Singular modelled distributions

We can repeat almost the same proof as above: Note that the only cases where we usedF PD^γwere (5.22), (5.29) and (5.31). While it is now enough to recall in (5.22) and (5.29) that now xPΩ, we can restrict the sum in (5.31) to α¹ P AVz|N^d|s since polynomial entries vanish in the paraproduct (Lemma 2.1.14) and (5.19) is proved.

The distance estimates for modelspΠ,Γq,pΠ,ˆ Γqˆ follow almost immediately from a repetition of the computations performed above, one only has to repeat the induction (5.21) for}F_y^α1 ´Γ^α_yx¹F_x´ pFˆ_y^α1´Γˆ^α_yx¹Fˆ_xq} instead.

5.3 Singular spaces and extensions

5.3.1 Singular modelled distributions

Throughout this subsection we will fix a regularity structure T “ pA,T, Gq (not necessarily satisfying Assumption 2.3.12) together with a scaling vector

s“ pθ,s¹q P p1,8q ˆ r1,8q^d´1

for some θ ą 1. We further consider a time horizon T P p0,1s and an associated family of sets

Ω^T_t :“ pt, Tq ˆR^d´1, Ω^T :“ p0, Tq ˆR^d´1 “ ď

tPp0,Tq

Ω^T_t

for t P p0, Tq. We will also, similarly as in [Hai14], introduce a set P^T :“␣

xPR^d : x₁ P t0, Tu( . so that Ω^T “`

r0, Ts ˆR^d˘

zP^T. The reason why we also exclude points x₁ “T is only technical and lies in the fact that we prefer to work on open sets.

In [Hai14, Definition 6.2] the author defines the notion of a singular, modelled distributionF :R^dÑV´^γ PD^γ,ηpΩ^T;Tq, where “typically” ηďγ, with norm

}F}D^γ,ηpΩ^T;Tq :“ sup

tPp0,1q

sup

αPA

t^pα´ηq_0θ sup

xPΩ^T_t

}F_x^α}Tα

` sup

tPp0,1q

sup

αPA

sup

x,yPΩ^T_t,}y´x}sďt

t^γ´ηθ }F_y^α´Γ^α_yxF_x}T_α

}y´x}^γ´α . (5.32)

In fact, the norm introduced in [Hai14] has a slightly different definition, but is equivalent to the expression in (5.32). D^γ,ηpΩ^T;Tq should be seen as the space of modelled distribution with a possible blow-up at tx₁ “0u. This space then shows nice behavior under multiplication and composition with smooth functions [Hai14, Proposition 6.12, 6.13]. It further has the following property for β P rη, γs, α ăβ and x, y PΩ^T_t,}y´x}s ďt

}F_y^α´Γ^α_yxF_x^ăβ}Tα À }F}D^γ,ηpΩ^T;Tqt^η´βθ }y´x}^β´α, (5.33) where F^ăβ :“ ř

α¹PA:αăβF^α1 , which means that “lower regularity comes with a weaker weight”. Alas, the space D^γ,ηpΩ^T;Tq is useless for our purposes due to the locality condition

}y´x} ďt (5.34)

fory, xP Ω^T in the second term in (5.32). Since a Fourier based approach is highly non-local a requirement like (5.34) is hard to translate. We here propose to re-quire instead 5.33 from the beginning (for general points not only those satisfying (5.34)). The estimates for multiplication and composition are then again true, com-pare Lemma 5.3.7 and 5.3.9 below (except for a slight increase in the weight which is irrelevant for our purposes).

Definition 5.3.1. Let V ĎT be a sector, letVzW be the complement of a sectorW withinV and letpΠ,Γqbe a model onT. Given parametersη, γ PRzA_N^d withη ďγ we say that F : Ω^T Ñ V is an element of D^rη,γspΩ^T;VzWq “ D^rη,γspΩ^T;VzW,Γq if the following semi-norm is finite

}F}_D^rη,γs_pΩ^T_;VzWq :“ sup

βPrη,γszA

Nd

sup

tPp0,Tq

t^β´ηθ }F^ăβ}_D^β_pΩ^T

t;VzWq ă 8, (5.35)

where F^ăβ :“ř

αPA:αăβF^α. Given two models pΠ,Γq, pΠ,ˆ Γqˆ and

F P D^rη,γspΩ^T;VzW,Γq, Fˆ P D^rη,γspΩ^T;VzW,Γqˆ we also introduce the notion of a

“distance”

}F; ˆF}_Drη,γspΩ^T;VzW,Γ,ˆΓq:“ sup

βPrη,γszA

Nd

sup

tPp0,Tq

t^β´ηθ }F^ăβ; ˆF^ăβ}_DβpΩ^T_t;VzW,Γ,Γqˆ ă 8. (5.36) IfW “ t0uwe simply writeD^rη,γspΩ;Vq:“D^rη,γspΩ;Vzt0uq, a similar remark applies for the distance (5.36).

124 5.3 Singular spaces and extensions Remark 5.3.2. _B A remark such as 2.3.15 applies: The notation “VzW” in F P D^rη,γspΩ^T;VzWq is supposed to indicate two things, first that F takes values in the sector V (and not only in VzW) and moreover that (5.35) is finite for its components in VzW. In particular we have

D^rη,γspΩ^T;Vq Ď D^rη,γspΩ^T;VzWq.

Note that a slight technical difference to (2.3.14)is that we allow a priori forF with values above γ, so that (5.35) is really only a semi-norm, even if W “ t0u.

Remark 5.3.3. The exclusion of the set A_N^d is actually not needed at this stage of our theory, it will however be convenient in Section 6.2 below.

Note that we do not require any lower bound for η so that β in the supremum in (5.35) might be below the lowest regularity ofV and in this case we simply have F^ăβ “0. We further extend by conventionF PD^rη,γspΩ^TqtoR^dby settingFpxq “ 0 for x P pΩ^Tq^c. If F P D^rη,γspΩ^Tq we have in fact also a bound in D^β for β ă η as shown by the subsequent lemma.

Lemma 5.3.4. Let F PD^rη,γspΩ^T;VzWqwith VzW, η, γ as in Definition 5.3.1. We then have for tP p0, Tq and β ďγ

}F^ăβ}D^βpΩ^T_t;VzWq À p1` }Γ}_ηqt^{pη´βq^0θ} }F}D^rη,γspΩ^T;VzWq

with F^ăβ as in Definition 5.3.1.

Proof. For β ě η this is just the Definition of D^rη,γspΩ^T;VzWq (without even the need of the constant p1` }Γ}ηq). For the case β ă η note first that F^ăη P D^ηpΩ^T;VzWq with }F}D^ηpΩ^T;VzWq ď }F}_D^rη,γs_pΩ^T_;VzWq. Remark 3.2 of [Hai14] then states that F^ăβ “ pF^ăηq^ăβ PD^βpΩ^T;VzWq with

}F^ăβ}D^βpΩ^T;VzWq À p1` }Γ}_ηq}F^ăη}D^ηpΩ^T;VzWq

and the claim follows.

The following lemma shows that the first term in (5.32) is bounded forF PD^rη,γs (up to an arbitrarily small lossκą0).

Lemma 5.3.5. Let F PD^rη,γspΩ^T;VzWqwith VzW, η, γ as in Definition 5.3.1. We then have for αPAVzW, tP p0, Tq, xPΩ^T_t and any κą0

}F_x^α}Tα Àt^{pη´α´κq^0θ} }F}D^rη,γspΩ^T;VzWq. (5.37) Given a second model pΠ,ˆ Γˆq and Fˆ PD^rη,γspΩ^T;VzWq we also have

}F_x^α´Fˆ_x^α}Tα Àt^{pη´α´κq^0θ} }F; ˆF}_Drη,γspΩ^T;VzW,Γ,Γqˆ . (5.38)

Proof. For both inequalities, (5.37) and (5.38), take β “ pα `κq _ η in (5.35) or (5.36) respectively, with κą0without loss of generality small enough such that βP rη, γszA<sub>N</sub>d. The claim then follows due toη´ pα`κq _η “ pη´α´κq ^ pη´ηq “ pη´α´κq ^0.

Lemma 5.3.5 implies in particular that D^rη,γspΩ^T;Tq Ď D^γ,η´κpΩ^T;Tq, so that we can apply results like [Hai14, Proposition 6.9] to have a reconstruction RF for F P D^rη,γspΩ^T;Tq, when F is extended by0 as explained above.

Before we show that our spaceD^rη,γspΩ^T;Tqessentially behaves likeD^γ,ηpΩ^T;Tq under multiplication let us recall the definition of a product on T.

Definition 5.3.6 (Definition 4.1, 4.6 from [Hai14]). A product ‹ is a continuous bilinear map from T ˆT to T such that τ₁‹τ₂ P T^α1`α2 for τ<sub>1</sub> P T<sup>α</sup>1, τ<sub>2</sub> P T<sup>α</sup>2 and α<sub>1</sub>, α<sub>2</sub> PA, where we set Tα1`α2 “ t0u if α₁`α₂ R A.

A pair of sectors V¹, V² ĎT is called γ-regular for γ P R if for any Γ PG, τ1 P V^α1, τ₂ PV^α2 with α₁, α₂ P A, α₁`α₂ ăγ we have Γpτ₁‹τ₂q “ Γτ₁‹Γτ₂.

Lemma 5.3.7. Let V¹, V² ĎT be two sectors of regularity α₁, α₂ and let

η₁, γ₁, η₂, γ₂ PRzA_N^dwithη₁ ďγ₁ andη₂ ďγ₂ be such thatγ “ pγ₁`α<sub>2</sub>q^pγ<sub>2</sub>`α₁q R A_N^d and letκ ą0be such thatη“γ^pη1`η2´κq^pη1`α2´κq^pη2`α1´κq RA_N^d. If the pair pV¹,V²q is γ-regular, and we are given some model pΠ,Γq and maps F P D^rη1,γ1spΩ^T;V1,Γq, GPD^rη2,γ2spΩ^T;V2,Γq, then we have the estimate

}F ‹G}D^rη,γspΩ^T;Tq À p1` }Γ}γ1_γ2q ¨ }F}D^rη1,γ1spΩ^T;V₁q}G}D^rη2,γ2spΩ^T;V₂q. (5.39) If we are given a second model pΠ,ˆ Γqˆ and

FˆP D^αrη11,γ1spΩ^T;V¹,Γq,ˆ Gˆ PD^αrη22,γ2spΩ^T;V²,Γq, we also haveˆ }F ‹G; ˆF ‹Gˆ}D^rη,γspΩ^T;Tq ÀK¨`

}F; ˆF}_D^rη1,γ1spΩ^T;V1,Γ,ˆΓq

` }G; ˆG}<sub>D</sub>rη2,γ2spΩ<sup>T</sup>;V<sub>2</sub>,Γ,Γqˆ ` }Γ´Γ}ˆ γ1_γ2

˘, (5.40) where K is a polynomial in the corresponding norms of F, F , G,ˆ G,ˆ Γ and Γ.ˆ Proof. Note that, without loss of generality, we can choose κą0 smaller than any ε ą 0 and we can assume that }F}D^rη1,γ1spΩ^T;V₁q ď 1 and }G}_D^rη2,γ2spΩ^T;V₂q ď 1. Fix βP rη, γszA_Nd and αPA with αăβ.

Consider for tP p0, Tqand x, y PΩ^T_t Γ^α_yxpF ‹Gq^ăβ_x ´ pF ‹Gq^α_y “Γ^α_yx

˜ ÿ

ν1`ν2ăβ

F_x^ν1‹G^ν_x²

¸

´ ÿ

µ1`µ2“α

F_y^µ1 ‹G^µ_y²

“ ÿ

µ1`µ2“α

˜ ÿ

ν1`ν2ăβ

Γ^µ_yx¹F_x^ν1 ‹Γ^µ_yx²G^ν_x² ´F_y^µ1 ‹G^µ_y²

¸ ,

126 5.3 Singular spaces and extensions where the sums run over ν₁ P AV₁, ν₂ P AV₂ and µ₁ PAV₁, µ₂ P AV₂. We will bound each term on the right hand side separately. Let us reshape first

ÿ Lemma 5.3.4 for the second factor. Up to a constant proportional top1` }Γ}_γ1qthis yields

}y´x}^ν_s^1´µ1t^{pη1´ν1θ´κq^0} ¨ }y´x}^β´ν_s ^1´µ2t^{pη2´pβ´νθ 1qq^0}

“ }y´x}^β´α_s t^{pη1´ν1´κq^0`pηθ 2´pβ´ν1qq^0} (5.43)

Note that the application of Lemma 5.3.4 was allowed since ν₁ ěα₁ and thus β´ν₁ ďγ´ν₁ ďγ´α₁ “ pγ₁`α<sub>2</sub>q ^ pγ<sub>2</sub>`α₁q ´α₁ ď pγ₂`α₁q ´α₁ “γ₂ Let us bound the exponent of tin (5.43) from below, by distinguishing between the 4 different cases that might occur

pη₁´ν₁´κq ^0` pη₂´ pβ´ν₁qq ^0

Applying once more Lemma 5.3.4 and 5.3.5 the term (5.42) can be bounded, up to a constant proportional top1` }Γ}_γ2q, as

}y´x}^β´µ2´µ1t^{pη1´pβ´µθ 2qq^0} t^{pη2´µ2θ´κq^0} “ }y´x}^γ´αt^{pη1´pβ´µ2qq^0`pηθ 2´µ2´κq^0} . The exponent oft can be bounded from below as above, so that we get altogether

}pF ‹Gq^α_y ´Γ^α_yxpF ‹Gq^ăβ_x }Tα À p1` }Γ}_{γ1_γ2}q }y´x}^β´α_s t^η´βθ .

To estimate the first term of the norm (2.46) note that for α and β as above we have by Lemma 5.3.5 for xPΩ^T_t

}pF ‹Gq^α_x}T_α “

›

›

› ÿ

µ1`µ2“α

F_x^µ1 ‹G^µ_x²

›

›

›Tα

À ÿ

µ1`µ2“α

t^{pη1´µθ1´κ{2q^0}¨t^{η2´µ2θ´κ{2^0} Àt^η´β,

which closes the proof for (5.39). To control (5.40) we use the same decomposition as above, that is

Γ^α_yxpF‹Gq^ăβ_x ´ pF‹Gq^α_y

“ ÿ

ν1ăβ´µ2

Γ^µ_yx¹F_x^ν1‹`

Γ^µ_yx²G^ăβ´ν_x ¹´G^µ_y²˘

´`

Γ^µ_yx¹F_x^ăβ´µ2´F_y^µ1˘

‹G^µ_y²

and similarly for Γ^α_yxpFˆ‹Gqˆ ^ăβ_x ´ pFˆ ‹Gqˆ ^α_y. Successive applications of the triangle inequality then yield with exactly the same estimates as above (5.40).

We also have the following embedding result for the spaces D^rη,γs.

Lemma 5.3.8. Given η, γ, η¹, γ¹ P RzA_Nd such that η ď γ, γ ě γ¹ and η ě η¹ we have the embeddingD^rη,γspΩ^T;VzWq ĎD^rη1,γ1spΩ^T;VzWq, more precisely if α is the regularity of VzW:

}F}_D^rη1,γ1s_pΩ^T_;VzWqÀ p1` }Γ}ηq ¨T^{η^α´η}

1

θ _0

}F}D^rη,γspΩ^Tq;VzWq. (5.44) Given a second model pΠ,ˆ Γˆq on T we further have

}F; ˆF}_Drη1,γ1spΩ^T;VzW,Γ,Γqˆ ÀT^{η^α´η}

1 θ _0`

}F}D^rη,γspΩ^Tq;VzWq}Γ´Γ}ˆ _η

` p1` }Γ}ˆ _ηq }F; ˆF}_Drη,γspΩ^Tq;VzW,Γ,Γqˆ

˘. (5.45)

128 5.3 Singular spaces and extensions (5.45) follows by exactly the same arguments.

We now consider again some product ‹on a regularity structureT “ pA,T, Gq which is equipped with a modelpΠ,Γq. Let V Ď T be some function like sector in the sense of Definition 2.3.4. We assume thatV is stable under ‹, by what we mean that τ₁‹τ₂ PV holds for τ₁, τ₂ P V. compare page 46). Note that this sum is infinite (and therefore it is actually not contained in the direct sumÀ

αPAT^α), but well-defined since for every homogeneity αPAV only finitely many terms contribute to (5.46). We will work with the object F^ăγ “ř

αPAV:αăγpFpVqq^α from now on. We have the following result, which can be seen as the analogue of [Hai14, Proposition 6.13] for the spaces D^rη,γs.

Lemma 5.3.9. Let γ, η, V be as above, let pΠ,Γq be some model and let F P C⁸pRⁿ;Rq be such that it has at most polynomially growing derivatives. We then have

F^ăγ : pD^rη,γspΩ^T;V,Γqqⁿ ÑD^rη´κ,γspΩ^T;V,Γq

for any κ ą0 and the norm of F^ăγpVq for V P pD^rη,γspΩ^T;V,Γqqⁿ is bounded by a polynomial in the norms of V₁, . . . , V_n,Γ. Given a second model pΠ,ˆ Γqˆ we further

have for V P pD^rη,γspΩ^T;V,Γqqⁿ, Vˆ P pD^rη,γspΩ^T;V,Γqqˆ ⁿ the bound

}F^ăγpVq;F^ăγpVˆq}_Drη,γspΩ^T;V,Γ,Γqˆ ÀK¨ p}V; ˆV}_Drη,γspΩ^T;V,Γ,Γqˆ ` }Γ´Γ}ˆ _γq, (5.47) where we write }V; ˆV}_Drη,γspΩ^T;V,ˆΓ,Γq “ řn

i“1}V_i; ˆV_i}_Drη,γspΩ^T;V,Γ,Γqˆ and where K is some polynomial in the corresponding norms of V₁,Vˆ₁, . . . , V_n,Vˆ_n,Γ and Γ.ˆ

Remark 5.3.10. If F has derivatives that grow faster than any polynomial we still have F : pD^rη,γspΩ^T;Vqqⁿ ÑD^rη´κ,γspΩ^T;Vq and (5.47), but now K and FpVq can be bounded uniformly for all V, Vˆ in any bounded set.

Proof. We will use the notation K for a polynomial, that might change from line to line, in the norms of V, Γ (and V ,ˆ Γˆ in the last part of the proof). Fix some βP rη´κ, γsand ζ P p0, ηqsuch that ζ ăminpAVzt0uq ^1. We will use throughout the proof that due to Lemma 5.3.4 and Lemma 2.1.23 we have by our choice ofζ

}V¹}_C^ζ

spΩ^Tq À }V^ăζ}D^ζpΩ^T;Vq À }V}D^rη,γspΩ^T;Vq “K . (5.48) We have to estimate for t P p0, Tq and x, y PΩ^T_t with }y´x}s ď1

ÿ

kPNⁿ

1

k!B^kFpV_y¹q ppV_y´V_y¹q^‹kq^α´Γ^α_yx ÿ

lPNⁿ

1

l!B^lFpV_x¹q ppV_x´V_x¹1q^‹lq^ăβ

“ÿ

kPNⁿ:|k|ďL

1

k!B^kFpV_y¹q ppV_y ´V_y¹1q^‹kq^α

´ÿ

lPNⁿ:|l|ďL

1

l!B^lFpV_x¹q ÿ

0ďmďl

ˆl m

˙

Γ^α_yxppV_xq^‹mq^ăβp´V_x¹q^pl´mq. (5.49)

We can choose anyLěβ{ζ in the second line, since above β{ζ the terms under consideration vanish by our choice ofζ. We take

L“rβ{ζs.

Let us emphasize that we really measure the size of the multi-indicesk, lin isotropic scaling, that is |k| “k₁`. . .`k_d.

By Lemma 5.3.7 we have (since V is function like) }Γ^α_yxppV_xq^‹mq^ăβ´`

V_y^‹m˘α

}Tα ÀK t^η´κ´βθ }y´x}^β´α_s ,

130 5.3 Singular spaces and extensions so that, due to (5.48), is enough to consider instead of (5.49)

ÿ

where we used the (isotropic) multidimensional Taylor formula in the last step. The only k that contribute satisfy |k| ď α{ζ, so that we can replace the last line by taking only Since further by the binomial theorem, (5.48) and once more Lemma 5.3.7

›

We now turn to the Lipschitz continuity of F. We split, once more, as above F^αpVyq ´Γ^α_yxFpVxq^ăβ “

ÿ

|k|ďα{ζ

AkpVqBkpVq ` ÿ

|l|ďL

ClpVq, F^αpVˆ_yq ´Γ^α_yxFpVˆ_xq^ăβ “ ÿ

|k|ďα{ζ

A_kpVˆqB_kpVˆq ` ÿ

|l|ďL

Cˆ_lpVˆq,

where

AkpVq “ 1

k!ppVy´V_y¹1q^‹kq^α, B_kpVq “

ÿ

|r|“L`1´|k|

L`1´ |k|

r!

ż1 0

dζp1´ζq^L´|k|B<sup>k`r</sup>FpV<sub>x`ζpy´xq</sub>¹ q pV_y¹ ´V_x¹q^r, C_kpVq “ ÿ

lPNⁿ:|l|ďL

1

l!B^lFpV_x¹q ÿ

0ďmďl

ˆl m

˙

“Γ^α_yxppV_xq^‹mq^ăβ ´ pV_y^‹mq^α‰

p´V_x¹q^pl´mq

(and similar forCˆ_kpVˆq). For A_k, B_k, C_k one easily sees by the arguments above }A_kpVq}T^α ďKt^η´κ´βθ , }A_kpVq ´A_kpVˆq}Tα ďKt^η´κ´βθ }V; ˆV}_Drη,γspΩ^T;V,Γ,Γqˆ , }B_kpVq}Tα ďK}y´x}^β´α_s ,

}CkpVq ´CˆkpVˆq}Tα ďKt^η´κ´βθ }y´x}^β´α_s p}V; ˆV}_D^rη,γs_pΩT;V,Γ,Γqˆ ` }Γ; ˆΓ}γq. (5.47) follows from these estimates as soon as we can show }B<sub>k</sub>pVq ´B<sub>k</sub>pVˆq}Tα ď K}y´x}<sup>β´α</sup><sub>s</sub> }V; ˆV}<sub>D</sub>rη,γspΩ<sup>T</sup>;V,Γ,Γqˆ , which is true since we chose L ěβ{ζ so that for rPN<sup>n</sup>,|r| “L`1´ |k| by the mean value theorem inRⁿ

|pV_y¹´V_x¹q^r´ pVˆ_y¹´Vˆ_x¹q^r| À p|V_y¹´V_x¹|^|r|´1` |Vˆ_y¹´Vˆ_x¹|^|r|´1q ¨sup

z

|V_z¹´Vˆ_z¹|

“sup

z

|V_z¹´Vˆ_z¹| ¨ p|V_y¹´V_x¹|^L´|k|` |Vˆ_y¹´Vˆ_x¹|^L´|k|q

À }V; ˆV}_Drη,γspΩ^T;V,Γ,Γqˆ K}y´x}^pL´|k|qζ_s ď }V; ˆV}_Drη,γspΩ^T;V,Γ,Γqˆ K}y´x}^β´α_s . We actually still have to bound the first terms in (2.46), (2.47) but this is once more an argument as in 5.51.

We can also reformulate the reconstruction theorem, Theorem 2.3.19, for our spacesD^rη,γspVq, which is a version of [Hai14, Proposition 6.9].

Lemma 5.3.11. Let V be a sector of regularity ´θ ă α ď 0. Let further γ P p0,8qzA_N^d and η P p´θ, γszA_N^d. Then, given some model pΠ,Γq, there is for F P

132 5.3 Singular spaces and extensions D^rη,γspΩ^T;V,Γqa unique distributionRF which coincides withinΩ^T_t with the unique distribution given by Theorem 2.3.19 and satisfies further for any κą0

}RF}_C^{α^η´κ}

s pR^d;Rq ÀK₁¨ }F}_D^rη,γs_pΩ^T_;Vq. (5.52)

where K₁ is a polynomial in |||pΠ,Γq|||_γ. Given a second model pΠ,ˆ Γqˆ and a corre-sponding operator Rˆ we further have

}RF ´RˆFˆ}_C^{α^η´κ}

s pR^d;Rq ÀK₂¨ p}F; ˆF}_Drη,γspΩ^T;V,Γ,Γqˆ ` |||pΠ,Γq;pΠ,ˆ Γq|||ˆ _γq. (5.53) where K₂ is a polynomial in the norms of F, F ,ˆ pΠ,Γq and pΠ,ˆ Γˆq.

Proof. Due to the continuous embedding D^rη,γspΩ^T;Vq Ď D^γ,η´κpΩ^T;Vq for κ ą 0 (page 125) this as consequence of [Hai14, Proposition 6.9].

Im Dokument Refinements of the Solution Theory for Singular SPDEs (Seite 134-144)