The Φ 43 measure via Girsanov ’ s theorem *

We construct the Φ3 measure on a periodic three dimensional box as an absolutely continuous perturbation of a random translation of the Gaussian free field. The shifted measure is constructed via Girsanov’s theorem and the relevant filtration is the one generated by a scale parameter. As a byproduct we give a self-contained proof that the Φ3 measure is singular wrt. the Gaussian free field.


Introduction
The Φ 4 3 measure on the three dimensional torus Λ = T 3 = (R/2πZ) 3 is the probability measure ν on distributions S (Λ) corresponding to the formal functional integral ν(dϕ) = 1 Z exp −λ Λ (ϕ 4 − ∞ϕ 2 )dx µ(dϕ) (1.1) where µ is the law of the Gaussian free field with covariance (1 − ∆) −1 on Λ, Z a normalization constant and λ the coupling constant. The ∞ appearing in this expression reminds us that many things are wrong with this recipe. The key difficulty can be traced to the fact that the measure we are looking for it is not absolutely continuous wrt. the reference measure µ. This fact seems part of the folklore even if we could not find a rigorous proof for it in the available literature apart from a work of Albeverio and Liang [1], which however refers to the Euclidean fields at time zero, and the work of Feldman and Osterwalder [11] in infinite volume. The singularity of the Φ 4 3 measure is indeed a major technical difficulty in a rigorous study of (1.1). Obtaining a complete construction of this formal object (both in finite and infinite volume) has been one of the main achievements of the constructive quantum field theory program [12,10,26,11,22,5,8].
In recent years the rigorous study of the Φ 4 3 model has been pursued from the point of view of stochastic quantization. In the original formulation of Parisi-Wu [25], stochastic quantization is a way to introduce additional degrees of freedom (in particular a dependence on a fictious time) in order to obtain an equation whose solutions describe a measure of interest, in this case the Φ 4 3 measure on Λ as in (1.1) or its counterpart in the full space. Rigorous analysis of stochastic quantization for simpler models like Φ 4 2 (the two-dimensional analog of eq. (1.1)) started with the work [20]. It has been only with the fundamental work of Hairer on regularity structures [19] that the three dimensional model could be successfully attacked, see also [9,21]. This new perspective on this and related problems led to a series of new results on the global space-time control of the stochastic dynamics [24,15,2,23] and to a novel proof of the construction of non-Gaussian Euclidean quantum field theories in three dimensions [14].
A conceptual advantage of stochastic quantization is that it is a method which is insensitive to questions of absolute continuity wrt. to a reference measure. This, on the other hand, is the main difficulty of the Gibbisan point of view as expressed in eq. (1.1). In order to explore further the tradeoffs of different approaches we have recently developed a variational method [4] for the construction and description of Φ 4 3 .
We were able to provide an explicit formula for the Laplace transform of Φ 4 3 in terms of a stochastic control problem in which the controlled process represents the scale-by-scale evolution of the interacting random field.
The present paper is the occasion to explore further this point of view by constructing a novel measure via a random translation of the Gaussian free field and by proving that the Φ 4 3 measure can be obtained as an absolutely continuous perturbation thereof. Without entering into technical details right now, let us give the broad outline of this construction. We consider a Brownian martingale (W t ) t 0 with values in S (Λ) and such that W t is a regularization of the Gaussian free field µ at (Fourier) scale t. Let us denote P its law and E the corresponding expectation. In particular, W t → W ∞ in law as t → ∞ and W ∞ has law µ. We can identify the Φ 4 3 measure ν as the weak limit ν T → ν as T → ∞ of the family of probability measures (ν T ) T 0 on S (Λ) defined as ν T (·) = P T (W T ∈ ·), where P T is the measure on paths (W t ) t 0 with density is a quartic polynomial in the field ϕ with (a T , b T ) T a family of (suitably diverging) renormalization constants. The scale parameter t ∈ R + allows to introduce a filtration and a measure Q v defined as the Girsanov transformation where ( L v t ) t 0 is the quadratic variation of the (scalar) local martingale (L v t ) t 0 and (v t ) t 0 is a progressively measurable process with values in L 2 (Λ). Let The Φ 4 3 measure via Girsanov's theorem be the density of P T wrt. Q v . We will show that it is possible to choose v in such a way that the family (D T ) T 0 is uniformly integrable under Q v and that D T → D ∞ weakly in L 1 (Q v ). With particular choice of v we call Q v the drift measure: it is the central object of this paper. By Girsanov's theorem the canonical process (W t ) t 0 satisfies the equation where (W t ) t 0 is a Gaussian martingale under Q v (and has law equal to that of (W t ) t 0 under P, i.e. it is a regularized Gaussian free field). We will show also that the drift v t can be written as a (polynomial) function of ( . Therefore we have an explicit description of the process (W t ) t 0 under the drift measure Q v as the unique solution of the path-dependent SDE Let us note that this formula expresses the "interacting" random field (W t ) t as a function of the "free" field (W t ) t . It is a formula which shares very similar technical merits with the stochastic quantization approach.
The drift measure Q v is half way between the variational description in [4] Here R ∈ C [0, ∞], C 1−δ Q v -almost surely andW [3] ∈ C [0, ∞], C 1/2−δ Q v -almost surely, can both be constructed formW , and have given law under Q v . This decomposition allows one to reduce many almost sure properer ties of Q v and so of Φ 4 3 to properties of the "free field"W . For the slightly less singular Hartree nonlinearity this has been exploited in [6,7] to prove local wellposedness almost surely for initial data distributed according to the Gibbs measure. Let us briefly sketch another application of (1.4): we can use it prove that the Wick square is well defined almost surely with respect to the Φ 4 3 measure. Indeed we can write converges to a well defined random distribution as T → ∞, which is the Wick square ofW T . It has been shown in [4] Lemma 4 and Lemma 25 thatW TW [3] T converges to a well defined random distribution. Finally since As another application of (1.4) we provide also a self-contained proof of the singularity of the Φ 4 3 measure ν wrt. the Gaussian free field µ. We have already remarked that the singularity of Φ 4 The Φ 4 3 measure via Girsanov's theorem all the details in order to provide a reference for this fact. The basic idea is to consider the observable Λ W 4 T where the brackets denotes Wick products and prove that it diverge with different speed as T → ∞ under the measure P and Q v because in the first case the process (W t ) t is a Brownian martingale and therefore by the properties of Wick products also the process Λ W 4 t t is a martingale with variance growing like T . Under the measure Q v however the presence of the drift (V t ) t 0 produces a deterministic contribution whose size is also T and which dominates the fluctuations of the observable. Therefore the singularity of Φ 4 3 can be directly linked with the pathwise properties of the scale-by-scale process (W t ) t 0 in the ultraviolet region and our proof of singularity shows also that the drift measure Q v is singular wrt. P. Intuitively, the drift (V t ) t 0 in the SDE (1.3) is not regular enough (as t → ∞) to be along Cameron-Martin directions for the law P of the process (W t ) t 0 and therefore the Girsanov transform (1.2) gives a singular measure when extended all the way to T = +∞.
Let us stress that the main contribution of the present paper remains that of describing the drift measure as a novel object in the context of Φ 4 3 and similar measures and pursuing the study of Euclidean quantum fields from the point of view of stochastic analysis.
Notations. Let us fix some notations and objects.
• For a ∈ R d we let a := (1 + |a| 2 ) 1/2 . B(x, r) ⊆ R denotes the open ball of center x ∈ R and radius r > 0. We write A B for A CB for some constant C and A B for A B and B A.
• The constant κ > 0 represents a small positive number which can be different from line to line.
• Denote with S (Λ) the space of Schwartz functions on Λ and with S (Λ) the dual space of tempered distributions. The notationf or F f stands for the space Fourier transform of f and we will write g(D) to denote the Fourier multiplier operator with symbol g : R n → R, i.e. F (g(D)f ) = gF f .
• B α p,q = B α p,q (Λ) denotes the Besov spaces of regularity α and integrability indices p, q as usual. C α = C α (Λ) is the Hölder-Besov space B α ∞,∞ , W α,p = W α,p (Λ) denote the standard fractional Sobolev spaces defined by the norm f W s,q := D s f L q and H α = W α,2 . The symbols ≺, , • denotes spatial paraproducts wrt. a standard Littlewood-Paley decomposition. The reader is referred to Appendix A for an overview of the various functional spaces and paraproducts.

The setting
The setting of this paper is the same of that in our variational study [4]. In this section we will briefly recall it and also state some results from that paper which will be needed below. They concern the Boué-Dupuis formula and certain estimates which are relevant to the analysis of absolute continuity.
Let Ω := C R + ; C −3/2−κ (Λ) and F be the Borel σ-algebra of Ω. On (Ω, F ) consider the probability measure P which makes the canonical process (X t ) t 0 a cylindrical Brownian motion on L 2 (Λ) and let (F t ) t 0 the associated filtration completed with respect to sets of P-measure 0. In the following E without any qualifiers will denote expectations wrt. P and E Q will denote expectations wrt. some other measure Q.
On the probability space (Ω, F , P) there exists a collection (B n The Φ 4 3 measure via Girsanov's theorem Fix some decreasing ρ ∈ C ∞ c (R + , R + ) such that ρ = 1 on B(0, 9/10) and supp ρ ⊂ B(0, 1). For x ∈ R 3 let ρ t (x) := ρ( x /t) and Denote J s = σ s (D) D −1 and consider the process (W t ) t 0 defined by It is a centered Gaussian process with covariance for any ϕ, ψ ∈ S (Λ) and t, s 0, by Fubini theorem and Ito isometry. By dominated convergence lim t→∞ E[ W t , ϕ W t , ψ ] = n∈Z 3 n −2φ (n)ψ(n) for any ϕ, ψ ∈ L 2 (Λ). For any finite "time" T the random field W T on Λ has a bounded spectral support and the stopped process W T t = W t∧T for any fixed T > 0, is in C(R + , C ∞ (Λ)). Furthermore (W T t ) t only depends on a finite subset of the Brownian motions (B n ) n∈Z 3 .
Observe that J t satisfies the following bound for any function f ∈ B s p,p or f ∈ W s,p with p ∈ [1, ∞] and s ∈ R and for any α ∈ R. We will denote by W n t , n = 1, 2, 3, the n-th Wick-power of the Gaussian random variable W t (under P) and introduce the convenient notations W 2 t := 12 W 2 t , W 3 t := 4 W 3 t . Furthermore we will write ( D −1/2 W t ) n , n ∈ N for the n-th Wick-power of D −1/2 W t . It exists for any 0 < t < ∞ and any n 1 since it is easy to see that D −1/2 W t has a covariance with a diagonal behavior which can be controlled by log t . These Wick powers converge as T → ∞ in spaces of distributions with regularities given in the following table: Table 1: Regularities of the various stochastic objects. The domain of the time variable is understood to be [0, ∞], CC α = C ([0, ∞]; C α ) and L 2 C α = L 2 (R + ; C α ). Estimates in these norms holds a.s. and in L p (P) for all p 1 (see [4]).
We denote by H a the space of (F t ) t 0 -progressively measurable processes which are P-almost surely in H := L 2 (R + × Λ). We say that an element v of H a is a drift. Below we will also need drifts belonging to H α := L 2 (R + ; H α (Λ)) for some α ∈ R where H α (Λ) is the Sobolev space of regularity α ∈ R and we will denote the corresponding space with H α a . For any v ∈ H a define the measure Q v on Ω by Denote with H c ⊆ H a the set of drifts v ∈ H a for which Q v (Ω) = 1, and set W v := W −I(v), We will need also the following objects. For all t 0 let θ t : R 3 → [0, 1] be a smooth function such that θ t (ξ)σ s (ξ) = 0 for s t, θ t (ξ) = 1 for |ξ| t/2 provided that t T 0 for some T 0 > 0. For example one can fix smooth functionsθ, η : Then letθ t (ξ) :=θ(ξ/t) and define where ζ(t) : R + → R is a smooth function such that ζ(t) = 0 for t 10 and ζ(t) = 1 for t 3. Then eq. (2.2) holds with T 0 = 3. Let for any f ∈ S (Λ).
Our aim here to study the measures µ T defined on C −1/2−κ as and suitable a T , b T → ∞. For convenience the measure µ T is not normalized and, wrt.
to the notations in the introduction we have Recall the following results of [4].
Theorem 2.1. For any a T , b T ∈ R, and f : where V T is given by (2.4). Then the variational formula holds for any finite T .
This is a consequence of the more general Boué-Dupuis formula. (2.6) We will use several times below eq. (2.6) in order to control exponential integrability of various functionals. By a suitable choice of renormalization and a change of variables in the control problem (2.5) we were able in [4] to control the functional in Theorem 2.1 uniformly up to infinity.
and the functionals Ψ f As a consequence we obtain the following corollary (cfr. Corollary 1 and Lemma 6 in [4])

Construction of the drift measure
We start now to implement the strategy discussed in the introduction: identify a translated measure sufficiently similar to Φ 4 3 . Intuitively the Φ 4 3 measure should give rise to a canonical process which is a shift of the Gaussian free field with a drift of the form given by eq. (2.7). Indeed this drift u should be the optimal drift in the variational formula. A small twist is given by the fact that the relevant Gaussian free field entering these considerations is not the process W = W (X) but that obtained from the shifted canonical process X u t = X t − t 0 u s ds which we denote by Moreover, to prevent explosion at finite time, we have to modify the drift in large scales and add a coercive term. This will also allow later to prove some useful estimates. As a consequence, we define the functional whereT > 0, n ∈ N are constants which will be fixed later on and where we understand all the Wick renormalizations to be given functions of W , i.e. polynomials in W where the constants are determined according to the law of W under P. We look now for the solution u of the equation Expanding the Wick polynomials appearing in Ξ(W − I(u), u) we obtain the equation for all s 0. This is an integral equation for t → u t with smooth coefficients depending smoothly on W and can be solved via standard methods. Since the coefficients are of polynomial growth the solution could explode in finite time. Note that for any finite time the process (u s ) s>0 has bounded spectral support. As a consequence we can solve the equation in L 2 and as long as t 0 u 2 L 2 ds is finite we can see from the equation that sup s t u s 2 L 2 is finite. By the existence of local solutions we have that, for all N 0, the stopping time is strictly positive P-almost surely and u exists up to the (explosion) time T exp := sup N ∈N τ N . The following lemma will help to show that P-almost surely T exp = +∞ and will also be very useful below.
uniformly in t 0, for any pair of adapted processes w, g ∈ L 2 (P, H) such that Integrating over the probability space and using Cauchy-Schwarz inequality, we obtain where g t is an arbitrary function. By Lemma 3.7 below, we have constants c, C and a random variable Q T (W ) such that As a consequence, we deduce And we can conclude by sending N → ∞ and using monotone convergence.
In particular, taking w = −1 t τ N u and g = −w, we have Then, for any s T we have provided n is chosen sufficiently large. Using Gronwall's inequality this gives which implies T exp = +∞. In addition and by construction, the process u N t := 1 {t τ N } u t satisfies Novikov's condition, so it is in H c and Girsanov's transformation allows us to define the probability measure Q u N on C R + , C −1/2−κ (Λ) given by s t 0 has the same law as (W t ) t 0 under P. We observe also that W u N s = W u s for 0 s τ N and that u satisfies the equation If we think of the terms containing W u as given (that is, we ignore their dependence on u), eq. (3.6) is a linear integral equation in u which can be estimated via Gronwall-type arguments. In order to do so, let us denote by U : H →û the solution map of the equation This last equation is linear and therefore has nice global solutions (let's say in C(R + , L 2 )) and by uniqueness and eq. (3.6) we have u t = U t (W u ) for t ∈ [0, T exp ). From this perspective the residual dependence on u will not play any role since under the shifted measure the law of the process W u does not depend on u. By standard paraproduct where we have crucially exploited the presence of the cutoff 1 {s T } to introduce the small factorT −κ and we have employed the notatioñ By Gronwall's lemma, Under Q u N , the terms inH τ N are in all the L p spaces by hypercontractivity and moreover for any p 1 one can chooseT large enough so that also the exponential term is in L p . Using eq. (3.6) it is then not difficult to show that E Q u N 1 [ u N2 p H −1/2−κ ] < ∞ for any p > 1 (again provided we takeT large enough depending on p) as long as N 1 > N 2 . By the spectral properties of J and the equation for u, the process t → 1 {t T } u t is spectrally supported in a ball of radius T , so we get in particular that uniformly for any choice of N 1 N 2 0.
Lemma 3.2. The family (Q u N ) N weakly converges to a limit Q u on C R + , C −3/2−κ . Under Q u it holds T exp = ∞ almost surely and Law Q u (X u ) = Law P (X). Moreover for any and therefore we also have Q u ({τ N T }) CT 1+κ N −1 which in turn implies Q u (T exp < T ) = 0. This proves that T exp = +∞ under Q u , almost surely. As a consequence we can extend Q u to all of F = ∨ T CF T since for any A ∈ F T we can set This establishes that Law Q u (X u ) = Law P (X). On the other hand if A ∈ F T we have, using the martingale property of the Girsanov density, by monotone convergence and the fact that T exp = +∞ P-almost surely. Therefore as claimed.
The following lemma will also be useful in the sequel and it is a consequence of the above discussion: Proof. This follows from the bound (3.8), after choosingT large enough, where we recall thatT has been introduce in the definition of the drift through eq. (3.1).

Proof of absolute continuity
In this section we prove that the measure µ T is absolutely continuous with respect to the measure Q u which we constructed in Lemma 3.2. First recall that the measures µ T defined on Ω as dµ T dP = e −V T (W T ) can be described, using Lemma 3.2, as a perturbation of Q u with density D T given by

Lemma 3.4.
There exists a p > 1, such that for any K > 0, in particular, the family (D T ) T is uniformly integrable under Q u .
Proof. The proof eq. (3.9) is given in Section 3.2 below. For the uniform integrability fix ε > 0. Our aim is to show that there there exists δ > 0 such that Q u (A) < δ implies A D T dQ u < ε. From Corollary 2.4, for any ε > 0 there exists a K > 0 such that The family of measures (µ T ) T 0 is sequentially compact w.r.t. strong convergence on (Ω, F ). Furthermore any accumulation point is absolutely continuous with respect to Q u .
Proof. We choose a sub-sequence (not relabeled) such that D T → D ∞ weakly in L 1 (Q u ), for some D ∞ ∈ L 1 (Q u ). It always exists by uniform integrability. We now claim that for It is enough to check this for A ∈ F S for any S ∈ R + since these generate F . But there we have for T S, Recall that the Φ 4 3 measure can be defined as a weak limit of the measuresμ T on C −1/2−κ given by  (3.10) for some D ∞ ∈ L 1 (Q u ).

L p bounds
Now we will prove local L p -bounds on the density D T . In the sequel we will denotẽ W = W u , with u satisfying (3.3), namely u = U (W ). Before we proceed let us study how the functional U (W ) behaves under shifts ofW , since later we will want to apply the Boué-Dupuis formula and this kind of behavior will be crucial. Let w ∈ L 2 ([0, ∞) × Λ) and denote u w := U (W + I(w)) and h w := U (W + I(w)) + w = u w + w.
The process h w satisfies h w − w = u w = Ξ(W + I(w), u w ).
More explicitly, for all s 0 we have we can write (3.12) The first two terms in (3.11) will be used for renormalization while the remainder r w contains terms of higher regularity which will have to be estimated in the sequel.
Proof. Proof of eq. (3.9) Observe that Combining these two facts we have where we have set h w = w + U (W + I(w)) as above. Recall now that from Theorem 2.3 there exists a constant C, independent of T , such that for each h w , Using eq. (3.11) we compute At this point we need a lower bound for Given that we need to take p > 1, this expression present a difficulty in the fact that the term T 0 w t 2 L 2 dt appears with a negative coefficient. Note that this term cannot easily be controlled via T 0 r w t + w t 2 L 2 dt since the contribution r w , see eq. (3.12), contains factors which are homogeneous in w of order up to 3. This is the reason we had to localize the estimate, introduce the "good" term W ∞ + I ∞ (h w ) n C −1/2−κ , and introduce the term J s D −1/2 ( ( D −1/2 W s ) n ) in (3.1) which will help us to control the growth of r w . Indeed in Lemma 3.6 below, a Gronwall argument will allow us to show that T 0 w t 2 L 2 dt can be bounded by a combination of the other "good" terms as This implies that for 1 < p 2, which gives the claim. Note that here we used the bound as well as the fact that I t (h w ) C −1/2−κ I ∞ (h w ) C −1/2−κ to conclude.
Proof. Let us recall the notation Write r w s =r w s + Aux s (W, w) and observe that by Lemma 3.1 we with g = r w we have Gronwall inequality allows to conclude.
Proof. We recall that (see eq. (3.4)) and since E sup T <∞ ( D −1/2 W T ) i p C −κ < ∞ for any p < ∞ and any ε > 0 it is enough to bound ( D −1/2 I T (w)) n+1−i q Proof. Note that The estimation for the other terms is easy but technical and postponed until Section 5.

Singularity of Φ 3 w.r.t. the free field
The goal of this section is to prove that the Φ 4 3 measure is singular with respect to the Gaussian free field. For this we have to find a set S ⊆ C −1/2−κ (Λ) such that P(W ∞ ∈ S) = 1 and Q u (W ∞ ∈ S) = 0. Together with (3.10), this will imply singularity.
We claim that setting for some suitable sequence (T n ) n such that T n → ∞, does the job. Here denotes the Wick ordering with respect to the Gaussian free field. For later use we define Before we proceed with the proof let us briefly motivate the choice of the event S and give a sketch of the proof below. By Ito's formula one can show that where we recall that under P X t is a cylindrical Wiener process. From this formula using the properties of W θ T ,3 t and Ito isometry we will deduce that and extracting a subsequence we get P(W ∞ ∈ S) = 1. On the other hand with R denoting a regular remainder we have and under Q u the process X u is a again a cylindrical Wiener process. Therefore, as in (4.2) we have a martingale whose variance is estimated as However now the additional drift term in (4.3) grows faster than T 1/2+δ since it behaves as the positive term whose average can be estimated exactly as in (4.3), that is proportional to T . To make the argument rigorous we need only to show that all the neglected terms cannot compensate for this divergence, this will be done by estimating their average size and then using Borel-Cantelli.
Let us start by proving that P(W ∞ ∈ S) = 1 for some sequence (T n ) n .
Proof. Wick products corresponds to iterated Ito integrals. Introducing the notation we can verify by Ito formula that Since θ T J t = 0 for t T , Ito isometry gives The next step of the proof is to check that Q u (W ∞ ∈ S) = 0. More concretely we will show that for a sub-sequence of (T n ) n (not relabeled) Q u almost surely. Observe that We expect the term to go to infinity faster than T 1−δ , Q u -almost surely. To actually prove it, we start by a computation in average.
Next we upgrade this average bound to almost sure divergence of the random variable at least as T 1−δ for some δ small.

Lemma 4.3.
There exists a δ 0 > 0 such that for any δ 0 δ > 0, there exists a sequence (T n ) n such that P − almost surely Proof. Define We will show that e −G T → 0 in L 1 (P), which implies that there exists a sub-sequence (T n ) n such that e −G Tn → 0 almost surely. From this our statement follows. By the where where have used that θ T J t = 0 for t T and introduced the notations, for 0 i 3, Our aim now to prove that the last three terms are bounded below uniformly as T → ∞ (while we already know that the first one diverges). For i ∈ {1, 2, 3} by Lemmas 5.4 and 5.6. Here Q t (W ) is a random variable only depending on W such that sup t E[|Q t (W )| p ] < ∞ for any p < ∞. Then Now for the first term we obtain For the second term we use that A 0 Next we obtain an estimate which will help with the proof of the main theorem.
Furthermore, there exists a (deterministic) sub-sequence (T n ) n such that Proof. Recall that under Q u we have W t = W u t + I t (u) where u is defined above by (3.3) and Law Q u (W u ) = Law P (W ). With this in mind we compute where, as above, By the computation from Lemma 4.4 we have then For the fourth term we proceed in the same way: which is bounded in expectation uniformly in T , so the fourth term goes to 0 in L 1 (Q u ) as well. It remains to analyze the second term. Again introducing the notation and applying Young's inequality with the exponents (32, 32/9, 32/22), we obtain Proof. We observe that since W 2 t is spectrally supported in a ball or radius ∼ t For the first estimate we know that (1 − θ t )I t (w) is supported in an annulus of radius ∼ t, so (1−θ t )I t (w) L 2 t −1+ε I t (w) H 1−ε and furthermore by interpolation I t (w) By definition t 1/2 J t is a uniformly bounded Fourier multiplier regularizing by 1, and putting everything together, by paraproduct estimates For the second term in addition observe that the function t 1/2 J t is spectrally supported in an annulus of radius ∼ t, and regularizes by 1 so again by estimates for the resonant For the third estimate again applying paraproduct estimates and the properties of J, J t ( W 2 s ≺ I t (w)) 2 EJP 26 (2021), paper 81. Now, the claim follows from interpolation and Young's inequality Lemma 5.4. Let f ∈ C [0, ∞], C −1/2−ε and g ∈ C([0, ∞], H 1 ) such that f t , g t have spectral support in a ball of radius proportional to t. There exists n ∈ N such that the following estimates hold: J t (f t g 2 t ) 2 J t (f t g 2 t ) 2 Proof. By the spectral properties of J t , J t (f t g 2 t ) 2 Applying Young's inequality with exponents n 2 , n/2 (n/2−1) with n such that 2n (n/2−1) 4 + ε where ε is chosen as in Lemma 5.2 we have t −3/2 f t n C −1/2−δ + g t n W −1/2,n + g t 2 H 1 . Now the second estimate follows from chosing n large enough (depending on δ) and using Besov embedding after taking f = g.
Proof. For the first estimate we again use the spectral properties of W, I, and J and obtain by paraproduct estimate J s (W t I t (w) I t (u)) 2 and the claim follows by Young's inequality. For the second J s ((I s (w)) 2 I s (u)) 2 L 2 t 2−2κ (I s (w)) 4 L 4 I t (u) 2 C −1/2−κ , and the claim follows again by Young's inequality.
Lemma 5.6. Let f t ∈ C [0, ∞], C −1/2−δ and g t ∈ C([0, ∞], H 1 ) such that f t , g t have spectral support in a ball of radius proportional to t. Then the following estimates hold (J t (f t g t )) 2 t −1+2δ f t (J t (f t g t )) 2 This proves the first estimate. For the second we continue Proof. This follows in the same fashion as Lemma 5.6.

A Besov spaces and paraproducts
In this section we will recall some well known results about Besov spaces, embeddings, Fourier multipliers and paraproducts. The reader can find full details and proofs in [3,16].