Mohar Dey

Frédéric Doumenc

Béatrice Guerrier

Numerical simulation of dip-coating in the evaporative regime

Keywords: 47, 54, -r Pattern selection; pattern formation -47, 85, mb Coating flows -47, 55, np Contact lines

A hydrodynamic model is used for numerical simulations of a polymer solution in a dip-coatinglike experiment. We focus on the regime of small capillary numbers where the liquid flow is driven by evaporation, in contrast to the well-known Landau-Levich regime dominated by viscous forces. Lubrication approximation is used to describe the flow in the liquid phase. Evaporation in stagnant air is considered (diffusion-limited evaporation), which results in a coupling between liquid and gas phases. Self-patterning due to the solutal Marangoni effect is observed for some ranges of the control parameters. We first investigate the effect of evaporation rate on the deposit morphology. Then the role of the spatial variations in the evaporative flux on the wavelength and mean thickness of the dried deposit is ascertained, by comparing the 2D and 1D diffusion models for the gas phase. Finally, for the very low substrate velocities, we discuss the relative importance of diffusive and advective components of the polymer flux, and consequences on the choice of the boundary conditions.

Introduction

Self patterning in drying processes with contact lines have been the focus of numerous experimental and theoretical studies during the last few years. Indeed, besides numerous industrial applications, configurations such as droplets or dip-coating processes involve complex physics due to multi-scale phenomena (see for instance [1][2][3][4]). In dipcoating like experiments, it is now well documented that several regimes appear depending on the capillary number Ca, that compares capillary to viscous forces (Ca = ηV sub /γ, η being the dynamic viscosity, V sub the substrate velocity and γ the surface tension). We consider only small velocities for which inertial effects can be neglected. For sufficient capillary number, viscous forces are large enough to drag a film from the bath. This regime is known as the Landau-Levich regime [5,6]. The thickness of the dragged film is proportional to Ca 2/3 . More recently, studies have focused on the regime encountered for smaller capillary numbers, for which the liquid flow into the meniscus is mainly driven by the evaporation [7,8]. The present work deals with this evaporative regime, for the dip-coating like configuration depicted in figure 1. We consider a receding meniscus on a moving substrate, fed by a rectangular inlet duct taking some polymer solution from an infinite and perfectly mixed reservoir at constant polymer volume fraction φ 0 . The bidimensional domain under consideration is composed of the inlet duct (-L p < x < 0), followed by a liquid film (0 < x < W ) surrounded by a Send offprint requests to: Frédéric Doumenc gas domain (0 < x < W and 0 < z < H). In contrast to the Landau-Levich regime, the dried film thickness is a decreasing function of the substrate velocity, since it is proportional to V -1 sub . This scaling may be derived by writing the mass balance for the solution and the solute [7,[9][10][11][12], under the following assumptions: steady regime, no flow in the dry deposit, imposed concentration φ 0 and negligible diffusive flux at the entrance of the duct (located at x = -L p in figure 1). The dry deposit thickness h p reads

h p ≃ F ev φ 0 V sub , (1)
with F ev = W 0 v ev (x)dx the total evaporation flux over the liquid free surface, where the solvent evaporates with the local rate v ev (x).

Moreover, for some values of the process parameters (substrate velocity, evaporation rate, reservoir solute concentration), the system may develop regular patterns induced by a periodic movement of the contact line. Despite the large number of experimental and theoretical studies published in the last decade, we are still lacking a complete understanding of the mechanisms responsible for this periodic regime (see Frastia et al. [13] for a complete review). Of particular interest are models predicting the periodic patterning without any artificial trigger. Such models were developed to describe the evaporation of dewetting liquid layers [14,15,13], of droplets [16], or the transfer of a surfactant monolayer over a moving substrate [17]. The present study follows a previous work dedicated to a dip-coating-like configuration [18]. A hydrodynamic model taking into account varying viscosity and Marangoni effect was developed and succeeded in capturing a selfpatterning phenomenon for some values of the substrate velocity. This same model is used to investigate two new points. First we focus on the the role of the evaporation rate on deposit morphology. In the configuration studied, solvent evaporates into ambient air. It is well known that evaporation flux close to a contact line or a meniscus exhibits strong spatial variations induced by 2D diffusion of the vapor in the gas phase. We address the importance of this spatial heterogeneity on the occurrence of the periodic regime by comparing a 2D diffusion model with a simplified 1D model.

In a second part, we discuss the relative importance of advection and diffusion on polymer transport. We show the importance of the inlet boundary condition at very low substrate velocity, where some unphysical results may be obtained due to backward diffusion polymer flow.

Modeling

This study examines a drying binary solution (one volatile solvent and one non-volatile solute) in a 2D meniscus. The geometry is represented in figure 1. The solution is in contact with a moving, flat substrate with a no-slip boundary condition. Gravity is neglected, and a uniform concentration over the thickness (z direction) is assumed in the liquid phase. The free surface is in contact with air at atmospheric pressure and we assume a local thermodynamic equilibrium at the interface. The whole system is supposed to be at a constant temperature (isothermal problem). Evaporation is limited by the diffusion of solvent vapor into stagnant air. Two cases are considered for the evaporation flux: the vapor diffusion in the gas phase is described by a bidimensional or a one-dimensional model.

We focus on the long-term behavior, when a steady state or a periodic regime has been reached, the latter case corresponding to self-patterning of the deposit.

Hydrodynamics in the film: lubrication approximation

The lubrication approximation is used to describe the liquid flow in the film (0 < x < W in figure 1). Previous simulations show that this domain may be divided into three parts schematically depicted in figure 1 [18]. The first part of the meniscus corresponds to the static solution at equilibrium, with constant curvature. At the opposite, close to the end of the simulation domain, the film is gelled due to the very high viscosity and is simply translated at the substrate velocity. In between is the transition region, with varying curvature and strong concentration gradients.

The direction parallel to the substrate (flow main direction) is denoted by x; the axis normal to the substrate is denoted by z. Assuming a Newtonian fluid, Stokes equations with small slope approximation lead to the following standard result [19]:

Q(x, t) = V sub h - h 3 3η(φ) ∂P ∂x + h 2 2η(φ) dγ dφ ∂φ ∂x , (2)
where V sub is the substrate velocity, h is the liquid height, P is the liquid pressure, η is the dynamic viscosity, γ is the surface tension, and φ is the polymer volume fraction.

The three terms on the right-hand side of equation ( 2) describe the flow produced by the substrate motion, the pressure gradient and the surface tension gradient (solutal Marangoni effect), respectively. The surface tension is assumed to vary linearly with the polymer volume fraction (non linear effect that may occur in polymer solutions [20] are neglected).

The pressure drop across the free surface includes the capillary pressure and the disjoining pressure and reads

P -P atm = -γ(φ) ∂ 2 h ∂x 2 -A/(6πh 3 ), (3)
with A the Hamaker constant. The expression of the disjoining pressure accounts for long range van der Waals interactions and corresponds to a complete wetting configuration [21,22]. The disjoining pressure impacts significantly on the results for very thin films only. This is not the case for most of the configurations considered in the following, except for some specific cases considered in section 4.2. Q(x, t) is the liquid volumic flux (by unit of width) at abscissa x and time t over a cross-section normal to x,

Q(x, t) = h(x) 0 u(x, z) dz, (4)
where u(x, z) is the solution velocity component in the x direction.

The flux of non-volatile solute Q p reads

Q p = φQ -Dh ∂φ ∂x . (5)
The second term on the right-hand side of equation ( 5) is the Fick's law integrated over the film thickness, which accounts for polymer/solvent relative motion. D is the mutual diffusion coefficient of the binary system. For polymer solutions, D keeps the same order of magnitude for polymer volume fraction less than about 0.7, and then decreases strongly (several orders of magnitude) with increasing polymer concentration (cf. for instance [23]). In the present model D is assumed constant. Indeed previous simulations have shown that the fall of D for φ > 0.7 did not change the results significantly. A more sophisticated approach would consist to compute D from the expression of the osmotic pressure, as done by U.Thiele and co-authors [24].

The global mass balance and the non-volatile solute mass balance respectively read

∂h ∂t + ∂Q ∂x = -v ev (x, t), (6)
∂(φh) ∂t + ∂Q p ∂x = 0, (7)
where the local evaporation velocity v ev (x, t) is positive for evaporation.

Hydrodynamic boundary conditions

At x = 0 we impose the liquid height h 0 and the curvature C 0 . The height h 0 is supposed to be small enough to ensure the validity of a small slope approximation. Imposing h 0 and C 0 at x = 0 creates a meniscus section of static length L st = 2h 0 /C 0 . (L st corresponds to the intersection of a parabolic meniscus with the horizontal axis with zero contact angle). Variations of curvature and height at x = 0, produced by the evaporation or the substrate motion, are supposed to be negligible. The hydrodynamic boundary conditions at x = 0 are then

h(x = 0) = h 0 , ∂ 2 h ∂x 2 (x = 0) = C 0 . (8)
Two different solutal boundary conditions are compared. The first one is similar to the one used in our previous study [18], i.e. a constant value of the polymer volume fraction is imposed at x = 0,

CL inlet1 : φ(x = 0) = φ 0 . (9)
However, in experiments such as the ones described in [9,10], the polymer volume fraction is not imposed at the meniscus, but at some distance upstream. Thus a more realistic boundary condition has also been considered. A long inlet duct of length L p and constant height h 0 is introduced upstream of the meniscus, with L p >> W . The reservoir concentration is now imposed at the entrance of the inlet duct,

CL inlet2 : φ(x = -L p ) = φ 0 . ( 10
)
There is no free surface and thus no evaporation in this inlet duct. The solute distribution is thus described by an advection-diffusion equation:

for -L p < x < 0 : ∂φ ∂t + Q 0 h 0 ∂φ ∂x = ∂ ∂x (D ∂φ ∂x
), (11) with

Q 0 = Q(x = 0).
Close to x = W , the deposit is assumed frozen. We thus impose the condition

for x ≥ W : ∂h ∂t + V sub ∂h ∂x = 0. ( 12
)
2.3 Gas phase and evaporation at the free surface 2.3.1 M G2D : 2D diffusion in the gas phase

The local evaporation velocity v ev (x, t) in equation ( 6) is calculated by solving a diffusion problem in the gas phase for a domain of length W and height H (H would roughly correspond to the boundary layer thickness for a process with a forced air-flow above the free surface). Because H is considerably larger than h 0 , we assume a rectangular domain for the gas phase. The liquid thickness variations are therefore neglected in the computation of the mass transfer in the gas phase. The bottom boundary of this domain corresponding to the liquid-gas interface is thus: z = 0 and 0 < x < W . Two models are compared in the following. In the first one, denoted M G2D , a 2D diffusion equation is solved in the gas phase. The solvent vapor is assumed to diffuse into stagnant air and the diffusion equation in the gas phase reads

∂c g ∂t = D g ( ∂ 2 c g ∂x 2 + ∂ 2 c g ∂z 2 ), for 0 < x < W and 0 < z < H , (13)
where c g is the solvent vapor concentration in the air, and D g is the vapor diffusion coefficient. The vertical walls are assumed to be impermeable. A Dirichlet condition with a zero solvent vapor concentration is imposed at the top. We thus obtain the following boundary conditions for the gas domain (see figure 1), ∂c g ∂x = 0 for (x = 0 or x = W ) and 0 < z < H, (14)

c g = c ∞ for 0 ≤ x ≤ W and z = H. (15)
Another boundary condition corresponding to the coupling between the liquid and the gas phases is obtained from the local thermodynamic equilibrium at the free surface.

Assuming that the vapor behaves as an ideal gas, the vapor chemical potential reads

µ v = µ v0 + RT ln P vs P vs0 , ( 16
)
where R is the ideal gas constant, T is the temperature, P vs is the saturating vapor pressure over the solution at pressure P , P vs0 is the saturating vapor pressure over the pure solvent at pressure P atm , and µ v0 is the vapor chemical potential when P vs = P vs0 . The solvent chemical potential in the incompressible liquid phase reads

µ S = µ s0 + vs [(P -P atm ) -Π], ( 17
)
where Π is the osmotic pressure, vs is the solvent molar volume, and µ s0 is the chemical potential of the pure solvent when P = P atm . Using equations (16)(17) along with the equation ( 3), the ideal gas law and the standard relation vs Π = -RT ln(a) where a is the solvent activity, the thermodynamic equilibrium at the interface reads

c g = a(φ, T )P vs0 (T ) M s RT exp - vs RT (γ ∂ 2 h ∂x 2 + A 6πh 3 ) , (18)
for 0 ≤ x ≤ W and z = 0, where M s is the solvent molar mass. The solvent activity is given by the classical Flory-Huggins theory [START_REF] Flory | Principles of polymer chemistry[END_REF]. For a polymer of high molar mass, a = (1 -φ) exp (φ + χφ 2 ) where χ is the polymer-solvent interaction parameter. Regarding the evaporation flux, it is worthwhile to note the difference between a polymer solution and a pure fluid, as considered for instance by Eggers and Pismen [START_REF] Eggers | [END_REF]. Indeed, unlike for pure fluid, the variation of solvent activity with concentration allows to regularize the problem for most of the configurations studied, without the need of disjoining pressure. It is only for some configurations where a very thin liquid film (∼ 1nm) is obtained that disjoining pressure must be considered (see section 4.2).

The last equation represents the mass flux conservation across the interface:

ρv ev = -D g ∂c g ∂z , ( 19
)
where ρ is the pure solvent density.

M G1D : 1D diffusion in the gas phase

In the second model, denoted M G1D in the following, we get the evaporation flux from the phenomenological expression

ρv ev = h m (c g (z = 0) -c ∞ ), (20)
where h m is a constant mass transfer coefficient. The relation (20) corresponds to 1D and steady transfer in the gas phase (∂ 2 c g /∂z 2 = 0). With these assumptions, h m is related to the boundary layer thickness δ g by the relation h m = D g /δ g . Notice that another interpretation of the relation (20) would be possible in the different configuration of an atmosphere of pure solvent (no inert gas). In that case, evaporation would be limited by kinetic effects, and the evaporation rate would be given by the relation (20), with h m computed from the Hertz-Knudsen theory [27].

Resolution

A part of the results have been obtained by solving the above equations with a homemade Fortran software (finite differences with pure implicit scheme of order 1 in time and 2 in space [28]), and the other part with Comsol Multiphysics, a commercial finite element software (BDF solver, quadratic Lagrange elements). Both softwares give the same results, with a lower computational time in the latter case. The first run is performed with an arbitrary initial condition. Then the evaporation flux is changed stepwise and the solution obtained for the previous step is used as initial condition. The effect of the initial conditions on the long time behavior has been analyzed in a previous study and was shown to have a negligible impact on the results. (cf. [18] for more details).

Numerical solutions provide the liquid film shape, the local evaporation flux, the concentration field in the gas (for M G2D ), and the concentration and velocity fields in the solution. The following geometrical values are used: Physico-chemical properties are those of Polyisobutylene (PIB)/toluene [START_REF] Mark | Polymer data Handbook[END_REF][START_REF] Marcus | The properties of solvents[END_REF][START_REF] Doumenc | [END_REF]. The toluene molar mass and diffusion coefficient in the gas phase are M s = 92.14g.mol -1 , and D g = 8.6 × 10 -6 m 2 .s -1 , respectively. The solution density and the diffusion coefficient are assumed to be constant: ρ = 900kg.m -3 and D = 10 -10 m 2 .s -1 . The surface tension γ is assumed to increase linearly with the solute volume fraction from γ = 28 × 10 -3 N.m -1 (pure toluene) to γ = 34 × 10 -3 N.m -1 (pure PIB). The viscosity η exhibits a strong increase with the solute volume fraction (12 orders of magnitude from pure toluene to pure PIB for a PIB molar mass of M W = 500kg.mol Although the results are presented in dimensional variables, we present here the main dimensionless parameters involved in this problem. We use h 0 and h 0 /C 0 for the vertical and horizontal scales, respectively, V sub for the velocity scale, η 0 = η(φ 0 ) and γ 0 = γ(φ 0 ) for the viscosity and surface tension scales. From the lubrication equation (2) we get the modified Capillary number δ and Marangoni number M a:

H = 3mm, W = 1mm, C 0 = 2mm -1 , h 0 = 0.
δ = 3η 0 V sub γ 0 (h 0 C 0 ) 3/2 , M a = γ solute -γ solvent 2η 0 V sub (h 0 C 0 ) 1/2 .
The Peclet number is obtained from the equation (5),

P e = V sub D (h 0 /C 0 ) 1/2 .
For most of the simulations, we have 6×10 -6 δ 4×10 -4 , 4×10 3 M a 3×10 5 , 10 P e 70.

An exception is figure 11, where the substrate velocity is varied over a large number of decades.

x / mm 4 Results and discussion

Evaporation flux

In this section we explore the role of the evaporation rate on the deposit morphology. Results are presented as a function of the global evaporation flux F ev = W 0 v ev (x)dx. Note that F ev is not a direct input of the model. In the 2D model (M G2D ), it is changed by modifying the saturated vapor pressure P vs0 (T ) in equation (18). To focus on evaporation rate only, all other properties are still taken at T = 25 o C. For M G1D , the evaporation rate is changed by modifying the mass transfer coefficient h m (cf. equation ( 20)).

It is well known that the evaporation flux close to a contact line or a meniscus exhibits strong spatial variations induced by 2D diffusion effect in the gas phase. Using electrostatic analogy, Deegan and co-authors derived a simplified expression of the evaporative flux that diverges at the contact line [32]. The model used here for an evaporating solution suppresses the divergence, but a significant peak in the profile of the evaporation rate still remains, as illustrated in figure 2 (continuous line).

First we focus on evaporation flux intensity, using M G2D and varying F ev . Then we analyze the impact of flux spatial inhomogeneity by comparing 1D and 2D models in the gas (M G1D and M G2D ). Simulations are done with the boundary condition CL inlet1 and disregarding disjoining pressure in the lubrication equation. These two assumptions are valid as the substrate velocity is high (V sub = 30µm/s, see next section) and the obtained film thickness is large enough to disregard disjoining pressure.

M G2D -Effect of evaporation flux intensity

The substrate velocity is fixed at V sub = 30µm/s, and the control parameters are the evaporation flux F ev and the bulk solute concentration φ 0 . Different behaviors may be observed, depending on these control parameters. Typical liquid film profiles are illustrated in semi-log scale in figure 3 for φ 0 = 0.01 and three values of F ev . All the profiles first overlap with the constant curvature meniscus and then enter the transition region, earlier for the highest value of the evaporation flux. For F ev = 1720µm 2 /s we obtain a steady solution, with a monotonic profile. The intermediate value of F ev also leads to a steady solution, but with a bump in the transition region. At last, for F ev = 400µm 2 /s no stationary solution is found. After formation, the bump moves away from the meniscus, and dries out with a strong increase of its viscosity, which prevents the possibility of a flow in the substrate framework. The bump is then advected to the outlet by the substrate motion. A new bump appears and grows in the transition region, and follows the same way (notice that the red curve in figure 3 is thus a "snapshot" of the liquid film at a given time). The formation of a bump by a solutal Marangoni flow has already been observed and discussed in the literature (see for instance [33] and [34] and the discussion in [18]). The periodic regime and the resulting deposit self-patterning are thus a consequence of the solutal Marangoni effect. Indeed, in the present model, a constant surface tension always results in a monotonic meniscus profile (no bump), and a steady state at long time (flat deposit).

The morphological phase diagram as a function of control parameters is shown in figure 4. It exhibits the different behaviors of the meniscus described above: steady solutions with or without formation of a bump or periodic behavior. The periodic sub-regime is only obtained for low bulk concentrations (φ 0 = 0.01 and 0.02) and for a small interval of evaporation flux (red circles in figure 4). Qualitatively, the morphological phase diagram and meniscus typical shapes are similar to those obtained when varying the substrate velocity [18]. The effect of F ev and V sub are thus qualitatively comparable.

As already said, after the transition region the system is frozen due to the very high viscosity of the solution. The pattern is thus simply translated at the substrate velocity V sub . It is thus possible to estimate the wavelength and amplitude of the deposit. Typical morphologies of the deposits are given in figure 5 for φ 0 = 0.01. For ease of comparison, two wavelengths are drawn for each value of F ev , and the local amplitude is scaled by the mean dry thickness. At low evaporation flux the periodic structure shows a main peak as well as a small secondary peak. This structure evolves towards a reverse morphology, with a single valley, as the evaporation flux decreases. The mean thickness of the dry film is proportional to F ev , which is the scaling expected from the simplified model (1).

The evolution of the wavelength as a function of the evaporation flux is shown in figure 6 (red circles for model M G2D ). At high evaporation flux we observe a strong increase of the wavelength, which corresponds to the transition from the periodic regime to the steady one when F ev exceeds a given value. Qualitatively we observe the same transition when the control parameter is the substrate velocity, the divergence of the wavelength being observed this time when V sub is smaller than a given value. This strong increase in wavelength indicates a global bifurcation as also hypothesized in [13] and shown for another line-deposition mechanism in [35].

Comparison of M G1D and M G2D

To infer the role of spatial variation of the evaporation rate induced by 2D diffusion in the gas phase, comparison is made between 2D and 1D models. For the same global evaporation flux F ev , figure 2 illustrates typical profiles of the evaporation rate for models M G2D and M G1D . Unlike the peak induced by 2D effects, M G1D leads to a constant evaporation rate. Indeed, for polymer solutions, the solvent activity is close to one for polymer volume fraction smaller than about 0.6. It is only when the solute concentration increases sufficiently at the end of the meniscus (for x > 0.42) that the evaporation rate falls down, due to the solvent activity decrease. All other things being equal, a periodic sub-regime is still observed for the M G1D model, but for higher values of the global evaporation flux F ev , as illustrated in figure 6. Wavelengths are of the same order than those obtained with M G2D . Deposit morphologies are also rather similar, although topographies are a little smoother with M G1D (figure 5). As a conclusion, the peak of the evaporation rate characteristic of 2D diffusion in the gas phase is not necessarily required to get the instability (this is in agreement with several studies in the literature, see for instance [13]). The periodic regime could also be observed for 1D diffusion but in a different range of evaporation flux.

Polymer transport: diffusion against advection

We see from equation ( 5) that the polymer flux Q p is the sum of an advection term Q p-ad = φQ and a diffu- sion term Q p-dif = -Dh∂φ/∂x. In this section, we investigate the relative importance of these two terms, and the consequences of diffusion on the choice of the solutal boundary condition at the system entrance. Simulations are performed with M G2D (T = 25 o C) and φ 0 = 0.01. Results are presented for V sub = 5µm/s and 10µm/s, and boundary conditions CL inlet1 (no inlet duct) and CL inlet2 (inlet duct of length L p = 1cm). For these values of the control parameters, we get a steady state at long time.

From equation ( 7) it follows that Q p is constant along the whole domain.

Figures 7 and8 show the polymer volume fraction φ(x) and the liquid height h(x), respectively. Typically, φ(x) increases sharply from the bulk value φ 0 to high concentration. For the higher velocity V sub = 10µm/s, the front occurs far from the entrance for boundary conditions CL inlet1 or CL inlet2 . Diffusion can thus be neglected over advection at x = 0 (figure 9), and both boundary conditions give very close results (see figures 7 and8).

Things are very different for the lower velocity V sub = 5µm/s. Using CL inlet2 , it can be observed that the concentration front enters into the inlet duct (figure 7) and thus diffusion flux is no more negligible at x = 0 (figure 10). However, the inlet duct is long enough to recover a negligible diffusion flux at the domain entrance located at x = -L p . But if φ = φ 0 is imposed at x = 0, the simulation converges towards a strongly different solution. Indeed, Q p-ad and Q p-dif compensate each other, and Q p is almost zero (figure 10). This very small polymer flux results in a very thin deposit, of nanometric thickness determined by the disjoining pressure term in equation ( 18) (figure 8).

A consequence of the importance of diffusion at the domain entrance is shown in figure 11, where the dried deposit thickness is plotted as a function of the substrate velocity, along with the prediction of the simplified model (1). When the boundary condition CL inlet2 is used, the deposit thickness follows the law (1) in the whole range of velocity. Conversely, the boundary condition CL inlet1 leads to a kind of dewetting transition for velocities V sub ≤ 5µm/s, where diffusion almost balances advection. This results in a strong departure from the model (1), con- sistent with the fact that this model has been derived assuming negligible diffusion. Such a dewetting has not been observed in experiments described in [9,10]. Therefore, it may be necessary to consider an inlet duct to reproduce the experimental conditions with the model.

Conclusion

The effect of the evaporation flux on the deposit characteristics have been investigated. We show that the evaporation intensity can be used to control the deposit characteristics, i.e. the mean thickness, the wavelength and the shape. Modifying the spatial variations of the local evaporation velocity v ev (x) shifts the self-patterning domain to other values of the total flux F ev , but results are qualitatively similar.

In the second part, we show after discussing the relative importance of diffusion and advection that the length of the zone preceding the meniscus, introduced in CL inlet2 is of major importance at very low substrate velocity. We have compared two extreme cases. In the first one, the inlet duct is long enough to ensure quasi null polymer diffusion at the entrance of the duct (x = -L p ). In the second one, we impose the polymer volume fraction at x = 0 (i.e. CL inlet1 ), and a dewetting transition is obtained at low velocity. Some intermediate cases should be investigated. Beyond the geometry studied, these results show that regime of very low contact line velocities may be strongly impacted by diffusion. We are currently performing experiments on polymer solutions in a dip-coating like configuration to test the approach presented in this paper.

Fig. 1 .

 1 Fig. 1. Geometry of the dip-coating like configuration.

  1mm, and L p = 10mm for the boundary condition CL inlet2 . At z = H the Dirichlet condition for concentration is c ∞ = 0.

  -1 ). The following empirical law gives η in Pa.s at T = 25 o C : Y = 8.22 + 13.2X + 5.2X 2 + 0.70X 3 , where Y = log 10 (η) and X = log 10 (φ). The saturated vapor pressure is P vs0 = 3792Pa at T = 25 o C. The interaction Flory parameter χ has been measured by gravimetric experiments, χ = 0.45 + 0.30φ at T = 25 o C. The Hamaker constant A is 10 -19 J. Temperature is 25 o C unless specified.

12 M 1 Fig. 2 .

 1212 Fig. 2. Local evaporation rate for MG1D and MG2D models and two close values of Fev (φ0 = 0.01 and V sub = 30µm/s).

Mohar 2 F 1 Fig. 3 .Fig. 4 .

 2134 Fig.3. Liquid height for different evaporation flux (φ0 = 0.01 and V sub = 30µm/s). From top to bottom: Fev = 1720µm 2 /s (steady sub-regime, monotonic profile); Fev = 980µm 2 /s (steady sub-regime, stationary bump); Fev = 400µm 2 /s (periodic sub-regime, moving bump)

FevFevFev = 2090µm 2 Fev = 2160µm 2 Fig. 5 .

 225 Fig. 5. Deposit height for the periodic sub-regime (φ0 = 0.01 and V sub = 30µm/s), for MG2D model (top) and MG1D (bottom). Abscissa and ordinate are scaled by the wavelength λ and mean thickness hpm, respectively.

F ev / µm 2 .Fig. 6 .

 26 Fig. 6. Pattern wavelength in the periodic sub-regime for MG1D and MG2D model (φ0 = 0.01 and V sub = 30µm/s).

1Fig. 7 .

 7 Fig. 7. Concentration variation in the meniscus. Solid lines correspond to the boundary condition CL inlet2 and dashed lines to the boundary condition CL inlet1 (φ0 = 0.01 and MG2D with T = 25 o C).

Vsub = 10 Fig. 8 .Vsub = 10 Fig. 9 .F lux /µm 2 .Vsub = 5 Fig. 10 .Fig. 11 .

 108109251011 Fig. 8. Meniscus profile (semi-log scale). Solid lines correspond to the boundary condition CL inlet2 and dashed lines to the boundary condition CL inlet1 (φ0 = 0.01 and MG2D with T = 25 o C) .

Acknowledgments: The authors thank ANR EVAPEC 13-BS09-0010-01 for funding.


Andreas Hartmann
email: hartmann@math.u-bordeaux.fr

Xavier Massaneda
email: xavier@mat.ub.es

Artur Nicolau

Pascal Thomas
email: pthomas@cict.fr

Departament De

INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS

Keywords: Mathematics Subject Classification. 30E05, 32A35 free interpolation, Smirnov and Nevanlinna class, harmonic majorants, Poisson balayage

We consider a free interpolation problem in Nevanlinna and Smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions. We also consider the related problem of characterizing positive functions in the disk having a harmonic majorant. An answer is given in terms of a dual relation which involves positive measures in the disk with bounded Poisson balayage. We deduce necessary and sufficient geometric conditions, both expressed in terms of certain maximal functions.

INTRODUCTION AND STATEMENT OF RESULTS

1.1. Interpolating sequences for the Nevanlinna Class. Let be a discrete sequence of points in the unit disk ¡ . For a space of holomorphic functions ¢ , the interpolation problem consists in describing the trace of ¢ on , i.e. the set of restrictions ¢ ¤£ , regarded as a sequence space. One approach is to fix a target space ¥ and look for conditions so that ¢ ¤£ §¦ ¥ . An alterna- tive approach, known as free interpolation, is to require that ¢ ¤£ be ideal, i.e. stable under multiplication by © . See [Nik02, Section C.3.1 (Volume 2)], in particular, Theorem C.3.1.4, for functional analytic motivations. This approach is natural for those spaces that are stable under multiplication by © , the space of bounded holomorphic functions on ¡ . For Hardy and Bergman spaces both definitions turn out to be equivalent, with the usual choice of ¥ as an space with an appropriate weight (see [ShHSh], [START_REF] Seip | Beurling type density theorems in the unit disk[END_REF]).

The situation changes for the non-Banach classes we have in mind, namely the Nevanlinna class ¦ ! #" %$ '& ¡ )( 10 $ 32 54 6 87 @9 BA C ED GF IH 'P Q $ 3" SR T £ U& WV YX acb ( d£ Ye gf ih qp sr and the related Smirnov class T ¦ t 0 $ 32 34 6 87 @9 A C ED F H 'P Q $ 3" SR T £ U& WV YX acb ( u£ Ee gf ¦ A C ED F H 'P Q $ 3" SR T £ U& vX acb ( d£ Ee gf wr yx

We briefly discuss the known results. Naftalevič [START_REF] Naftalevič | On interpolation by functions of bounded characteristic (Russian), Vilniaus Valst[END_REF] described the sequences for which the trace £ coincides with the sequence space ¥ Na ¦ § & W ( 0 y & A @ £ U£ ( $ 5" SR T £ £ h p (we state the precise result after Proposition 1.12). The choice of ¥ Na is motivated by the fact that y & A £ d £ d( $ 3" SR T £ U& ( d£ eh fp for , and this growth is attained. Unfortunately, the growth condition imposed in ¥ Na forces the sequences to be confined in a finite union of Stolz angles. Consequently a big class of Carleson sequences (i.e. sequences such that B© g£ f¦ ä© ), namely those containing a subsequence tending tangentially to the boundary, cannot be interpolating in the sense of Naftalevič. This does not seem natural, for h© is in the multiplier space of . In a sense, the target space ¥ Na is "too big". Further comments on Naftalevič's result can be found in [START_REF] Hartmann | Interpolating sequences for holomorphic functions of restricted growth[END_REF] and below, after Proposition 1.12.

For the Smirnov class, Yanagihara [START_REF] Yanagihara | Interpolation theorems for the class £ ¥¤[END_REF] proved that in order that T £ contains the space ¥ Ya ¦ & W g ( 0 ji & A k £ d l£ d( $ 3" SR T £ £ jh mp , it is sufficient that is a Carleson sequence. However there are Carleson sequences such that T £ does not embed into ¥ Ya [Ya74, Theorem 3] : the target space ¥ Ya is "too small".

We now turn to the definition of free interpolation.

Definition.

A sequence space ¥ is called ideal if d© j¥ )n ¥ , i.e. whenever & W o ( o p ¥ and & rq so ( o t ä© , then also & rq so S So ( o k ¥ . Definition. Let ¢ be a space of holomorphic functions in ¡ . A sequence n u¡ is called free interpolating for ¢ if ¢ v£ is ideal. We denote f tw yx z ¢ . Remark 1.1. For any function algebra ¢ containing the constants, ¢ v£ is ideal if and only if

¨© n v¢ ¤£ x

The inclusion is obviously necessary. In order to see that it is sufficient notice that, by assump- Our characterization of interpolating sequences for the Nevanlinna class is as follows. Note that the existence of a harmonic majorant occurs at two junctures: first, to decide which sequences of points are free interpolating, second, to identify the trace space that arises for those sequences which are indeed free interpolating.

Theorem 1.2. Let be a sequence in ¡ . The following statements are equivalent:

(a) is a free interpolating sequence for the Nevanlinna class . (b) The trace space is given by: £ ¦ ¥ FE v0 ¦ & W ( 0 HG I& ! 0) 21 T & ¡ ( such that & & ( P' $ 3" SR T £ £ , t gx (c) ! ¤ admits a harmonic majorant. (d) There exists Q SR % such that for any sequence of nonnegative numbers UT ,

V W¢ ¤ T ! ¤ & ( ¦ V £¢ ¤ T $ 3" SR £ ¦ e& ( d£ 9 PX Q s y Y ¢ ¥ba V £¢ ¥¤ T 4 e& 7 e( x
We recall that any positive harmonic function on the unit disk is the Poisson integral of a positive measure on the unit circle, Let e ¡ denotes the normalized Lebesgue measure in ¢ ¡ . Also, for a nonnegative function ! on the unit disk, let £ ! denote the associated non-tangential maximal function (see (1.1) below).

Theorem 1.3. Let be a sequence in ¡ . The following statements are equivalent:

(a) is a free interpolating sequence for the Smirnov class T .

(b)

The trace space is given by The classical Carleson condition characterizing interpolating sequences for bounded analytic functions in the unit disk is w a ! ¤ h p , hence statements (c) in both results above can be viewed as Carleson-type conditions.

T £ ¦ ¥ E ¥¤ 0 ¦ u & W g ( 0 HG I& t! 0) 21
In view of Theorems 1.2 and 1.3, it seems natural to ask whether the measure e such that ! ¤ X 4 dc De Cg can be obtained from in a canonical way. We do not have an answer to this question, but with Propositions 1.12 and 1.13 it is easy to construct examples that discard natural candidates, such as the (weighted) sum of Dirac masses Problem. Which functions ! 0 %¡ 8 @9 T admit a (quasi-bounded) harmonic majorant?

Answers to this problem lead to rather precise theorems about the permissible decrease of the modulus of bounded holomorphic functions, e.g. Corollary 1.5 below. See [Hay], [START_REF] Lyubarskii | A uniqueness theorem for bounded analytic functions[END_REF]; [EiEs] also provides a survey of such results. The existence of harmonic majorants is relevant as well to the study of zero-sequences for Bergman and related spaces of holomorphic functions [START_REF] Luecking | Zero sequences for Bergman spaces[END_REF].

An answer to the problem of positive harmonic majorants can be given in dual terms (see [BNT] for another characterization). This can be applied to study the decrease of a non-zero bounded analytic function in the disk along a given non-Blaschke sequence. , if and only if is the union of a Blaschke sequence and a sequence £ for which there exists a universal

constant Q ¦ Q & £ ( such that V ¤ ¢ ¦¥ T ¤ $ 3" SR §¡ 9 ¤ X Q s w Y ¢ ¥` a V ¤ ¢ ¦¥ T ¤ 4 ¤ & 7 ( for any sequence of nonnegative numbers & T ¤ ( ¤ ¢ ¦¥ .
In a similar way, Theorem 1.3 (d), (e) are obtained as an application of the following analogue of Theorem 1.4 for quasi-bounded harmonic functions (i.e. for the Smirnov class).

Theorem 1.6. Let ! be a nonnegative Borel function on the unit disk ¡ . The following statements are equivalent: For the problem of harmonic majorants it is desirable to obtain criteria which, although only necessary or sufficient, are more geometric and easier to check than the duality conditions of Theorems 1.4 and 1.6.

Recall that the Stolz angle with vertex 7 )¢ ¡ and aperture is defined by (a) If ! admits a harmonic majorant, then £ ! £ 9

£ ¢¡ & 7 e( 0 ¦ u ¡ u0 £ 7 £ X & A £ £ H ( gx
¤ & ©¢ ¡ )( . (b) If ! admits a positive quasi-bounded harmonic majorant, then £ ! £ 9

¤ §¥ Q & ©¢ ¡ )( . (c) If £ ! £ 9 & ©¢ ¡ ( , then the function ! admits 4 dc £ ! g 0 ¦ 4 dc £ ! e ¡ g as a quasi-bounded harmonic majorant.

As far as necessary conditions are concerned, there is a way to improve the previous result by using the Hardy-Littlewood maximal function. Given ' % , this is defined as

©& ( )0 ¦ w A & ( F 5
where the supremum is taken over all arcs containing .

For ! ' % define ! & 7 e( 0 ¦ w ¢ a ! & ( ¨ & 7 e( ¦ w ¢ a ! & ( w Y ¢ & "! # ( & ( 5
where %$ is the characteristic function of a set & and is the "Privalov shadow" interval

(1.2) 0 ¦ 7 ¢ ¡ u0 g £ & 7 ( gx Proposition 1.8. (a) If ! admits a harmonic majorant, then ! £ 9 ¤ & ¢ ¡ )( . (b) If ! admits a quasi-bounded harmonic majorant, then ! £ 9 ¤ §¥ Q & ¢ ¡ ( .
We will give some examples in Proposition 7.4 that show that this is indeed stronger than the necessary condition given in the first part of Proposition 1.7, but still falls short of giving a sufficient condition for the existence of a harmonic majorant. 1.3. Geometric criteria for interpolation. We would like to obtain some geometric implications of the analytic conditions given in Theorems 1.2 and 1.3. To begin with, we would like to state the maybe surprising result that separated Blaschke sequences (with respect to the hyperbolic distance) are interpolating for the Smirnov class (and hence the Nevanlinna class). Recall that a sequence is called

separated if 1 & W ( 0 ¦ 2 3x ¡ £¢ ¤ ¦¥ ¦ § & 5 ©¨( PR % , where § & 65 1( 0 ¦ £ © & j( u£ ¦ A 5
is the pseudo-hyperbolic distance.

For such sequences, the values $ 3" %R £ ¦ e& ( d£ 9 can always be majorized by the values at of the Poisson integral of an integrable function (see Proposition 4.1), thus the following corollary is immediate from Theorem 1.3.

Corollary 1.9. Let be a separated Blaschke sequence. Then f tw yx z T (hence q tw yx z

).

More precise conditions can be deduced from Propositions 1.7, 1.8 and (c) in Theorems 1.2 and 1.3. Corollary 1.10. Let be a sequence in ¡ .

(a) If f tw yx z (1.5) then q tw yx z T (and so f Gw yx gz as well).

then ! ¤ £ 9 ¤ & ©¢ ¡ j( . If f tw yx z T then ! ¤ £ 9 ¤ ¦¥ Q & ¢ ¡ ( . (b) If £ ! ¤ t
Condition (1.3) already appeared in [Ya74, Theorem 1] as a necessary condition for the sequence space ¥ Ya (as defined in the beginning of Section 1.1) to be included in the trace of T .

Condition (1.4) is discussed in Proposition 1.12 and the corollary thereafter.

In some situations the conditions above are indeed a characterization of interpolating sequences. For instance, the weak £ 9 -condition characterizes interpolating sequences lying on a radius, while for sequences approaching the unit circle very tangentially the characterization is given by the strong £ 9 -condition. This is collected in the next results. Proposition 1.12. Assume that n v¡ lies in a finite union of Stolz angles.

( has bounded balayage implies in particular that approaches the circle tangentially. In Section 8, we will see more concrete conditions of geometric separation which are sufficient to imply that e ¤ has bounded balayage (Proposition 8.2).

When e ¤

has bounded balayage, the trace space will embed into Yanagihara's target space. More precisely, the following result holds.

Proposition 1.14. The following are equivalent:

(a) £ n q¥ Ya , (b) T £ n q¥ Ya , (c) e ¤ has bounded balayage, i.e. w Y ¢ £ba i & A £ d l£ d( 4
& 7 e( h qp . Yanagihara considered the sequences such that T £ ¥ Ya . These are automatically in w yx z T , since for any Blachke sequence ¥ Ya ¨© . Conversely, Lemma 8.1 (see Section 8) implies that ¥ Ya n ¥ E ¥¤ , thus if w yx z T , then by Theorem 1.3(b) T £ ¡ ¥ Ya . Therefore Theorem 1.3 characterizes in particular the sequences studied by Yanagihara.

Altogether, free interpolation for the Nevanlinna and Smirnov classes can be described in terms of the intermediate target spaces ¥ E and ¥ E ¥¤ . Notice first that always T £ n ¥ E ¤ and £ n ¥ FE (this is proved at the beginning of Section 5). So, q w yx z T if and only if T £ ¥ E ¤ , and f tw yx z if and only if £ ¥ E . Observe also that ¥ Ya n q¥ E ¤ n q¥ FE vn q¥ Na .

The paper is organized as follows. The next section is devoted to collecting some basic results on functions in the Nevanlinna class. In Section 3 we prove the sufficiency for interpolation of the conditions (c) of Theorems 1.3 and 1.2. We essentially use a result by Garnett allowing interpolation by © functions on sequences which are denser than Carleson sequences, under some decrease assumptions on the interpolated values. In Section 4 we study the necessity of these conditions. We first observe that in the product ¦ & ( appearing in Theorem 1.2, only the fac- tors © & ¨( with ¨close to are relevant. Then we split the sequence into four pieces, thereby reducing the interpolation problem, in a way, to that on separated sequences. The trace space characterization will be discussed in Section 5. In Section 6 we consider measures with bounded balayage, show that they operate against positive harmonic functions and prove Theorems 1.4 and 1.6. In Section 7, we prove Proposition 1.8, and provide examples to show that the sufficient condition is not necessary, and the necessary condition not sufficient. Section 8 is devoted to the proofs of Corollary 1.11, Propositions 1.12, 1.14, and 1.13, as well as the deduction of Naftalevič's result from Theorem 1.2. Also, we give examples of measures with bounded balayage. In the final section, we exploit the reasoning of Section 3 to construct non-Carleson interpolating sequences for "big" Hardy-Orlicz classes.

Acknowledgements. The authors wish to express special thanks to Jean-Baptiste Hiriart-Urruty for introducing them to Farkas' Lemma, to Stephen Gardiner for pointing out an efficient characterization of quasi-bounded harmonic functions, and to Alexander Borichev for Lemma 6.6, and his discussions with us about harmonic majorants.

PRELIMINARIES

We next recall some standard facts about the structure of the Nevanlinna and Smirnov classes (general references are e.g. [START_REF] Garnett | Bounded analytic functions[END_REF], [START_REF] Nikolski | Operators, functions, and systems: an easy reading[END_REF] or [RosRov]). for some positive Borel measure e singular with respect to Lebesgue measure.

According to the Riesz-Smirnov factorization, any function t T is represented as ¦ ¦ 9 H 5 where 9 5 H are outer with 9 © 5 H © X A , is singular inner, ¦ is a Blaschke product and £ £ ¦ A . Similarly, functions t are represented as p¦ ¦ 9 9 H H 5 with `outer, ` © X A , `singular inner, ¦ is a Blaschke product and £ £ ¦ A .

In view of the Riesz-Smirnov factorization described above, the essential difference between Nevanlinna and Smirnov functions is the extra singular factor appearing in the denominator in the Nevanlinna case. This is reflected in the corresponding result for free interpolation in by the fact that ! ¤ is bounded by a harmonic function, not necessarily quasi-bounded.

FROM HARMONIC MAJORANTS TO INTERPOLATION

For a given Blaschke sequence n ¡ set 1 p¦ £ D¦ y& ( u£ . The key result to the proof of the sufficient condition is the following theorem by Garnett [START_REF] Garnett | Two remarks on interpolation by bounded analytic functions, Banach spaces of analytic functions[END_REF], that we cite for our purpose in a slightly weaker form (see also [START_REF] Nikolski | Operators, functions, and systems: an easy reading[END_REF] as a general source, in particular C.3.3.3(g) (Volume 2) for more results of this kind).

Theorem. Let ü0 c % 5 p f( 8 c % 5 p f( be a decreasing function such that ¡ © Q ¨& ( ( we ( h p . If a sequence ¦ & W g ( satisfies £ £ X 1 ¨& $ 3" SR X 1 ( 5 G 5
then there exists a function t © such that £ ¦ .

Observe that according to our former notation we have $ 5" SR & vX 1 ( ¦ A @ ! ¤ & ( .

As we have already noted in Remark 1.1, in order to have free interpolation in the Nevanlinna and Smirnov classes, it is sufficient that ü© n £ and © n T £ respectively. Our aim will be to accommodate the decrease given in Garnett's result by an appropriate function in or T . This is the crucial step in the proof given hereafter of the sufficiency of conditions (c) in both Theorems 1.3 and 1.2.

Proof of sufficiency of 1.3 (c) and 1.2 (c)

. The proof will be presented for the more difficult case 7 e e & 7 e( has positive real part in the disk. By Smirnov's theorem, ¡ is an outer function in some , ¢ h A , and therefore in T (see [START_REF] Nikolski | Operators, functions, and systems: an easy reading[END_REF], in particular A.4.2.3 (Volume 1)). Also ¡ ¢ & ¡ ( is in the Nevanlinna class. By assumption we have $ 3

" SR & A 1 ( X Re ¡ & ( , t
. Take now ¨& ( ( ¦ & A @ ( ( 8 H , which obviously satisfies the hypothesis of Garnett's theorem, and set ¦ & C @ ¡ ( H , which is still outer in T . We have the estimate

£ d & ( u£ ¦ £ C @ ¡ & ( d£ H ' & C @ Re ¡ & ( ( H ' & A @ $ 5" SR X 1 ( H ¦ A ¨& W$ 5" SR & WX 1 ( ( 5 hence the sequence & ¡ y ( defined by e 1¦ A & ( ¨& $ 3" SR & WX 1 ( ( 5 t 5 is bounded by A .
In order to interpolate an arbitrary q ¦ & rq U ( ¨© by a function in , split

q s j¦ £¢ q s ¤ e ¡ £¢ & ¡ & ( ( 1 1 ¨& W$ 3" %R X 1 ( ¦¥ ¨ § & ( ¡ £¢ & ¡ & ( ( x Since by hypothesis & q ¤ e ¡ £¢ & ¡ & ( ( 1 (
is bounded, we can apply Garnett's result to interpolate the sequence

j¦ q ¤ e ¡ £¢ & ¡ & ( ( 1 1 ¨& W$ 5" SR X 1 ( 5 t 5 by a function © . Now © ¦ ¡ £¢ & ¡ ( is a function in with © p£ ¦ qq .
The proof for the Smirnov case is obtained by observing that if the measure e is absolutely continuous, then ¡ ¢ & ¡ ( is in the Smirnov class and so is the interpolating function © .

FROM INTERPOLATION TO HARMONIC MAJORANTS

We first show that in order to construct the appropriate function estimating $ 3" SR £ ¦ e& ( d£ 9 we only need to consider the factors of ¦ given by points ¨ which are close to . This is in accordance with the results for some related spaces of functions [HaMa01, Theorem 1], and it obviously implies Corollary 1.9. In what follows we will argue on one sequence, say 9 . The arguments are the same for the other sequences.

Our first observation is that, by construction, for ¥ 5 £ ¥ 9

, ¥ ¦ £ , § & ¥ 5 £ ( )0 ¦ 2 5x ¢ ¥ ¤ ¢ § & 65 1( P' 1 R % 5
for some fixed 1 . In what follows, the letters , £ ... will stand for indices in

¡ H of the form & 5 ¥ W( 5 % X ¥ h C o
. The closed rectangles ¥ are compact in ¡ so that 9 ! ¥ can only contain a finite number of points (they contain at least one point, by assumption). Therefore % h ! " @0 ¦ 4 g2 5x $ 5" SR £ D¦ e& ( u£ . The proof for T goes along the same lines, except that singular inner factors do not occur in (4.3), and so will not appear in (4.5) either.

THE TRACE SPACES

In this short section we prove the trace space characterization of free interpolation given in Theorems 1.2 and 1.3.

In order to see that (b) in each theorem implies free interpolation it suffices to observe that ä© n q¥ E ¤ n q¥ FE and use Remark 1.1.

For the proof of the converse, we will only consider the situation in the Nevanlinna class, since the case of the Smirnov class is again obtained by removing the singular part of the measure and the singular inner factors.

w Y ¢ ¥ba V o ¥ ¢ T o ¥ ¢ 4 ¢¡ ¤£ & 7 ( X A 5 then (6.3) V o ¥ ¢ T o ¥ ¢ w ¡ ¥£ ! X £ ¨ x Furthermore, Q 9 £ dX £ ¨ X Q £ , where Q R A is an absolute constant. Proof of Lemma 6.3. Pick ö ¥ ¢ ¥ o ¥ ¢ such that ! & ö ¥ ¢ ( P' & y ¡ ¤£ ! ( C and define the measure e 0 ¦ i o ¥ ¢ T o ¥ ¢ 1 §¦ ¡ ¤£ . Then, if UT o ¥ ¢ satisfies (6.2), ¦ & e ( & 7 e( ¦ F a 4 & 7 e( we e & ( ¦ V o ¥ ¢ T o ¥ ¢ 4 ¦ ¡ ¤£ & 7 e( X V o ¥ ¢ T o ¥ ¢ 4 ¡ ¤£ & 7 e( X tx
So if ! satisfies (6.1),

V o ¥ ¢ T o ¥ ¢ w ¡ ¥£ ! X C V o ¥ ¢ T o ¥ ¢ ! & ö ¥ ¢ ( ¦ C F a ! e e X C £ x
The converse direction is easier, and left to the reader (it also follows from the proof of the theorem, below).

We now need a classical result in convex analysis. Recall that the convex hull of a subset ¨n 9 © is defined as

" Sx & ¨( )0 ¦ E V `¤ 9 ` `0 ` ¨5 `' % 5 V ` `¦ A r yx
If we write 9 T ¨0 ¦ u 0 e ' % 5 G ¨ , then the conical convex hull of ¨is defined as

" Sx ¡ & ¨( 0 ¦ " Sx & 9 T ¨( ¦ E V `¤ 9 ` `0 ` ¨5 `' % r x
When ¨is a finite set, the conical convex hull is equal to its closure:

" Sx ¡ & ¨( ¦ " Sx ¡ & ¨(
(for this and other facts, see [HULL]). The key fact for us will be the generalized form of the Minkowski-Farkas Lemma (see [ We will use the following special case. For a vector ¥ 9 © , the coordinates are denoted by ¥ `, A X X e . Also, 9 © T denotes the set of points of 9 © with nonnegative coordinates. Corollary 6.5. Given 9 © , A X X

, let ¢ T 0 ¦ 9 © T 0 ¡ 5 ¢ X A , and suppose that ¢ T ¦ ¦

. Then the following properties are equivalent for

¥ 9 © T : (a) For any t ¢ T , ¡ ¥ 5 ¢ X A . (b) There exist A' % 5 A X X
such that i E ¤ 9 ¦ A and for any ¦ A 5 ax x x ¥5 e , ¥ `X E V ¤ 9 ` x Proof. Let EX ` 9 ©¨`¨© be the canonical basis of 9 © and consider ¨0 ¦ & v W5 A ( 5 A X X ¡ & X `5 % ( 5 A X X e w gx Then ¢ T corresponds to the ¢ in Theorem 6.4, from what we see that (a) implies that there exist

0' % 5 `' % , A X X , A X X e , such that & ¥ 5 A ( ¦ E V ¤ 9 & W U5 A ( © V `¤ 9 `& vX `5 % ( x When applied to each coordinate, this yields A ¦ i E ¤ 9 and ¥ `¦ E V ¤ 9 ` `X E V ¤ 9 ` x
The converse implication is immediate.

Proof of Theorem 6.2. Suppose that ! satisfies (6.1). For each nonnegative integer , we define

#0 ¦ ¢ 4 ¡ ¤£ & ¡ £¢ & 4 § C C ED ( ( ¦¥ Q ¨o ¨ Q ¨¢ ¨H ¡ 9 for % X X C fA 5 e g0 ¦ i o ¤ Q C o and ¢ T 0 ¦ y UT o ¥ ¢ Q ¨o ¨ Q ¨¢ ¨H ¡ 9 9 © T 0 V Q ¨o ¨ Q ¨¢ ¨H ¡ 9 T o ¥ ¢ 4 ¡ ¤£ & ¡ £¢ & 4 § C C dD ( ( X A 5 for A X X C fA r x
Obviously, ¢ T is not empty: for instance T Q ¥ Q ¦ A and T o ¥ ¢ ¦ % for ' A gives a point in ¢ T . We claim that any UT o ¥ ¢ ¢ T will satisfy (6.2) up to a constant. Indeed, for any f c % 5 C ED ( , there is an index vh C so that § C C ED X f h & @ A ( § C C ED , therefore by Harnack's inequality, for any such that £ £

X A C , 4 & vX `b ( ¦ 4 ¡ £¢ ¥¤ ` §¦ H © H 'P b & ¡ £¢ & § C C ED ( ( X 4 & ¡ £¢ & § C C ED ( x Therefore 9 T o ¥ ¢ satisfies (6.
2), and by Lemma 6.3 and the hypothesis, ! satisfies (6.3) with constant £ ¨ . Corollary 6.5 then implies the existence of positive coefficients &

( H © 9 ¤ Q with sum equal to £ ¨ , such that w ¡ ¤£ ! X H © 9 V ¤ Q 4 ¡ ¥£ & ¡ £¢ & 4 § C C dD ( ( ¦ F ba 4 ¡ ¥£ e 07 5
where 7 is the discrete measure on the circle given by the following combination of Dirac masses:

7 ¦ H © 9 V ¤ Q 1 £¢ ¥¤ ` ¦ H © H 'P x
Since the mass of 7 is uniformly bounded by £ ¨ , we can take a weak* limit 7 of this sequence of measures, so that for any & 5 £ ( , w ¡ ¤£ ! X F ` a 4 ¡ ¤£ e 7 ¦ & & o ¥ ¢ ( 5 where & 0 ¦ 4 dc 7 g . Harnack's inequality now implies that there is an absolute constant Q 9 such that Q 9 & & S( ' ! & ( for any ¡ . This proves the theorem, with the inequality

2 5x & & % ( )0 & ! 0) 1 & ¡ )( 5 & ' ! X Q 9 £ ¨ X Q HQ 9 £ x
The constants Q , and Q 9 only depend on the discretization we have chosen. Picking a dis- cretization with smaller "squares", we may make all three constants as close to A as we wish. Now we can prove Corollary 1.5.

Proof of Corollary 1.5. Given a non-Blaschke sequence , arguing as in [NPT] We finish this section with the proof of Theorem 1.6. where & v o ( is an increasing sequence of positive numbers tending to infinity, to be determined later, and & © o ( is given recursively by © Q ¦ % and ¨o & @ A ( ¦ ¨o T 9 & @ A ( . Observe that each ¨o is defined on the whole real line (they give supporting hyperplanes for the polygonal convex graph of ¨). We shall also use ¥ T ¦ 4 d) ¢ & ¥ 5 % ( for ¥ 9 . We prove that ¨© ! admits a harmonic majorant using Theorem 1.

V o # Q F a c ¨T o © ! o & ( ¨T o © ! o T 9 & ( g e e & ( ¦ F a ¨Q © ! & ( we e & S( @ V o # 9 F a c ¨T o © ! o w& ( ¨T o 9 © ! o & ( g e e & ( X q Q F a ! & S( we e & ( @ V o # 9 F a & v So So 9 ( & ! o & ( ( T e e & S( X q Q ¡ Q @ V o 6# 9 & v
£ & & % ( ¦ V o £ & o & % ( X Q & Q & % ( V o A % ¨o h vp 5
and defines therefore a quasi-bounded harmonic majorant of ! .

Proof of Lemma 6.6. Proof. It is well known and easy to see that there exists a constant T such that 4 ' T ¡ ¥£ for any ¥ o ¥ ¢ (the constant T depends on the aperture of the Stolz angle). Therefore, for any

Set £ 0 ¦ 4 d) ¢ & ! © 5 C & & % ( ( .
o ¥ ¢ n ¢ ¡ 0 ¦ ¡ ¦ go & such that e & o ¥ ¢ ( R £ & o ¥ ¢ ( x Note that & o cannot contain two contiguous intervals such that o ¥ ¢ ¡ o ¥ ¢ T 9 ¦ o 9 ¥ ¢ ¥ , because then o 9 ¥ ¢ ¥ n ¦ ¡ ¦ go & , a contradiction. Therefore, if o ¥ ¢ n & o , then e & o ¥ ¢ ( X e & o 9 ¥ ¢ ¥ ( X £ & o 9 ¥ ¢ ¥ ( ¦ C £ & o ¥ ¢ ( h C e &
¥ o ¥ ¢ 4 dc 7 g & ( ' T F ` a ¡ ¥£ & 7 e( e 7 & 7 e( ' T 7 & o ¥ ¢ ( & o ¥ ¢ ( ' T ! o ¥ ¢ ¦ T y ¡ ¥£ ! 5
which proves that

4 dc & A 2T ( 7 g
is the harmonic majorant we are looking for.

To see that the condition is not necessary, consider any In the same way as in Corollary 1.

y V ¢ ¢ ¡¥ & A £ & ¥ ( d£ d( q w £¢ ¥¤ )% $ 3" SR £ D¦ e& ( d£ 9 r h fp 5
where the supremum is taken over all ¨¨n ¨such that & ¥ ( 5 ¥ ¨¨ is a disjoint family, then is interpolating for the Nevanlinna class.

We move next to the proof of the necessary condition in terms of the Hardy-Littlewood maximal function.

Proof of Proposition 1.8. (a) The problem can be localized, so we may work on the upper half plane, £ T 0 ¦ @ ¥¤ B0 ¦¤ R % , with ¨ § T © 0 ¦ & ¤ 5 @ ¤ y( , restricting ourselves to positive harmonic functions which are Poisson integrals of positive measures with finite mass. Here the Poisson kernel is given by

4 § T © & ( ¦ A D ¤ & ( H @ ¤ H
x For convenience we shall write here £ & t£ for the Lebesgue measure of a measurable set & n 9 . Also, we only need to look at boundary points in a fixed bounded interval, say A X X A .

For any

( R % , let & & 0 ¦ c A 5 A g 0 ! & ( R ( . For any & & , there exists ¦ & ( and ¦ & ( such that (7.2) ! & ( F R ( £ £ 5 i.e. ! & ( u£ £ R ( £ £ x
By Vitali's covering lemma, there exist an absolute constant T 9 f& % 5 A ( and a disjoint family of intervals

@0 ¦ & a( , A X X
, such that i £ S£ ' T 9 £ & & £ . Write 0 ¦ & ( ¦ 0 @ ¤ . Note that since the point is contained in the "tent" over ( (therefore in the tent over ) the points are separated in the hyperbolic metric. Now define new points in the following way : let ¤ 0 ¦ £ £ C ¦ ¤ S£ S£ £ ( £ ' ¤ and 0 ¦ @ ¤ . Note that £ ! ¥ ( £ ' £ £ C . We claim that & & ( P' ( , where & is a harmonic majorant of ! . Indeed,

writing & ¦ 4 c De Cg , & & ( ¦ A D ¤ F ¡ A A @ ¢ & § ( © ¥ ( ¥ H e e & ( ( ' A D ¤ F ¡ A A @ ¢ & § ( © ( ¥ H e e & ( ( ¦ ¤ ¤ & & ( 5 and, by (7.2), & & ( ' ! & ( PR ( £ £ £ ( £ ¦ ( ¤ ¤ . Therefore, since £ & £ 9 ¤ & 9 ( , T 9 C £ & & £ X A C V £ S£ X V £ ! ¥ ( £ X £ £ & R ( £ X Q £¢ ( x (b) Similarly.
We now give two examples showing that the necessary condition of Proposition 1.8 is strictly stronger than that of Proposition 1.7 but still not sufficient.

Proposition 7.4.

(a) There are functions ! such that ! £ 9

¤ & ¢ ¡ ( , but that do not admit a harmonic majorant.

(b) There are functions

! such that £ ! £ 9 ¤ & ©¢ ¡ ( , but ! £ 9 ¤ & ©¢ ¡ )( .
Proof. The proof rests on the following family of examples. Note that it is easy to turn those examples into examples of sequences which are (or are not) interpolating for the Nevanlinna class.

Again we will work on £ T . Our functions ! will vanish everywhere on the upper half plane, except on the sequence ¢ 0 ¦ ¢ @ ¥¤ ¢ , where ¢ ¦ £ ¡ and ¤ ¢ ¦ £ ¤ . To ensure that ¤ ¢ X & ¢ T 9

¢ ( H we take ' C & @ A ( . With this choice, it can be deduced from Proposition 8.2 (or the remark before Lemma 8.4), that a necessary and sufficient condition for the existence of a harmonic majorant is that £ ! £ 9 , that is,

(7.3) V ¢ ! & ¢ ( ¤ ¢ h qp x
We note that

¨ ¨ & ( ¦ C A @ 4 ) ¢ & A 5 4 § § 4
© ( x Henceforth we only study data ! ¢ 0 ¦ ! & ¢ ( which are increasing sequences of positive numbers tending to infinity. We also assume that & ! ¢ ¤ ¢ @ ! ¢ T 9 ¤ ¢ T 9 ( w& ¢ ¢ T 9 ( ¢ forms an increasing sequence. Let £ Q & ( ( 0 ¦ 4 g2 5x £ 0 ( h ! ¢ . The necessary condition arising from the fact that £ ! £ 9

¤ & 9 ( reads (7.4)

V ¢ # ¢ ¦¥ & ¤ ¢ £ 9 ¤ Q & ( ( X Q ( 5 § ( R % x
This condition will be assumed for both examples. Now, for

£ ' £ Q & ( ( , define ¢ 0 ¦ 0 ! ¢ ¨¢ & ( 0R ( & ¢ ¤ ¢ ! ¢ ( 5 ¢ @ ¤ ¢ ! ¢ ( ( , and let £ 9 & ( ( )0 ¦ 4 I2 3x £ 0 ¢ ! ¢ T 9 ¦ ¦ . Then, ¢ # ¢ # & ¢ ¦ & % 5 ¢ # & @ ¤ ¢ # & ! ¢ # & ( ( ¦ & % 5 £ ¡ 9 & ( ( @ £ ¤ 9 & ( ( ! ¢ # & ( ( and
(7.5) 0 ! & ( PR ( £ ¡ 9 & ( ( @ £ ¤ 9 & ( ( ! ¢ # & ( @ C ( ¢ # & 9 V ¢ ¤ ¢ ¥ & ! ¢ £ ¤ £ ¡ 9 & ( ( @ C ( ¢ # & V ¢ ¤ ¢ ¥ & ! ¢ £ ¤ x
In order to prove (a), choose ! ¢ 0 ¦ ¡ ¢ £ ¤ 9 . Since ( ¡ ¢ ¥ & £ Q & ( ( ¤ 9 , condition (7.4) becomes that & ¢¡ ¢ ( ¢ remains bounded above, while the necessary and sufficient condition (see (7.3)) is does not have bounded balayage, even though another measure associated with the sequence will (see the proof).

0 ! & ( PR ( A ( $ 3" SR ( @ C ( ¢ # & V ¢ ¤ ¢ ¥ & A £ $ 3" SR £ A ( $ 5" SR ( @ C ( $ 3" SR 9 $ 3" %R £ 9 & ( ( $ 3" %R £ Q & ( ( B X A ( $ 3" SR ( @ Q ( X Q
£ ¢¡ 1 R & & ¥ ( ( ¡ 1 R & & £ ( ( d£ R ¡ 9 & A £ & ¥ (
In order to prove Proposition 8.2 consider the "Carleson window" ¥ & vX `b 5 V ( centered at X cb , of side V On the other hand, we can see that for the above lemma to hold, we must have ¡ 9 & ( ( AR R ( . More precisely, take the sequence in the upper half-plane given by ¢ 0 ¦ X ¢ @ £ 9 3 H X ¢ x

: ¥ & vX cb 5 V ( 0 ¦ u ¡ s0 A £ d £ X qV 5 %£ ¢¡ 1 R & S( f £ X fV gx
Then, Re ¢ Re ¢ T 9 X ¢ , so the sequence ¢ ¢ verifies the separation condition in Lemma 8.4 with ¡ 9 & ( s& W$ 5" SR 9

§ ( 9 3 H , but V ¢ & Im ¢ ( 4 & % ( V ¢ 9 Re ¢ Im ¢ B H ¦ V ¢ A £ ¦ p x
Proof of Lemma 8.4. Let f c % 5 C ED ( . By hypothesis, there is at most one t such that if and only if the last sum converges. Furthermore, when this is the case, the function & can always be taken quasi-bounded (see Lemma 8.1), so that interpolation can actually be performed in the Smirnov class. 9. HARDY-ORLICZ CLASSES Let © t0 9 8 c % 5 p f( be a convex, nondecreasing function satisfying (i) $ 32 34 & 7 © © & ( ( ( ¦ p (ii) H -condition: © & ( @ C ( X £ © & ( ( h@ , ( ' ( Q for some constants £ 5 ' % and ( Q 9 . Such a function is called strongly convex (see [RosRov]), and one can associate with it the corresponding Hardy-Orlicz class ¦ Y t T 0 F ba © & $ 3" SR £ U& 7 e( u£ ( we ¡ & 7 e( lh qp 5 where U& 7 e( is the non-tangential boundary value of at 7 )¢ ¡ , which exists almost everywhere.

f 0 ¦ ¢ ¡ 1 R & ( A C ¡ 9 & A £ U£ ( 5 ¡ 1 R & ( @ A C ¡ 9 & A £ U£ ( ¥ x Let e 0 ¦ i ¦¥ ¢ ¤ & A £ d
In [START_REF] Hartmann | Free interpolation in Hardy-Orlicz spaces[END_REF], the following result was proved.

Theorem. Let © be a strongly convex function satisfying (i), (ii) and the H -condition:

where R % is a suitable constant and

( 9 9 . Then n ¡ is free interpolating for if and only if is a Carleson sequence, and in this case £ ¦ E ¦ & v ( 0 £ £ ¦ V £¢ ¤ & A £ U£ ( ! & W$ 5" SR £ g £ d( h fp gx The conditions on © imply that there exist ¢ 5 ¡ & % 5 p f( such that n n £¢ . In particular, the H -condition implies the inclusion n for some ¢ R % . This H -condition has a strong topological impact on the spaces. In fact, it guarantees that metric bounded sets are also bounded in the topology of the space (and so the usual functional analysis tools still apply in this situation; see [START_REF] Hartmann | Free interpolation in Hardy-Orlicz spaces[END_REF] for more on this and for further references). It was not clear whether this was only a technical problem or if there existed a critical growth for © (below exponential growth © & ( ( ¦ X & corresponding to spaces) giving a breakpoint in the behavior of interpolating sequences for .

We can now affirm that this behavior in fact changes between exponential and polynomial growth. Let © be a strongly convex function with associated Hardy-Orlicz space . Assume moreover that © satisfies . Since go 8 p the elements of the sequence ¦ Q ¡ 9 are arbitrarily close and cannot be a Carleson sequence. By construction, $ 3" %R £ ¦ e& ( d£ 9 X 4 c g & ( (as before, we may possibly have to multiply ¥ with some constant T to have that condition also in the points ¨o , but this operation conserves the integrability condition), and therefore q tw yx z .

© & v @ © ( X

Corollary 1. 5 .

 5 Let be a separated non-Blaschke sequence and & ¢¡ g ( £¢ ¥¤ a sequence of positive values. Then there exists a non-zero function © & ¡ ( with £ U& ( d£ h ¡ Y , B

( a )

 a There exists a (positive) quasi-bounded harmonic function & such that ! & S( X & & ( for all ¡ . (b) There is a convex increasing function ¨0 c % 5 p f(

  A function is called outer if it can be written in the form e. on ¢ ¡ and $ 3" SR ©¥ g £ 9 & ©¢ ¡ j( . Such a function is the quotient ¦ 9 S H of two bounded outer functions 9 . In particular, the weight ¥ is given by the boundary values of £ 9 S H £ . Setting ¦ $ 3" %R ¥ , we have $ 5" SR £ & ( u£ ¦ 4 c g & S( ¦ F ` This formula allows us to freely switch between assertions about outer functions and the associated measures @e ¡ . Another important family in this context are inner functions:h© such that £ £

  of the Nevanlinna class. So, assume that & s! 3) 21 T & ¡ )( majorizes ! ¤ . Then & is the Poisson integral of a positive measure e on the circle and the function (3.1) ¡ & ( ¦ F ba 7 A@

  one can show that there exists a function © & ¡ ( in the unit disk with £ U& ( u£ h ¡ Y for any if and only if is the union of a Blaschke sequence and a sequence £ for which there exists a positive harmonic function & in the unit disk with & & ¡ ( PR $ 3" %R ¡ ¤ for all £ . Then the result follows from Theorem 1.4.

  4. Let ! o k¦ ! !# go and ¡ do 0 ¦ w ¢ & ¡ a ! o e e ( . If e % , then F a ¨© ! & ( we e & S( ¦ V o # Q F vo ¨ §¦ go T 9 ¨o © ! o w& ( e e & ( ¦

  t0 © ¡ . Then the intervals o ¥ d9 are all disjoint; however 4 ¡ ¤£ # & A ( ¡ C o & o ¥ d9 ( 9 , so that condition (7.1) will fail (the sum is comparable to ¢ ¨).

  & W$ 5" SR £ ( 8 9 , this condition fails, so that ! admits no harmonic majorant. However, £ Q & ( ( & ( $ 5" SR ( ( 9 ¡ ¢ # & w& ( £ 9 & ( ( ( , thus £ 9 & ( ( & ( ¡ ¢ # & ( 9 3 ¡ , and £ 9 & ( ( & ( $ 5" SR ( ( 9 3 ¡ . Therefore equation (7.5) becomes

£ 9

 9 ( 5 and this choice of ! does satisfy the necessary condition given in Proposition 1.8.To prove the second statement in the Lemma, choose ¡ ¢ 0 ¦ A . With similar but easier calcula- condition fails for ! , even though ! satisfies the necessary condition in Propo- sition 1.7.8. PROOFS OF THE GEOMETRIC CONDITIONSProof of Corollary 1.11. Since n 7 ¢ ¡ 0 0£ ! ¤ & 7 e( ' $ 5" SR £ D¦ e& ( u£ 9 r 5 t 5 to prove (a) and (b) it suffices to apply condition (a) of Corollary 1.10. Statement (c) follows from the next Lemma applied to ! ¤ . Lemma 8.1. Let ! 0 S¡ 8 @9 T satisfy i W¢ a & A w £ U£ ( ! & ( h fp . Then ! admits a quasi-bounded harmonic majorant. Proof. Let ¥ ¦ i ! & ( ¨. By assumption ¥ £ 9 & ©¢ ¡ j( and obviously £ ! X ¥ , hence the result follows from Corollary 1.10(b) and Theorem 1.3. Parts (b) and (c) also follow directly from Theorem 1.2(d), by a simple argument based on the

  tion, for any & rq ( I ä© there exists ¡ ¢ such that ¡ & ( ¦ uq U . Thus, if & v U& ( ( I ¢ v£ , the sequence of values & rq U E U& ( ( can be interpolated by ¡ ¢ . It is then clear that w yx gz T n w yx z . Free interpolation for these classes entails the existence of nonzero functions vanishing on all except a given Q . Hence the Blaschke condition i £¢ ¥¤ & A £ d l£ d( h fp is necessary and will be assumed throughout this paper. We say that a Borel measurable function ! defined on the unit disk admits a positive harmonic majorant if and only if there exists a positive harmonic function & on the unit disk such that & & ( (' ! & ( for any ¡ . Let ! 0) 21 a& ¡ ( denote the space of harmonic functions in ¡ and ! 3) 21 T & ¡ ( the subspace of its positive functions. Consider also the Poisson kernel in ¡ :

	¦	Given the Blaschke product ¦ ¤ v¦ ¨ § £¢ ¥¤ © ¤ © . Here © 1¦ & £ d l£ ( & ( & A ( 9 . Define then with zero-sequence , denote ¦ t¦ ! ¤ & ( )0 ¦ " # $ $ 3" SR £ ¦ e& ( d£ 9 if ¦ t	¦ ¤ ¦
		4 & 65 87 e( ¦ 4 & 7 e( ¦	Re 9 7 A@ 7 CB	¦ A £ d £ H £ D7 £ H x

% if G Definition.

  T & ¡ ( quasi-bounded 0 H& & ( P' $ 5" SR gT £ £ 5 U t gx

	(c) ! ¤ admits a quasi-bounded harmonic majorant. (d) $ 32 54 o 7 © y §¦ ©¨ ¢ ¤ V ! " $# go T ! ¤ & ( ¦ % , where % & W ( denotes the set of nonegative sequences UT such that w Y ¢ £ba i £¢ ¤ T 4 e& 7 e( X A . (e) (i) y & $' Q ( & y 7 )¢ ¡ 0 0£ ! ¤ & 7 e( ' ( Y( )h qp , and (ii) $ 32 54 o 7 © i £¢ ¤ T o ! ¤ & ( ¦ % for any sequence of sequences of nonnegative numbers UT o % & W ( such that $ 32 54 o 7 © i £¢ ¥¤ T o 4 & 7 e( ¦

% almost everywhere on ¢ ¡ .

  The Poisson balayage (or swept-out function) of a finite positive measure We will be interested in the class of measures having bounded balayage. Recall that Carleson measures are those finite positive measures whose balayage has bounded mean oscillation (see & e ( such that F a & & ( e e & ( X Q & & % ( for any positive harmonic function in the unit disk ¡ . Let ! be a nonnegative Borel function on the unit disk ¡ . The following statements

	Define	% q0 ¦ u e positive Borel measures on ¡ such that w Y ¢ ¥ba ¦ & e ( & 7 e( X	A gx
	Theorem 1.4. are equivalent: (a) There exists a (positive) harmonic function & such that ! & S( X (b) There exists a constant Q ¦ Q & ! ( such that w ¢ F a ! & ( we e & ( X Q x	& & S( for all ¡ .
		e in the closed unit disk is defined as ¦ & e ( & 7 e( ¦ F a 4 & 7 e( e e & S( 7 )¢ ¡ @x

[Gar81, Theorem VI.1.6, p. 229]); this is also an easy consequence of the 9 -BMO duality (see

[START_REF] Garnett | Bounded analytic functions[END_REF] Theorem VI.4.4, p. 245]

). Hence positive measures with bounded balayage form a subclass of the usual Carleson measures. It is easy to see (cf. Section 6) that positive measures with bounded balayage are precisely those which operate against positive harmonic functions, that is, those measures e for which there exists a constant Q ¦ Q The necessity of condition (b) is obvious (e. g. Q ¦ & & % ( ), while the sufficiency follows from a convenient version of a classical result in Convex Analysis, known as Minkowski-Farkas Lemma. The characterization of interpolating sequences in the Nevanlinna class in dual terms given by condition (d) in Theorem 1.2 follows from this result.

  Condition (b) is inspired by a characterization of quasi-bounded harmonic functions given in Armitage and Gardiner's book [ArGa, Theorem 1.3.9, p. 10].

	(d) (i) y & $' Q ( & y (ii) $ 32 54 o 7 © F a ! e e o I¦ % for any sequence e o gn % such that $ 32 54 7 )¢ ¡ 0 0£ ! & 7 e( ' ( Y( h vp , and o 7 © ¦ & e o ( & everywhere on ¢ ¡ ,	7 e( ¦ % almost
	that © ! admits a harmonic majorant on ¡ ; (c) $ 32 54 o 7 © w ¢ F # go W ! e e ¦ % .	8 c % 5 p f( with $ 32 34 & 7 © ¨& ( ( ( ¦	@ p such

  £ 9 & ¢ ¡ ( then Gw yx gz T (and hence Gw yx gz

		).
	Notice that the necessary conditions obtained by replacing ! ¤ by £ ! ¤ in (a) also hold. This in an immediate consequence of the estimate ! ¤ ' £ ! ¤ .
	This result implies the following Carleson-type conditions. Corollary 1.11. (a) If q tw yx z T , then $ 52 34 4 4 7 @9 & (1.3) A £ d l£ d( $ 3" SR £ D¦ e& ( u£ 9 ¦ % x (b) If q tw yx z , then w W¢ ¤ & A £ d l£ d( $ 3" SR £ D¦ e& ( u£ 9 h qp x (1.4)
	(c) If is Blaschke and	V W¢ ¤ & A £ d l£ d( $ 3" SR £ D¦ e& ( u£ 9 h fp 5

  Observe that the hyperbolic diameter of each Whitney square ¥ o ¥ ¢ is bounded between two absolute constants.We split the sequence into four pieces: ¤¦ §¦ ©¨`¤ 9 `such that each piece `lies in a union of dyadic squares that are uniformly separated from each other. More precisely, set

	Proof of the necessity of 1.3 (c) and 1.2 (c). We will use a dyadic partition of the disk: for any in ¡ , let (4.1) o ¥ ¢ 0 ¦ s EX acb 0 gf c C ED ¤£ wC o o ( 5 % X £ h C o x 5 C ED & £ @ A ( C and the associated Whitney partition in "dyadic squares": (4.2) ¥ o ¥ ¢ 0 ¦ u dV YX cb 0 X cb o ¥ ¢ 5 A C o X fV h A C o 9 gx
	9 ¦ ! ¥ 9 5 o ¥ H is given by ¥ H where the family ¥ 9 ¢ o ¥ ¢ (for the remaining three sequences we respectively choose ¥ H o ¥ H ¢ T 9 o ¥ ¢ , ¥ H o T 9 ¥ H ¢ o ¥ ¢ and ¥ H o T 9 ¥ H ¢ T 9 o ¥ ¢ ). In order to avoid technical difficulties we count only those ¥ containing points of (in case ! ¥ is empty there is nothing to prove).
	bounded ¡ ( and ¡ is given by (3.1) with & & ( 5 ¡ 5 , £ 9 & ©¢ ¡ ( , such that positive harmonic function & ¦ 4 c g $ 3" %R ¥ $# £ © & ( u£ X and therefore an outer function T , where ¦ ¡ £¢ & e e ¦ 1e ¡ , such that ¥ $# £ © e& ( d£ ' £ & ( u£ 5 ¡ x
	Proof. We shall use the intervals	introduced in (1.2). In [NPT, p. 124, lines 3 to 17], it is
	proved that the function given by	& 7 e( ¦ T Q V £¢ ¥¤ ¨& 7 e( 5
	where

Proposition 4.1. Let be a Blaschke sequence. For any 1 & % 5 A ( , there exists a quasi-T Q is an appropriate positive constant, is suitable. At this juncture, the separation hypoth- esis made in [NPT, Lemma 4] is no longer used.

  Again, 9 is a quotient of two bounded outer functions and we can suppose that 9 is outer in t© with 9 © and 9 is singular inner.Construct in a similar way functions ¡ `, `for the sequences `, ¦ C 5 ¤£ 65 ¦¥ , and define the products ¡ ¦ § ¨`¤ 9 ¡ `and ¦ § ¨`¤ 9 `. Of course ¡ is outer in © , and is singular inner. So, whenever ¨ t Therefore, the positive harmonic function & ¦ $ 3" %R £ ¡ 1£ satisfies & & ( P'

	Since & H 5 H does not vanish and is bounded above by 1, the function $ 3" SR £ f& H 5 H harmonic function. By Harnack's inequality, there exists an absolute constant T £ is a negative ' A such that A T £ $ 5" SR £ f& H & 9 ¢ ( 5 H & 9 ¢ ( u£ 3£ X £ $ 3" %R £ f& H & S( 5 H & ( d£ 5£ X T £ $ 5" SR £ f& H & 9 ¢ ( 5 & 9 ¢ ( u£ 3£ 5 ¥ ¢ 5 H hence £ D& H & 9 ¢ ( 5 H & 9 ¢ ( d£ ¦ X £ D& H & ( 5 H & ( d£ X £ f& H & 9 ¢ ( 5 & 9 ¢ ( u£ 9 3 ¦ 5 ¥ ¢ x H This yields £ & & H 5 H ( ¦ & ¨( u£ X £ & & H 5 H ( & 9 ¢ ( u£ X £ ¦ ¤ # ( 4( & 9 ¢ ( u£ (4.4) for every ¨ t 9 ! ¥ ¢ . Let us now exploit Proposition 4.1. By construction, the sequence 9 3 In 9 is separated. Therefore, there exists an outer function 9 in the Smirnov class such that £ ¦ # ( 4( # & 9 ¢ ( d£ ' £ 9 & 9 ¢ ( d£ 5 £ ¡ x
	associated with A . Also, we can use Harnack's inequality as above to get X £ 9 & 9 ¢ ( d£ ' £ ¦ 9 & ¨( u£ for every ¨ t 9 ¢¡ ¥ ¢ . This together with (4.4) and our definition of 9 ¢ give £ D¦ ¤ ¥ & ¨( u£ ' £ ¦ ¤ # & 9 ¢ ( d£ ¦ £ D¦ ¤ # ( 4( & 9 ¢ ( u£ § w£ D¦ # ( 4( # & 9 ¢ ( u£ ' £ & & H 5 H ( ¦ & ¨( d£ § w£ ¦ 9 & ¨( d£ for every ¨ ¥ ¢ and ¥ ¢ ¥ 9 . Set ¡ 9 ¦ & & H 9 ( ¦ and 9 ¦ 5 ¦ H ; by construction, ¡ 9 is outer with ¡ 9 © 10 ). Take 9 ¥ 2 such that " ¦ £ D¦ # ( & 9 ( d£ . Assume now that f tw yx z . Since © n £ , there exists a function 9 such that 9 & ( ¦ " 3 % if 9 3 dx # $ A if 9 , there exists £ A 5 C 5 ¤£ 65 ¦¥ such that ¨ G ¢ , and hence (4.5)
	By the Riesz-Smirnov factorization we have (4.3) 9 ¦ ¦ ¤ # ( 4( & 9 & H 5 where 5 H is singular inner, & 9 is some function in © and & H 5 H is outer in t© . Again, we can assume & ` © X A , ¦ A 5 C . Hence A ¦ £ 9 & 9 ¢ ( d£ X £ D¦ ¤ # ( 4( & 9 ¢ ( d£ § A £ D& H & 9 ¢ ( 5 & 9 ¢ ( d£ 5 H and £ ¦ ¤ # ( 4( & 9 ¢ ( d£ ' £ D& H & 9 ¢ ( 5 H & 9 ¢ ( d£ 5 £ ¡ x

£¢ ¤ $# &% ' )( £ D¦ & ( d£ (note that we consider the entire Blaschke product ¦ X A £ D¦ e& ( u£ ' £ ¡ ¢ & ( ¢ & ( d£ ' £ ¡ & ( & ( d£ x

  ¡ H@ he e ¢ which obviously satisfies 4 dc fe Cg & ( ' $ 3" %R T £ £ , t %R T £ £ . The Radon-Nikodym decomposition of e is given by e e ¦ e ¡ @ fe e £ , where £ 9 & ¢ ¡ ( is positive and e ¡ By definition, is outer inT and © ¦ . Clearly, $ 3" %R T £ £ X $ 3" SR £ © & ( u£ , thus £ £ X £ © & ( d£. Since £ is ideal by assumption, there exists Q interpolating & v g ( . 6. HARMONIC MAJORANTS AND MEASURES WITH BOUNDED BALAYAGE Let us start by proving that positive measures with bounded balayage are precisely those which operate against positive harmonic functions. Recall that % & e ( & The next result is a refined version of Theorem 1.4 stated in the introduction. Let ! be a nonnegative Borel function on the unit disk. Then there exists a harmonic function & such that & & ( P' ! & ( for any ¡ if and only if

	Assume that & W g ( £ and that written as ¦ 9 w& H H ( , where 9 t© , 9 © is such that & ( ¦ g X A , H is singular inner with associated , f . Since can be singular measure e ¡ , and H t© is an outer function with H © X A , we can define the positive finite measure e ¦ $ 5" SR & A £ H £ ( we . Conversely, suppose that & W e ( is such that there is a positive finite measure e with 4 c De Cg & ( ' Theorem 6.2. (6.1) £ 0 ¦ w ¢ F a ! e e h fp x $ 3" is a positive finite singular measure. Let be the singular inner function associated with e ¡ , and let Furthermore, £ i¦ 2 5x & & % ( )0 & ! 3) 21 u& ¡ )( 5 & ' ! x be the function defined by U& ( ¦ ¡ ¢ 9 F ` a 7 A@ 7 & 7 e( we ¡ & 7 e( B 5 ¡ x 7 e( ¦ ¡ a 4 & 7 ( e e & That (6.1) is necessary is clear from the above considerations. In order to prove that it is sufficient, we will reduce ourselves to a discrete version of it. We will use the dyadic squares introduced in (4.2). As in the previous section, choose a point o ¥ ¢ in each square, say & ¥ o ¥ ¢ ( ¦ o ¥ ¢ 0 ¦ & C o ( ¡ £¢ & C dD £ wC o ( x A Observe that by Harnack's inequality, there exists a universal constant such that : if 65 ¨lie in the same Whitney square ¥ o ¥ ¢ (as defined in (4.2)), then 9 4 ¥ & 7 ( X 4 & 7 ( X 4 ¥ & 7 e( , for any 7 )¢ ¡ . 7 e( and % q0 ¦ u e positive Borel measures on ¡ such that w Y ¢ ¥ba ¦ & e ( & 7 e( X Lemma 6.3. The function ! satisfies condition (6.1) if and only if there exists a constant £ ¨ such that for any sequence of nonnegative coefficients UT ao ¥ ¢ such that A gx (6.2)
	Proposition 6.1. Let balayage uniformly bounded by w Y ¢ £ba ¦ & e ( & 7 e( ¦ y T . Furthermore, the relevant constants are related: T R % such that ¤ F a & e e 0 & t! 0) 21 T & ¡ ( 5 8& & % ( ¦ A ¦¥ 5	e has
	and for any positive harmonic function & , & & % ( ¦ 4 ) ¢ ¢ F a & e e x Proof. Let & ¦ 4 dc 7 g , where 7 ' % is a measure on ¢ ¡ . If e F a & & ( e e & ( ¦ F ba F a 4 & 7 e( e e & ( e 07 & 7 ( X T 7 & ©¢ ¡ ( ¦ T has balayage bounded by & & % ( x Conversely, since ¨ § 8 4 & 7 e( is a harmonic function for any fixed 7 , ¡ a 4 & 7 e( we e & ( is pointwise T , defined. Pick a sequence 7 o such that $ 52 34 o 7 © F a 4 & 7 o ( we e & S( ¦ w Y ¢ £ba F a 4 & 7 e( we e & S( 5 where the supremum on the right hand side might a priori be infinite. Since the set & 0 ¦ & ! 0) 1 T & ¡ )( t0 & & % ( ¦ A is uniformly bounded on compact sets in ¡ , a normal family argument shows that y ¡ a & e e 0 6& & I h p . Observe that the mapping © § 8 4 & 7 o ( is in & for every 7 o , ¡ . Hence w o ¡ a 4 & 7 o ( e e & ( )h fp .
	This proves that

e be a positive Borel measure on the disk. Then ¡ a & e e is finite for any positive harmonic function & on the disk if and only if there exists some e has bounded balayage, and the equalities between constants that we had announced.

  ( . Part (i) holds whenever ! admits a harmonic majorant, be it quasi-bounded or not (see Proposition 1.7), while (ii) follows from the dominated convergence theorem.Since e o t % , their Carleson norms are uniformly bounded by some Q Q R % . We apply the direct part of [Gar81, Lemma I.5.5, p. 32] to ! ; the lemma is stated for harmonic functions, but harmonicity plays no role in the proof of the direct part. We obtain

	Proof of Theorem 1.6. & a( ¡ &
	& d( ¢ & c( . We proceed by contradiction. Suppose that there exist 1 R % and a sequence of measures e o % o such that (6.4) F !# go W ! e e o ' 1 ux Let £ e o ¦ !# go e o . Then F ba ¦ & £ e o ( & 7 ( e ¡ & 7 e( ¦ £ e o y& ¡ ( ¦ e o & y ! ' Y( x
	£ e o w& ¡ ( ¦ e o & y ! ' Y( X T 9 Q Q & 8 £ ! ' s Y( X T 9 Q Q Q by (d) (i). Since the sequence & ¦ & £ e o ( ( o tends to % in £ 9 & ¢ ¡ )( , some subsequence must tend to % 5 almost everywhere, and applying (d) (ii) to that subsequence, we find a contradiction with (6.4). & c( ¤ & b( . We define a function ¨on 9 T by ¨& ( ( ¦ ¨o & ( ( ¦ So ( @ © o for ( c 5 @ A g 5

d

  Since $ 32 34 o ¡ Eo ¦ % , we can choose an increasing sequence & W eo ( such that $ 52 34 io l So ¤¦ p , but ¨o R % for ' A , Q ¦ % and & ¡ go ( o # 9 is an increasing sequence of positive numbers tending to infinity fast enough so that i o 6# 9 A % ¨o h fp . In order to prove (a) let & Q be a harmonic majorant of ¨© ! . Each ! o

	Define ! o 0 ¦ ! ¤ ¡ ¨ §¦ ¤ ¡ ¤	#	; thus £ ¨© ! ¦ i o £ ¨o ©	! o .
	The following Lemma is due to Alexander Borichev.
	Lemma 6.6. There exists an absolute constant Q SR % such that whenever ! ' % is bounded and ! X & for some & ! 0) 1 T & ¡ ( , then there exists £ & & ! 0) 21 T & ¡ ( quasi-bounded such that ! X £ and F ` a £ & & 7 e( e ¡ & 7 e( ¦ £ & & % ( X Q & & % ( x
	is then bounded and quasi-bounded such that majorized by & Q % ¨o , hence by applying the previous lemma we find £ & o ! o X £ & o and £ & o w& % ( X Q & Q & % ( % ¨o . The series £ & 0 ¦ i £ & o converges in & o ' % for £ 9 & ©¢ ¡ 1( , since £ all and
			So	So	9 ( ¡ Eo x
	i o 6# 9 & v So & b( ¨ & a( . First notice that ¨can be replaced by a function £ So 9 ( ¡ do h qp , and we are done. ¨X ¨with the same properties
	as ¨and the additional explicit description: £ ¨& ( ( ¦ £ ¨o & ( ( ¦ So ( @ © o	' ¨o ( for ( c go	5 eo T 9 g

where So '

  o ¥ ¢ ( x Any positive function ! such that ! o ¥ ¢ 0 ¦ w

	¡ ¥£ ! satisfies (7.1) admits a harmonic majorant. On the other hand, the positive harmonic function § 8 4 & A ( does not satisfy (7.1) for certain choices of
	For any interval	, let £
	. We write % is to be chosen. This measure is absolutely continuous with respect to arc length. e £ e 0 ¦ Q H £ £¢ $ e ¡ & 7 e( C@ ` a $ e e ¦ e £ e 9 @ e £ e 5 H where Q H R The function we are looking for is £ & 0 ¦ 4 dc £ e g . Indeed, let ¡ and suppose that there exist a dyadic interval n & , maximal among the dyadic intervals contained in & , such that (6.5) F ¢ 4 & 7 e( e & 7 e( ' A Q H x Then clearly £ & & S( ' £ ' ! & ( . We claim that if is such that (6.5) does not hold for any maximal dyadic interval n & , then £ & & S( ' & & S( , which will finish the proof.
	Under that assumption, since the level sets of the Poisson integral in (6.5) are arcs of circles connecting the extremities of £

be the interval of same center and triple length, and let £ & 0 ¦ ¡ £ , where the union is taken over all the dyadic intervals included in & 0 ¦ ¡ o & o , where they make a fixed angle with ¢ ¡ depending on Q H , we must have £ 7 £ h' T ¥¤ & ( for any 7 and any maximal dyadic subinterval of & , so that Corollary 7.2.

¨.

  11, Corollary 7.2 and Proposition 1.8 imply the following result. For ¥ ¦ ¥ o ¥ ¢ , write & ¥ ( ¦ o ¥ ¢ (the radial projection of the square to an arc of the circle). Assume that is contained in a union ¨of Whitney squares ¥ of center & ¥ (

	Corollary 7.3. and that

  ¨9 , ä© duality. %R £ ¦ e& ( d£ 9 behaves asymptotically like $ 5" SR £ D¦ ¤ ¢ & ( u£ 9 in £ Y ¡¢ (here `). Also, we can assume that the sequence is radial (this means that we replace the initial sequence by one which is in a uniform pseudo-hyperbolic neighborhood of the initial one; by Harnack's inequality such a perturbation does not change substantially the behavior of positive harmonic functions). This and Theorem 1.3 prove the assertion. The proof for the Nevanlinna class is even simpler. Set e e ¦¥ ¦ 1 9 , the Dirac mass on A ¢ ¡ . SR T £ © & ( d£ ¦ F a $ 5" SR T £ © & ( u£ e e ¤ & ( X F a & & ( we e ¤ & ( h qp x Finally, to prove that (b) implies (c), suppose that (c) doesn't hold, i.e. , there exists £ 9 & ¢ ¡ ( such that ¡ ba ¡ ¤ ¦ p . Taking an outer function © T with $ 3" SR £ © p£ ¦ 4 c g Proof of Naftalevič's theorem. Assume that is contained in a finite union of Stolz angles and (1.4) holds. By Proposition 1.12, w yx z , hence the trace £ is given by the majorization condition of Theorem 1.2(b). Taking as majorizing function the Poisson integral of the sum of the Dirac masses at the vertices, we see that £ ¥ Na . Conversely, if £ ¦ ¥ Na then the trace is ideal, so is free interpolating and by Corollary 1.11(b) (1.4) holds. According to Theorem 1.2(b) and the definition of ¥ Na , the function This prevents & from being the Poisson integral of a finite positive measure. Assume that n v¡ is contained in the union of a family ¨of Whitney squares such that

	Proof of Proposition 1.12. It is enough to consider the case where is contained in only one Stolz angle. Indeed, if ¦ ¦ o `¤ 9 `with n £ Y ¡ , ¥ ¦ A 5 x x x 5 , and 7 ` ¦ 7 , then $ 32 54 7 Y ¡¢ ¥ ¢ ¦¥ £ ¢ £ ¦ ¤ ( Y& ( u£ ¦ A 5 ¦ b5 harmonic function & so that $ 5" SR T £ © p£ X & . Thus, taking e as in (1.6), V £¢ ¤ & A £ d l£ d( $ 3" ¡ ¤ 0 ¦ i & A £ d l£ d( 4 is unbounded. Since ¡ ¤ is lower semi-continuous, this implies that ¡ ¤ £ © & ¢ ¡ )( . Since £ © is the dual of £ 9 we see that V £¢ ¤ & A £ d l£ d( $ 3" SR £ © & ( u£ ¦ V £¢ ¤ & A £ d l£ d( F ` a 4 E t¦ F ba ¡ ¤ g¦ p 5 so (b) doesn't hold. so that $ 3" According to Corollary 1.11 it is enough to prove the sufficiency of the conditions. Let us first show that condition (1.3) implies interpolation in T . In order to construct a function £ 9 & ¢ ¡ j( meeting the requirement of Theorem 1.3(c) assume that ¦ o o n c % 5 A ( is arranged in increasing order and set £ ¡ Yo f¦ & A £ d o £ ( $ 5" SR £ D¦ ¡ & o ( d£ 9 . Clearly there exists a decreasing sequence & ¢¡ Yo ( o with £ ¡ do X ¡ do , ¡ , and $ 32 54 o ¡ do v¦ % . Now, if o v¦ ¡ , set o ¦ o 0 o T 9 , o ¦ ¡ do ¡ do T 9 , and set & 7 e( ¦ V o o & o ( ¡ & 7 e( 5 7 )¢ ¡ ix Then £ 9 & ¢ ¡ 1( , and 4 dc g & o ( ' F ¡ 4 & o 5 87 e( V ¢ ¢ & o ( & 7 e( we ¡ & 7 e( ¤ V ¢ # go ¢ & o ( A & A £ o £ ( F e ¡ & 7 ( ¦ i ¢ # go ¢ A # £ d o £ ¦ ¡ do A £ o £ ' £ ¡ do ¦ $ 5" SR £ D¦ ¡ & o ( d£ 9 x ! & ( ¦ " # $ & A £ U£ ( 9 if ¦ t % if G admits a harmonic majorant & . Let & & ( ¦ 4 dc fe hg & ( and consider the intervals ¡ ¦ u 7 )¢ ¡ 0 S £ ¡ & 7 ( gx There exist constants and Q ¡ such that e & ¡ ( PR Q ¡ for any such that & & ( ' & A £ d £ ( 9 . If is not contained in a finite union of Stolz angles, then there is an accumulation point 7 ¢ ¡ of ¨n such that ¨ n £ ¤ & 7 e( for any . Pick R ; then for ¨ ¨, ¡ ¦¥ 7 and we can construct an infinite subsequence ¨¨n ¨such that the Privalov shadows ¡ ¥ ¦¥ ¢ ¥¤ ¥ ¥ are A £ d o £ disjoint. We now give an example of a concrete separation condition implying that e ¤ has bounded
	From (1.4) we get and we finish by applying Theorem 1.2. $ 3" SR £ ¦ ¡ & o ( d£ 9 £ balayage. Proposition 8.2.	A A £ d o £	£ 4 c De §¥ g & o ( 5
	Proof of Proposition 1.13. By Corollary 1.11(c), we already know that (1.5) is a sufficient condi-tion for to be interpolating for T . Conversely, suppose that is interpolating for , that is, ! ¤ admits a harmonic majorant. Since e ¤ has bounded balayage, then ¡ a ! ¤ e e ¤ h qp , which is exactly (1.5).
	Proof of Proposition 1.14. It is obvious that (a) implies (b). If we assume (c),	e ¤ will act against
	any positive harmonic function. Suppose © . As seen in Section 5, there exists a positive

  d£ d( , where & ¥ ( is the center of ¥ and ¡ is a positive function, with ¡ & (

	Note that this covers some cases where	e ¤
	for any ¥ 5 £ ¨, ¥ ¦ £ decreasing and Then q tw yx z if and only if f tw yx z T , and if and only if F Q ¡ & ( H h fp x V C¢ ¢ & A £ d & ¥ ( u£ ( v w £¢ ¥¤ % ' $ 3" SR £ ¦ e& ( d£ 9 h fp

  The next result is a Carleson-type condition which implies boundedness of the balayage. , let & & f S( 0 ¦ ¡ 0 4 & vX cb ( I' ( . This is a disk, tangent to the unit circle at the point X `b , of radius A y& ( @ A ( . Therefore & & f S( n ¥ & vX `b 5 8Q ( ( for ( ' A , say, withWe will now compare measures satisfying the condition in Lemma 8.3, measures with bounded balayage and Carleson measures. Each set is included in the next, and the examples will show that both inclusions are strict. In particular, if we take ¢ ¦ £ ¤ , with A h X C , e Suppose that ¡ is a positive valued function such that ¡ & ( Let ¡ 9 stand for the inverse function of ¡ . Then, if we have a sequence n v¡ such that

	Lemma 8.3. Suppose that with Then One can check that e is a Carleson measure if and only if there exists a constant such that e & ¥ & vX cb 5 V ( ( X ¡ & WV ( , where ¡ is a nondecreasing function on c % 5 C ( F Q ¡ & ( H e h qp x F 9 & ( e X )1 5 § 1 R % 9 and e is a measure with bounded balayage if and only if it satisfies the condition in Lemma 8.3, which happens if and only if F 9 & ( e h fp x A is a measure with bounded balayage but it does not satisfy the condition in Lemma 8.3; if we take & ( ¦ is a Carleson A , e measure, but it does not have bounded balayage. e is a measure with bounded balayage. A discrete version of this condition is V o C o w b ¢ ¡ e & ¥ & WX cb 5 C o ( ( h vp 5 as can be checked by writing a Riemann sum. Proof. For any ( R In view of Proposition 1.13, among other things, it is interesting to understand for which separated sequences the corresponding measure e ¤ has bounded balayage. It is easy to see that this is the case when £ £ U£ ¨ £ ©Ẅ£ 3£ ' & H , ©¨ ¦ , but more is true. A £ d l£ d( 9 3 Lemma 8.4. is increasing and F Q ¡ & ( H e h qp x % Q SR % an absolute constant. Using the distribution function F a 4 & vX `b ( e e & ( ¦ F © Q e & & & f S( ( e ( X Q 9 @ F © 9 e & & & f %( ( e ( X Q 9 @ F © 9 e & ¥ & WX cb 5 8Q ( ( ( e ( X Q 9 @ F © 9 ¡ & Q ( ( e ( X Q 9 @ Q F 9 Q ¡ & ( H e h fp x £ d l£ £ ¨£ ' ¡ 9 & A £ U£ ( 5 § ¨ ¦ 5 the measure e ¤ has bounded balayage. Examples of such functions ¡ are given by s& W$ 5" SR 9 § ( 8 9 ¡ , with ¡ R % . In that case, ¡ 9 & ( ( ( & W$ 5" SR 9 & ( 9 T .
	Example 1. Let concentrated on the circles centered at the origin of radius A ¦ o be a sequence of nonnegative reals. Let e ¢¡ o C given in dual terms by be the measure F a U& S( e e ¡ & ( 0 ¦ V o 6# 9 o A C ED F IH 'P Q & & C o ( X cb ( e gf ex A One can check that e ¡ is a Carleson measure if and only if it has bounded balayage and this happens if and only if i o o h sp . Also e ¡ satisfies the condition in Lemma 8.3 if and only if i o i ¢ # go ¢ h fp . Example 2. Let be a nonnegative-valued function on the interval be the c % 5 A ( . Let e measure concentrated on the ray from the origin to A given by F a & ( e e & ( )0 ¦ F 9 Q U& ( & ( e x

e & & & f %( ( and the fact that the measure e is bounded, we get the following estimate for the balayée of e :

  v£ ( 1 ¦¥ By the proof of Lemma 8.3 for this specific value of f , we see that it will be enough to check that for some absolute constants Q 9 5 Q H Consider 6 0 ¦ s ¨ ¦ t0 e ¨ ¥ & vX cb 5 V ( . For any ¨ 6 we have& ¦¥ ( ¦ ¡ 9 & A £ d ¨£ ( X C £ f ¡ 1 R & ¨( d£ X C V f £ V 5 f @ £ V g . Sincethey are disjoint, i ¦¥ ¢ ¢¡ ¤£ & ¦¥ ( X ¦¥ %V Proof of Proposition 8.2. For each Whitney square ¥ in ¨, let & ¥ ( be the point in ! ¥ such that $ 5" SR £ D¦ & & ¥ ( ( d£ 9 ¦ q4 ) ¢ d$ 5" SR £ D¦ e& ( u£ 9 0 e t ! ¥ gx Let be the sequence formed by & ¥ ( 10 ¥ ¨ . By Lemma 8.4 the corresponding measure e ¡ has bounded balayage. Therefore, there exists a positive harmonic function & with & & & ¥ ( ( P' $ 3" %R £ ¦ & & ¥ ( ( u£ 9 if and only if V & A £ & ¥ ( d£ d( $ 3" SR £ ¦ & & ¥ ( ( u£ 9 h qp x According to condition (c) in Theorem 1.2 one deduces that f w yx z

					5
	so the intervals Using that ¡ & ( ¥ are all contained in is increasing we have w ¥ ¢ §¡ ¨£ A £ ¨£ & ¦¥ ( X	¡ ¢ w ¦¥ ¢ ¢¡ £ & ¥ ( ¥ w ¦¥ ¢ ¢¡ £ & ¥ (	X ¡ & C V ( C V x	.
	Finally,	e ¨& ¥ & vX cb	5 V ( ( ¦ V ¥ ¢ §¡ ¤£ A £ d ¨£ X w ¥ ¢ ¢¡ ¤£ A £ ©¨£ & ¦¥ (	V ¥ ¢ ¢¡ ¤£ & ¦¥ ( X ¡ & C V ( C V ¥ %V ¦ ¥ ¡ & C V ( x
	F a 4 & vX `b ( e e ¤ & ( ¦ &	. Then A £ U£ ( 4 e& vX cb ( @ F a 4 & WX cb ( e e ¨& ( X	Q @ F a 4 & vX cb ( e e ¨& ( x
			e ¨& ¥ & vX cb	, one has 5 V ( ( X Q 9 ¡ &	Q H	V ( 5 for % h V h C x

c

  Q . The standard example in this setting is © & ( ( ¦ ( for ¢ R A . We have the following result. Let © t0 9 8 c % 5 p f( be a strongly convex function such that (9.1) holds. If there exists a positive weight £ 9 & ©¢ ¡ ( such that © © £ 9 & ¢ ¡ ( and ! ¤ X 4 c g , then Gw yx gz The reasoning carried out in Section 3 leads to an interpolating function of the form ¡ £¢ & ¡ ( , with © , and ¦ & C @ ¡ ( H outer in for all ¢ h A (note that the measure e defining ¡ here is absolutely continuous, in fact e ¦ e ¡ ). Also, n SR £ ¡ £¢ ¡ £ ( ¦ ¡ © © h p so that ¡ ¢ & ¡ ( We give an example of an interpolating sequence for which is not Carleson, thus justifying our claim that there is a breakpoint between Hardy-Orlicz spaces verifying the H -condition and those that do not. Consider the functions © and let Q ¦ u o o n v¡ be a Carleson sequence verifying o ©! ¢ ¦ , where ao are the arcs defined in (1.2). Since i o & A £ o £ d( hh p , there exists a strictly increasing sequence of positive numbers & ¡ yo ( o ( go h p and £ 9 & ¢ ¡ j( since ¢ R A . Associate with Q a second Carleson sequence 9 ¦ ¨o o such that the pseudo-hyperbolic distance between 30 A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS corresponding points satisfies £ © ¥ ¡ & o ( d£ ¦ X

	# ¡
	T %& Theorem 9.1. ¡ & ( ¦ F ba 7 A@ (9.1) for some fixed constant T ' A and for all 7 & © & v ( @ © & © ( ( 5 5 © ' 7 ( e ¡ & 7 e( x on © . By construction, algebra, we deduce that ¡ £¢ & ¡ ( such that i o & for any ¢ R % by our conditions . Since is an A k £ o £ d( go h p and $ 32 54 o 7 © go ¦ p . Setting ¦ V o 9 3 o ¡ 5 we obtain ¡ © © ¦ i o & A £ o £ d¤ ¢ ¡

( . Proof. Note first that (9.1) implies that is an algebra contained in T , hence it is sufficient to interpolate bounded sequences (see Remark 1.1). As in Section 3, we set ¡ © & W$ 5" . Example 9.2. ¦ , ¦ £

& & ( ¦ 4 dc fe hg & ( ¦ F ba 4 & 7 ( we e & 7 e( x

We will say that a harmonic function is quasi-bounded if and only if it admits an absolutely continuous boundary measure (for the reasons for this terminology, see [He69,). The analogous result for the Smirnov class will, as can be expected, involve quasi-bounded harmonic functions.

All authors supported by the PICS program no 1019 of Generalitat de Catalunya and CNRS. First and third author also supported by European Commission Research Training Network HPRN-CT-2000-00116. Second author also supported by the DGICYT grant BFM2002-04072-C02-01 and the CIRIT grant 2001-SGR00172. Third author also supported by DGICYT grant BFM2002-00571 and the CIRIT grant 2001-SGR00431. 1 C © & ( ( X © & ( @ U( 5 ( ' ( 9

WEAKER CONDITIONS FOR THE EXISTENCE OF HARMONIC MAJORANTS

In this section we state first a sufficient condition implied by a result of Borichev on a similar problem. On the other hand, we also prove the necessary condition of Proposition 1.8 and show that it is not sufficient.

Theorem 7.1 (BNT). Given a collection of nonnegative data ! o ¥ ¢ In 9 T , there exists a finite positive measure 7 on

if and only if

This is an analogue of the discretized version of Theorem 1.2(d), (as in Lemma 6.3) obtained by considering only measures of type A £ U£ ( ? Theorem 7.1 together with the estimate £ 4 provide a sufficient (but not necessary) con- dition for domination by true harmonic functions, which is clearly less restrictive than requiring that £ ! £ 9 & ©¢ ¡ ( , but easier to check in concrete examples than the characterizing condition of Theorem 1.4.