘 𝑣 ( 𝑃 𝑡 𝑓 − 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 ) ∥ 𝒑 = ∥ ∇ 𝑘 𝑣  𝑡 𝑠 𝜕 𝑟 Γ 𝑟 𝑔 𝑟 ∗ 𝐹 𝑡 𝑑 𝑟 ∥ 𝒑 ≲  𝑡 𝑠 ∥ ∇ 𝑘 𝑣 𝜕 𝑟 Γ 𝑟 𝑔 𝑟 ∥ 𝒒 ∥ 𝐹 𝑡 ∥ 𝒑 ′ 𝑑 𝑟 ≲ ∥ 𝑓 ∥ 𝒑 ′  𝑡 𝑠 𝑟 𝑑 ( 3 2 𝑞 𝑥 + 1 2 𝑞 𝑣 − 2 ) − 𝑘 2 − 1 𝑑 𝑟 = ∥ 𝑓 ∥ 𝒑 ′  𝑡 𝑠 𝑟 𝑑 ( 3 2 ( 1 𝑝 𝑥 − 1 𝑝 ′ 𝑥 ) + 1 2 ( 1 𝑝 𝑣 − 1 𝑝 ′ 𝑣 ) ) − 1 − 𝑘 2 𝑑 𝑟 ≲ ∥ 𝑓 ∥ 𝒑 ′ | 𝑡 − 𝑠 | 𝛿 𝑠 𝑑 ( 3 2 ( 1 𝑝 𝑥 − 1 𝑝 ′ 𝑥 ) + 1 2 ( 1 𝑝 𝑣 − 1 𝑝 ′ 𝑣 ) ) − 𝑘 2 − 𝛿 for any 𝛿 ∈ [ 0 , 1 ] . This shows ( 3.9 ). □ W e conclude the current section with some analytic estimates on 𝑃 𝑡 . 10 Lemma 3.4. Let 𝛽 ∈ ℝ ; 𝒑 1 , 𝒑 2 ∈ [ 1 , ∞] 2 such that 𝒑 1 ⩽ 𝒑 2 . There exists a constant 𝐶 = 𝐶 ( 𝑑 , 𝛽 , 𝒑 1 , 𝒑 2 ) , such for any 𝑡 > 0 , ℎ ∈ 𝔹 𝛽 𝒑 1 ; 𝒂 and 𝑓 ∈ 𝕃 𝒑 1 , ∥ 𝑃 𝑡 ℎ ∥ 𝒑 2 ⩽ 𝐶 ( 1 ∧ 𝑡 ) 𝛽 2 − 1 2 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) ∥ ℎ ∥ 𝔹 𝛽 𝒑 1 ; 𝒂 if 𝛽 ≠ 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) , (3.11) ∥ ∇ 𝑣 𝑃 𝑡 ℎ ∥ 𝒑 2 ⩽ 𝐶 ( 1 ∧ 𝑡 ) 𝛽 2 − 1 2 − 1 2 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) ∥ ℎ ∥ 𝔹 𝛽 𝒑 1 ; 𝒂 if 𝛽 ≠ 1 + 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) , (3.12) ∥ 𝑃 𝑡 𝑓 ∥ 𝒑 2 ⩽ 𝐶 ( 1 ∧ 𝑡 ) − 1 2 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) ∥ 𝑓 ∥ 𝒑 1 , (3.13) 1 𝛽 + 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) ≠ 0 ∥ 𝑃 𝑡 𝑓 ∥ 𝔹 𝛽 , 1 𝒑 2 ; 𝒂 + ∥ 𝑃 𝑡 𝑓 ∥ 𝔹 𝛽 𝒑 2 ; 𝒂 ⩽ 𝐶 ( 1 ∧ 𝑡 ) − 𝛽 2 − 1 2 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) ∥ 𝑓 ∥ 𝒑 1 . (3.14) Proof. W e put 𝜅 : = 𝒂 · ( 𝑑 𝒑 1 − 𝑑 𝒑 2 ) . Let 𝛽 1 , 𝛽 2 be two real numbers. It is sho wn in [ HRZ23 , Lemma 2.16, (2.35)] that for any 𝑓 ∈ 𝔹 𝛽 1 𝒑 1 ; 𝒂 , 1 𝛽 2 ≠ 𝛽 1 − 𝜅 1 ∥ 𝑃 𝑡 𝑓 ∥ 𝔹 𝛽 2 , 1 𝒑 2 ; 𝒂 + ∥ 𝑃 𝑡 𝑓 ∥ 𝔹 𝛽 2 𝒑 2 ; 𝒂 ≲ ( 1 ∧ 𝑡 ) − ( 𝛽 2 − 𝛽 1 + 𝜅 ) 2 ∥ 𝑓 ∥ 𝔹 𝛽 1 𝒑 1 ; 𝒂 . (3.15) By cho osing ( 𝛽 1 , 𝛽 2 ) = ( 𝛽 , 0 ) and noting that ∥ 𝑃 𝑡 ℎ ∥ 𝒑 2 ≲ ∥ 𝑃 𝑡 ℎ ∥ 𝔹 0 , 1 𝒑 2 ; 𝒂 (by embe dding ( 2.2 ) ), we obtain ( 3.11 ) from ( 3.15 ) . By Theorem 2.2 , ∥ ∇ 𝑣 𝑃 𝑡 ℎ ∥ 𝒑 2 ≲ ∥ ∇ 𝑣 𝑃 𝑡 ℎ ∥ 𝔹 1 , 1 𝒑 2 ; 𝒂 , we can obtain ( 3.12 ) from ( 3.15 ) in a similar way . When 𝒑 1 ≠ 𝒑 2 , we take 𝛽 = 0 in ( 3.11 ) and apply the inequality ∥ 𝑓 ∥ 𝔹 0 𝒑 1 ; 𝒂 ≲ ∥ 𝑓 ∥ 𝒑 1 (by embedding ( 2.2 ) ), to obtain ( 3.13 ) . In the case when 𝒑 1 = 𝒑 2 , ( 3.13 ) is straightforward from denitions. Similarly , ( 3.14 ) is a conse quence of emb edding ( 2.2 ) and ( 3.15 ) upon taking ( 𝛽 1 , 𝛽 2 ) = ( 0 , 𝛽 ) . □ 3.2. Exponential estimates. The next result allows applications of Girsanov transform. Proposition 3.5. Let 𝒑 0 ∈ [ 1 , ∞] 2 be such that 𝒂 · 𝑑 𝒑 0 < 2 . Let ∥ 𝑓 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 0 ) < ∞ . Let ( 𝑓 𝑛 ) 𝑛 be a sequence of b ounded functions on [ 0 , 1 ] × ℝ 2 𝑑 such that sup 𝑛 ∥ 𝑓 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 0 ) < ∞ and lim ℎ ↓ 0 sup 𝑛 ( 𝑛 − 1 ∧ ℎ ) ∥ 𝑓 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 ∞ ) = 0 . (3.16) Then for any real numb er 𝑐 , sup 𝑧 𝔼 exp  𝑐  1 0 𝑓 ( 𝑟 , 𝑀 𝑟 ( 𝑧 ) ) 𝑑 𝑟  < ∞ , (3.17) sup 𝑛 sup 𝑧 𝔼 exp  𝑐  1 0 𝑓 𝑛 ( 𝑟 , 𝑀 𝑘 𝑛 ( 𝑟 ) ( 𝑧 ) ) 𝑑 𝑟  < ∞ . (3.18) Proof. ( 3.17 ) follows from the Kr ylov’s estimate [ RZ24 , Lemma 4.1] and the similar way of showing ( 3.18 ). Therefor e we only need to show ( 3.18 ). Without loss of generality we can assume that ( 𝑓 𝑛 ) 𝑛 and 𝑐 are non-negative . W e x ( 𝑠 , 𝑡 ) ∈ [ 0 , 1 ] 2 ⩽ and put 𝐼 𝑠 , 𝑡 : =  𝑡 𝑠 𝑓 𝑛 ( 𝑟 , 𝑀 𝑘 𝑛 ( 𝑟 ) ( 𝑧 ) ) 𝑑 𝑟 . Denoting 𝑠 : = 𝑘 𝑛 ( 𝑠 ) + 1 𝑛 , we have 𝐼 𝑠 , 𝑡 =   𝑠 ∧ 𝑡 𝑠 +  𝑡 𝑠 ∧ 𝑡  𝑓 𝑛 ( 𝑟 , 𝑀 𝑘 𝑛 ( 𝑟 ) ( 𝑧 ) ) 𝑑 𝑟 = : 𝐼 1 + 𝐼 2 . 11 W e use ( 3.16 ) to have 𝔼 𝑠 𝐼 1 ≲ sup 𝑟 ∈ [ 𝑠 , ¯ 𝑠 ∧ 𝑡 ] ∥ 𝑓 𝑛 ( 𝑟 ) ∥ ∞ ( 𝑠 ∧ 𝑡 − 𝑠 ) ≲ ( 𝑛 − 1 ∧ ( 𝑡 − 𝑠 ) ) ∥ 𝑓 𝑛 ∥ ∞ . When 𝑟 ⩾ 𝑠 , 𝑘 𝑛 ( 𝑟 ) ⩾ 𝑠 . By Markov property and ( 3.8 ), we have 𝔼 𝑠 𝐼 2 =  𝑡 𝑠 ∧ 𝑡 𝑃 𝑘 𝑛 ( 𝑟 ) − 𝑠 𝑓 𝑛 ( 𝑟 , 𝑀 𝑠 ( 𝑧 ) ) 𝑑 𝑟 ≲  𝑡 𝑠 ∧ 𝑡 | 𝑘 𝑛 ( 𝑟 ) − 𝑠 | − 𝒂 · 𝑑 2 𝒑 0 ∥ 𝑓 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 0 ) 𝑑 𝑟 . It is straightforward to verify that  𝑡 𝑠 ∧ 𝑡 | 𝑘 𝑛 ( 𝑟 ) − 𝑠 | − 𝒂 · 𝑑 2 𝒑 0 𝑑 𝑟 ≲ ( 𝑡 − 𝑠 ) 1 − 𝒂 · 𝑑 2 𝒑 0 ⩽ ( 𝑡 − 𝑠 ) 1 − 𝒂 · 𝑑 2 𝒑 0 (for details, see [ LL25 , Lemma 3.10]). For any ℎ > 0 , we denote 𝑜 𝑠 𝑐 ( ℎ ) : = sup 𝑛 ( 𝑛 − 1 ∧ ℎ ) ∥ 𝑓 𝑛 ∥ ∞ + ℎ 1 − 𝒂 · 𝑑 2 𝒑 0 sup 𝑛 ∥ 𝑓 𝑛 ∥ 𝕃 ∞ 𝑇 𝕃 𝒑 0 . W e have shown that there is a nite constant 𝜅 such that 𝔼 𝑠 𝐼 𝑠 , 𝑡 ⩽ 𝜅 𝑜 𝑠𝑐 ( 𝑡 − 𝑠 ) for every ( 𝑠 , 𝑡 ) ∈ [ 0 , 1 ] 2 ⩽ . This implies (see similar arguments in [ Lê22 , Theorem 2.3] and [ LL25 , Lemma 3.5]) that for all 𝑚 ∈ ℕ and ( 𝑠 , 𝑡 ) ∈ [ 0 , 1 ] 2 ⩽ , 𝔼 𝑠 𝐼 𝑚 𝑠 , 𝑡 ⩽ 𝑚 ! 𝜅 𝑚 𝑜 𝑠 𝑐 ( 𝑡 − 𝑠 ) 𝑚 . Using ( 3.16 ) , we can choose a ℎ 0 > 0 such that 𝑜 𝑠 𝑐 ( ℎ ) ⩽ 1 2 𝑐𝜅 for all ℎ ⩽ ℎ 0 . Then, we have for any 𝑡 − 𝑠 ⩽ ℎ 0 , by T aylor expansion, 𝔼 𝑠 exp ( 𝑐 𝐼 𝑠 , 𝑡 ) ⩽ ∞  𝑚 = 0 2 − 𝑚 = 2 , 𝑎 . 𝑠 . Thus, for 𝑁 : = ⌊ ℎ − 1 0 ⌋ , we have 𝔼 exp  𝑐  1 0 𝑓 𝑛 ( 𝑟 , 𝑀 𝑘 𝑛 ( 𝑟 ) ) 𝑑 𝑟  = 𝔼 𝑁 − 1  𝑘 = 0 𝔼 𝑘 ℎ 0 exp  𝑐  ( 𝑘 + 1 ) ℎ 0 𝑘 ℎ 0 𝑓 𝑛 ( 𝑟 , 𝑀 𝑘 𝑛 ( 𝑟 ) ) 𝑑 𝑟  𝔼 𝑁 ℎ 0 exp  𝑐  1 𝑁 ℎ 0 𝑓 𝑛 ( 𝑟 , 𝑀 𝑘 𝑛 ( 𝑟 ) ) 𝑑 𝑟  ⩽ 2 𝑁 + 1 . Since 𝑁 is independent of 𝑛 , this completes the proof. □ 4. P r o o f o f m a i n r e s u l t s W e provide the proof of Theorem 1.3 . Let S be the space of Schwartz functions on ℝ 2 𝑑 . W e begin with a dual identity . Lemma 4.1. Let 𝒓 ∈ ( 1 , ∞) 2 , 𝒓 ′ be the Hölder conjugate of 𝒓 and 𝑓 be a function in 𝕃 𝒓 . Then ∥ 𝑓 ∥ 𝒓 = sup 𝑔 ∈S : ∥ 𝑔 ∥ 𝒓 ′ = 1  ℝ 2 𝑑 𝑓 ( 𝑧 ) 𝑔 ( 𝑧 ) 𝑑 𝑧 . (4.1) 12 Proof. W e assume without loss of generality that 𝑓 ≠ 0 . Dene 𝑔 ( 𝑥 , 𝑣 ) = ∥ 𝑓 ∥ 1 − 𝑟 𝑣 𝒓 1 ( ∥ 𝑓 ( · ,𝑣 ) ∥ 𝑟 𝑥 > 0 ) ∥ 𝑓 ( · , 𝑣 ) ∥ 𝑟 𝑣 − 𝑟 𝑥 𝑟 𝑥 | 𝑓 ( 𝑥 , 𝑣 ) | 𝑟 𝑥 − 1 sgn ( 𝑓 ( 𝑥 , 𝑣 ) ) . It is straightfor ward to verify that ∥ 𝑓 ∥ 𝒓 =  𝑓 ( 𝑧 ) 𝑔 ( 𝑧 ) 𝑑 𝑧 and that ∥ 𝑔 ∥ 𝒓 ′ ⩽ 1 . T ogether with Hölder inequality , this shows (analogously to classical arguments [ Fol99 , 6.13 Proposition]) that ∥ 𝑓 ∥ 𝒓 = sup 𝑔 ∈ 𝕃 𝒓 ′ : ∥ 𝑔 ∥ 𝒓 ′ ⩽ 1  ℝ 2 𝑑 𝑓 ( 𝑧 ) 𝑔 ( 𝑧 ) 𝑑 𝑧 . Since S is dense in 𝕃 𝒓 ′ , this implies ( 4.1 ). □ Throughout the r est of the section, Theorems 1.1 and 1.2 are always enfor ced thr oughout the section. W e x 𝑡 ∈ ( 0 , 1 ] and 𝜑 ∈ S . By applying Itô ’s formula to 𝑟 → 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) and 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) , one sees that 𝔼 𝜑 ( 𝑍 𝑡 ) = 𝔼 𝑃 𝑡 𝜑 ( 𝑍 0 ) + 𝔼  𝑡 0 𝑏 ( 𝑟 , 𝑍 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) 𝑑 𝑟 , and 𝔼 𝜑 ( 𝑍 𝑛 𝑡 ) = 𝔼 𝑃 𝑡 𝜑 ( 𝑍 0 ) + 𝔼  𝑡 0 Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟 . This implies that | ⟨ 𝜑 , 𝜌 𝑡 − 𝜌 𝑛 𝑡 ⟩ | ⩽    𝔼  𝑡 0 ( 𝑏 − 𝑏 𝑛 ) ( 𝑟 , 𝑍 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) 𝑑 𝑟    +    𝔼  𝑡 0 𝑏 𝑛 ( 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑟 ) − 𝑏 𝑛 ( 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟    +    𝔼  𝑡 0  𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑟 ) − Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) )  · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟    = : 𝐼 𝑛 1 ( 𝑡 ) + 𝐼 𝑛 2 ( 𝑡 ) + 𝐼 𝑛 3 ( 𝑡 ) . (4.2) In the following we estimate 𝐼 𝑛 𝑖 , 𝑖 = 1 , 2 , 3 separately . 4.1. T aming error. W e derive estimates for 𝐼 𝑛 1 from ( 4.2 ) . As an intermediate step, we hav e the following result. Lemma 4.2. For any ¯ 𝒑 , ¯ 𝒒 ∈ [ 2 , ∞] 2 with 1 / ¯ 𝒑 + 1 / ¯ 𝒒 ⩽ 1 , there are constants 𝐶 𝑖 > 0 , 𝑖 = 1 , 2 , such that for any 0 < 𝑠 ⩽ 𝑡 ⩽ 1 and 𝑓 1 ∈ 𝕃 ¯ 𝒑 , 𝑓 2 ∈ 𝕃 ¯ 𝒒 , 𝔼 | 𝑓 1 ( 𝑍 𝑠 ) 𝑓 2 ( 𝑍 𝑡 ) | ⩽ 𝐶 1 𝑠 − 1 2 ( 𝒂 · 𝑑 ¯ 𝒑 + 𝒂 · 𝑑 ¯ 𝒒 ) ∥ 𝑓 1 ∥ ¯ 𝒑 ∥ 𝑓 2 ∥ ¯ 𝒒 , (4.3) 𝔼 | 𝑓 1 ( 𝑍 𝑛 𝑠 ) 𝑓 2 ( 𝑍 𝑛 𝑡 ) | ⩽ 𝐶 2 𝑠 − 1 2 ( 𝒂 · 𝑑 ¯ 𝒑 + 𝒂 · 𝑑 ¯ 𝒒 ) ∥ 𝑓 1 ∥ ¯ 𝒑 ∥ 𝑓 2 ∥ ¯ 𝒒 . (4.4) In particular , there is a nite constant 𝐶 > 0 such that for any 𝑡 ∈ ( 0 , 1 ] ∥ 𝜌 𝑛 𝑡 ∥ ¯ 𝒑 ′ ⩽ 𝐶 𝑡 − 1 2 ( 𝒂 · 𝑑 ¯ 𝒑 ) where 1 / ¯ 𝒑 ′ + 1 / 𝒑 = ( 1 , 1 ) . (4.5) 13 Proof. Let us rst show ( 4.4 ) . ( 4.3 ) then follows in a similar and even easier manner . Let 𝑀 𝑡 ( 𝑧 ) be dened as ( 3.1 ). Let 𝜌 = 𝜌 𝑛 ( 𝑧 ) : = exp  −  1 0 Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑠 , 𝑀 𝑘 𝑛 ( 𝑠 ) ( 𝑧 ) ) 𝑑𝑊 𝑠 − 1 2  1 0   Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑠 , 𝑀 𝑘 𝑛 ( 𝑠 ) ( 𝑧 ) )   2 𝑑 𝑠  . Note that 𝑓 𝑛 ( 𝑡 , 𝑥 ) : = | Γ 𝑡 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑡 , 𝑥 ) | 2 satises ( 3.16 ) with 𝒑 0 : = 𝒑 / 2 . Applying Theorem 3.5 , we see that for any 𝜆 > 0 , sup 𝑛 sup 𝑧 𝔼 exp  𝜆  1 0   Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑠 , 𝑀 𝑘 𝑛 ( 𝑠 ) ( 𝑧 ) )   2 𝑑 𝑠  < ∞ . This implies that 𝔼 𝜌 = 1 and sup 𝑛 sup 𝑧 𝔼 ( 𝜌 ) 𝜆 < ∞ for any 𝜆 > 0 . By Girsanov theorem, we see that 𝑍 𝑛 ( 𝑧 ) shares the same law under ℙ as 𝑀 ( 𝑧 ) under ℚ , where 𝑑 ℚ : = 𝜌 𝑑 ℙ . In particular , we have 𝔼 | 𝑓 1 ( 𝑍 𝑛 𝑠 ) 𝑓 2 ( 𝑍 𝑛 𝑡 ) | = 𝔼 ℚ | 𝑓 1 ( 𝑀 𝑠 ) 𝑓 2 ( 𝑀 𝑡 ) | . By Hölder inequality 𝔼 | 𝑓 1 ( 𝑍 𝑛 𝑠 ) 𝑓 2 ( 𝑍 𝑛 𝑡 ) | ≲ ∥ 𝑓 1 ( 𝑀 𝑠 ) 𝑓 2 ( 𝑀 𝑡 ) ∥ 𝐿 2 ( Ω ) =   ℝ 4 𝑑 | 𝑓 1 | 2 ( 𝑧 ) | 𝑓 2 | 2 ( Γ 𝑡 − 𝑠 𝑧 + 𝑧 ′ ) 𝑔 𝑡 − 𝑠 ( 𝑧 ′ ) 𝑔 𝑠 ( 𝑧 ) 𝑑 𝑧𝑑 𝑧 ′  1 2 , which, upon applying Hölder inequality to the integral with respect to 𝑧 , yields 𝔼 | 𝑓 1 ( 𝑍 𝑛 𝑠 ) 𝑓 2 ( 𝑍 𝑛 𝑡 ) | ≲ ∥ 𝑓 1 ∥ ¯ 𝒑 ∥ 𝑓 2 ◦ Γ 𝑡 − 𝑠 ∥ ¯ 𝒒 ∥ 𝑔 𝑡 − 𝑠 ∥ 1 ∥ 𝑔 𝑠 ∥ 𝒓 ≲ 𝑠 − 𝒂 · 𝑑 2 𝒓 ∥ 𝑓 1 ∥ ¯ 𝒑 ∥ 𝑓 2 ∥ ¯ 𝒒 where 1 / 𝒓 = 1 / ¯ 𝒑 + 1 / ¯ 𝒒 , which is ( 4.4 ) . ( 4.5 ) is simply obtained by duality and ( 4.4 ) with taking 𝑓 1 therein to be some constant function and ¯ 𝒒 = ( ∞ , ∞) . □ Remark 4.3 . The condition ( 1.8 ) , which rules out 𝜅 = 1 / 2 in ( 1.11 ) , is crucial for the application of Girsanov theorem in the pr evious proof. It may be possible to employ parametrix method (as in [ JM24 ]), avoiding usage of the Girsanov theorem, to treat the case of ( 1.11 ) with 𝜅 = 1 / 2 . Lemma 4.4. Recall that 𝐼 𝑛 1 ( 𝑡 ) is dened in ( 4.2 ) . For any 𝒑 ⩽ 𝒒 , 𝑡 ∈ ( 0 , 1 ] , we have | 𝐼 𝑛 1 ( 𝑡 ) | ≲ 𝑛 − 𝜗 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 ) ∥ 𝜑 ∥ 𝒒 . (4.6) Proof. For any 𝑠 ∈ ( 0 , 𝑡 ] , we put ℎ 𝑠 : = ( 𝑏 𝑛 ( 𝑠 ) − 𝑏 ( 𝑠 ) ) · ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 . Applying Itô’s formula to 𝑟 ↦→ 𝑃 𝑠 − 𝑟 ℎ 𝑠 ( 𝑍 𝑟 ) , we have 𝔼 ℎ 𝑠 ( 𝑍 𝑠 ) = 𝔼 𝑃 𝑠 ℎ 𝑠 ( 𝑍 0 ) + 𝔼  𝑠 0 𝑏 ( 𝑟 , 𝑍 𝑟 ) · ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ 𝑠 ( 𝑍 𝑟 ) 𝑑 𝑟 , which by ( 4.3 ) implies that | 𝔼 ℎ 𝑠 ( 𝑍 𝑠 ) | ≲ ∥ 𝑃 𝑠 ℎ 𝑠 ∥ ∞ +  𝑠 0 𝑟 − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ 𝑠 ∥ 𝒑 𝑑 𝑟 . 14 W e note that ℎ belongs to 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) , hence, 𝔼  1 0 | ℎ 𝑠 ( 𝑍 𝑠 ) | 𝑑 𝑠 is nite in view of ( 4.3 ) . Thus, by Fubini and the previous estimate , we have 𝐼 𝑛 1 ( 𝑡 ) ⩽  𝑡 0 ∥ 𝑃 𝑠 ℎ 𝑠 ∥ ∞ 𝑑 𝑠 +  𝑡 0  𝑠 0 𝑟 − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ 𝑠 ∥ 𝒑 𝑑 𝑟 𝑑 𝑠 . (4.7) Using Bony’ paraproduct (see Section B ), we decompose ℎ 𝑠 = ℎ ≺ 𝑠 + ℎ ≽ 𝑠 where ℎ ≺ 𝑠 : = ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ≺ ( 𝑏 𝑛 − 𝑏 ) ( 𝑠 ) and ℎ ≽ 𝑠 : = ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ≽ ( 𝑏 𝑛 − 𝑏 ) ( 𝑠 ) . W e estimate each term in ( 4.7 ) accordingly . First, by ( B.2 ) and Theorem 2.2 , we have for 𝛼 ∈ { 0 , − 𝛿 } ∥ ℎ ≺ 𝑠 ∥ 𝔹 𝛼 𝒑 ; 𝒂 ≲ ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ ∞ ∥ 𝑏 𝑛 − 𝑏 ∥ 𝔹 𝛼 𝒑 ; 𝒂 ≲ ∥ 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 1 , 1 ∞ ; 𝒂 ∥ 𝑏 𝑛 − 𝑏 ∥ 𝔹 𝛼 𝒑 ; 𝒂 . Applying ( 3.14 ), we have ∥ 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 1 , 1 ∞ ; 𝒂 ≲ ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 . These estimates imply that ∥ ℎ ≺ 𝑠 ∥ 𝔹 𝛼 𝒑 ; 𝒂 ≲ 𝑛 𝛼 𝜗 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 . (4.8) T ogether with ( 3.11 ), this implies that ∥ 𝑃 𝑠 ℎ ≺ 𝑠 ∥ 𝕃 ∞ ≲ min ( 𝑠 − 𝛿 2 − 𝒂 · 𝑑 2 𝒑 ∥ ℎ ≺ 𝑠 ∥ 𝔹 − 𝛿 𝒑 ; 𝒂 , 𝑠 − 𝒂 · 𝑑 2 𝒑 ∥ ℎ ≺ 𝑠 ∥ 𝔹 0 𝒑 ; 𝒂 ) ≲ 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 𝑛 − 𝜗 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 . (4.9) Concerning the term of ℎ ≽ 𝑠 , we dene 𝒓 via the relation 1 / 𝒓 : = 1 / 𝒑 + 1 / 𝒒 , then apply ( B.8 ) and Theorem 2.2 to see that ∥ ℎ ≽ 𝑠 ∥ 𝒓 ≲ min  ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 𝛿 , 1 𝒒 ; 𝒂 ∥ 𝑏 𝑛 − 𝑏 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 , ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 0 , 1 𝒒 ; 𝒂 ∥ 𝑏 𝑛 − 𝑏 ∥ 𝒑  ≲ min  ∥ 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 1 + 𝛿, 1 𝒒 ; 𝒂 ∥ 𝑏 𝑛 − 𝑏 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 , ∥ 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 1 , 1 𝒒 ; 𝒂 ∥ 𝑏 𝑛 − 𝑏 ∥ 𝒑  . W e use ( 3.14 ) and Theorem 1.2 to estimate each term on the right-hand side ab ov e, this and embedding ( 2.2 ) yield ∥ ℎ ≽ 𝑠 ∥ 𝔹 0 𝒓 ; 𝒂 ≲ ∥ ℎ ≽ 𝑠 ∥ 𝒓 ≲ ∥ 𝜑 ∥ 𝒒 ( 𝑡 − 𝑠 ) − 1 2  [ ( 𝑡 − 𝑠 ) 𝑛 2 𝜗 ] − 𝛿 2 ∧ 1  . (4.10) Then, applying ( 3.13 ), we have ∥ 𝑃 𝑠 ℎ ≽ 𝑠 ∥ ∞ ≲ 𝑠 − 𝒂 · 𝑑 2 𝒓 ∥ ℎ ≽ 𝑠 ∥ 𝒓 ≲ 𝑠 − 𝒂 · 𝑑 2 𝒓 ∥ 𝜑 ∥ 𝒒 ( 𝑡 − 𝑠 ) − 1 2  [ ( 𝑡 − 𝑠 ) 𝑛 2 𝜗 ] − 𝛿 2 ∧ 1  . T ogether with ( 4.9 ) and ( A.2 ), we obtain that  𝑡 0 ∥ 𝑃 𝑠 ℎ 𝑠 ∥ ∞ 𝑑 𝑠 ⩽  𝑡 0 ∥ 𝑃 𝑠 ℎ ≺ 𝑠 ∥ ∞ 𝑑 𝑠 +  𝑡 0 ∥ 𝑃 𝑠 ℎ ≽ 𝑠 ∥ ∞ 𝑑 𝑠 ≲ ∥ 𝜑 ∥ 𝒒  𝑛 − 𝜗  𝑡 0 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝑞 𝑑 𝑠 +  𝑡 0 𝑠 − 𝒂 · 𝑑 2 𝒓 ( 𝑡 − 𝑠 ) − 1 2  [ ( 𝑡 − 𝑠 ) 𝑛 2 𝜗 ] − 𝛿 2 ∧ 1  𝑑 𝑠  15 ≲ ∥ 𝜑 ∥ 𝒒 𝑛 − 𝜗 𝑡 − 𝒂 · 𝑑 2 𝒓 . (4.11) On the other hand, by ( 3.12 ), we have ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ 𝑠 ∥ 𝒑 ⩽ ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ ≺ 𝑠 ∥ 𝒑 + ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ ≽ 𝑠 ∥ 𝒑 ≲ min  ( 𝑠 − 𝑟 ) − 1 + 𝛿 2 ∥ ℎ ≺ 𝑠 ∥ 𝔹 − 𝛿 𝒑 ; 𝒂 , ( 𝑠 − 𝑟 ) − 1 2 ∥ ℎ ≺ 𝑠 ∥ 𝔹 0 𝒑 ; 𝒂  + ( 𝑠 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ ℎ ≽ 𝑠 ∥ 𝔹 0 𝒓 ; 𝒂 , which, by ( 4.8 ), ( 4.10 ), implies that ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ 𝑠 ∥ 𝒑 ≲ ∥ 𝜑 ∥ 𝒒 ( 𝑠 − 𝑟 ) − 1 2 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒   [ ( 𝑠 − 𝑟 ) 𝑛 2 𝜗 ] − 𝛿 2  ∧ 1  + ∥ 𝜑 ∥ 𝒒 ( 𝑠 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ( 𝑡 − 𝑠 ) − 1 2  [ ( 𝑡 − 𝑠 ) 𝑛 2 𝜗 ] − 𝛿 2 ∧ 1  . Using the previous estimate and ( A.2 ), w e have  𝑡 0  𝑠 0 𝑟 − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 𝑃 𝑠 − 𝑟 ℎ 𝑠 ∥ 𝒑 𝑑 𝑟 𝑑 𝑠 ≲ ∥ 𝜑 ∥ 𝒒   𝑡 0  𝑠 0 𝑟 − 𝒂 · 𝑑 𝒑 ( 𝑠 − 𝑟 ) − 1 2 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒   [ ( 𝑠 − 𝑟 ) 𝑛 2 𝜗 ] − 𝛿 2  ∧ 1  𝑑 𝑟 𝑑 𝑠 +  𝑡 0  𝑠 0 𝑟 − 𝒂 · 𝑑 𝒑 ( 𝑠 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ( 𝑡 − 𝑠 ) − 1 2  [ ( 𝑡 − 𝑠 ) 𝑛 2 𝜗 ] − 𝛿 2 ∧ 1  𝑑 𝑟 𝑑 𝑠  ≲ ∥ 𝜑 ∥ 𝒒  𝑛 − 𝜗  𝑡 0 𝑠 − 𝒂 · 𝑑 𝒑 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 𝑑 𝑠 +  𝑡 0 𝑠 1 2 − 𝒂 · 𝑑 𝒑 − 𝒂 · 𝑑 2 𝒒 ( 𝑡 − 𝑠 ) − 1 2  [ ( 𝑡 − 𝑠 ) 𝑛 2 𝜗 ] − 𝛿 2 ∧ 1  𝑑 𝑠  ≲ ∥ 𝜑 ∥ 𝒒 𝑛 − 𝜗 𝑡 1 2 − 𝒂 · 𝑑 𝒑 − 𝒂 · 𝑑 2 𝒒 . (4.12) Plugging ( 4.11 ) and ( 4.12 ) into ( 4.7 ), we obtain ( 4.6 ). □ 4.2. Quadrature error. T o estimate the quadrature err or , the term 𝐼 𝑛 3 in ( 4.2 ) , we will make use of the following results. Lemma 4.5. For any 𝒒 ∈ [ 1 , ∞] 2 with 1 / 𝒑 + 1 / 𝒒 ⩽ 1 , then there is a constant 𝐶 > 0 such that for all 𝑡 ∈ ( 𝑛 − 1 , 1 ) , 𝑓 ∈ S and ℎ ∈ 𝕃 𝒑 ,    𝔼 ℎ ( 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) )  Γ 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑓 ( 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) ) − 𝑓 ( 𝑍 𝑛 𝑡 )     ⩽ 𝐶 𝑛 − 1 ∥ ℎ ∥ 𝒑  ( 𝑘 𝑛 ( 𝑡 ) ) − 𝒂 · 𝑑 𝒑  𝑛 − 1 ∥ ∇ 𝑥 𝑓 ∥ 𝕃 ∞ + ∥ ∇ 𝑣 𝑓 ∥ 𝕃 ∞  + ( 𝑘 𝑛 ( 𝑡 ) ) − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 ) ( ∥ 𝑓 ∥ 𝔹 2 𝒒 ; 𝒂 + ∥ ∇ 2 𝑣 𝑓 ∥ 𝕃 𝒒 )  . (4.13) Proof. First w e dene 𝐴 𝑛 ( 𝑡 , 𝑧 ) : =  𝑡 𝑘 𝑛 ( 𝑡 )  𝑠 𝑘 𝑛 ( 𝑡 ) Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑧 ) 𝑑 𝑟 𝑑 𝑠 , 𝐵 𝑛 ( 𝑡 , 𝑧 ) : =  𝑡 𝑘 𝑛 ( 𝑡 ) Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑠 , 𝑧 ) 𝑑 𝑠 . 16 Note that 𝑍 𝑛 𝑡 − 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) =   𝑡 𝑘 𝑛 ( 𝑡 ) 𝑉 𝑛 𝑟 𝑑 𝑟 , 𝐵 𝑛 ( 𝑡 , 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) ) + 𝑊 𝑡 − 𝑊 𝑘 𝑛 ( 𝑡 )  =  ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) ) 𝑉 𝑛 𝑘 𝑛 ( 𝑡 ) + 𝐴 𝑛 ( 𝑡 , 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) ) +  𝑡 𝑘 𝑛 ( 𝑡 ) ( 𝑊 𝑠 − 𝑊 𝑘 𝑛 ( 𝑡 ) ) 𝑑 𝑠 , 𝐵 𝑛 ( 𝑡 , 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) ) + 𝑊 𝑡 − 𝑊 𝑘 𝑛 ( 𝑡 )  . Since  𝑊 𝑡 − 𝑊 𝑘 𝑛 ( 𝑡 ) ,  𝑡 𝑘 𝑛 ( 𝑡 ) ( 𝑊 𝑠 − 𝑊 𝑘 𝑛 ( 𝑡 ) ) 𝑑 𝑠  ( 𝑑 ) =  𝑊 𝑡 − 𝑘 𝑛 ( 𝑡 ) ,  𝑡 − 𝑘 𝑛 ( 𝑡 ) 0 𝑊 𝑠 𝑑 𝑠  = 𝑀 𝑡 − 𝑘 𝑛 ( 𝑡 ) is independent of 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) , recalling that 𝑀 𝑡 and 𝑍 𝑛 𝑡 admit densities 𝑔 𝑡 and 𝜌 𝑛 𝑡 , respectively , we have ℐ 𝑛 ( 𝑡 ) : =    𝔼 ℎ ( 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) )  Γ 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑓 ( 𝑍 𝑛 𝑘 𝑛 ( 𝑡 ) ) − 𝑓 ( 𝑍 𝑛 𝑡 )     =     ℝ 4 𝑑  𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑣 , 𝑣 ) − 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑣 ) + 𝐴 𝑛 ( 𝑡 , 𝑧 ) + 𝑥 ′ , 𝑣 + 𝐵 𝑛 ( 𝑡 , 𝑧 ) + 𝑣 ′ )  ℎ ( 𝑧 ) 𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ′ ) 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ) 𝑑 𝑧𝑑 𝑧 ′    , where 𝑧 = ( 𝑥 , 𝑣 ) and 𝑧 ′ = ( 𝑥 ′ , 𝑣 ′ ) in the integral. W e note that 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑣 , 𝑣 ) − 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑣 ) + 𝐴 𝑛 ( 𝑡 , 𝑧 ) + 𝑥 ′ , 𝑣 + 𝐵 𝑛 ( 𝑡 , 𝑧 ) + 𝑣 ′ ) = − 𝛿 ( 𝐴 𝑛 ( 𝑡 ,𝑧 ) ,𝐵 𝑛 ( 𝑡 ,𝑧 ) ) 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑣 ) + 𝑥 ′ , 𝑣 + 𝑣 ′ ) − 𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) ) 𝑣 , 𝑣 ) , where 𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 ( 𝑥 , 𝑣 ) : = 𝑓 ( 𝑥 + 𝑥 ′ , 𝑣 + 𝑣 ′ ) − 𝑓 ( 𝑥 , 𝑣 ) . W e note that | 𝛿 ( 𝐴 𝑛 ( 𝑡 ,𝑧 ) ,𝐵 𝑛 ( 𝑡 ,𝑧 ) ) 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) 𝑣 ) + 𝑥 ′ , 𝑣 + 𝑣 ′ ) | ⩽ | 𝐴 𝑛 ( 𝑡 , 𝑧 ) | ∥ ∇ 𝑥 𝑓 ∥ 𝕃 ∞ + | 𝐵 𝑛 ( 𝑡 , 𝑧 ) | ∥ ∇ 𝑣 𝑓 ∥ 𝕃 ∞ , which implies that ℐ 𝑛 ( 𝑡 ) ⩽  ℝ 4 𝑑 | ℎ ( 𝑧 ) | ( | 𝐴 𝑛 ( 𝑡 , 𝑧 ) | ∥ ∇ 𝑥 𝑓 ∥ 𝕃 ∞ + | 𝐵 𝑛 ( 𝑡 , 𝑧 ) | ∥ ∇ 𝑣 𝑓 ∥ 𝕃 ∞ ) 𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ′ ) 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ) 𝑑 𝑧𝑑 𝑧 ′ +     ℝ 4 𝑑  𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) ) 𝑣 , 𝑣 )  ℎ ( 𝑧 ) 𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ′ ) 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ) 𝑑 𝑧𝑑 𝑧 ′    = : ℐ 1 𝑛 ( 𝑡 ) + ℐ 2 𝑛 ( 𝑡 ) . Note that by ( 1.5 ), ∥ 𝐴 𝑛 ( 𝑡 ) ∥ 𝒑 ≲ ∥ 𝑏 𝑛 ∥ 𝒑  𝑡 𝑘 𝑛 ( 𝑡 )  𝑠 𝑘 𝑛 ( 𝑠 ) 𝑑 𝑟 𝑑 𝑠 ⩽ 𝑛 − 2 and ∥ 𝐵 𝑛 ( 𝑡 ) ∥ 𝒑 ≲ 𝑛 − 1 . By Hölder inequality and heat kernel estimate of 𝜌 𝑛 𝑡 , ( 4.5 ), we have ℐ 1 𝑛 ( 𝑡 ) ≲ ∥ ℎ ∥ 𝒑 ( ∥ 𝐴 𝑛 ( 𝑡 ) ∥ 𝒑 ∥ ∇ 𝑥 𝑓 ∥ 𝕃 ∞ + ∥ 𝐵 𝑛 ( 𝑡 ) ∥ 𝒑 ∥ ∇ 𝑣 𝑓 ∥ 𝕃 ∞ ) ∥ 𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ∥ 1 ∥ 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ∥ ( 𝒑 / 2 ) ′ ≲ ( 𝑘 𝑛 ( 𝑡 ) ) − 𝒂 · 𝑑 𝒑 ∥ ℎ ∥ 𝒑 ( 𝑛 − 2 ∥ ∇ 𝑥 𝑓 ∥ 𝕃 ∞ + 𝑛 − 1 ∥ ∇ 𝑣 𝑓 ∥ 𝕃 ∞ ) . For ℐ 2 𝑛 ( 𝑡 ) , it follows from ( A.4 ) that for any 𝑧 ′ : = ( 𝑥 ′ , 𝑣 ′ ) ∈ ℝ 2 𝑑 , ∥ 𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 − 𝑣 ′ ∇ 𝑣 𝑓 ∥ 𝕃 𝒒 ≲ ∥ 𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 − 𝛿 ( 0 ,𝑣 ′ ) 𝑓 ∥ 𝕃 𝒒 + ∥ 𝛿 ( 0 ,𝑣 ′ ) 𝑓 − 𝑣 ′ ∇ 𝑣 𝑓 ∥ 𝕃 𝒒 ≲ | 𝑥 ′ | 2 3 ∥ 𝑓 ∥ 𝔹 2 𝒒 ; 𝒂 + | 𝑣 ′ | 2 ∥ ∇ 2 𝑣 𝑓 ∥ 𝕃 𝒒 17 ≲ | 𝑧 ′ | 2 𝒂 ( ∥ 𝑓 ∥ 𝔹 2 𝒒 ; 𝒂 + ∥ ∇ 2 𝑣 𝑓 ∥ 𝕃 𝒒 ) . Then by the symmetry , Hölder ine quality we hav e ℐ 2 𝑛 ( 𝑡 ) =     ℝ 4 𝑑  𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) ) 𝑣 , 𝑣 ) − 𝑣 ′ · ∇ 𝑣 𝑓 ( 𝑥 + ( 𝑡 − 𝑘 𝑛 ( 𝑡 ) ) 𝑣 , 𝑣 )  ℎ ( 𝑧 ) 𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ′ ) 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ) 𝑑 𝑧𝑑 𝑧 ′    =     ℝ 4 𝑑  𝛿 ( 𝑥 ′ ,𝑣 ′ ) 𝑓 ( 𝑥 , 𝑣 ) − 𝑣 ′ · ∇ 𝑣 𝑓 ( 𝑥 , 𝑣 )  𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ′ ) ( ℎ 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ) ( Γ 𝑘 𝑛 ( 𝑡 ) − 𝑡 𝑧 ) 𝑑 𝑧𝑑 𝑧 ′    ≲ ∥ ℎ ∥ 𝒑 ( ∥ 𝑓 ∥ 𝔹 2 𝒒 ; 𝒂 + ∥ ∇ 2 𝑣 𝑓 ∥ 𝕃 𝒒 ) ∥ 𝜌 𝑛 𝑘 𝑛 ( 𝑡 ) ∥ 𝒓  ℝ 𝑑 | 𝑧 ′ | 2 𝒂 𝑔 𝑡 − 𝑘 𝑛 ( 𝑡 ) ( 𝑧 ′ ) 𝑑 𝑧 ′ ≲ ( 𝑘 𝑛 ( 𝑡 ) ) − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 ) ∥ ℎ ∥ 𝒑 𝑛 − 1 ( ∥ 𝑓 ∥ 𝔹 2 𝒒 ; 𝒂 + ∥ ∇ 2 𝑣 𝑓 ∥ 𝕃 𝒒 ) , where 1 / 𝒓 = 1 − 1 / 𝒑 − 1 / 𝒒 . This completes the proof. □ Lemma 4.6. For and 𝑓 ∈ S and any 2 / 𝑛 ⩽ 𝑠 ⩽ 𝑡 ⩽ 1 | 𝔼 𝑓 ( 𝑍 𝑛 𝑡 ) − 𝔼 Γ 𝑡 − 𝑠 𝑓 ( 𝑍 𝑛 𝑠 ) | ≲ ∥ 𝑓 ∥ 𝒑 | 𝑡 − 𝑠 | 1 2 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 . (4.14) Proof. Let 𝑡 ∈ ( 0 , 1 ] . For any 𝑓 ∈ S , 𝑟 → 𝑃 𝑟 𝑓 solves the following kinetic PDE 𝜕 𝑟 𝑃 𝑟 𝑓 = ( Δ 𝑣 + 𝑣 · ∇ 𝑥 ) 𝑃 𝑟 𝑓 , 𝑃 0 𝑓 = 𝑓 , (4.15) which by Itô ’s formula to 𝑟 → 𝑃 𝑡 − 𝑟 𝑓 ( 𝑍 𝑛 𝑟 ) implies that 𝔼 𝑓 ( 𝑍 𝑛 𝑡 ) = 𝔼 𝑃 𝑡 𝑓 ( 𝑍 0 ) + 𝔼  𝑡 0 Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝑓 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟 . For any 0 < 𝑠 < 𝑡 , by replacing 𝑓 with Γ 𝑡 − 𝑠 𝑓 , w e also have 𝔼 Γ 𝑡 − 𝑠 𝑓 ( 𝑍 𝑛 𝑠 ) = 𝔼 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 ( 𝑍 0 ) + 𝔼  𝑠 0 Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) · ∇ 𝑣 𝑃 𝑠 − 𝑟 Γ 𝑡 − 𝑠 𝑓 ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟 . Hence, we hav e | 𝔼 𝑓 ( 𝑍 𝑛 𝑡 ) − 𝔼 Γ 𝑡 − 𝑠 𝑓 ( 𝑍 𝑛 𝑠 ) | (4.16) ⩽ ∥ 𝑃 𝑡 𝑓 − 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 ∥ ∞ + 𝔼  𝑡 𝑠 | Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝑓 ( 𝑍 𝑛 𝑟 ) | 𝑑 𝑟 + 𝔼  𝑠 0 | Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) ∇ 𝑣 ( 𝑃 𝑡 − 𝑟 − 𝑃 𝑠 − 𝑟 Γ 𝑡 − 𝑠 ) 𝑓 ( 𝑍 𝑛 𝑟 ) | 𝑑 𝑟 = : 𝐻 1 + 𝐻 2 + 𝐻 3 . For 𝐻 1 , taking ( 𝛿 , 𝑘 , 𝒑 , 𝒑 ′ ) = ( 1 2 , 0 , ∞ , 𝒑 ) in ( 3.9 ), we have ∥ 𝑃 𝑡 𝑓 − 𝑃 𝑠 Γ 𝑡 − 𝑠 𝑓 ∥ ∞ ≲ ∥ 𝑓 ∥ 𝒑 | 𝑡 − 𝑠 | 1 2 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 . (4.17) Applying ( 4.4 ), ( 3.12 ), ( 2.2 ) and the fact that 𝑘 𝑛 ( 𝑟 ) ⩾ 𝑘 𝑛 ( 𝑠 ) ⩾ 𝑠 / 2 , it follows that 𝐻 2 ≲ ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 )  𝑡 𝑠 ( 𝑘 𝑛 ( 𝑟 ) ) − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑟 𝑓 ∥ 𝒑 𝑑 𝑟 18 ≲ 𝑠 − 𝒂 · 𝑑 𝒑 ∥ 𝑓 ∥ 𝔹 0 𝒑 ; 𝒂  𝑡 𝑠 ( 𝑡 − 𝑟 ) − 1 2 𝑑 𝑟 ≲ 𝑠 − 𝒂 · 𝑑 𝒑 | 𝑡 − 𝑠 | 1 2 ∥ 𝑓 ∥ 𝒑 . T o estimate 𝐻 3 , w e decompose the integral on [ 0 , 𝑠 ] into regions [ 0 , 𝑛 − 1 ] and [ 𝑛 − 1 , 𝑠 ] . Applying ( 3.9 ) and ( 1.7 ), we have 𝔼  𝑛 − 1 0 | Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) ∇ 𝑣 ( 𝑃 𝑡 − 𝑟 − 𝑃 𝑠 − 𝑟 Γ 𝑡 − 𝑠 ) 𝑓 ( 𝑍 𝑛 𝑟 ) | 𝑑 𝑟 ⩽ sup 𝑡 ∈ [ 0 , 𝑛 − 1 ] ∥ 𝑏 𝑛 ( 𝑡 , ·) ∥ ∞  𝑛 − 1 0 𝔼 | ∇ 𝑣 ( 𝑃 𝑡 − 𝑟 − 𝑃 𝑠 − 𝑟 Γ 𝑡 − 𝑠 ) 𝑓 | ( 𝑍 𝑛 𝑟 ) 𝑑 𝑟 ≲ ∥ 𝑓 ∥ 𝒑 | 𝑡 − 𝑠 | 1 2 𝑛 𝜁  𝑛 − 1 0 ( 𝑠 − 𝑟 ) − 1 𝑟 − 𝒂 · 𝑑 2 𝒑 𝑑 𝑟 ≲ | 𝑡 − 𝑠 | 1 2 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 ∥ 𝑓 ∥ 𝒑 . T o obtain the last ine quality ab ov e, we have use d the estimate ( 𝑠 − 𝑟 ) − 1 ≲ 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 ( 1 𝑛 − 𝑟 ) − 1 2 + 𝒂 · 𝑑 2 𝒑 for each 𝑟 ∈ [ 0 , 𝑛 − 1 ] and the assumption that 𝜁 ⩽ 1 / 2 . For the other integration over the other region, we apply ( 4.4 ), ( 3.9 ), inequality 𝑘 𝑛 ( 𝑟 ) ⩾ 𝑟 / 2 for any 𝑟 > 𝑛 − 1 and ( A.2 ) to get that 𝔼  𝑠 𝑛 − 1 | Γ 𝑟 − 𝑘 𝑛 ( 𝑟 ) 𝑏 𝑛 ( 𝑟 , 𝑍 𝑛 𝑘 𝑛 ( 𝑟 ) ) ∇ 𝑣 ( 𝑃 𝑡 − 𝑟 − 𝑃 𝑠 − 𝑟 Γ 𝑡 − 𝑠 ) 𝑓 ( 𝑍 𝑛 𝑟 ) | 𝑑 𝑟 ⩽ ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 )  𝑠 𝑛 − 1 ( 𝑘 𝑛 ( 𝑟 ) ) − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 ( 𝑃 𝑡 − 𝑟 − 𝑃 𝑠 − 𝑟 Γ 𝑡 − 𝑠 ) 𝑓 ∥ 𝒑 𝑑 𝑟 ≲ ∥ 𝑓 ∥ 𝒑  𝑠 0 𝑟 − 𝒂 · 𝑑 𝒑 ( 𝑠 − 𝑟 ) − 1 2 [ ( ( 𝑡 − 𝑠 ) ( 𝑠 − 𝑟 ) − 1 ) ∧ 1 ] 𝑑 𝑟 ≲ ∥ 𝑓 ∥ 𝒑 | 𝑡 − 𝑠 | 1 2 𝑠 − 𝒂 · 𝑑 2 𝒑 . Combining all the estimates above, w e obtain ( 4.14 ). □ Lemma 4.7. For any 𝒒 ⩾ 𝒑 , 𝑡 ∈ ( 0 , 1 ] 𝐼 𝑛 3 ( 𝑡 ) ≲ 𝑛 − 1 2  𝑡 − 𝒂 · 𝑑 2 𝒒 − 𝜁 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒒 + 𝒂 · 𝑑 𝒑 )  ∥ 𝜑 ∥ 𝒒 , (4.18) where 𝐼 𝑛 3 ( 𝑡 ) is dened in ( 4.2 ) . Proof. W e rst make an observation that taking 𝛽 = 0 in ( 3.12 ) and using the emb edding 𝕃 𝒒 ↩ → 𝔹 0 𝒒 ; 𝒂 (see ( 2.2 )), we hav e ∥ ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ∥ ∞ ≲ ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 and ∥ ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ∥ 𝒒 ≲ ( 𝑡 − 𝑟 ) − 1 2 ∥ 𝜑 ∥ 𝒒 . (4.19) In the case when 𝑡 ⩽ 3 / 𝑛 , using The or em 1.2 and applying ( 4.4 ) and ( 4.19 ), we obtain that 𝐼 𝑛 3 ( 𝑡 ) ⩽ sup 𝑠 ⩽ 𝑛 − 1 ∥ 𝑏 𝑛 ( 𝑠 ) ∥ ∞  𝑡 ∧ 𝑛 − 1 0 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ∥ ∞ 𝑑 𝑟 +  𝑡 𝑡 ∧ 𝑛 − 1 ( 𝑘 𝑛 ( 𝑟 ) ) − 𝒂 · 𝑑 2 𝒑 ∥ 𝑏 𝑛 ( 𝑟 ) ∥ 𝒑 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ∥ ∞ 𝑑 𝑟 ≲ 𝑛 𝜁  𝑡 0 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 / 𝒒 2 ∥ 𝜑 ∥ 𝒒 𝑑 𝑟 +  𝑡 𝑡 ∧ 𝑛 − 1 ( 𝑘 𝑛 ( 𝑟 ) ) − 𝒂 · 𝑑 2 𝒑 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 / 𝒒 2 ∥ 𝜑 ∥ 𝒒 𝑑 𝑟 ≲ 𝑛 𝜁 𝑡 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 +  𝑡 𝑡 ∧ 𝑛 − 1 𝑟 − 𝒂 · 𝑑 2 𝒑 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 / 𝒒 2 ∥ 𝜑 ∥ 𝒒 𝑑 𝑟 19 ≲  𝑛 𝜁 𝑡 1 2 − 𝒂 · 𝑑 2 𝒒 + 𝑡 1 2 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝑑 𝒒 )  ∥ 𝜑 ∥ 𝒒 . Since 𝑛 ⩽ 3 𝑡 − 1 , this implies ( 4.18 ). In the case when 𝑡 > 3 / 𝑛 , we have 𝐼 𝑛 3 ( 𝑡 ) ⩽    𝔼  𝑡 0 Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑠 , 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) ·  Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ( 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) − ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ( 𝑍 𝑛 𝑠 )  𝑑 𝑠    +    𝔼  𝑡 0 Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) ( 𝑏 𝑛 ( 𝑠 ) · ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ) ( 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) − ( 𝑏 𝑛 ( 𝑠 ) · ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ) ( 𝑍 𝑛 𝑠 ) 𝑑 𝑠    = : 𝑆 1 + 𝑆 2 . (4.20) W e dene 1 𝒓 : = 1 𝒑 + 1 𝒒 and 𝑄 𝑛 ( 𝑠 ) : = 𝔼 Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑏 𝑛 ( 𝑠 , 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) ·  Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ( 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) − ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ( 𝑍 𝑛 𝑠 )  . W e have 𝑆 1 =  𝑡 0 | 𝑄 𝑛 ( 𝑠 ) | 𝑑 𝑠 ⩽   1 𝑛 0 +  𝑡 1 𝑛    𝑄 𝑛 ( 𝑠 )   𝑑 𝑠 = : 𝑆 11 + 𝑆 12 . Using ( 3.12 ) (with 𝛽 = 0 ) and ( 2.2 ), we have 𝑆 11 ⩽ 2 𝑛 𝜁  1 𝑛 0 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ ∞ 𝑑 𝑠 ≲ 𝑛 𝜁  1 𝑛 0 ( 𝑡 − 𝑠 ) − 1 2 − 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 𝑑 𝑠 . Noting that 𝑛 − 1 ⩽ 𝑡 / 3 and ( 𝑡 − 𝑠 ) − 1 ⩽ 3 2 𝑡 − 1 , we have 𝑆 11 ≲ 𝑛 𝜁  1 𝑛 0 𝑡 − 1 2 − 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 𝑑 𝑠 ≲ 𝑛 𝜁 − 1 𝑡 − 1 2 − 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 ≲ 𝑛 − 1 2 𝑡 − ( 𝜁 + 𝑑 2 𝒒 ) ∥ 𝜑 ∥ 𝒒 . Concerning 𝑆 12 , on one hand, when 𝑠 > 1 / 𝑛 , 𝑘 𝑛 ( 𝑠 ) > 𝑠 / 2 so that by ( 4.4 ) and ( 4.19 ), we have | 𝑄 𝑛 ( 𝑠 ) | ≲ 2 ( 𝑘 𝑛 ( 𝑠 ) ) − 𝒂 · 𝑑 2 𝒓 ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 𝑇 ( 𝕃 𝒑 ) ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝒒 ≲ 𝑠 − 𝒂 · 𝑑 2 𝒓 ( 𝑡 − 𝑠 ) − 1 2 ∥ 𝜑 ∥ 𝒒 . (4.21) On the other hand, by ( 4.13 ), Theorem 2.2 , ( 3.14 ), together with the fact 𝒒 ⩾ 𝒑 we have | 𝑄 𝑛 ( 𝑠 ) | ≲ ∥ 𝑏 𝑛 ∥ 𝐿 ∞ 1 ( 𝕃 𝒑 )  𝑛 − 2 ( 𝑘 𝑛 ( 𝑠 ) ) − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 3 , 1 ∞ ; 𝒂 + 𝑛 − 1 ( 𝑘 𝑛 ( 𝑠 ) ) − 𝒂 · 𝑑 𝒑 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 1 , 1 ∞ ; 𝒂 + 𝑛 − 1 ( 𝑘 𝑛 ( 𝑠 ) ) − 𝒂 · 𝑑 2 𝒓 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝔹 2 , 1 𝒒 ; 𝒂  ≲ ∥ 𝜑 ∥ 𝒒  𝑛 − 2 𝑠 − 𝒂 · 𝑑 𝒑 ( 𝑡 − 𝑠 ) − 2 − 𝑑 2 𝒒 + 𝑛 − 1 𝑠 − 𝒂 · 𝑑 𝒑 ( 𝑡 − 𝑠 ) − 1 − 𝑑 2 𝒒 + 𝑛 − 1 𝑠 − 𝒂 · 𝑑 2 𝒓 ( 𝑡 − 𝑠 ) − 3 2  . Combining the above estimate with ( 4.21 ), noting that 𝒂 · 𝑑 2 𝒒 < 1 2 , we have | 𝑄 𝑛 ( 𝑠 ) | ≲ ∥ 𝜑 ∥ 𝒒  𝑠 − 𝒂 · 𝑑 2 𝒓 ( 𝑡 − 𝑠 ) − 1 2 min ( 𝑛 − 1 ( 𝑡 − 𝑠 ) − 1 , 1 ) + 𝑠 − 𝒂 · 𝑑 𝒑 ( 𝑡 − 𝑠 ) − 1 2  min ( 𝑛 − 2 ( 𝑡 − 𝑠 ) − 2 , 1 ) + min ( 𝑛 − 1 ( 𝑡 − 𝑠 ) − 1 , 1 )   20 ≲ ∥ 𝜑 ∥ 𝒒  𝑠 − 𝒂 · 𝑑 2 𝒓 ( 𝑡 − 𝑠 ) − 1 2 min ( 𝑛 − 1 ( 𝑡 − 𝑠 ) − 1 , 1 ) + 𝑠 − 𝒂 · 𝑑 𝒑 ( 𝑡 − 𝑠 ) − 1 2 min ( 𝑛 − 1 ( 𝑡 − 𝑠 ) − 1 , 1 )  , which by Theorem A.1 implies that 𝑆 12 ⩽  𝑡 1 / 𝑛 | 𝑄 𝑛 ( 𝑠 ) | 𝑑 𝑠 ≲ ∥ 𝜑 ∥ 𝒒  𝑡 1 / 𝑛  𝑠 − 𝒂 · 𝑑 2 𝒓 ( 𝑡 − 𝑠 ) − 1 2 min ( 𝑛 − 1 ( 𝑡 − 𝑠 ) − 1 , 1 ) + 𝑠 − 𝒂 · 𝑑 𝒑 ( 𝑡 − 𝑠 ) − 1 2 min ( 𝑛 − 1 ( 𝑡 − 𝑠 ) − 1 , 1 )  𝑑 𝑠 ≲ 𝑛 − 1 2 𝑡 − 𝒂 · 𝑑 2 𝒓 ∥ 𝜑 ∥ 𝒒 . Therefore , combining the estimate for 𝑆 11 , we have 𝑆 1 ≲ 𝑛 − 1 2  𝑡 − 𝒂 · 𝑑 2 𝒒 − 𝜁 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒒 + 𝒂 · 𝑑 𝒑 )  ∥ 𝜑 ∥ 𝒒 . (4.22) Next we estimate 𝑆 2 . By dividing the inter val [ 0 , 𝑡 ] into [ 0 , 1 / 𝑛 ] ∪ [ 1 / 𝑛, 2 / 𝑛 ] ∪ [ 2 / 𝑛, 𝑡 ] , following from ( 4.4 ), w e have 𝑆 2 ≲  1 𝑛 0 ∥ 𝑏 𝑛 ( 𝑠 ) ∥ ∞ ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ ∞ 𝑑 𝑠 +  2 𝑛 1 𝑛 ( 𝑘 𝑛 ( 𝑠 ) ) − 𝒂 · 𝑑 2 𝒑 ∥ 𝑏 𝑛 ( 𝑠 ) ∥ 𝒑 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ ∞ 𝑑 𝑠 +  𝑡 2 𝑛 | 𝔼 Γ 𝑠 − 𝑘 𝑛 ( 𝑠 ) 𝑓 ( 𝑠 , 𝑍 𝑛 𝑘 𝑛 ( 𝑠 ) ) − 𝔼 𝑓 ( 𝑠 , 𝑍 𝑛 𝑠 ) | 𝑑 𝑠 , where 𝑓 ( 𝑠 ) : = 𝑏 𝑛 ( 𝑠 ) · ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 . Then using conditions ( 1.7 ) , ( 1.6 ) , inequalities ( 4.19 ) and ( 4.14 ) , we obtain that 𝑆 2 ≲ 𝑛 𝜁  1 𝑛 0 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 𝑑 𝑠 ∥ 𝜑 ∥ 𝒒 +  2 𝑛 1 𝑛 ( 𝑘 𝑛 ( 𝑠 ) ) − 𝒂 · 𝑑 2 𝒑 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 𝑑 𝑠 ∥ 𝜑 ∥ 𝒒 +  𝑡 2 𝑛 | 𝑠 − 𝑘 𝑛 ( 𝑠 ) | 1 2 ( 𝑘 𝑛 ( 𝑠 ) ) − 1 2 − 𝒂 · 𝑑 2 𝒑 ∥ 𝑏 𝑛 ( 𝑠 ) · ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝒑 𝑑 𝑠 . Noting that ( 𝑡 − 𝑠 ) − 1 ≲ 𝑡 − 1 for 𝑠 < 2 𝑛 < 2 𝑡 3 , 𝑘 𝑛 ( 𝑠 ) ⩾ 𝑠 2 for 𝑠 > 1 𝑛 , and ∥ 𝑏 𝑛 ( 𝑠 ) · ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ 𝒑 ⩽ ∥ 𝑏 𝑛 ( 𝑠 ) ∥ 𝒑 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑠 𝜑 ∥ ∞ ≲ ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜑 ∥ 𝒒 , one sees that 𝑆 2 ≲  𝑛 𝜁 − 1 𝑡 − 1 2 − 𝒂 · 𝑑 2 𝒒 +  2 𝑛 1 𝑛 𝑠 − 𝒂 · 𝑑 2 𝒑 𝑡 − 1 2 − 𝒂 · 𝑑 2 𝒒 𝑑 𝑠 + 𝑛 − 1 2  𝑡 2 𝑛 𝑠 − 1 2 − 𝒂 · 𝑑 2 𝒑 ( 𝑡 − 𝑠 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 𝑑 𝑠  ∥ 𝜑 ∥ 𝒒 ≲  𝑛 𝜁 − 1 𝑡 − 1 2 − 𝒂 · 𝑑 2 𝒒 + 𝑛 − 1 + 𝒂 · 𝑑 2 𝒑 𝑡 − 1 2 − 𝒂 · 𝑑 2 𝒒 + 𝑛 − 1 2 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 )  ∥ 𝜑 ∥ 𝒒 ≲ 𝑛 − 1 2  𝑡 − 𝜁 − 𝒂 · 𝑑 2 𝒒 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒒 + 1 2 𝒂 · 𝑑 𝒑 )  ∥ 𝜑 ∥ 𝒒 , provided that 𝑛 − 1 ≲ 𝑡 . These estimates and ( 4.22 ) yield ( 4.18 ). □ 21 4.3. Proof of Theorem 1.3 . Proof of Theorem 1.3 . W e hinge on the inequality ( 4.2 ) . Let 𝒑 ′ be the Holder conjugate of 𝒑 . Using ( 3.12 ), it is easy to see that 𝐼 𝑛 2 ( 𝑡 ) =     𝑡 0 ⟨ 𝑏 𝑛 ( 𝑟 ) · ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 , 𝜌 𝑟 − 𝜌 𝑛 𝑟 ⟩ 𝑑 𝑟    ⩽ ∥ 𝑏 𝑛 ∥ 𝒑  𝑡 0 ∥ ∇ 𝑣 𝑃 𝑡 − 𝑟 𝜑 ∥ ∞ ∥ 𝜌 𝑟 − 𝜌 𝑛 𝑟 ∥ 𝒑 ′ 𝑑 𝑟 ≲ ∥ 𝜑 ∥ 𝒒  𝑡 0 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜌 𝑟 − 𝜌 𝑛 𝑟 ∥ 𝒑 ′ 𝑑 𝑟 . The terms 𝐼 𝑛 1 and 𝐼 𝑛 3 are estimated by ( 4.6 ) and ( 4.18 ) respectively . By Theorem 4.1 , w e can take supremum o ver all smooth and bounded 𝜑 with ∥ 𝜑 ∥ 𝒒 = 1 to obtain from ( 4.2 ) that ∥ 𝜌 𝑡 − 𝜌 𝑛 𝑡 ∥ 𝒒 ′ ≲ 𝑛 − ( 1 2 ∧ 𝜗 )  𝑡 − 𝜁 − 𝒂 · 𝑑 2 𝒒 + 𝑡 − 1 2 ( 𝒂 · 𝑑 𝒑 + 𝒂 · 𝑑 𝒒 )  +  𝑡 0 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒒 ∥ 𝜌 𝑟 − 𝜌 𝑛 𝑟 ∥ 𝒑 ′ 𝑑 𝑟 . (4.23) T aking 𝒒 = 𝒑 yields ∥ 𝜌 𝑡 − 𝜌 𝑛 𝑡 ∥ 𝒑 ′ ≲ 𝑛 − 𝜗 𝑡 − 𝒂 · 𝑑 𝒑 +  𝑡 0 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒑 ∥ 𝜌 𝑟 − 𝜌 𝑛 𝑟 ∥ 𝒑 ′ 𝑑 𝑟 + 𝑛 − 1 2  𝑡 − 𝜁 − 𝒂 · 𝑑 2 𝒑 + 𝑡 − 𝒂 · 𝑑 𝒑  . Since 𝜁 ⩽ 1 / 2 and 𝒂 · 𝑑 𝒑 < 1 2 , we have  𝑡 0 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒑  𝑟 − 𝜁 − 𝒂 · 𝑑 2 𝒑 + 𝑟 − 𝒂 · 𝑑 𝒑  𝑑 𝑟 ≲  𝑡 0 ( 𝑡 − 𝑟 ) − 1 2 − 𝒂 · 𝑑 2 𝒑 𝑟 − 1 2 − 𝒂 · 𝑑 2 𝒑 𝑑 𝑟 ≲ 𝑡 − 𝒂 · 𝑑 𝒑 . Applying a Grönwall’s inequality of V olterra’s type (see [ Zha10 , Lemma 2.2] and [ Hao23 , Lemma A.4]), we obtain that ∥ 𝜌 𝑡 − 𝜌 𝑛 𝑡 ∥ 𝒑 ′ ≲ 𝑛 − 𝜗 𝑡 − 𝒂 · 𝑑 𝒑 + 𝑛 − 1 2  𝑡 − 𝜁 − 𝒂 · 𝑑 2 𝒑 + 𝑡 − 𝒂 · 𝑑 𝒑  ≲ 𝑛 − ( 𝜗 ∧ 1 2 )  𝑡 − 𝜁 − 𝒂 · 𝑑 2 𝒑 + 𝑡 − 𝒂 · 𝑑 𝒑  . Using the previous estimates in ( 4.23 ) yields ( 1.9 ), completing the pr oof. □ A p p e n d i x A. S o m e t e c h n i c a l l e m m a s The following elementary lemma is similar as [ HRZ23 , Lemma A.2]. Lemma A.1. Let 𝜀 > 0 , 𝛼 1 , 𝛼 2 ∈ [ 0 , 1 ) ; 𝛾 1 , 𝛾 2 ⩾ 0 be some xed numb ers. For any 𝛼 , 𝛾 ⩾ 0 , dene ℓ 𝛼 , 𝛾 ( 𝜀 ) =          𝜀 ∧ 1 if 𝛾 < 1 − 𝛼 , ( 𝜀 ( 1 + | log 𝜀 | ) ) ∧ 1 if 𝛾 = 1 − 𝛼 , 𝜀 1 − 𝛼 𝛾 ∧ 1 if 𝛾 > 1 − 𝛼 . (A.1) There there is a nite constant 𝐶 = 𝐶 ( 𝛼 1 , 𝛼 2 , 𝛾 1 , 𝛾 2 ) such that for all 𝑡 > 0 ,  𝑡 0 𝑠 − 𝛼 1 ( 𝑡 − 𝑠 ) − 𝛼 2 min ( 𝜀𝑠 − 𝛾 1 ( 𝑡 − 𝑠 ) − 𝛾 2 , 1 ) ⩽ 𝐶 𝑡 1 − 𝛼 1 − 𝛼 2 ( ℓ 𝛼 1 , 𝛾 1 ( 𝜀 𝑡 − 𝛾 1 − 𝛾 2 ) + ℓ 𝛼 2 , 𝛾 2 ( 𝜀 𝑡 − 𝛾 1 − 𝛾 2 ) ) . 22 In particular , when 𝛾 1 = 0 , 𝛾 2 = 𝛾 > 1 − 𝛼 2 and 𝜀 = 𝜆 − 𝛾 , we have  𝑡 0 𝑠 − 𝛼 1 ( 𝑡 − 𝑠 ) − 𝛼 2 ( [ ( 𝜆 ( 𝑡 − 𝑠 ) ) − 𝛾 ] ∧ 1 ) 𝑑 𝑠 ⩽ 𝐶  𝜆 − 1 + 𝛼 2 𝑡 − 𝛼 1  ∧ 𝑡 1 − 𝛼 1 − 𝛼 2 . (A.2) Proof. Let 𝐼 ( 𝑡 , 𝜀 ) denote the integral on the left-hand side. By a change of variable, we have 𝐼 ( 𝑡 , 𝜀 ) = 𝑡 1 − 𝛼 1 − 𝛼 2 𝐼 ( 1 , 𝜀 𝑡 − 𝛾 1 − 𝛾 2 ) . Thus, it suces to estimate 𝐼 ( 1 , 𝜀 ) for 𝜀 > 0 . W e have 𝐼 ( 1 , 𝜀 ) =   1 / 2 0 +  1 1 / 2  𝑠 − 𝛼 1 ( 1 − 𝑠 ) − 𝛼 2 min ( 𝜀𝑠 − 𝛾 1 ( 1 − 𝑠 ) − 𝛾 2 , 1 ) 𝑑 𝑠 ≲  1 / 2 0 𝑠 − 𝛼 1 min ( 𝜀𝑠 − 𝛾 1 , 1 ) 𝑑 𝑠 +  1 1 / 2 ( 1 − 𝑠 ) − 𝛼 2 min ( 𝜀 ( 1 − 𝑠 ) ) − 𝛾 2 , 1 ) 𝑑 𝑠 =  1 / 2 0 𝑠 − 𝛼 1 min ( 𝜀𝑠 − 𝛾 1 , 1 ) 𝑑 𝑠 +  1 / 2 0 𝑠 − 𝛼 2 min ( 𝜀𝑠 − 𝛾 2 , 1 ) 𝑑 𝑠 . It is straightforward to verify that  1 / 2 0 𝑠 − 𝛼 min ( 𝜀𝑠 − 𝛾 , 1 ) 𝑑 𝑠 ≲ ℓ 𝛼 , 𝛾 ( 𝜀 ) for all 𝜀 > 0 , 𝛼 ∈ [ 0 , 1 ) and 𝛾 ⩾ 0 . This completes the proof. □ Lemma A.2. Let 𝛼 > 𝛽 > 0 . Then there is a constant 𝐶 = 𝐶 ( 𝛼 , 𝛽 ) > 0 such that for any 𝜀 > 0 ,  𝑗 = − 1 2 − 𝛽 𝑗  1 ∧ ( 2 𝛼 𝑗 𝜀 )  ⩽ 𝐶 𝜀 𝛽 𝛼 . (A.3) Proof. It is easy to see that ∞  𝑗 = − 1 2 − 𝛽 𝑗  1 ∧  𝜀 2 𝛼 𝑗   ⩽ ∞  𝑗 = − 1 2 𝛽 + 1  2 𝑗 + 1 2 𝑗 𝑠 − 𝛽 − 1 ( 1 ∧ ( 𝜀 𝑠 𝛼 ) ) 𝑑 𝑠 ≲  ∞ 0 𝑠 − 𝛽 − 1 ( 1 ∧ ( 𝜀 𝑠 𝛼 ) ) 𝑑 𝑠 ≲ 𝜀 𝛽 𝛼  ∞ 0 𝑠 − 𝛽 − 1 ( 1 ∧ 𝑠 𝛼 ) 𝑑 𝑠 ≲ 𝜀 𝛽 𝛼 . □ Lemma A.3. Let 𝑠 ∈ ( 0 , 1 ) and 𝒑 ∈ [ 1 , ∞] 2 . There is a constant 𝐶 = 𝐶 ( 𝑑 , 𝑠 , 𝒑 ) > 0 such that for all ℎ ∈ ℝ 𝑑 and 𝑓 ∈ 𝔹 3 𝑠 𝒑 ; 𝒂 , ∥ 𝑓 ( · + ℎ, ·) − 𝑓 ( · , ·) ∥ 𝕃 𝒑 ⩽ 𝐶 | ℎ | 𝑠 ∥ 𝑓 ∥ 𝔹 3 𝑠 𝒑 ; 𝒂 . (A.4) Proof. W e note that for any 𝑥 , 𝑣 ∈ ℝ 𝑑 , | 𝑓 ( 𝑥 + ℎ , 𝑣 ) − 𝑓 ( 𝑥 , 𝑣 ) | ⩽ ∞  𝑗 = − 1 | R 𝒂 𝑗 𝑓 ( 𝑥 + ℎ, 𝑣 ) − R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 ) | 23 ⩽ ∞  𝑗 = − 1  | R 𝒂 𝑗 𝑓 ( 𝑥 + ℎ, 𝑣 ) | + | R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 ) |  ∧  ℎ ·  1 0 ∇ 𝑥 R 𝒂 𝑗 𝑓 ( 𝑥 + 𝑠 ℎ, 𝑣 ) 𝑑 𝑠  . Then it follows from Bernstein’s inequalities (see ( 2.4 )) that ∥ 𝑓 ( · + ℎ, ·) − 𝑓 ( · , ·) ∥ 𝕃 𝒑 ≲ ∞  𝑗 = − 1  | ℎ | ∥ ∇ 𝑥 R 𝒂 𝑗 𝑓 ∥ 𝕃 𝒑  ∧ ∥ R 𝒂 𝑗 𝑓 ∥ 𝕃 𝒑 ≲ ∞  𝑗 = − 1   | ℎ | 2 3 𝑗  ∧ 1  ∥ R 𝒂 𝑗 𝑓 ∥ 𝕃 𝒑 ≲ ∞  𝑗 = − 1   | ℎ | 2 3 𝑗  ∧ 1  2 − 3 𝑠 𝑗 ∥ 𝑓 ∥ 𝔹 3 𝑠 𝒑 ; 𝒂 ≲ | ℎ | 𝑠 ∥ 𝑓 ∥ 𝔹 3 𝑠 𝒑 ; 𝒂 , provided by ( A.3 ), and the proof completes. □ Lemma A.4. Let 𝑓 be a measurable function dened on ℝ 2 𝑑 , dene 𝑓 𝑛 : = 𝑓 ∗ 𝜑 𝑛 where 𝜑 𝑛 ( 𝑥 , 𝑣 ) : = 𝑛 4 𝑑 𝜗 𝜑 ( 𝑛 3 𝜗 𝑥 , 𝑛 𝜗 𝑣 ) , 𝜗 ⩾ 0 and 𝜑 is a probability density function on ℝ 2 𝑑 with 𝜑 ( 𝑥 , 𝑣 ) = 𝜑 ( 𝑥 , − 𝑣 ) . Then for any 𝑠 ∈ ℝ , 𝛽 ∈ ( 0 , 2 ) , 𝒑 ∈ [ 1 , ∞) 2 , we have ∥ 𝑓 − 𝑓 𝑛 ∥ 𝔹 𝑠 , 1 𝒑 ; 𝒂 ≲ 𝑛 − 𝜗 𝛽 ∥ 𝑓 ∥ 𝔹 𝑠 + 𝛽 𝒑 ; 𝒂 . (A.5) Proof. By Theorem 2.1 , ∥ 𝑓 − 𝑓 𝑛 ∥ 𝔹 𝑠 𝒑 ; 𝒂 = sup 𝑗 2 𝑠 𝑗 ∥ R 𝒂 𝑗 ( 𝑓 − 𝑓 𝑛 ) ∥ 𝒑 . Since 𝜑 𝑛 is a probability density function, for any ( 𝑥 , 𝑣 ) ∈ ℝ 2 𝑑 , we have R 𝒂 𝑗 ( 𝑓 − 𝑓 𝑛 ) ( 𝑥 , 𝑣 ) =  ℝ 2 𝑑 𝜑 𝑛 ( 𝑦, 𝑤 )  R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 ) − R 𝒂 𝑗 𝑓 ( 𝑥 − 𝑦 , 𝑣 − 𝑤 )  𝑑𝑦𝑑 𝑤 =  ℝ 2 𝑑 𝜑 𝑛 ( 𝑦, 𝑤 )  R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 ) − R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 − 𝑤 )  𝑑𝑦𝑑 𝑤 +  ℝ 2 𝑑 𝜑 𝑛 ( 𝑦, 𝑤 )  R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 − 𝑤 ) − R 𝒂 𝑗 𝑓 ( 𝑥 − 𝑦 , 𝑣 − 𝑤 )  𝑑𝑦𝑑 𝑤 = : 𝐼 1 ( 𝑗 ) + 𝐼 2 ( 𝑗 ) . Using the symmetry 𝜑 𝑛 ( 𝑥 , 𝑣 ) = 𝜑 𝑛 ( 𝑥 , − 𝑣 ) and Bernstein’s inequalities (see ( 2.4 )) one sees that ∥ 𝐼 1 ( 𝑗 ) ∥ 𝒑 =      ℝ 2 𝑑 𝜑 𝑛 ( 𝑦, 𝑤 )  R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 + 𝑤 ) + R 𝒂 𝑗 𝑓 ( 𝑥 , 𝑣 − 𝑤 ) − 2 R 𝑎 𝑗 𝑓 ( 𝑥 , 𝑣 )  𝑑𝑦𝑑 𝑤     𝒑 ≲  ℝ 2 𝑑 | 𝜑 𝑛 ( 𝑦, 𝑤 ) | | 𝑤 | 2 ∥ ∇ 2 R 𝒂 𝑗 𝑓 ∥ 𝕃 𝒑 𝑑𝑦𝑑 𝑤 ≲ 2 2 𝑗 ∥ R 𝒂 𝑗 𝑓 ∥ 𝕃 𝒑 𝑛 − 2 𝜗 . In view of ( A.4 ), we have ∥ 𝐼 2 ( 𝑗 ) ∥ 𝒑 ≲  ℝ 2 𝑑 | 𝜑 𝑛 ( 𝑦, 𝑤 ) | | 𝑦 | 2 3 ∥ R 𝒂 𝑗 𝑓 ∥ 𝔹 2 𝒑 ; 𝒂 𝑑𝑦𝑑 𝑤 ≲ ∥ R 𝒂 𝑗 𝑓 ∥ 𝔹 2 𝒑 ; 𝒂 𝑛 − 2 𝜗 ≲ sup 𝑘 2 2 𝑘 ∥ R 𝑎 𝑘 R 𝒂 𝑗 𝑓 ∥ 𝒑 𝑛 − 2 𝜗 ≲ sup 𝑘 ∼ 𝑗 2 2 𝑘 ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑 𝑛 − 2 𝜗 ≲ 2 2 𝑗 ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑 𝑛 − 2 𝜗 . 24 Moreover , it is easy to see that ∥ R 𝒂 𝑗 ( 𝑓 − 𝑓 𝑛 ) ∥ 𝒑 ⩽ ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑 + ∥ R 𝒂 𝑗 𝑓 𝑛 ∥ 𝒑 ≲ ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑 . Therefore , we get ∥ 𝑓 − 𝑓 𝑛 ∥ 𝔹 𝑠 , 1 𝒑 ; 𝒂 = ∞  𝑗 = − 1 2 𝑠 𝑗 ∥ R 𝒂 𝑗 ( 𝑓 − 𝑓 𝑛 ) ∥ 𝒑 ≲ ∞  𝑗 = − 1 2 𝑠 𝑗 ∥ R 𝒂 𝑗 𝑓 ∥ 𝒑   2 2 𝑗 𝑛 − 2 𝜗  ∧ 1  ≲ ∞  𝑗 = − 1 2 − 𝛽 𝑗   2 2 𝑗 𝑛 − 2 𝜗  ∧ 1  ∥ 𝑓 ∥ 𝔹 𝑠 + 𝛽 𝒑 ; 𝒂 , which by ( A.3 ) implies ( A.5 ). □ Lemma A.5. Let 𝑁 > 1 and 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑣 ) ∈ ( 1 , ∞] 2 such that 𝒑 ≠ ( ∞ , ∞) . For any 𝛿 ∈ ( 0 , ( ( 𝑝 𝑥 ∧ 𝑝 𝑣 ) − 1 ) ( 𝒂 · 𝑑 𝒑 ) ) , there is a constant 𝐶 = 𝐶 ( 𝑑 , 𝒑 , 𝛿 ) > 0 such that for any 𝑓 , 𝑔 ∈ 𝕃 𝒑 satisfying | 𝑔 | ⩽ | 𝑓 | 1 | 𝑓 | > 𝑁 , one has ∥ 𝑔 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ⩽ 𝐶 𝑁 − 𝛿 ( 𝒂 · 𝑑 / 𝒑 ) − 1 ∥ 𝑓 ∥ 1 + 𝛿 ( 𝒂 · 𝑑 / 𝒑 ) − 1 𝕃 𝒑 . (A.6) Proof. W e consider three cases. Case (i) ( 𝑝 𝑥 = ∞ , 𝑝 𝑣 ≠ ∞ ). Let 𝑞 ∈ [ 1 , 𝑝 𝑣 ) be such that 𝛿 + 𝑑 𝑝 𝑣 = 𝑑 𝑞 . Using embe dding 𝔹 0 , 1 ( ∞ , 𝑞 ) ; 𝒂 ↩ → 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ([ HRZ23 , Appendix B]) and ( 2.2 ), we have ∥ 𝑔 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ≲ ∥ 𝑔 ∥ 𝔹 0 , 1 ( ∞ ,𝑞 ) ; 𝒂 ≲ ∥ 𝑔 ∥ 𝕃 ( ∞ ,𝑞 ) ⩽ ∥ 𝑓 1 | 𝑓 | > 𝑁 ∥ 𝕃 ( ∞ ,𝑞 ) . Note that ∥ 𝑓 ( · , 𝑣 ) 1 | 𝑓 ( · ,𝑣 ) | > 𝑁 ∥ 𝐿 ∞ ⩽ ∥ 𝑓 ( · , 𝑣 ) ∥ 𝐿 ∞ 1 { ∥ 𝑓 ( · ,𝑣 ) ∥ 𝐿 ∞ > 𝑁 } . Setting ℎ ( 𝑣 ) : = ∥ 𝑓 ( · , 𝑣 ) ∥ 𝐿 ∞ , we have ∥ 𝑓 1 | 𝑓 | > 𝑁 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ≲ ∥ ℎ 1 | ℎ | > 𝑁 ∥ 𝐿 𝑞 ⩽ 𝑁 − 𝑝 𝑣 − 𝑞 𝑞 ∥ ℎ ∥ 𝑝 𝑣 𝑞 𝐿 𝑝 𝑣 = 𝑁 − 𝛿 𝑝 𝑣 𝑑 ∥ 𝑓 ∥ 𝑝 𝑣 𝑞 𝕃 𝒑 . Case (ii) ( 𝑝 𝑣 = ∞ ). Let 𝑞 ∈ [ 1 , 𝑝 𝑥 ) be such that 𝛿 + 3 𝑑 𝑝 𝑥 = 3 𝑑 𝑞 . Using embe dding 𝔹 0 , 1 ( 𝑞, ∞) ; 𝒂 ↩ → 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ([ HRZ23 , Appendix B]) and ( 2.2 ), we have ∥ 𝑔 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ≲ ∥ 𝑔 ∥ 𝔹 0 , 1 ( 𝑞, ∞ ) ; 𝒂 ≲ ∥ 𝑓 1 | 𝑓 | > 𝑁 ∥ 𝕃 ( 𝑞, ∞ ) . For each 𝑣 ∈ ℝ 𝑑 , we have ∥ 𝑓 ( · , 𝑣 ) 1 | 𝑓 ( · ,𝑣 ) | > 𝑁 ∥ 𝐿 𝑞 ⩽ 𝑁 − 𝛿 𝑝 𝑥 3 𝑑 ∥ 𝑓 ( · , 𝑣 ) ∥ 𝑝 𝑥 𝑞 𝐿 𝑝 𝑥 . This gives ( A.6 ) upon taking supremum ov er 𝑣 . Case (iii) ( 𝑝 𝑥 , 𝑝 𝑣 ≠ ∞ ). W e put 𝑘 = 1 + 𝛿 ( 𝒂 · 𝑑 𝒑 ) − 1 . Since 𝛿 ∈ ( 0 , ( ( 𝑝 𝑥 ∧ 𝑝 𝑣 ) − 1 ) ( 𝒂 · 𝑑 𝒑 ) ) , we see that 𝒒 : = ( 𝑞 𝑥 , 𝑞 𝑣 ) = ( 𝑝 𝑥 𝑘 , 𝑝 𝑣 𝑘 ) ∈ [ 1 , ∞) 2 and 𝛿 + 3 𝑑 𝑝 𝑥 + 𝑑 𝑝 𝑣 = 3 𝑑 𝑞 𝑥 + 𝑑 𝑞 𝑣 . 25 Then by embedding 𝔹 0 , 1 𝒒 ; 𝒂 ↩ → 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ([ HRZ23 , Appendix B]) and ( 2.2 ), we have ∥ 𝑔 ∥ 𝔹 − 𝛿 , 1 𝒑 ; 𝒂 ≲ ∥ 𝑔 ∥ 𝔹 0 , 1 𝒒 ; 𝒂 ≲ ∥ 𝑔 ∥ 𝕃 𝒒 ≲ ∥ 𝑓 1 | 𝑓 | > 𝑁 ∥ 𝕃 𝒒 . It is straightforward to see that ∥ 𝑓 1 | 𝑓 | > 𝑁 ∥ 𝕃 𝒒 ⩽ 𝑁 − 𝑝 𝑥 − 𝑞 𝑥 𝑞 𝑥   ℝ 𝑑 ∥ 𝑓 ( · , 𝑣 ) ∥ 𝑝 𝑥 𝑞 𝑣 𝑞 𝑥 𝕃 𝑝 𝑥 𝑑 𝑣  1 𝑞 𝑣 = 𝑁 − 𝑝 𝑥 − 𝑞 𝑥 𝑞 𝑥 ∥ 𝑓 ∥ 𝑝 𝑣 𝑞 𝑣 𝕃 𝒑 . Noting that 𝑝 𝑥 − 𝑞 𝑥 𝑞 𝑥 = 𝑘 − 1 = 𝛿 ( 𝒂 · 𝑑 𝒑 ) − 1 , the proof completes. □ A p p e n d i x B. P a r a p r o d u c t e s t i m at e s i n a n i s o t r o p i c B e s o v s p a c e s Recall the Bony decomp osition 𝑓 𝑔 = 𝑓 ≺ 𝑔 + 𝑓 ◦ 𝑔 + 𝑓 ≻ 𝑔 = : 𝑓 ≺ 𝑔 + 𝑓 ≽ 𝑔, (B.1) where 𝑓 ≺ 𝑔 = 𝑔 ≻ 𝑓 : = ∞  𝑘 = 1 𝑆 𝒂 𝑘 − 1 𝑓 R 𝒂 𝑘 𝑔, 𝑆 𝒂 𝑘 𝑓 : = 𝑘  𝑗 = 0 R 𝒂 𝑗 𝑓 , 𝑓 ◦ 𝑔 : =  | 𝑖 − 𝑗 | ⩽ 2 R 𝒂 𝑖 𝑓 R 𝒂 𝑗 𝑔 . Lemma B.1 (Paraproduct estimates) . Let 𝒑 , 𝒒 , 𝒓 ∈ [ 1 , ∞] 2 with 1 / 𝒑 + 1 / 𝒒 = 1 / 𝒓 , 𝛼 ∈ ℝ and 𝛽 < 0 . Then ∥ 𝑓 ≺ 𝑔 ∥ 𝔹 𝛼 𝒓 ; 𝒂 ≲ ∥ 𝑓 ∥ 𝒒 ∥ 𝑔 ∥ 𝔹 𝛼 𝒑 ; 𝒂 (B.2) and ∥ 𝑓 ≻ 𝑔 ∥ 𝔹 𝛼 + 𝛽 𝒓 ; 𝒂 ≲ ∥ 𝑓 ∥ 𝔹 𝛼 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝔹 𝛽 𝒒 ; 𝒂 . (B.3) When 𝛼 + 𝛽 > 0 , ∥ 𝑓 ◦ 𝑔 ∥ 𝔹 𝛼 + 𝛽 𝒓 ; 𝒂 ≲ ∥ 𝑓 ∥ 𝔹 𝛼 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝔹 𝛽 𝒒 ; 𝒂 . (B.4) Moreover , we have ∥ 𝑓 ◦ 𝑔 ∥ 𝕃 𝒓 ≲ ∥ 𝑓 ∥ 𝔹 𝛼 , 1 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝔹 − 𝛼 𝒒 ; 𝒂 , (B.5) ∥ 𝑓 ≻ 𝑔 ∥ 𝕃 𝒓 ⩽ ∥ 𝑓 ∥ 𝔹 − 𝛽, 1 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝔹 𝛽 , 1 𝒒 ; 𝒂 , (B.6) ∥ 𝑓 ≻ 𝑔 ∥ 𝕃 𝒓 ⩽ ∥ 𝑓 ∥ 𝔹 0 , 1 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝒒 . (B.7) Consequently , ∥ 𝑓 ≽ 𝑔 ∥ 𝒓 ≲ min ( ∥ 𝑓 ∥ 𝔹 0 , 1 𝒒 ; 𝒂 ∥ 𝑔 ∥ 𝒑 , ∥ 𝑓 ∥ 𝔹 − 𝛽, 1 𝒒 ; 𝒂 ∥ 𝑔 ∥ 𝔹 𝛽 , 1 𝒑 ; 𝒂 ) . (B.8) 26 Proof. The estimates ( B.2 ) , ( B.3 ) and ( B.4 ) are standard (see [ HZZZ24 , Lemma 2.11] for instance). W e only show ( B.5 ) - ( B.7 ) , which implies ( B.8 ) through the embedding 𝕃 𝒒 ↩ → 𝔹 0 𝒒 ; 𝒂 (see ( 2.2 ) ). Concerning ( B.5 ), by denition and Hölder inequality , we have that ∥ 𝑓 ◦ 𝑔 ∥ 𝒓 ≲ ∞  𝑖 = 0 2  ℓ = − 2 ∥ R 𝒂 𝑖 𝑓 ∥ 𝒑 ∥ R 𝒂 𝑖 + ℓ 𝑔 ∥ 𝒒 ≲ ∞  𝑖 = 0 2 𝛼 𝑖 ∥ R 𝒂 𝑖 𝑓 ∥ 𝒑 ∥ 𝑔 ∥ 𝔹 − 𝛼 𝒒 ; 𝒂 ≲ ∥ 𝑓 ∥ 𝔹 𝛼 , 1 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝔹 − 𝛼 𝒒 ; 𝒂 . As for ( B.7 ) and ( B.6 ), applying Hölder inequality , we have ∥ 𝑓 ≻ 𝑔 ∥ 𝕃 𝒓 ⩽ ∞  𝑘 = 1 ∥ 𝑆 𝒂 𝑘 − 1 𝑔 R 𝒂 𝑘 𝑓 ∥ 𝒓 ⩽ ∞  𝑘 = 1 ∥ 𝑆 𝒂 𝑘 − 1 𝑔 ∥ 𝒒 ∥ R 𝒂 𝑘 𝑓 ∥ 𝒑 . From ( 2.1 ) , we have  𝑆 𝒂 𝑘 𝑔 ( 𝜉 ) = 𝜒 𝒂 0 ( 2 − 𝑘 𝒂 𝜉 ) ˆ 𝑔 ( 𝜉 ) . This implies, through Y oung convolution inequal- ity , that ∥ 𝑆 𝒂 𝑘 𝑔 ∥ 𝒑 ⩽ ∥ 𝑔 ∥ 𝒑 . Hence, we hav e ∥ 𝑓 ≻ 𝑔 ∥ 𝕃 𝒓 ⩽ ∞  𝑘 = 1 ∥ 𝑔 ∥ 𝒒 ∥ R 𝒂 𝑘 𝑓 ∥ 𝒑 ⩽ ∥ 𝑓 ∥ 𝔹 0 , 1 𝒑 ; 𝒂 ∥ 𝑔 ∥ 𝒒 , which shows ( B.7 ). Furthermore , since 𝛽 < 0 , we hav e ∥ 𝑆 𝒂 𝑘 𝑔 ∥ 𝒑 ⩽ 2 − 𝛽 𝑘 ∥ 𝑔 ∥ 𝔹 𝛽 , 1 𝒒 ; 𝒂 . 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