Modelled distributions are paracontrolled

nota-tion “VzW” in F P P^γpR^d;VzWq indicates two things, first that F takes values in the sector V (and not only in VzW) and moreover that (5.15) (and the structure condition) is true for its components in VzW. In particular we have

P^γpR^d;Vq Ď P^γpR^d;VzWq. Remark 5.1.7. Since for αP A_VzW and x, y PR^d we have Γ^α_yxF_x “ ř

α¹PA_VzW:α¹ěαΓ^α_yxF_x^α1 the paraproduct PpF,Γ^αq is really independent of the components of F which are not in VzW. As we did require F to have bounded components inVzW we see that the paraproductsPpF,Γ^αqin (5.15)are well-defined.

In [GIP15] an operator like PpF,Πq as well as a notion like (5.16) were already introduced. The authors further give an alternative construction for the reconstruc-tion operatorR (compare Theorem 2.3.19) based on Littlewood-Paley theory, with R satisfying a slightly weaker bound. By uniqueness these two objects coincide however for γ ą 0, see [GIP15, Lemma 6.7]. The authors also show the following result, which we will use occasionally.

Lemma 5.1.8. Given F P D^γpR^d;Tq with γ ą 0 the distribution RF is paracon-trolled by F, more precisely

}RF ´PpF,Πq}C_s^γpR^d;Rq À }F}D^γpR^d;Tq.

As we already pointed out above, what we presented in Definition 5.1.1 is not a strict generalization of the approach from [GIP15] or Chapter 4 as we use instead of a modified paraproduct ăă, as in Definition 4.1.4, a space-time paraproduct (5.8).

A construction in space-time might seem more natural, but we have to pay a price:

Since the solutions to SPDEs such as (5.7) are only defined on a compact time inter-val, we have to extend them in order to make sense of the space-time paraproduct.

We postpone the presentation of the corresponding tools and their application to Sections 5.3 and 6.2 and turn now to the main result of this chapter. With the help of the operatorsPpF,Γ^αqand the structure condition (5.13) we are now able to give a complete correspondence between the concepts from paracontrolled analysis (as modified above) and regularity structures, this will be the content of Theorem 5.2.1 in the next section.

5.2 Modelled distributions are paracontrolled

The following theorem is the main result of this chapter, it reveals a deep connection between the paraproduct approach and the theory of regularity structures. We show

that the spacesD^γpR^d,VzWqand P^γpR^d;VzWqare identical. For technical reasons we have to exclude the case that γ PR is contained in the locally finite set

A_Nd :“A` |N^d|s (5.17)

This is necessary since we want to apply for the spaces Cs^γ´α appearing in the Definition of 5.1.5 the Hölder characterization from Lemma 2.1.23. If one sees the following theorem as a generalization of Lemma 2.1.23 then the exclusion of (5.17) corresponds to the restrictionγ R |N^d|s required there.

Theorem 5.2.1. Let T “ pA,T, Gq be a regularity structure with a subsector V and a complement VzW of a subsector W Ď V within V. Let further pΠ,Γq be a model on T. We then have for any γ P RzA_Nd

D^γpR^d;VzW,Γq “P^γpR^d;VzW,Γq (5.18) with equivalent norms (where the equivalence constants can be chosen proportional to some polynomial in }Γ}γ). Moreover, given a second model pΠ,ˆ Γˆq and modelled distributions F P D^γpR^d;VzW,Γq, FˆP D^γpR^d;VzW,Γqˆ we also have

}F; ˆF}_P^γ_pR^d_;VzW,Γ,Γqˆ ÀK₁ ¨ }F; ˆF}_D^γ_pR^d_;VzW,Γ,Γqˆ

}F; ˆF}_D^γ_pR^d_;VzW,Γ,Γqˆ ÀK₁ ¨ }F; ˆF}_P^γ_pR^d_;VzW,Γ,Γqˆ

` }Γ´Γ}ˆ γp}F}D<sup>γ</sup>pR<sup>d</sup>;VzW,Γ` }Fˆ}_DγpR^d;VzW,Γqˆ q, where K₁ ą0 is a polynomial in the norms of F, F ,ˆ Γ and Γ.ˆ

We have further the following local embedding propertyfor polynomials: Suppose the regularity structure satisfies Assumption (2.3.12)and we are given an open setΩ and a mapF `P :R^dÑV´^γ that satisfies the structure condition (5.13) on Ω forγ and αPAV X |N^d|s with αăγ. IfF, P are chosen such that F :R^d Ñ pVzTq^γ_´, P : R^dÑT^γ´, we have

}F `P}D^γpΩ;Vq ÀK₂¨

”

}F}_DγpR^d;VzTq (5.19)

` sup

αPAVX|N^d|s

´

}P^α}CbpΩ;Tαq` }P^α´PpF,Γ^αq}_C^γ´α_{s pΩ;Tαq}

¯ ı , (5.20) whereK₂ ą0is some polynomial in}F}D^γpR^d;VzTq, }P^α}CbpΩ;Tαq, }Γ}γMoreover given two such functionsF `P,Fˆ`Pˆ for the models pΠ,Γq and pΠ,ˆ Γˆq we have

}pF `Pq;pFˆ`Pˆq}_DγpΩ;V,Γ,Γqˆ ÀK₃ ¨

”

}F; ˆF}_DγpR^d;VzT,Γ,Γqˆ ` }Γ´Γ}ˆ _γ

` sup

αPAVX|N^d|s

´

}P^α´Pˆ^α}CbpΩ;T_αq` }P^α´Pˆ^α´PpF´F ,ˆ Γ^αq}_C^γ´α

s pΩ;Tαq

¯ ı ,

118 5.2 Modelled distributions are paracontrolled

Proof. We will include polynomial powers of}Γ}γ in the constant indicated by “À”.

We assume without loss of generality thatA_VzW contains only elements below γ.

To showD^γpR^d;VzWq Ď P^γpR^d;VzWqnote first thatF PD^γpR^d;VzWqalready We then follow similar ideas as in [GIP15, Subsection 6.2.]: We can rewrite for xPR^d and αPAVzW the first term on the right hand side. Since the spectral support of the summands is contained in an annulus scaled by 2^js it is sufficient to bound each term by 2^´jpγ´αq}F}D^γpR^d;VzWqdue to Lemma 2.1.19. And indeed we have, usingş so that we have with Lemma 2.1.14

}

Note that it is sufficient to take }x´y}_s ď1, compare Remark 2.3.16. Let us start our induction with α “ maxAVzW (which exists since we assumed maxAVzW ă γ ă 8). By definition of Γ_yx we have Γ^α_yxF_x “ F_x^α and thus PpF,Γ^αq “ 0 due

where we applied Lemma 2.1.20 (together with}y´x}_s ď1).

Let us now assume that we already know for some α P A_VzW that for any α¹ PAVzW, α¹ ąα

Since (5.22) does already decay in the right order due to Lemma 2.1.23 (and the assumption γ R A_N^d), we are only left with the last line which we identify as the limit forN Ñ 8 (in T^α for every x, y) of

120 5.2 Modelled distributions are paracontrolled Indeed: observe the following three limits

ÿ

where we used in the last line for both terms thatş

Ψ^j_v1“0 for j ą0to cancel the α¹ “αcomponents. We can therefore reshape D^N_j as (withR^γ´α_x;y´x being the Taylor

1A short computation shows that Definition 2.3.9 already implies (Hölder) continuity of the mapsyÞÑΓ^α_yxτ forτ PT and xPR^d.

remainder as in (2.22))

where we used in the second line spectral support properties to restrict the inner sum toi„j and the convolution-like structure to move in the last term the Taylor remainder to the Littlewood-Paley block. The last term can be estimated by }x´ y}^γ´α_s via Lemma 2.1.20 and 2.1.14 if one uses that fork PN^d there is aC ą0 such which can be easily checked by direct computation. To handle the term (5.28) we first estimate the sum in the square brackets by the induction hypothesis and Lemma 2.1.14: The rest of the estimate for (5.28) then follows via the exact same proceeding as in Lemma 2.1.23: Pick j₀ such that 2^´j0´1 ă }x´y}s ď2^´j0 and bound the sum up

where we applied Lemma 2.1.20 in the low-frequency case and in both cases the scaling ofΨ^j from Lemma 2.1.14. In total

}F_y^α´Γ^α_yxF_x}T_α À }x´y}^γ´α_s ,

122 5.3 Singular spaces and extensions