Julien Fausty

Nathalie Bozzolo

Marc Bernacki

A Level Set Finite Element Anisotropic Grain Growth Study

Application

The turbine disks of aircraft engines, obtained by hot forging, must withstand extreme pressures at extreme temperatures and resist to fatigue crack propagation.

Material

Fig. 2: Experimental EBSD map of an Inconel 718 billet with twin boundaries in red and grain boundaries in black, furnished by Alexis Nicolay.

The material of choice for these disks are often Ni based superalloys whose compositions and microstructures have to be optimized. Annealing twins, boundaries of lower energy, are often present in these microstructures and play an important role in the in-service properties [TCV16].

Numerics

Fig. 3: A part of a numerical microstructure.

To accelerate the material development process, industrial players need numerical tools to simulate microstructural evolutions during forging operations. None of the existing frameworks are able to handle strong anisotropy of grain boundary energy correctly.

Thermodynamics

Fig. 4: Schema of the creation of a hump at a grain boundary.

If is the grain boundary energy and P is the pressure felt by the grain boundary. At equilibrium, neglecting the torque terms:

P =  (1)
where  is the curvature of the grain boundary [Her99]. The velocity of the grain boundary ṽn is defined as:

ṽn = µP ñ (2)
where µ is the mobility of the grain boundary and ñ is the unitary normal to the boundary. The grain boundary network is represented by a set of N signed distance functions such that for (X, t)

LS-FE

2 ⌦ ⇥ [0, t end ] [BLC11]: ⇢ i (X, t) = ±d(X, ) || r i || = 1 8i 2 {0, ..., N} (3)
In order to resolve the grain growth problem, a transport equation is used:

@ i @t + ṽn • r i = 0 (4)

Anisotropic Adaptation

Extension of the field to the entire ⌦ domain:

(X, t) = calc , 8X 2 = 0 8X 2 ⌦ (5)
using a FE solver. A supplemental convection term due to the weak formulation is taken into account in the anisotropic model using a Convection-Di↵usion assembly of the FE matrix. Equation ( 4) becomes:

@ i @t µ i = 0 (6)
with a weak formulation using ' 2 H 1 0 (⌦) as a test function:

Z ⌦ @ i @t ' + Z ⌦ µ r • r i ' + Z ⌦ µ r' • r i = 0 (7)
The Grim Reaper

In [GNS99] an academic case is described: If 23 = 13 , the system achieves a stationary state where:

ij sin ' k = K (8) cos ' 3 2 = 23 2 12 = r 2 (9) v x = 12 µ(⇡ ' 3 ) a ( 10
)
with K a constant.

In this work:

23 = 1 a = 2 µ = 1
and 12 is varied with r.

Conclusions

• The equilibrium angles at the triple junction can not be obtained naturally without considering torque terms. • The kinetics of the triple junction and its surroundings are completely dependent upon the equilibrium angles.

Bibliography

[BLC11] M. Bernacki, R.E. Logé, and T. Coupez. Level set framework for the finiteelement modelling of recrystallization and grain growth in polycrystalline materials. Scripta. Materialia, 64:525-528, 2011.

[GNS99] Harald Garcke, Britta Nestler, and Barbara Stoth. A multi phase field concept: Numerical simulations of moving phase boundaries and triple junctions. with t = 0.01 and r = 2.

• P tr used as a convergence criteria.

• Small dependence of the solution with respect to mesh size.

• Convergence is obtained starting at: • P tr used as a convergence criteria.

• Large dependence of solution on time step.

• Convergence is obtained for t starting at:

t = 0.01
The financial support of this work via the industrial Chair OPALE co-financed by SAFRAN and the French National Research Agency (ANR) is acknowledged by the authors.

Varying Anisotropy

Equilibrium Angles

• As r increases, the solutions are closer and closer together.

• The angles for r 6 = 1 are clearly much lower than predicted by the theory.

• Equation (9)/Young's equilibrium is not respected.

• Although convergence of the method is ensured, the simulation does not arrive at the correct solution.

• The question must be revisited in the state of the art.

Velocities

• Relative velocities of the triple junction get lower as r increases.

• A stationary state is obtained (v x = cnst) for t = 0.14.

• Kinetics are around twice as fast as they should be for r = 2.

• Obtained velocities at the triple junction are much higher than those predicted using the imposed grain boundary energies for every r 6 = 1.

• Kinetics of the triple junction are coherent with obtained angles.

• Kinetics of the profile completely dependent on the equilibrium angles.

Fig. 1 :

 1 Fig. 1: A diagram of an generic aircraft engine.

Fig. 5 :

 5 Fig. 5: Le level-set distance function of a circular grain.

Fig. 6 :

 6 Fig. 6: Before and after an extension operation on .

Fig

  Fig. 7: Diagram of the Grim Reaper case.

SIAM

  Journal of Applied Mathematics, 1999. [Her99] Conyers Herring. Surface Tension as a Motivation for Sintering. Springer Berlin Heidelberg, 1999. [TCV16] Damien Texier, Jonathan Cormier, and Patrick Villechaise. Crack initiation sensitivity of wrought direct aged alloy 718 in the very high cycle fatigue regime: the role of non-metallic inclusions. Material Science and Engineering, 678:122-136, 2016.

Fig. 8 :Fig. 9 :

 89 Fig. 8: Zoomed up image of the triple junction at t end for di↵erent mesh sizes.

Fig. 12 :

 12 Fig. 12: Zoomed up image of the triple junction at t end for di↵erent anisotropic ratios.


Antonio Costanzo

Valeria Loscrì
email: valeria.loscri@inria.fr

Sandra Costanzo

Software Defined Platforms for Visible Light Communication: State of Art and New Possibilities

come   L'archive ouverte pluridisciplinaire

Introduction

The requirement of connected devices exponentially increases day by day as well as the bandwidth needed for novel applications. As a consequence of this, radio frequency spectrum becomes more and more crowded, thus leading to serious limitations in the diffusion of novel technologies when considering challenging environments such as hospitals and airplanes, where EM interferences are particularly unwanted. The integration of Visible Light Communication (VLC) in the existing infrastructures could provide a valid tool to address these issues. In this context, Software Defined concepts could significantly simplify the integration process, leading to the development of low cost and flexible architectures. To this end, Universal Software Radio Peripheral (USRP), designed and commercialized by Ettus Company and National Instruments, is emerging as a comparatively low cost hardware platform for software defined architectures, allowing rapid prototyping and test bed validations. The aim of the present work is to exploit the current literature in the framework of Software Defined Systems based on USRP platform for VLC purposes, while briefly highlighting some useful potentialities for the development of solid and flexible architectures.

Visible Light Communication: from Photophone to Software Defined VLC

The adoption of visible light for message transmission is one of the most ancient and intuitive techniques adopted in long distance communications. In the Roman Empire, a system of mirrors and shutters located on apposite towers was employed for communicating messages up to 20 km. The first modern communication in the visible spectrum, however, dates back to 1880, when Charles Sumner Tanter and Alexander Graham Bell invented the photophone. This device allowed speech transmission through the adoption of a light beam modulated by switching concavity and convexity of a reflecting screen [START_REF] Bell | The Photophone[END_REF]. In the first decade of 20 th century, the German Siemens & Halske Company improved modulation using carbon arc lamps and provided commercial units for German Navy, while British Almery adopted high infrared sensitive molybdenite cells. Before the second world war, many governments (Italy, Germany, US, Japan) attempted to develop optical wireless communication for military purposes. Then, advances in radio frequency communication and, further, the adoption of low loss optical fibers, caused a progressive decrease of interest in free space VLC. Due to the exponential increase of bandwidth needed by novel applications and the overpopulation of RF spectrum, in the last fifteen years, this topic has gained a new interest in the scientific community. The first significant work, addressing a visible light communication through a white Light Emitting Diode (LED), and discussing the main requirements of the devices, was done in 2003 at University of Keyo, Japan [START_REF] Komine | Fundamental Analysis for Visible-Light Communication System using LED Lights[END_REF], while significant studies have been carried out by Haas et al. [START_REF] Afgani | Visible Light Communication Using OFDM, Testbeds and Research Infrastructures for the Development of Networks and Communities[END_REF][START_REF] Elgala | Indoor broadcasting via white LEDs and OFDM[END_REF]. Further, according to the rapid diffusion of efficient and low cost LED's, many authors started to deal with different aspects characterizing visible light communication, especially with respect to modulation and dimming support, until a first standardization of the physical and MAC layer, ratified by IEEE (802.15.7) and included in Wireless Personal Area Networks [START_REF] Rajagopal | IEEE 802.15.7 Visible Light Communication: Modulation Schemes and Dimming Support[END_REF]. However, since novel advances are frequently addressed in last years, new standards should be provided. Nowadays, research in VLC follows two main directions: the first one implies an advance in the design of hardware components used for transmitting and receiving the optical signal (led, photodiodes, driving circuits, amplifiers etc.), with the aim to enlarge the bandwidth of the RF front-end and to allow the possibility of using a larger part of the huge unlicensed electromagnetic spectrum in the visible light frequencies (770THz-400THz). A second way, instead, mainly deals with a practical and low-cost integration of VLC elements into existing infrastructures. With reference to this latter scenario, the development of a solid Software Defined paradigm for VLC seems to be an attractive field in the improvement of this technology. As most of the signal operations are performed by software, low cost hardware could be used and an easier interface with other communication paradigms could be allowed for a practical integration in the existing infrastructures. An interesting low cost platform (OPEN VLC) based on a Beagle Bone Black module, has been recently developed by Wang et al. [START_REF] Wang | OpenVLC: Software-Defined Visible Light Embedded Networks[END_REF]. The platform, composed of a simple optical front-end, a set of essential medium access primitives and a basic interaction with internet protocols, allows a transmission up to 1 meter with a rate of 12.5 Kbit/s in its basic configuration. Despite the slow rate and the limited communication distance, this architecture could be extended in order to be available for various applications, such as the passive ambient light recognizer described in [START_REF] Wang | Passive Communication with Ambient Light[END_REF]. An interesting preliminary work, employing the software defined platform USRP is provided in [START_REF] Gavrincea | Rapid prototyping of standard-compliant visible light communications system[END_REF], where GNU Radio open source library is employed for signal processing and two simple applications (a chat service and a video download) are developed. The same authors provided a detailed description of the system in [START_REF] Baranda | An SDR Implementation of a Visible Light Communication System Based on the IEEE 802.15.7 Standard[END_REF], where the standard 802.15.7 PHY I is achieved with a data transmission of 100kbit/s. An audio streaming transmission with similar performances is designed in [START_REF] Hussain | Software Defined VLC System: Implementation and Performance Evaluation[END_REF]. In both cases, OOK modulation is adopted. Preliminary studies and a proof of concept on OFDM and PPM modulation using USRP platform and MatLab elaboration are provided in [START_REF] Thomas | Adaptation of OFDM under Visible Light Communications and Illumination Constraints[END_REF], while in [START_REF] Mirvakili | Wireless Access Test-bed through Visible Light and Dimming Compatible OFDM[END_REF] the same authors developed a driver topology able to drive an array of LEDs using a RPO-OFDM signal coming from USRP. A deeper investigation about Software Defined systems for VLC using USRP platform is provided in the next section.

Visible Light Communication through Universal Software Radio Peripheral

The first Universal Software Defined Radio, initially proposed by Ettus in 2006 [13], was obtained from USB2 interface. It owned a 30 MHz instantaneous bandwidth and it was well integrated into GNU Radio, an open source framework developed in the same period [START_REF] Grayver | Implementing Software Defined Radio[END_REF]. This transceiver, however, does not allow an easy design for a compact device; moreover, a full adaptability of the software code on the fly for different scenarios can be roughly achieved [START_REF] Costanzo | High Resolution Software Defined Radar System for Target Detection[END_REF]. The second generation USRP (NI2920, NI2921 and NI2922), currently adopted in most of recent implementations, is provided by Gigabit Ethernet interface, owning a 50 MHz instantaneous bandwidth and easily interfaced through commercial software such as MatLab or LabView. It is also equipped with advanced FPGA, with an SD card for standalone operations. The main components of this architecture are shown in the block diagram of Fig. 1 [16].

Figure 1. USRP NI2920 Architecture Clock generation and synchronization, AD and DA conversions, host processor interfacing, and power regulation are provided in the motherboard, while operations such as amplifying, filtering, up and down conversion are implemented in the RF front end, or eventually in modular daughterboards, selected for the specific applications. Since a visible light source is typically incoherent and intensity modulation with direct detection (IM/DD) is mainly used, a low frequency or band-base daughterboard may be employed [START_REF] Hussain | Software Defined VLC System: Implementation and Performance Evaluation[END_REF], such as the LFTX NI device [START_REF]NI USRP-2921 National Instruments Data-sheet[END_REF], whose operative range ranges from DC up to 30 MHz, thus covering the entire bandwidth of most commercial LEDs. In terms of dedicated hardware, only the optical Rx and Tx front-ends are required, instead of antennas used in radio or radar applications [START_REF] Costanzo | High Resolution Software Defined Radar System for Target Detection[END_REF], while most of the operations are directly performed and controlled on the fly by software. This aspect is particularly important to easily adapting existing signal modulation schemes to VLC transmission. Feeding and amplifying stage, led driving circuitry and the led itself in transmission, the photodiode and the trans-impedance amplifying in the receiver front end, could be chosen in order to find the appropriate trade-off between costs and performances. So, using the USRP and offthe-shelf components, it is possible to achieve reasonable results in terms of data rate, bit error rate and maximum transmission distance, as shown in Fig. 2 for the USRP based Software Defined VLC platforms described in the previous section. RPO-OFDM Matlab-Simulink 400Ksps < 1m 10 -3 Figure 2. Performance of Software Defined VLC using USRP NI2920 Architecture Results achieved in [START_REF] Baranda | An SDR Implementation of a Visible Light Communication System Based on the IEEE 802.15.7 Standard[END_REF][START_REF] Hussain | Software Defined VLC System: Implementation and Performance Evaluation[END_REF][START_REF] Mirvakili | Wireless Access Test-bed through Visible Light and Dimming Compatible OFDM[END_REF] prove the possibility of using a Software Defined System for short and medium range applications which do not require very high rate, thus avoiding the need of expensive dedicated hardware, and providing a flexible simple architecture. Since multiple USRPs could be connected to obtain a MIMO system, some authors recently considered also this scenario. An interesting paper deals with the possibility of using multiple USRP schemes for a full duplex OFDM link [START_REF] Zhou | Full Duplex Prototype of OFDM on GNURadio and USRPs[END_REF], while a 2x2 MIMO Software Defined VLC is addressed in [START_REF] Deng | Adaptive Real-Time Software Defined MIMO Visible Light Communications using Spatial Multiplexing and Spatial Diversity[END_REF]. These two-latter works demonstrate how a software defined platform could be expanded to achieve a more sophisticated and performing architecture. Many other several advantages can be provided by USRP, which are not yet addressed in current architectures, such as for example the possibility of adapting on the fly the equalization based on a bit error rate indicator. Another interesting aspect could be the possibility of regulating in real time the minimum transmitting power necessary for closing the VLC link within a dynamic environment. In the context of LI-FI paradigm, in fact, where the same lamp will be used for both lighting and providing signal communication, a software defined control of transmitted power could be useful for implementing a switchable modulation scheme allowing to provide a very low light intensity without turning off the link when high levels of environmental illumination are not required. Considering the nonlinearity of the optical components, different new adaptive techniques could be employed for limiting distortion, and various typologies of software defined filtering could be addressed, especially for future outdoor applications, such as in vehicular network applications. The possibility of designing a RF-VLC hybrid communication systems, using a USRP for a radio frequency communication and another one for the visible light link, could be an interesting issue for many applications, such as indoor geo-localization, Wi-Fi performance improving, or other future scenarios where the coexistence of VLC and RF link, or an adaptive handover between them, could be employed. A possible basic architecture is proposed in Fig. 3. Figure 3. Hybrid Software Defined VLC -Wi-Fi using USRP NI292x Device In this basic example, a XCVR2450 transceiver daughterboard is used for interfacing the dual band RF front-end covering Wi-Fi ranges and performing up-down conversion, filtering and amplification. Two baseband daughterboards, a LFRX in the receiving VLC path and a LFTX in the transmitting VLC path, are employed for filtering and other signal conditioning. Clock generation and synchronization, ADCs, DACs, and power regulation are performed independently by the two motherboards, through a centralized software coordination. The scheme proposed in Fig. 3 could be considered as a basic architecture for easily implementing a test bed for novel algorithms which employ both directional broadcast VLC channels and omnidirectional Wi-Fi links, recently proposed in literature [START_REF] Rahaim | A Hybrid Radio Frequency and Broadcast Visible Light Communication System[END_REF][START_REF] Li | Cooperative Load Balancing in Hybrid Visible Light Communications and WiFi[END_REF] for providing additional capacity and reducing contentions in a crowded environment. Even if a proof of concept of a hybrid RF-VLC link has been addressed in [START_REF] Shao | Design of a visible-light-communication enhanced WiFi system[END_REF], no software defined platform, for the best of our knowledge, has yet been considered with this aim. Providing a flexible and generic platform, however, could significantly speed up the development of novel applications using both VLC and RF links; a hybrid system, for instance, could be used for providing a reliable internet access point in running subway cars. A VLC broadcast channel, in fact, could be provided in each car, while Wi-Fi link could be still used in uplink. Since there is not a significant light interference between different cars, there could be an efficient frequency reuse allowing the system to work even in a crowded environment. Similar systems could be used in rail tunnels and similar environment. Other interesting applications could be developed in the field of indoor positioning, where an adaptive handover between Radio Frequency and VLC channels could be performed to benefit from the accuracy of the optical link where the environmental conditions are conducive, and switching in real time to RF localization where optical link fails. Similar ideas could be also developed in the field of aviation, underwater communications, hospitals and healthcare, where a Software Defined flexible platform like the one briefly discussed in this work could significantly speed up both experimental tests and the development of real systems.

Conclusions

A brief survey on Software Defined Visible Light Communication systems, based on USRP platform, has been outlined in this work. Some considerations about possible future applications of the above platform in the framework of Visible Light Communication scenarios have been discussed, and a basic example, showing a Hybrid VLC-LIFI architecture, has been provided. Based on the architecture we have considered in this work, we envisaged different potential future directions that could impose VLC as an effective communication technology in coexistence with other classical communication technologies based on RF signals.

Acknowledgements

This work was partially supported by CPER DATA and by "21 eme Fonds Unique Interministériel", in the framework of the "Store Connect" project.


Petru Mironescu
email: mironescu@math.univ-lyon1.fr

Sum-intersection property of Sobolev spaces

If 1 < p < ∞ and 0 < λ < 1, we may always decompose any function f ∈ W 1,p (R N ) as f = g + h with g ∈ W λ,p/λ and h ∈ W p,1 . A stronger property, natural in the context of functional calculus in Sobolev spaces, is that we may choose g ∈ W λ,p/λ ∩ W 1,p and h ∈ W p,1 ∩ W 1,p . We address here the question of the validity of a similar result for three Sobolev spaces W s 1 ,p 1 , W s,p , W s 2 ,p 2 satisfying the proportionality relations

for some θ ∈ (0, 1).

For most of s 1 , . . ., p 2 satisfying (1), we prove that

In some exceptional situations, this equality does not hold, and we derive an alternative decomposition.

We also establish the validity of (2) when the first equality in (1) is replaced by the suboptimal condition s > θs 1 + (1 -θ)s 2 .

Introduction

In connection with the factorization of unimodular Sobolev maps, Haim Brezis and the author observed the following property of Sobolev spaces [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]. Let 1 < p < ∞ and 0 < λ < 1. Then every function f ∈ W 1,p (R N ) can be decomposed as f = g + h, with g ∈ (W λ,p/λ ∩ W 1,p )(R N ) and h ∈ (W p,1 ∩ W 1,p )(R N ).

(1.1)

We will present in appendix a proof of this fact using factorization. We will also explain there how (1.1) is related to functional calculus (superposition operators) in Sobolev spaces.

Decomposition (1.1) has a flavor of interpolation, and indeed we have for example when p = 2 the equality [20, Section 2.4.3, Theorem, p. 66] W 1,2 = [W λ,2/λ , F 2 1,1 ] θ,2 , with θ := 1/(2 -λ).

(1.2)

[We will recall in the next section the definition of the Triebel-Lizorkin spaces F s p,q .] Using (1.2) and the embedding F 2 1,1 → W 2,1 (see the next section), we find that W 1,2 ⊂ W λ,2/λ + W 2,1 . However, this does not yield the stronger conclusion W 1,2 ⊂ (W λ,2/λ ∩ W 1,2 ) + (W 2,1 ∩ W 1,2 ). Actually, one cannot derive the equality Z = (X ∩ Z) + (Y ∩ Z) merely from the inclusion Z ⊂ X + Y (take e.g.

X = R × {0}, Y = {0} × R and Z = {(x, x); x ∈ R)}.
We address here the following question. Let 0 ≤ s, s 1 , s 2 < ∞, and 1 ≤ p 1 , p, p 2 ≤ ∞. Assume that W s,p (R N ) ⊂ W s 1 ,p 1 (R N ) + W s 2 ,p 2 (R N ) for any N.

(1.3)

Is it true that W s,p (R N ) = (W s 1 ,p 1 ∩ W s,p )(R N ) + (W s 2 ,p 2 ∩ W s,p )(R N ) for any N?

(1.4)

We emphasize the fact that we ask for N-independent properties. For example, by the Sobolev embeddings we have W 1,1 ⊂ L 2 when N = 1 or 2, but not for N ≥ 3, and thus (1.3) does not hold for s 1 = s 2 = 0, s = 1, p 1 = p 2 = 2, p = 1.

Our first results characterize most of the triples T = (W s 1 ,p 1 ,W s,p ,W s 2 ,p 2 ) such that (1.3) and (1.4) hold.

Proposition 1.1. Assume that (1.3) holds. Then there exists some θ ∈ [0, 1] such that s ≥ θs 1 + (1 -θ)s 2 ,

(1.5)

1 p = θ p 1 + (1 -θ) p 2 .
(1.6) Proposition 1.2. Assume that for some θ ∈ [0, 1] we have (1.6) and s > θs 1 + (1 -θ)s 2 . Then both (1.3) and (1.4) hold.

On the other hand, (1.3) and (1.4) trivially hold when (1.5)-(1.6) are satisfied with θ = 0 or 1, since we then have either W s,p → W s 2 ,p 2 , or W s,p → W s 1 ,p 1 . We next investigate the case where

s = θs 1 + (1 -θ)s 2 , 1 p = θ p 1 + 1 -θ p 2
for some θ ∈ (0, 1).

(1.7)

In this case, (1.3) holds most of the time, but not always. For example, when N = 1 we have

W 1/2,2 (R) ⊂ W 1,1 (R) + L ∞ (R), (1.8)
i.e., (1.3) does not hold for the triple T = (W 1,1 ,W 1/2,2 , L ∞ ). Indeed, for N = 1 we have W 1,1 → L ∞ , and thus W 1,1 + L ∞ = L ∞ . However, W 1/2,2 ⊂ L ∞ .

Definition 1.3. A triple T = (W s 1 ,p 1 ,W s,p ,W s 2 ,p 2 ) is admissible if it satisfies (1.7).

An admissible triple T is irregular if s 1 = s 2 , 1 < p < ∞ and (exactly) one of the spaces W s 1 ,p 1 , W s 2 ,p 2 is of the form W k,∞ with k ∈ N. T is regular otherwise.

Thus T = (W 1,1 ,W 1/2,2 , L ∞ ) (which corresponds to the example occurring in (1.8)) is irregular.

Our main result is the following Theorem 1.4. Let T be a regular triple. Then both (1.3) and (1.4) hold.

Equivalently, for every regular triple T we have

W s,p (R N ) = (W s 1 ,p 1 ∩ W s,p )(R N ) + (W s 2 ,p 2 ∩ W s,p )(R N ), ∀ N.
(1.9)

For most of the regular triples, (1.4) follows automatically from (1.3), as explained in Proposition 1.6 below. Thus, in particular, the conclusion of the theorem follows whenever T is as in Proposition 1.6 and W s,p can be obtained by interpolation from W s 1 ,p 1 and W s 2 ,p 2 . However, when T is admissible W s,p need not be an interpolation space between W s 1 ,p 1 and W s 2 ,p 2 , at least for the standard real and complex methods [20, Sections 2.4.2-2.4.7, p. 64-73]; thus one cannot derive Theorem 1.4 directly from Proposition 1.6. We will present, in Section 3, a proof of Theorem 1.4 which does not rely on interpolation and establishes simultaneously (1.3) and (1.4). Definition 1.5. A Sobolev space W s,p is exceptional if s ∈ N and either p = 1 or p = ∞. It is ordinary otherwise. Proposition 1.6. Assume that W s,p , W s 1 ,p 1 and W s 2 ,p 2 are all three ordinary Sobolev spaces. Assume that for some (fixed

) N we have W s,p (R N ) ⊂ W s 1 ,p 1 (R N ) + W s 2 ,p 2 (R N ). Then for such N we have W s,p (R N ) = (W s 1 ,p 1 ∩ W s,p )(R N ) + (W s 2 ,p 2 ∩ W s,p )(R N ).
We now turn to irregular T's. At least in some special cases (see (1.8) and, more generally, the triples T = (W 1,1 ,W 1/p,p , L ∞ ), with 1 < p < ∞), (1.3) does not hold for such triples. We do not know the characterization of irregular triples T for which (1.3) and/or (1.4) do not hold. For irregular triples, we were only able to establish a weaker form of (1.4), in which the space W k,∞ is replaced by a slightly larger space, modeled on bmo (the local BMO space whose definition will be recalled in the next section).

Theorem 1.7. Let T be an irregular triple, and assume e.g. that p 2 = ∞ (and thus s 2 is an integer). Let

1 < q 2 < ∞. Then W s,p (R N ) = (W s 1 ,p 1 ∩ W s,p )(R N ) + (F s 2 ∞,q 2 ∩ W s,p )(R N ). (1.10)
In particular, when s 2 = 0 (and thus W s 2 ,p 2 = L ∞ ) we have

W s,p (R N ) = (W s 1 ,p 1 ∩ W s,p )(R N ) + (bmo ∩ W s,p )(R N ). (1.11) When s 2 > 0, we have W s,p (R N ) = (W s 1 ,p 1 ∩ W s,p )(R N ) + ({ f ∈ W s 2 -1,∞ ; D s 2 -1 f ∈ bmo } ∩ W s,p )(R N ).
(1.12)

In the special case s 2 = 0, s ∈ N, p 1 = 1, Theorem 1.7 was established in [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]Chapter 6].

Remark 1.8. The question of the validity of (1.3)-(1.4) is somewhat dual to the one of the validity of the Gagliardo-Nirenberg inequalities. There, one asks whether the inclusion

W s 1 ,p 1 (R N ) ∩ W s 2 ,p 2 (R N ) ⊂ W s,p (R N ) (1.13)
leads, for some appropriate θ ∈ [0, 1], to the estimate

f W s,p (R N ) f θ W s 1 ,p 1 (R N ) f 1-θ W s 2 ,p 2 (R N ) . (1.14)
In the spirit of our Proposition 1.1, one may prove that the validity of (1.13) for every N requires

s ≤ θs 1 + (1 -θ)s 2 , (1.15) 1 p = θ p 1 + (1 -θ) p 2 (1.16)
for some θ ∈ [0, 1]. If we have either "<" in (1.15) or θ ∈ {0, 1}, then we have both (1.13) and (1.14); this follows from the main result in [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: the full story[END_REF]. As in our situation, the interesting case is the one of admissible triples. In that case, (1.15) and (1.16) hold when s 1 , s, s 2 are integers, as established in the seminal contributions of Gagliardo [START_REF] Gagliardo | Ulteriori proprietà di alcune classi di funzioni in più variabili[END_REF] and Nirenberg [START_REF] Nirenberg | On elliptic partial differential equations[END_REF]. It turns out that (1.15) and (1.16) hold for most of the admissible triples, but not all of them. A characterization of the admissible triples for which (1.15) and (1.16) hold has been obtained in [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: the full story[END_REF]; see also [START_REF] Kašin | Remarks on the estimation of Lebesgue functions of orthonormal systems[END_REF], [START_REF] Oru | Rôle des oscillations dans quelques problèmes d'analyse non-linéaire[END_REF], [START_REF] Cohen | Harmonic analysis of the space BV[END_REF] for older partial results.

Remark 1.9. As one may expect, whenever it is possible to decompose f = f 1 + f 2 with f 1 ∈ (W s1 ,p 1 ∩ W s,p )(R N ) and f 2 ∈ (W s2 ,p 2 ∩ W s,p )(R N ), we also have a norm control for f 1 and f 2 in terms of f W s,p . A simple example of such decomposition with norm control is the following. For

f ∈ L 2 (R N ), set f 1 := f 1 {x;| f (x)|> f L 2 (R N ) } and f 2 := f 1 {x;| f (x)|≤ f L 2 (R N ) } . Then clearly f 1 ∈ (L 1 ∩ L 2 )(R N ) and f 2 ∈ (L ∞ ∩ L 2 )(R N )
, and in addition we have the norm controls

f 1 L 1 (R N ) ≤ f L 2 (R N ) , f 1 L 2 (R N ) ≤ f L 2 (R N ) , f 2 L ∞ (R N ) ≤ f L 2 (R N ) , f 2 L 2 (R N ) ≤ f L 2 (R N ) .
Note however that the map f → ( f 1 , f 2 ) is not linear. Likewise, in general we will construct nonlinear decompositions.

Our text is organized as follows. In Section 2, we recall some basic facts on function spaces, instrumental for our purposes. The proofs of Propositions 1.1, 1.2 and 1.6 and of Theorems 1.4 and 1.7 are presented in Section 3. A final appendix presents the factorization theory and its connections with the sum-intersection property and with the functional calculus in Sobolev spaces.

Basic properties of Triebel-Lizorkin spaces

Definition 2.1. Let ψ ∈ C ∞ c (R N ) be such that ψ = 1 in B 1 (0) and suppψ ⊂ B 2 (0). Define ψ 0 = ψ and, for j ≥ 1, ψ j (x) := ψ(x/2 j ) -ψ(x/2 j-1 ). Set ϕ j := F -1 ψ j ∈ S . 1 Then for each temperate distribution f we have

f = j f j in S ′ , with f j := f * ϕ j .
(2.1)

f = f j is "the" Littlewood-Paley decomposition of f ∈ S ′ .
Note that F f j = ψ j F f is compactly supported, and therefore f j ∈ C ∞ for each j.

Definition 2.2. Starting from for Littlewood-Paley decomposition, we define the Triebel-Lizorkin spaces F s p,q as follows: for -∞ < s < ∞, 0 < p < ∞ and 0 < q ≤ ∞, we let f F s p,q := 2 s j f j (x)

j≥0 l q (N) L p (R N ) , F s p,q := { f ∈ S ′ ; f F s p,q < ∞}.
Same definition when p = q = ∞. This definition has to be changed when p = ∞ and 1 < q < ∞ [20, Section 2.3.4, p. 50]: we let

f F s ∞ ,q = inf esssup x∈R N 2 s j f j (x) j≥0 l q (N) L ∞ (R N ) ; f j ∈ L ∞ (R N ), f = f j * ϕ j in S ′ .
Most of the Sobolev spaces can be identified with 1. If s > 0 is not an integer and 1 ≤ p ≤ ∞, then W s,p (R N ) = F s p,p .

2. If s ≥ 0 is an integer and 1 < p < ∞, then W s,p (R N ) = F s p,2 . When s ≥ 0 is an integer and either p = 1 or p = ∞, the Sobolev space W s,p cannot be identified with a Triebel-Lizorkin space.

Theorem 2.3 is usually used in conjunction with Lemma 2.5 below. The reason is that, in practice, we do not know the Littlewood-Paley decomposition of f , but only a Nikol'skij decomposition of f .

Definition 2.4. A Nikol'skij decomposition of f ∈ S ′ is a representation of the form f = f j in S ′ , with supp F f j ⊂ B 2 j+1 (0) \ B 2 j-1 (0), if j ≥ 1 B 2 (0), if j = 0 .
Note that in particular the Littlewood-Paley decomposition f = f j is a Nikol'skij decomposition.

Lemma 2.5.

1. Let 1 < p < ∞, 1 < q ≤ ∞ and s ∈ R. Consider a sequence ( f j ) such that 2 s j f j (x) j≥0 l q (N) L p (R N ) < ∞. Then f = f j * ϕ j converges in S ′ and f F s p,q 2 s j f j (x) j≥0 l q (N) L p (R N ) . (2.2) 2. Same conclusion if 1 ≤ p = q ≤ ∞.
Proof. It suffices to consider finite sums, and to establish (2.2) in this case. We start with item 2, which is easier. Note that f ∈ L p (R N ), and thus f ∈ S ′ . Let f = j≥0 f j be the Littlewood-Paley decomposition of f . Since

ϕ j * ϕ k = 0 if | j -k| ≥ 2, we find that f j = f * ϕ j = k f k * ϕ k * ϕ j = |k-j|≤1 f k * ϕ k * ϕ j , (2.3)
and thus

f j L p (R N ) ≤ |k-j|≤1 f k * ϕ k * ϕ j L p (R N ) ≤ |k-j|≤1 f k L p (R N ) ϕ k * ϕ j L 1 (R N ) ≤ C |k-j|≤1 f k L p (R N ) .
(2.4)

We obtain (2.2) with p = q from (2.4). We now consider item 1. From (2.3), we find that

| f j (x)| ≤ |k-j|≤1 | f k * ϕ k * ϕ j (x)| ≤ C |k-j|≤1 M f k (x). (2.5)
Here, M is the standard maximal operator, and we used the inequality [18, Proposition, p. 24]

| f * ρ ε (x)| ≤ C ρ M f (x), ∀ ρ ∈ S , ∀ ε > 0.
Using (2.5), we find that

f F s p,q 2 s j M f j (x) j≥0 l q (N) L p (R N ) 2 s j f j (x) j≥0 l q (N) L p (R N )
, the latter inequality being the Fefferman-Stein vectorial maximal inequality [START_REF] Fefferman | Some maximal inequalities[END_REF].

Definition 2.6. We define, for f ∈ L 1 loc (R N ),

f bmo := sup |B|≤1 ˆB | f | + sup |B|≤1 B B | f (x) -f (y)| dxd y,
the sup being taken over the balls of volume ≤ 1. We set bmo := { f ∈ L 1 loc (R N ); f bmo < ∞}. With its natural norm, bmo is the local BMO space.

Then we have [21, Theorem, p. 47] bmo = F 0 ∞,2 . Using this equality, Definition 2.2 and the embedding ℓ q → ℓ 2 , 0 < q < 2, we obtain the following

Corollary 2.7. If f = j≥0 f j * ϕ j in S ′ and 0 < q < 2, then f 2 bmo ≤ C esssup x∈R N j | f j (x)| 2 ≤ C esssup x∈R N j | f j (x)| q 2/q
, for some C independent of the f j 's. (2.6)

Corollary 2.8. For 1 < q < 2, we have F 0 ∞,q → bmo .

As we noticed above, when s ∈ N the space W s,1 is not a Triebel-Lizorkin space. However, we have the following Lemma 2.9.

1. When s ≥ 0, we have F s 1,1 → W s,1 (R N ).
2. More generally, for every s ≥ 0 and 1 ≤ p < ∞ we have F s p,1 → W s,p (R N ). The same holds when p = ∞ and s > 0 is not an integer.

3. When k > 0 is an integer and 1 < q ≤ 2, we have

F k ∞,q → { f ∈ W k-1,∞ (R N ); D k-1 f ∈ bmo }.
Proof. We start with p = 1. When s is not an integer, we actually have equality. When s = 0 and

f ∈ F 0 1,1 , we have f L 1 (R N ) ≤ j≥0 f j L 1 (R N ) = f F 0 1,1 < ∞.
When s ≥ 1 is an integer, we use the fact that [20, Section 2.3.8, Theorem (ii), pp. 58-59]

f F s 1,1 ∼ s j=0 D j f F s-j 1,1 ≥ s j=0 D j f F 0 1,1 ≥ s j=0 D j f L 1 (R N ) = f W s,1 (R N ) .
When 1 < p < ∞, the desired inclusion follows from

F s p,1 → F s p,q = W s,p (R N ) (with q = 2 or q = p, according to s).
Similarly if p = ∞ and s is not an integer. Finally, if p = ∞ and s is an integer, we argue as for p = 1, relying on Corollary 2.8 and [20, Section 2.3.8, Remark 2, p. 60].

We now briefly recall the characterization of Triebel-Lizorkin spaces in terms of wavelets. Let ψ 0 , ψ 1 be respectively a father and mother (sufficiently smooth) wavelets. For

G ∈ {0, 1} N , j ∈ N and m ∈ Z N , let ψ j G,m (x) := 2 N j/2 N r=1 ψ G r (2 j x r -m r ), x ∈ R N . Let, for f ∈ S ′ , λ j G,m := 0, if j > 0 and G = {0} N 2 N j/2 ( f , ψ j G,m ), otherwise . Recall [22, Section 3.1.3] that f = j,G,m 2 -N j/2 λ j G,m ψ j G,m in the sense of S ′ . Conversely, if f = G,m µ 0 G,m ψ 0 G,m + j>0,G ={0} N , m 2 -N j/2 µ j G,m ψ j G,m in the sense of S ′ , then the wavelet coefficients λ j G,m of f are given by λ j G,m = 0, if j > 0 and G = {0} N µ j G,m , otherwise . Let, for j ∈ N and m ∈ Z N , Q j,m be the cube N r=1 [2 -j (m r -1), 2 -j (m r + 1)]. Set, for 0 < q < ∞, s ∈ R, g(x) = g s p,q (x) := 2 sq j |λ j G,m | q 1 Q j,m (x) 1/q . (2.7)
When q = ∞, we replace the ℓ q norm by the sup norm.

Then one may read the smoothness of f in terms of the integrability properties of g. The following statement is a rephrasing of [22, Theorem 1.64, p. 33].

Theorem 2.10.

1. Let -∞ < s < ∞, 1 ≤ p < ∞, 0 < q ≤ ∞. Then f F s p,q ∼ g s p,q L p (R N ) .
2. Same conclusion if p = q = ∞.

3. In particular, if s > 0 is not an integer and

1 ≤ p ≤ ∞, then f W s,p (R N ) ∼ g s p,p L p . 4. If s ≥ 0 is an integer and 1 < p < ∞, then f W s,p (R N ) ∼ g s,2 L p .
Let us note that when p = q, this norm equivalence takes a particularly simple form. More specifically, we have

f p F s p,p ∼ j,G,m 2 (s p-N) j |λ j G,m | p , -∞ < s < ∞, 1 ≤ p < ∞ (2.8) f F s ∞,∞ ∼ sup j,G,m 2 s j |λ j G,m |, -∞ < s < ∞.
(2.9)

Our next result relies on properties of the Besov spaces B s p,q . In order to keep this section short, we will be rather sketchy. Lemma 2.11. Let 0 ≤ s < ∞, 1 ≤ p ≤ ∞ and ε > 0. Then, with g s p,q as in (2.7), we have

f W s,p (R N ) g s+ε p,p L p (R N ) , (2.10)
g s-ε p,p L p (R N ) f W s,p (R N ) . (2.11)
Sketch of proof. The above estimates are equivalent to the embeddings

F s+ε p,p → W s,p (R N ) → F s-ε p,p . (2.12)
When s is not an in an integer, we have W s,p (R N ) = F s p,p , and the conclusion is clear. When s is an integer and 1 ≤ p ≤ ∞, the Littlewood-Paley decomposition f = j f j of f satisfies [7, Lemma 2.1.1]

f 0 L p (R N ) ≤ f L p (R N ) , 2 s j f j L p (R N ) D s f L p (R N ) , ∀ j ≥ 1. (2.13)
Thus sup j 2 s j f j L p (R N ) f W s,p (R N ) , i.e., we have the embedding

W s,p (R N ) → B s p,∞ . (2.14)
On the other hand, we have [19, Chapter 5, Lemma 3.14]

D s f j L p (R N ) 2 s j f j L p (R N ) , ∀ j ≥ 0,
and thus

f W s,p (R N ) j f j L p (R N ) + D s f j L p (R N ) j 2 s j f j L p (R N ) .
Equivalently, we have the embedding

B s p,1 → W s,p (R N ). ( 2
F s+ε p,p = B s+ε p,p → B s p,1 → W s,p (R N ) → B s p,∞ → B s-ε p,p = F s-ε p,p .

Proofs

Proof of Proposition 1.1. In order to prove the existence of some θ such that (1.6) holds, we have to establish the double inequality

min{p 1 , p 2 } ≤ p ≤ max{p 1 , p 2 }. ( 3.1)
We argue by contradiction. Assume first that p > max{p 1 , p 2 }. Let

f (x) = 2 (1 + x 2 ) (1+ε)/(2p) , ∀ x ∈ R.
Clearly, f ∈ L p (R), and more generally f ∈ W k,p (R) for every integer k. It follows that f ∈ W s,p (R) for every s ≥ 0. On the other hand, for every f 1 , f 2 such that f = f 1 + f 2 and every x we have either

| f 1 (x)| ≥ f (x)/2 or | f 2 (x)| ≥ f (x)/2. We find that | f 1 (x)| p 1 + | f 2 (x)| p 2 f (x) p 1 + f (x) p 2 := g(x).
Since, for sufficiently small ε, we have g ∈ L 1 (R), we find that f ∈ L p 1 (R) + L p 2 (R). Therefore, f ∈ W s 1 ,p 1 (R) + W s 2 ,p 2 (R), which is a contradiction.

Assume next that p < min{p 1 , p 2 }. Let p < r < min{p 1 , p 2 }. Let N be sufficiently large such that W s,p (B) ⊂ L r (B); here, B is a ball in R N . By a standard extension argument, there exists some f ∈ W s,p c (R N ) such that f ∈ L r (R N ). Such an f does not belong to L r loc (R N ), and thus does not belong to

L p 1 (R N ) + L p 2 (R N ). We find that f ∈ W s 1 ,p 1 (R N ) + W s 2 ,p 2 (R N ), again a contradiction.
We thus know that (3.1) holds, or equivalently, that (1.6) holds for some θ.

We next proceed to the proof of (1.5). Assume first that p 1 = p 2 = p. Then θ is not determined by (1.6), and its existence is equivalent to s ≥ min{s 1 , s 2 }. Arguing by contradiction, assume that

s < min{s 1 , s 2 }. Let s < ρ < min{s 1 , s 2 }. If f ∈ W s,p (R) \ W ρ,p (R), then f ∈ W s 1 ,p (R) + W s 2 ,p (R) = W min{s 1 ,s 2 },p (R) ⊂ W ρ,p (R), a contradiction.
Assume next that p 1 = p 2 . Then θ is determined by (1.6). Argue again by contradiction and assume that s < θs 1 +(1-θ)s 2 . Set σ := θs 1 +(1-θ)s 2 > s. Consider some ε > 0 such that s+ε < σ-ε. In view of Lemma 2.11, in order to contradict (1.3) it suffices to establish, for some appropriate N, the non inclusion

F s+ε p,p ⊂ F s 1 -ε p 1 ,p 1 + F s 1 -ε p 2 ,p 2 . (3.2)
With no loss of generality, we may assume that

1 ≤ p 1 < p 2 ≤ ∞. (3.3)
We will treat separately the cases p 2 < ∞ and p 2 = ∞.

Set, in all cases,

α := s 1 -ε p 2 - s 2 -ε p 1 1 p 1 - 1 p 2 = s 1 p 2 - s 2 p 1 1 p 1 - 1 p 2 + ε. (3.4)
Proof of (3.2) when p 2 < ∞. We rely on the following

Claim. For appropriate C 1 , C 2 > 0, we have [a + b = S, S ≥ C 1 2 α j ] =⇒ [2 (s 1 -ε) j p 1 |a| p 1 + 2 (s 2 -ε) j p 2 |b| p 2 ≥ C 2 2 (σ-ε) j p ]. (3.5)
Granted the claim , we conclude as follows. Consider some f ∈ S ′ such that for every j, G and m we have either λ j G,m = 0 or |λ j G,m | ≥ C 1 2 α j , with C 1 as in (3.5). The claim combined with (2.8) implies that for every possible decomposition f = f 1 + f 2 we have

f 1 p 1 F s 1 -ε p 1 ,p 1 + f 2 p 1 F s 2 -ε p 2 ,p 2 f p F σ-ε p,p . (3.6)
We are now in position to obtain a contradiction. Let N be sufficiently large such that (σ

-ε + α)p < N. Let δ := N -(σ -ε + α)p > 0. Fix some G 0 ∈ {0, 1} N \ {0} N . For every j ∈ N, consider a set M j ⊂ Z N such that #M j ∼ 2 δ j . Set f := j, m∈M j 2 -N j/2 C 1 2 α j ψ j G 0 ,m .
By (2.8), we have

f p F s+ε p,p ∼ j 2 ((s+ε+α)p-N+δ) j = j 2 -((σ-ε)-(s+ε)) j p < ∞, while f p F σ-ε p,p ∼ j 2 ((σ-ε+α)p-N+δ) j = j 1 = ∞.
We complete the proof of (3.2) when p 2 < ∞ using the two above inequalities and (3.6).

Proof of (3.2) when p 2 = ∞ and θ ∈ (0, 1]. This time we have α = -(s 2 -ε). We modify the definition of f by setting

f := j, m∈M j 2 -N j/2 j2 α j ψ j G 0 ,m . Assume, by contradiction, that f = f 1 + f 2 for some f 1 ∈ F s 1 -ε p 1 ,p 1 and f 2 ∈ F s 2 -ε ∞,∞ . Write f 1 = j,G,m 2 -N j/2 a j G,m ψ j G,m , f 2 = j,G,m 2 -N j/2 b j G,m ψ j G,m . Since f 2 ∈ F s 2 -ε ∞,∞ , we have |b j G,m | ≤ C2 -(s 2 -ε) j = C2 α j , ∀ j,G, m.
Since a j G 0 ,m + b j G 0 ,m = j2 α j , ∀ j, ∀ m ∈ M j , for sufficiently large j 0 we have

|a j G 0 ,m | ≥ 1 2 j2 α j , ∀ j ≥ j 0 , ∀ m ∈ M j .
Inserting this into (2.8) and using the fact that

(s 1 -ε + α)p 1 -N + δ = (s 1 -s 2 )p 1 -θ(s 1 -s 2 )p = (s 1 -s 2 )(p 1 -θ p) = 0 (since p 1 = θ p),
we find that

f p 1 F s 1 -ε p 1 ,p 1
j≥ j 0 , m∈M j j p 1 2 ((s 1 -ε+α)p 1 -N) j ∼ j≥ j 0 j p 1 2 ((s 1 -ε+α)p 1 -N+δ) j = j≥ j 0 , m∈M j j p 1 = ∞.

On the other hand, we have

f p F s+ε p,p ∼ j p 2 ((s+ε+α)p-N+δ) j = j p 2 -((σ-ε)-(s+ε)) j p < ∞.
This leads to a contradiction and completes the proof of (3.2) when p 2 = ∞ and θ ∈ (0, 1]. Proof of (3.2) when p 2 = ∞ and θ = 0. This is similar to the case p 2 = ∞ and θ ∈ (0, 1]. We have

α = -(s 2 -ε) = -(σ -ε) < -(s + ε). Consider f := j,m 2 -N j/2 j2 α j ψ j G 0 ,m . [This time, the sum in m is over all m ∈ Z N .] We then have f ∈ F s+ε ∞,∞ .
Arguing by contradiction, we obtain that f cannot be decomposed as

f = f 1 + f 2 with f 1 ∈ F s 1 -ε p 1 ,p 1 and f 2 ∈ F s 2 -ε ∞,∞ . Indeed, as in the previous case, if f 2 ∈ F s 2 -ε
∞,∞ then for large j 0 we have

f 1 p 1 F s 1 -ε p 1 ,p 1 j≥ j 0 m∈Z N j p 1 2 (s 1 αp 1 -N) j = ∞.
Proof of the claim. Let S > 0. The function [0, ∞) ∋ t → g(t) := 2 (s 1 -ε) j (1 -t)S + 2 (s 2 -ε) j p 2 /p 1 t p 2 /p 1 S p 2 /p 1 is convex, and its derivative at the origin is negative. Thus g has a global minimum at the point t 0 where g ′ (t 0 ) = 0. Solving the equation g ′ (t) = 0, we find that t 0 = C 1 2 α j S -1 , with C 1 > 0 independent of j. Provided that S ≥ C 1 2 α j , we have t 0 ≤ 1, and therefore the first term in g(t) is non negative. For such S, we thus have g(t) ≥ g(t 0 ) ≥ 2 (s 2 -ε) j p 2 /p 1 (t 0 ) p 2 /p 1 S p 2 /p 1 = c 2 (s 2 -ε+α)p 2 /p 1 j = c2 (σ-ε)p/p 1 j , ∀ t ≥ 0, with c > 0 independent of S.

Let now a, b be such that a + b = S ≥ C 1 2 α j . Then

2 s 1 j p 1 |a| p 1 + 2 s 2 j p 2 |b| p 2 ≥ 2 s 1 j p 1 a p 1 + 2 s 2 j p 2 b p 2 ,
where

a :=        a, if 0 ≤ a, b ≤ S 0, if a < 0 and b > S S, if a > S and b < 0 , b :=        b, if 0 ≤ a, b ≤ S S, if a < 0 and b > S 0, if a > S and b < 0 . ( 3.7)
Therefore, it suffices to prove (3.5) under the extra assumption that 0

≤ a, b ≤ S. Write a = (1 -t)S, b = tS, with t ∈ [0, 1]. We then have 2 (s 1 -ε) j p 1 a p 1 + 2 (s 2 -ε) j p 2 b p 2 ∼ 2 (s 1 -ε) j a + 2 (s 2 -ε) j p 2 /p 1 b p 2 /p 1 p 1 = [g(t)] p 1 ≥ [g(t 0 )] p 1 ≥ c p 1 2 (σ-ε) j p .
Proof of Proposition 1.2 assuming Theorem 1.4. As already noticed in the proof of Proposition 1.1, when p 1 = p 2 = p or when θ ∈ {0, 1}, properties (1.3) and (1.4) are trivially true. We may thus assume that p 1 = p 2 and θ ∈ (0, 1). Set

λ := s -(θs 1 + (1 -θ)s 2 ) > 0. For 0 < ε < λ θ , let δ > 0 satisfy θε + (1 -θ)δ = λ.
Then we may pick ε such that neither s 1 + ε nor s 2 + δ is an integer. Thus the triple T := (W s 1 +ε,p 1 ,W s,p ,W s 2 +δ,p 2 ) is regular. Granted Theorem 1.4, this implies

(W s 1 ,p 1 ∩ W s,p )(R N ) + (W s 2 ,p 2 ∩ W s,p )(R N ) ⊂ W s,p (R N ) = (W s 1 +ε,p 1 ∩ W s,p )(R N ) + (W s 2 +δ,p 2 ∩ W s,p )(R N ) ⊂ (W s 1 ,p 1 ∩ W s,p )(R N ) + (W s 2 ,p 2 ∩ W s,p )(R N ). Proof of Proposition 1.6. Decompose f ∈ W s,p (R N ) as f = f 1 + f 2 , with f 1 ∈ W s 1 ,p 1 (R N ) and f 2 ∈ W s 2 ,p 2 (R N ). Write, in the sense of S ′ , f = j,G,m 2 -N j/2 λ j G,m ψ j G,m , f 1 = j,G,m 2 -N j/2 a j G,m ψ j G,m , f 2 = j,G,m 2 -N j/2 b j G,m ψ j G,m . In the spirit of (3.7), define a j G,m :=        a j G,m , if 0 ≤ a j G,m , b j G,m ≤ λ j G,m 0, if a j G,m < 0 and b j G,m > λ j G,m λ j G,m , if a j G,m > λ j G,m and b j G,m < 0 , f 1 := j,G,m 2 -N j/2 a j G,m ψ j G,m b j G,m :=        b j G,m , if 0 ≤ a j G,m , b j G,m ≤ λ j G,m λ j G,m , if a j G,m < 0 and b j G,m > λ j G,m 0, if a j G,m > λ j G,m and b j G,m < 0 , f 2 := j,G,m 2 -N j/2 b j G,m ψ j G,m .
Then

f = f 1 + f 2
, and Theorem 2.10 implies that

f 1 W s,p (R N ) f W s,p (R N ) < ∞, f 1 W s 1 ,p 1 (R N ) f 1 W s 1 ,p 1 (R N ) < ∞, f 2 W s,p (R N ) f W s,p (R N ) < ∞, f 2 W s 2 ,p 2 (R N ) f 2 W s 2 ,p 2 (R N ) < ∞.
Proof of Theorem 1.4. The case where p 1 = p 2 is trivial, since we then have W s,p ⊂ W min{s 1 ,s 2 },p . We may thus assume that

1 ≤ p 1 < p < p 2 ≤ ∞. (3.8)
We further distinguish between the cases s 1 = s 2 and s 1 = s 2 , and also between p 2 < ∞ and p 2 = ∞.

Given

f ∈ W s,p (R N ), we write f = j,G,m 2 -N j/2 λ j G,m ψ j G,m . Case 1. s 1 = s 2 = s ∈ N. Set f 1 := j,G,m 2 -N j/2 a j G,m ψ j G,m , f 2 := j,G,m 2 -N j/2 b j G,m ψ j G,m , with a j G,m := λ j G,m , if |λ j G,m | ≥ 2 -s j 0, if |λ j G,m | < 2 -s j , b j G,m := 0, if |λ j G,m | ≥ 2 -s j λ j G,m , if |λ j G,m | < 2 -s j . Since p 1 < p, we have |a j G,m | p 1 ≤ 2 s j(p 1 -p) |λ j G,m | p .
(3.9) Using (3.9), the fact that s is not an integer and (2.8), we find that

f 1 W s,p (R N ) f W s,p (R N ) , f 1 p 1 W s 1 ,p 1 (R N ) f p W s,p (R N ) . (3.10)
Similarly, if p 2 < ∞ then we have

f 2 W s,p (R N ) f W s,p (R N ) , f 2 p 2 W s 2 ,p 2 (R N ) f p W s,p (R N ) . (3.11)
On the other hand, if p 2 = ∞ then

f 2 W s,p (R N ) f W s,p (R N ) , f 2 W s 2 ,∞ (R N ) 1. (3.12)
We complete this step via (3.10)-(3.12).

Remark 3.1. The estimates (3.10)-(3.12) are nonlinear, while one would expect linear estimates. Actually, it is possible to obtain linear estimates by cutting the coefficients λ j G,m at height A2 -s j instead of 2 -s j , with A := f W s,p (R N ) . The corresponding decomposition satisfies

f 1 W s,p (R N ) + f 1 W s 1 ,p 1 (R N ) + f 2 W s,p (R N ) + f 2 W s 2 ,p 2 (R N ) f W s,p (R N ) .
Similar observations apply to all the other cases.

Case 2. s 1 = s 2 = s ∈ N. In this case, we follow the ideas of DeVore and Scherer [START_REF] Devore | Interpolation of linear operators on Sobolev spaces[END_REF] concerning the interpolation theory of classical spaces, in the form presented in Bennett and Sharpley [1, Section 5.5, pp. 347-362].

We claim that it suffices to decompose every f

∈ (W s,p ∩ C ∞ )(R N ) as f = f 1 + f 2 , with f 1 W s,1 (R N ) f W s,p (R N ) , f 1 W s,p (R N ) f W s,p (R N ) , (3.13)
f 2 W s,∞ (R N ) f W s,p (R N ) , f 2 W s,p (R N ) f W s,p (R N ) . (3.14)
Indeed, if this holds then Hölder's inequality implies that

f 1 W s,p 1 (R N ) f W s,p (R N ) , f 2 W s,p 2 (R N ) f W s,p (R N ) , 1 ≤ p 1 < p 2 ≤ ∞, (3.15)
and then a density argument shows that (3.13)-(3.15) hold without the extra assumption f ∈ C ∞ ; this settles this case.

We next proceed to the construction of f 1 and f 2 . Let M denote the standard maximal (uncentered) operator. Set

H ℓ (x) := |α|=ℓ |∂ α f (x)| and H(x) := k ℓ=0 H ℓ (x). Let Ω := {x ∈ R N ; M H(x) > τ} and M := R N \ Ω. Thus M is closed and H(x) ≤ τ, ∀ x ∈ M. Let c be such that M g L p (R N ) ≤ c g L p (R N ) , ∀ g ∈ L p (R N ). If τ := c H L p (R N ) ∼ f W s,p (R N ) , then |Ω| ≤ 1 τ p ˆΩ(M H) p (x) dx ≤ 1 τ p M H p L p (R N ) ≤ 1. (3.16)
We then let f 2 be the Whitney extension of f |M and set f 1 := ff 2 . More specifically, let (Q j ) be a Whitney covering of Ω with cubes of size ℓ j and centers y j . Let Q j,t denote the cube of center y j and size tℓ j . Recall the following properties of the Whitney covering:

(Q j,9/8 ) is a covering of Ω, Q j,4 intersects M, ∀ j, j 1 Q j (x) ≤ C(N), ∀ x ∈ Ω. (3.17)
Let (φ j ) be an adapted Whitney partition of unity in Ω, i.e., supp φ j ⊂ Q j,9/8 , ∀ j, and

|∂ α φ j | (ℓ j ) -α , ∀ α ∈ N N . ( 3.18)
Let x j ∈ M ∩ Q j,4 and set

T j (x) := |α|≤s-1 ∂ α f (x j ) (x -x j ) α α! ,
the Taylor expansion of order s -1 of f around x j . Then we set

f 2 := f , in M T j φ j , in Ω . This f 2 satisfies [1, Theorem 5.10, p. 355] f 2 ∈ W s,∞ (R N ) and
f 2 W s,∞ (R N ) τ ∼ f W s,p (R N ) . (3.19)
On the other hand, using the fact that |Ω| ≤ 1 (by (3.16)), we find that for every 1 ≤ r ≤ p the function f 1 satisfies

f 1 W s,r (R N ) = f -f 2 W s,r (Ω) ≤ f W s,r (Ω) + f 2 W s,r (Ω) f W s,r (Ω) + τ f W s,p (Ω) + τ f W s,p (R N ) . (3.20)
Combining (3.19)-(3.20), we also have

f 2 W s,p (R N ) f W s,p (R N ) . (3.21)
We obtain (3.13) (and complete this case) from (3.19)-(3.21).

Case 3. s 1 = s 2 and p 2 < ∞. This is somewhat the general case. We will prove below that

F s p,q = (F s 1 p 1 ,q 1 ∩ F s p,q ) + (F s 2 p 2 ,q 2 ∩ F s p,q ), (3.22)
under the assumptions

-∞ < s 1 , s, s 2 < ∞, s 1 = s 2 , 0 < p 1 < p < p 2 < ∞ such that (1.7) holds, 0 < q 1 , q, q 2 < ∞. (3.23)
In view of Theorem 2.3 and Lemma 2.9, this is stronger than the conclusion of Theorem 1.4. We now proceed to the proof of (3.22). Throughout the calculations we perform in this case, we assume (3.23).

Define, in the spirit of (3.4),

α := s 1 p 2 - s 2 p 1 1 p 1 - 1 p 2 . ( 3.24)
Let us first note that the proportionality condition (1.7) leads to the following identities

α = s p 2 - s 2 p 1 p - 1 p 2 = s 1 p - s p 1 1 p 1 - 1 p (3.25)
and

(s 1 + α)p 1 = (s + α)p = (s 2 + α)p 2 . (3.26)
In addition, we have either s 1 + α, s + α, s 2 + α > 0, or s 1 + α, s + α, s 2 + α < 0.

(3.27) Given a sequence (x j ) of nonnegative numbers, set, for i = 1, 2, S i (x) := 2 s i jq i (x j ) q i , g i (x) := [S i (x)] p i /q i , T(x) := 2 s jq (x j ) q , h(x) := [T(x)] p/q . (3.28) Lemma 3.2. There exists some finite constant C such that

[x j ≤ 2 α j , ∀ j] =⇒ g 2 (x) ≤ Ch(x).

Lemma 3.3.

There exists some finite constant C such that

[∀ j, x j ≥ 2 α j or x j = 0] =⇒ g 1 (x) ≤ Ch(x).
Granted the two lemmas, we proceed to the proof of (3.22). Let f ∈ F s p,q and write, in the sense of

S ′ , f = j,G,m 2 -N j/2 λ j G,m ψ j G,m . Set f 1 := |λ j G,m |>2 α j 2 -N j/2 λ j G,m ψ j G,m , f 2 := |λ j G,m |≤2 α j 2 -N j/2 λ j G,m ψ j G,m . (3.29)
Clearly, f 1 , f 2 ∈ F s p,q . We next note that, for each x and j, there exists some subset M( j, x) of Z N , say M( j, x) = {m ℓ j,x } k ℓ=1 (with k := 3 N independent of j and x), such that m ∈ M( j, k) =⇒ x ∈ Q j,m . This implies that for all x ∈ R N we have j,G,m

2 σ jρ a j G,m ρ 1 Q j,m (x) ∼ j,G,ℓ 2 σ jρ a j G,m ℓ j,x ρ 1 Q j,m ℓ j,x (x), ∀ σ, ∀ ρ, ∀ a j G,m .
(3.30)

Applying Lemmas 3.2 and 3.3 with

x j := λ j G,m ℓ j,x 1 Q j,m ℓ j,x (x)
and using (3.29)-(3.30), we find that

f 1 p 1 F s 1 p 1 ,q 1 f p F s p,q , f 2 p 2 F s 2 p 2 ,q 2 f p F s p,q . (3.31)
It thus remains to prove Lemmas 3.2 and 3.3.

Proof of Lemma 3.2. Define A := (s 2 + α)q 2 , B := (s + α)q. By (3.26), we have either A, B > 0, or A, B < 0. Set a j := 2 -α j x j ∈ [0, 1]. Then S 2 (x) = S 2 (a) := 2 A j (a j ) q 2 , g 2 (x) = g 2 (a) := S 2 (a) p 2 /q 2 , T(x) = T(a) := 2 B j (a j ) q , h(x) = h(a) := T(a) p/q .

Let J be an arbitrary nonnegative integer, and set A 2 J := {a = (a j ) j≥0 ; a j ∈ [0, 1], ∀ j, and a j = 0, ∀ j > J}.

(3.32)

In order to establish the lemma, it suffices to prove that

g 2 (a) ≤ C h(a), ∀ a ∈ A 2 J , (3.33)
provided C does not depend on J.

Fix J. For a ∈ A 2 J , a ≡ 0, set f 2 (a) := g 2 (a) h(a)
. Since f 2 is homogeneous of degree p 2 -p > 0, it attains its maximum at some a such that at least one of the a j 's equals 1. For this a, set

Λ 1 := { j ≤ J; a j = 0}, Λ 2 := { j ≤ J; a j = 1}, Λ 2 := { j ≤ J; 0 < a j < 1}.
By the above, we have Λ 2 = . Set m := min Λ 2 and M := maxΛ 2 .

Step 1. Proof of the lemma when Λ 3 = . Assume first that A, B > 0. Then

S 2 (a) = j∈Λ 2 2 A j ≤ j≤M 2 A j 2 AM , T(a) = j∈Λ 2 2 B j ≥ 2 BM .
We find that

f 2 (a) (2 AM ) p 2 /q 2
(2 BM ) p/q = 1, since A p 2 q 2 = B p q (by (3.26)).

If A, B < 0, we have similarly S 2 (a) 2 Am and T(a) ≥ 2 Bm , and therefore f 2 (a) 1.

Step 2. Proof of the lemma when Λ 3 = . Set ℓ := min Λ 3 , L := maxΛ 3 .

If j ∈ Λ 3 , then ∂ ∂a j f 2 (a) = 0, and thus p 2 2 A j [ S 2 (a)] p 2 /q 2 -1 (a j ) q 2 -1 [ T(a)] p/q = p2 B j [ T(a)] p/q-1 (a j ) q-1 [ S 2 (a)] p 2 /q 2 , which implies that (a j ) q 2 -q = C 1 2 (B-A) j , ∀ j ∈ Λ 2 , with C 1 = C 1 (a) constant.

(3.34)

Step 2.1. Proof of the lemma when Λ 3 = and q 2 = q. By (3.34), the quantity 2 (B-A) j does not depend on j ∈ Λ 3 . On the other hand, since q 2 = q we have B -A = (ss 2 )q = 0. Thus Λ 3 contains only one element, Λ 3 = {ℓ} = {L}. We find that

g 2 (a) = j∈Λ 2 2 A j + 2 Aℓ (a ℓ ) q , h(a) = j∈Λ 2 2 B j + 2 Bℓ (a ℓ ) q .
As in Step 1, when A, B > 0 we find that f 2 (a) 2 AM + 2 Aℓ (a ℓ ) q p 2 /q 2 BM + 2 Bℓ (a ℓ ) q p/q 2 A p 2 /qM + 2 A p 2 /q (a ℓ ) p 2 2 Bp/qM + 2 Bp/q (a ℓ ) p ≤ 1, the latter inequality following from A p 2 q 2 = B p q , p 2 > p and 0 < a ℓ < 1.

The case where A, B < 0 is handled similarly.

Step 2.2. Proof of the lemma when Λ 3 = and q 2 = q. Define γ := B -A q 2 -q . It follows from (3.34) that

a j = C 2 2 γ j , ∀ j ∈ Λ 3 . (3.35)
Let us note that

A + γq 2 = A + B -A q 2 -q q 2 = Bq 2 -Aq q 2 -q = qq 2 s -s 2 q 2 -q = 0.
We therefore have the following four possibilities:

1. A, B > 0, A + γq 2 > 0. 2. A, B > 0, A + γq 2 < 0. 3. A, B < 0, A + γq 2 > 0. 4. A, B < 0, A + γq 2 < 0.
We complete Step 2.2 in one of these cases, and let to the reader the three other ones, which are similar. Assume e.g. that A, B > 0 and A +γq 2 < 0. In this case we obtain an information on C 2 by letting, in (3.35), j = ℓ. [If A + γq 2 > 0, we take j = L.] Since 0 < a ℓ < 1, we have 0 < C 2 2 γℓ < 1, and thus C 2 = C 3 2 -γℓ , with 0 < C 3 < 1. We find that a j = C 3 2 γ( j-ℓ) , ∀ j ∈ Λ 3 , for some C 3 ∈ (0, 1).

(3.36)

Since A > 0 and A + γq 2 < 0, we find that

S 2 (a) ≤ j≤M 2 A j + j≥ℓ 2 (A+γq 2 ) j (C 3 ) q 2 2 -γq 2 ℓ 2 AM + 2 (A+γq 2 )ℓ (C 3 ) q 2 2 -γq 2 ℓ = 2 AM + 2 Aℓ (C 3 ) q 2 , while T(a) ≥ 2 BM + 2 Bℓ (C 3 ) q .
We find that

f 2 (a) 2 AM + 2 Aℓ (C 3 ) q 2 p 2 /q 2 2 BM + 2 Bℓ (C 3 ) q p/q 2 A p 2 /q 2 M + 2 A p 2 /q 2 ℓ (C 3 ) p 2 2 Bp/qM + 2 Bp/qℓ (C 3 ) p ≤ 1, since A p 2 q 2 = B p q , 0 < C 3 < 1 and p 2 > p.
The proof of Lemma 3.2 is complete.

Sketch of proof of Lemma 3.3. This is very much similar to the proof of Lemma 3.2. This time, we have a j ∈ {0} ∪ [1, ∞). With C := (s 1 + α)q 1 , we set S 1 (a) := 2 C j (a j ) q 1 and A 1 J := {a = (a j ) j≥0 ; a j = 0 or a j ≥ 2 α j , ∀ j, and a j = 0, ∀ j > J}.

If a ∈ A 1 J , a ≡ 0, we set f 1 (a) :=

[ S 1 (a)] p 1 /q 1 [T(a)] p/q . We have to prove that f 1 (a) 1, ∀ J, ∀ a ∈ A 1 J , a ≡ 0. This is obtained following the same strategy as in the proof of Lemma 3.2, considering, for a maximum point a of f 1 , the sets Λ 1 := { j ≤ J; a j = 0}, Λ 2 := { j ≤ J; a j = 1}, Λ 3 := { j ≤ J; 1 < a j < ∞}.

The key ingredients are that C and B are either both positive or both negative, respectively the fact that, when q = q 1 , the quantity C + B -C q 1 -q q 1 does not vanish.

Details are left to the reader. Case 4. s 1 = s 2 and p 2 = ∞. This is very much similar to Case 3. We prove the equality

F s p,q = (F s 1 p 1 ,q 1 ∩ F s p,q ) + (F s 2 ∞,∞ ∩ F s p,q ) (3.37) under the assumptions -∞ < s 1 , s, s 2 < ∞, s 1 = s 2 , 0 < p 1 < p < p 2 = ∞ such that (1.7) holds, 0 < q 1 , q < ∞. (3.38)
[For an improvement of (3.37) under more restrictive conditions of p 1 , see the proof of Theorem 1.7.] In view of Theorem 2.3 and Lemma 2.9, this implies Case 4. In order to prove (3.37), we decompose f ∈ F s p,q as in (3.29). By Theorem 2.10 and Lemma 3.3, we have f 1 ∈ F s 1 p 1 ,q 1 ∩ F s p,q . On the other hand, since p 2 = ∞ we have α = -s 2 , and then clearly (3.29) implies that f 2 ∈ F s 2 ∞,∞ ∩F s,p . The proof of Theorem 1.4 is complete.

Proof of Theorem 1.7. We will prove the following version of (3.37): we have F s p,q = (F s 1 p 1 ,q 1 ∩ F s p,q ) + (F s 2 ∞,q 2 ∩ F s p,q ) (3.39) under one of the following assumptions -∞ < s 1 , s, s 2 < ∞, s 1 = s 2 , 1 < p 1 < p < p 2 = ∞ such that (1.7) holds, 0 < q < ∞, 1 < q 1 , q 2 < ∞ (3.40) or -∞ < s 1 , s, s 2 < ∞, s 1 = s 2 , 1 = p 1 < p < p 2 = ∞ such that (1.7) holds, 0 < q < ∞, q 1 = 1, 1 < q 2 < ∞.

(3.41) Granted (3.39), we obtain the conclusion of Theorem 1.7 via Theorem 2.3, Corollary 2.8 and Lemma 2.9.

We now proceed to the proof of (3.39). Let f ∈ F s p,q , and let f = f j be the Littlewood-Paley decomposition of f . Set f j := |k-j|≤1 f k = |k-j|≤1 f * ϕ k * ϕ j . Taking into account the fact that ϕ j * ϕ k = 0 if | j -k| ≥ 2 and that k ϕ k = δ in the sense of S ′ , we find that Given x ∈ R N , let h(x) := 2 s jq | f j (x)| q p/q , so that h < ∞ a.e. Whenever h(x) < ∞, define J = J(x) as follows: J is the least non negative integer such that 2 J ≥ [h(x)] δ .

Lemma 3.4. Let δ and J be as above.

1. If δ > 0, then j<J 2 s 1 jq 1 | f j (x)| q 1 p 1 /q 1 h(x) and j≥J 2 s 2 jq 2 | f j (x)| q 2 1/q 2 1.

(3.46)

2. If δ < 0, then j>J 2 s 1 jq 1 | f j (x)| q 1 p 1 /q 1 h(x) and j≤J 2 s 2 jq 2 | f j (x)| q 2 1/q 2 1.

(3.47) Granted Lemma 3.4, we complete the proof of Theorem 1.7 as follows. Assume e.g. that δ > 0, the case δ < 0 being similar. Define, for a.e. x ∈ R N , g j (x) := f j (x), if j < J(x) 0, if j ≥ J(x) , h j (x) := 0, if j < J(x) f j (x), if j ≥ J(x) .

(3.48)

Combining (3.42), (3.43), Lemma 3.4 and Lemma 2.5, we find that the series f 1 := g j * ϕ j and f 2 := h j * ϕ j converge in S ′ , that f = f 1 + f 2 , and that f 1 ∈ F s 1 p 1 ,q 1 ∩ F s p,q , f 2 ∈ F s 2 ∞,q 2 ∩ F s p,q .

Proof of Lemma 3.4. We consider only the case δ > 0, the case δ < 0 being similar. Set M := [h(x)] δ . We let to the reader the case where M < 1 and thus J = 0 and the first sum in (3.46) vanishes.

Assuming that M ≥ 1, we have 2 J ∼ M and | f j (x)| ≤ 2 -s j [h(x)] 1/p = 2 -s j M 1/(δp) , ∀ j ≥ 0.

(3.49)

Since δ > 0, we have s > s 2 , and thus s 1 > s > s 2 . Using (3.49), we find that j<J 2 s 1 jq 1 | f j (x)| q 1 M q 1 /(δp) j<J 2 (s 1 -s) jq 1 M q 1 /(δp) 2 (s 1 -s)J q 1 ∼ M q 1 [1/(δp)+(s 1 -s)] .

(3.50) Combining (3.45) and (3.50), we find that j<J 2 s 1 jq 1 | f j (x)| q 1 p 1 /q 1 M 1/δ p 1 /p+(s-1-s)δ = M 1/δ = h(x),

i.e., the first inequality in (3.46) holds.

For the second inequality, we note that (3.49) leads to M q 2 /(δp) j≥J 2 (s 2 -s) jq 2 M q 2 /(δp) 2 (s 2 -s)J q 2 ∼ M q 2 [1/(δp)+(s 2 -s)] = 1, the latter equality following from the definition of δ.

  j f j * ϕ j = j,k f * ϕ k * ϕ j = j f * ϕ j = f . (3.42)On the other hand, we clearly have2 s j f j (x) j≥0 l q (N) L p (R N )2 s j f j (x) j≥0 l q (N) L p (R N ) that (1.7) and (3.44) imply the identity p 1 p + (s 1 -s)δp 1 = 1.(3.45)

j≥J 2 s 2

 2 jq 2 | f j (x)| q 2

Theorem 2.3. The

  following equalities of spaces hold, with equivalence of norms:

	Triebel-Lizorkin spaces [20, Section 2.3.5],
	[17, Section 2.1.2].

Equivalently, we have ϕ 0 = F -1 ψ and, for j ≥ 1, ϕ j (x) =

N j ϕ 0 (2 j x) -2 N( j-1) ϕ 0 (2 j-1 x).

Acknowledgments

The author thanks N. Badr, H. Brezis and B. Matei for useful discussions. He was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Appendix. Factorization, functional calculus, sum-intersection

The lifting problem for S 1 -valued Sobolev maps is the following. Let B be a ball in R N . Let s > 0 and 1 ≤ p ≤ ∞. Is it possible to lift every map u ∈ W s,p (B; S 1 ) as u = e ıϕ with ϕ ∈ W s,p (B; R)? This question has been completely answered in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]. The answer depends on s, p and N. For example, in W 1,p (B; S 1 ) the answer is positive if

Factorization is a substitute to lifting, but is also valid and relevant if the answer to the lifting problem is positive. Special cases of factorization were announced in [START_REF] Mironescu | Decomposition of S 1 -valued maps in Sobolev spaces[END_REF]. The general case is presented in [START_REF] Brezis | Sobolev maps with values into the circle[END_REF] and asserts the following. Let s > 0 and 1 ≤ p < ∞. Then every map u ∈ W s,p (B; S 1 ) can be factorized as u = e ıϕ v, with ϕ ∈ W s,p (B; R) and v ∈ F s p 1,1 (B; S 1 ). Factorization has the following application announced in the introduction. Let p > 1 and consider some f ∈ W 1,p (B; S 1 ). Set u := e ı f ∈ W 1,p (B; S 1 ). Let 0

(3.51)

We next invoke the following delicate result [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]. If

Our Theorem 1.4 yields the same conclusion without factorization.

Let us also note that not only factorization leads to a sum-intersection property, but sumintersection is necessary for factorization to hold. Indeed, let p ≥ 2 and u ∈ W 1,p (B; S 1 ). Then we may write u = e ı f with f ∈ W 1,p (B; R) [START_REF] Bethuel | Density of smooth functions between two manifolds in Sobolev spaces[END_REF]. Assume that we want to factorize u = e ıϕ v with ϕ ∈ W λ,p/λ (B; R) and v ∈ W p,1 (B; S 1 ). The first step consists of splitting (assuming this is possible) f = ϕ + ψ, with ϕ ∈ W λ,p/λ and ψ ∈ W p,1 . However, this decomposition does not imply that v := e ıψ ∈ W p,1 (B; S 1 ). Indeed, if s > 1 and ρ > 1, then a map g ∈ W σ,ρ satisfies e ıg ∈ W σ,ρ if and only if g satisfies the extra-assumption g ∈ W 1,σρ [START_REF] Brezis | Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces[END_REF]. In our case, this implies that factorization in W λ,p/λ (B; S 1 ) requires the sum-intersection property of the triple T = (W 1,p ,W λ,p/λ ,W p,1 ). However, one cannot reduce factorization to sum-intersection, since in general W s,p (B; S 1 ) does not have the lifting property.

Sum-intersection property has the following implication related to lifting, presented in [START_REF] Brezis | Sobolev maps with values into the circle[END_REF]. If sp < 1, then maps in W s,p (B; S 1 ) can be lifted within W s,p [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]. Factorization leads to a better result. Indeed, let u ∈ W s,p (B; S 1 ) and let ϕ ∈ W s,p (B; R) be a lifting of u. Write, as in Theorem 1.7, ϕ = ϕ 1 + ϕ 2 , with ϕ 1 ∈ BMO ∩W s,p and ϕ 2 ∈ W s p,1 ∩W s,p . Set v := e ıϕ 2 ∈ W s p,1 . Then v has a lifting ϕ 3 ∈ W s p,1 ∩L ∞ [START_REF] Mironescu | Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces[END_REF]. By Gagliardo-Nirenberg, we also have ϕ 3 ∈ W s,p , and clearly ϕ 3 ∈ BMO (since ϕ 3 ∈ L ∞ ). Finally, u = e ıψ , where ψ := ϕ 1 + ϕ 3 satisfies the improved regularity ψ ∈ W s,p ∩ BMO .