Serigrafía-Almacén

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, T q ˆ R^d´1, Ω^T :“ p0, T q ˆ R^d´1“ ď

tPp0,T q

Ω^T_t for t P p0, T q. We will also, similarly as in [Hai14], introduce a set

P^T :“␣x P R^d : x₁ P t0, T u( . so that Ω^T “`

r0, T s ˆ R^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 distribution F : R^dÑV´^γ PD^γ,ηpΩ^T;T q, where “typically” η ď γ, with norm

}F }D^γ,ηpΩ^T;T q :“ 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;T q 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 ă β and x, y P Ω^T_t, }y ´ x}s ď t

}F_y^α´ Γ^α_yxF_x^ăβ}Tα À }F }D^γ,ηpΩ^T;T qt^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;T q is useless for our purposes due to the locality condition

}y ´ x} ď t (5.34)

for y, x P 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. LetV Ď T be a sector, let VzW be the complement of a sector W withinV and let pΠ, Γq be a model on T . Given parameters η, γ P RzAN^d with η ď γ 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,T q

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, p ˆΠ, ˆΓq and

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

“distance”

}F ; ˆF }_Drη,γspΩ^T;VzW,Γ,ˆΓq:“ sup

βPrη,γszA

Nd

sup

tPp0,T q

t^β´ηθ }F^ăβ; ˆF^ăβ}_DβpΩ^T_t;VzW,Γ,ˆΓq ă 8 . (5.36) IfW “ t0u we simply write D^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 for F 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 convention F PD^rη,γspΩ^Tq to R^dby setting F pxq “ 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;VzWq with VzW, η, γ as in Definition 5.3.1. We then have for t P p0, T q and β ď γ

}F^ăβ}D^βpΩ^T_t;VzWq À p1 ` }Γ}ηq t

pη´βq^0

θ }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 }^DηpΩ^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 for F PD^rη,γs (up to an arbitrarily small loss κ ą 0).

Lemma 5.3.5. Let F PD^rη,γspΩ^T;VzWq with VzW, η, γ as in Definition 5.3.1. We then have for α P AVzW, t P p0, T q, x P Ω^T_t and any κ ą 0

}F_x^α}Tα À t

pη´α´κq^0

θ }F }D^rη,γspΩ^T;VzWq. (5.37)

Given a second model p ˆΠ, ˆΓq and ˆF PD^rη,γspΩ^T;VzWq we also have }F_x^α´ ˆF_x^α}Tα À t

pη´α´κq^0

θ }F ; ˆF }_Drη,γspΩ^T;VzW,Γ,ˆΓq. (5.38)

Proof. For both inequalities, (5.37) and (5.38), take β “ pα ` κq _ η in (5.35) or (5.36) respectively, with κ ą 0 without loss of generality small enough such that β P rη, γszA<sub>N</sub><sup>d</sup>. The claim then follows due to η ´pα `κq_η “ pη ´α ´κq^pη ´ηq “ pη ´ α ´ κq ^ 0.

Lemma 5.3.5 implies in particular that D^rη,γspΩ^T;T q Ď D^γ,η´κpΩ^T;T q, so that we can apply results like [Hai14, Proposition 6.9] to have a reconstruction RF for F PD^rη,γspΩ^T;T q, when F is extended by 0 as explained above.

Before we show that our spaceD^rη,γspΩ^T;T q essentially behaves like D^γ,ηpΩ^T;T q 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 ˆ T 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> P A, 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 P G, τ¹ P V^α1, τ₂ PV^α2 with α₁, α₂ P A, α1` α2 ă γ 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 P RzA_N^dwith η₁ ď γ1 and η₂ ď γ2 be such that γ “ pγ1`α2q^pγ2`α1q R A_N^d and let κ ą 0 be such that η “ γ ^pη1`η2´κq^pη1`α2´κq^pη2`α1´κq R A_N^d. If the pair pV¹,V²q is γ-regular, and we are given some model pΠ, Γq and maps F PD^rη1,γ1spΩ^T;V1, Γq, G PD^rη2,γ2spΩ^T;V2, Γq, then we have the estimate

}F ‹ G}D^rη,γspΩ^T;T qÀ 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 p ˆΠ, ˆΓq and

F Pˆ D^αrη11,γ1spΩ^T;V¹, ˆΓq, ˆG PD^αrη22,γ2spΩ^T;V², ˆΓq, we also have }F ‹ G; ˆF ‹ ˆG}_Drη,γspΩ^T;T qÀ K ¨`

}F ; ˆF }_Drη1,γ1spΩ^T;V1,Γ,ˆΓq

` }G; ˆG}<sub>D</sub>rη2,γ2spΩ<sup>T</sup>;V2,Γ,ˆΓq` }Γ ´ ˆΓ}γ1_γ2˘ , (5.40) where K is a polynomial in the corresponding norms of F, ˆF , G, ˆG, Γ and ˆΓ.

Proof. Note that, without loss of generality, we can choose κ ą 0 smaller than any ε ą 0 and we can assume that }F }Drη1,γ1spΩ^T;V1q ď 1 and }G}Drη2,γ2spΩ^T;V2q ď 1. Fix β P rη, γszA_N^d and α P A with α ă β.

Consider for t P p0, T q and x, y P Ω^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 P AV₁, 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 to p1 ` }Γ}γ1q this yields

Note that the application of Lemma 5.3.4 was allowed since ν₁ ě α1 and thus β ´ ν1 ď γ ´ ν1 ď γ ´ α1 “ pγ1` α2q ^ pγ2` α1q ´ α1 ď pγ2` α1q ´ α1 “ γ2

Let us bound the exponent of t in (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 to p1 ` }Γ}γ2q, as

}y ´ x}^β´µ2´µ1tpη1´pβ´µ2qq^0

θ tpη2´µ2´κq^0

θ “ }y ´ x}^γ´αtpη1´pβ´µ2qq^0`pη2´µ2´κq^0

θ .

The exponent of t 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_γ2q }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 x P Ω^T_t

}pF ‹ Gq^α_x}T_α “

›

›

› ÿ

µ1`µ2“α

F_x^µ1 ‹ G^µ_x²

›

›

›Tα

À ÿ

µ1`µ2“α

tpη1´µ1´κ{2q

θ ^0

¨ t^{η2´µ2´κ{2θ ^0} À t^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_yxp ˆF ‹ ˆGq^ăβ_x ´ p ˆF ‹ ˆGq^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_N^d such that η ď γ, γ ě γ¹ and η ě η¹ we have the embeddingD^rη,γspΩ^T;VzWq Ď D^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´η1

θ _0

}F }D^rη,γspΩ^Tq;VzWq. (5.44) Given a second model p ˆΠ, ˆΓq on T we further have

}F ; ˆF }_Drη1,γ1spΩ^T;VzW,Γ,ˆΓqÀ T

η^α´η1 θ _0`

}F }D^rη,γspΩ^Tq;VzWq}Γ ´ ˆΓ}η

` p1 ` }ˆΓ}ηq }F ; ˆF }_Drη,γspΩ^Tq;VzW,Γ,ˆΓq˘ . (5.45)

128 5.3 Singular spaces and extensions Proof. Using that for β ă η we have

}F^ăβ}D^ηpΩ^T_t;VzWq À p1 ` }Γ}ηq}F^ăη}D^ηpΩ^T_t;VzWq we get (5.44) by splitting }F }_Drη1,γ1spΩ^T;VzWq ď sup

tPp0,T q

sup

βPrη¹_α,ηszA

Nd

t^β´η1θ }F^ăβ}D^βpΩ^T_t;VzWq

` sup

tPp0,T q

sup

βPrη,γ¹szA

Nd

t^β´η1θ }F^ăβ}D^βpΩ^T_t;VzWq

À T

α´η1 θ _0

p1 ` }Γ}ηq sup

tPp0,T q

}F^ăη}_D^η_pΩ^T

t;VzWq

` T

η´η1

θ }F }_D^rη,γs_pΩ^T_;VzWq À p1 ` }Γ}ηq ¨ T

η^α´η1

θ _0

}F }D^rη,γspΩ^Tq;VzWq. (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 model pΠ, Γq. LetV 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.

For γ P p0, 8qzA_N^d, η P p0, γszA_N^d such that pV, Vq is γ-regular in the sense of Definition 5.3.6 define the composition of a smooth F P C⁸pRⁿ, Rq, n ě 1 with the vector V “ pV1, . . . , V_nq P pD^rη,γspVqqⁿ as in [Hai14, Subsection 4.2] by

F pV q “ÿ

k

1

k!B^kF pV¹q pV ´ V¹1q^‹k, (5.46) where we wrote V¹ “ pV₁¹, . . . , V_n¹q and use the notation v^‹k :“ v₁^‹k1 ‹ . . . ‹ v_n^‹kn for v P Vⁿ and k P Nⁿ (recall the definition of τ¹ as “the coefficient of τ before 1”, 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 α P AV only finitely many terms contribute to (5.46). We will work with the object F^ăγ “ř

αPAV: αăγpF pV qq^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 ą 0 and the norm of F^ăγpV q 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 p ˆΠ, ˆΓq we further

have for V P pD^rη,γspΩ^T;V, Γqqⁿ, ˆV P pD^rη,γspΩ^T;V, ˆΓqqⁿ the bound

}F^ăγpV q; F^ăγp ˆV q}_Drη,γspΩ^T;V,Γ,ˆΓqÀ K ¨ p}V ; ˆV }_Drη,γspΩ^T;V,Γ,ˆΓq` }Γ ´ ˆΓ}γq , (5.47) where we write }V ; ˆV }_Drη,γspΩ^T;V,ˆΓ,Γq “ řn

i“1}Vi; ˆV_i}_Drη,γspΩ^T;V,ˆΓ,Γq and where K is some polynomial in the corresponding norms of V₁, ˆV₁, . . . , V_n, ˆV_n, Γ and ˆΓ.

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 F pV q can be bounded uniformly for all V, ˆV 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 ˆV , ˆΓ in the last part of the proof). Fix some β P rη ´ κ, γs and ζ P p0, ηq such that ζ ă minpAVzt0uq ^ 1. We will use throughout the proof that due to Lemma 5.3.4 and Lemma 2.1.23 we have by our choice of ζ

}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, T q and x, y P Ω^T_t with }y ´ x}s ď 1

ÿ

kPNⁿ

1

k!B^kF pV_y¹q ppVy´ V_y¹q^‹kq^α´ Γ^α_yx ÿ

lPNⁿ

1

l!B^lF pV_x¹q ppVx´ V_x¹1q^‹lq^ăβ

“ÿ

kPNⁿ: |k|ďL

1

k!B^kF pV_y¹q ppVy ´ V_y¹1q^‹kq^α

´ÿ

lPNⁿ: |l|ďL

1

l!B^lF pV_x¹q ÿ

0ďmďl

ˆ l m

˙

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

We can choose any L ě β{ζ in the second line, since above β{ζ the terms under consideration vanish by our choice of ζ. We take

L “ rβ{ζs .

Let us emphasize that we really measure the size of the multi-indices k, l in isotropic scaling, that is |k| “ k1` . . . ` kd.

By Lemma 5.3.7 we have (since V is function like)

}Γ^α_yxppVxq^‹mq^ăβ´`V_y^‹m˘α

}Tα À K t

η´κ´β

θ }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 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, Γq a unique distribution RF which coincides within Ω^Tt with the unique distribution given by Theorem 2.3.19 and satisfies further for any κ ą 0

}RF }C^{α^η´κ}_s pR^d;Rq À K1¨ }F }_D^rη,γs_pΩ^T_{;V q}. (5.52) where K₁ is a polynomial in |||pΠ, Γq|||γ. Given a second model p ˆΠ, ˆΓq and a corre-sponding operator ˆR we further have

}RF ´ ˆR ˆF }_C^{α^η´κ}

s pR^d;Rq À K2¨ p}F ; ˆF }_Drη,γspΩ^T;V,Γ,ˆΓq` |||pΠ, Γq; p ˆΠ, ˆΓq|||γq . (5.53) where K2 is a polynomial in the norms of F, ˆF , pΠ, Γq and p ˆΠ, ˆΓq.

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].

In document Lineamientos para un sistema de prevención y combate de incendios en una empresa de plásticos. (página 34-38)