Pierre Ronco
email: pierreronco@yahoo.fr

Hanna Debiec

A podocyte view of membranous nephropathy: From Heymann nephritis to the childhood human disease

Keywords: membranous nephropathy (MN), podocyte, neutral endopeptidase (NEP), cationic bovine serum albumin (cBSA), complement pathways, regulated intramembrane proteolysis (RIP)

Membranous nephropathy (MN) is characterized by an accumulation of immune deposits on the subepithelial side of the glomerular basement membrane, which results in complement activation and proteinuria. Since 2002, several major antigens of the podocyte have been identified in human MN, the first one being neutral endopeptidase (NEP), the alloantigen involved in neonatal cases of MN that occur in newborns from NEP-deficient mothers. This discovery opened the field to the major advances that have occurred since then in the pathophysiology and treatment of MN. It is remarkable that experimental models such as Heymann nephritis and cationic bovine serum albumin-induced MN in the rabbit predicted the pathomechanisms of the human glomerulopathy.

The podocyte is at the center of the pathogenesis of MN either by providing a source of endogenous antigens or by creating an environment favorable to deposition and accumulation of immune complexes containing exogenous (non-podocyte) antigens. The podocyte is also a victim of complement activation and antibody blocking activity against enzymes or receptors. A search for innovative drugs aimed at protecting this cell against complement activation and the effects of prolonged ER stress, has become a priority.

INTRODUCTION

Membranous nephropathy (MN) is a rare disease (annual incidence, 1/100,000), but a major cause of nephrotic syndrome in the adult [START_REF] Ronco | Pathophysiological advances in membranous nephropathy: time for a shift in patient's care[END_REF]. It is characterized by an accumulation of immune deposits on the outer aspect of the glomerular capillary wall, by the podocyte. The immune deposits consist of IgG, mainly IgG4 and IgG1, of antigens that have long eluded identification, and of the membrane attack complex of complement C5b-9 (MAC). The formation of subepithelial immune deposits results in complement activation which is responsible for functional impairment of the glomerular capillary wall causing proteinuria. The so-called "idiopathic" (or primary) MN, without any identified etiology, is an autoimmune disease involving podocyte antigens, while secondary forms involve exogenous antigens such as viral and tumoral antigens.

Advances in medical science often proceed by steps which are highly interdependent. New, groundbreaking findings with important clinical implications often result from the combination of faithful experimental models and careful clinical observations. This is well illustrated by the story of MN which started more than 50 years ago. It is remarkable that in this disease, experimental models predicted the pathophysiology of the human glomerulopathy. In the past 15 years, great progress has been made in the understanding of the molecular pathomechanisms of human MN. These advances were inspired by studies of experimental models of MN, such as Heymann nephritis [START_REF] Couser | Experimental glomerulonephritis in the isolated perfused rat kidney[END_REF][START_REF] Heymann | Production of nephrotic syndrome in rats by Freund's adjuvants and rat kidney suspension[END_REF][START_REF] Van Damme | Experimental glomerulonephritis in the rat induced by antibodies directed against tubular antigens. V. Fixed glomerular antigens in the pathogenesis of heterologous immune complex glomerulonephritis[END_REF] and MN induced by cationic bovine serum albumin (BSA) in the rabbit [START_REF] Adler | Electrical charge. Its role in the pathogenesis and prevention of experimental membranous nephropathy in the rabbit[END_REF][START_REF] Border | Induction of membranous nephropathy in rabbits by administration of an exogenous cationic antigen[END_REF]. Studies of Heymann nephritis led to the concept that a podocyte antigen, megalin, could serve as a target of circulating antibodies leading to the in situ formation of immune complexes [START_REF] Kerjaschki | The pathogenic antigen of Heymann nephritis is a membrane glycoprotein of the renal proximal tubule brush border[END_REF][START_REF] Kerjaschki | Immunocytochemical localization of the Heymann nephritis antigen (gp330) in glomerular epithelial cells of normal Lewis rats[END_REF], which put the podocyte at the center of disease mechanisms. On the other hand, cationic BSArelated MN first illustrated the case of 'planted' antigen [START_REF] Debiec | Early-childhood membranous nephropathy due to cationic bovine serumalbumin[END_REF]. In humans, progress in this area started in 2002 with the identification of neutral endopeptidase (NEP; also known as neprilysin) as the responsible antigen in a rare subset of patients with alloimmune neonatal MN [START_REF] Debiec | Antenatal membranous glomerulonephritis due to anti-neutral endopeptidase antibodies[END_REF][START_REF] Debiec | Role of Truncating mutations in MME gene in feto-maternal alloimmunization and neonatal glomerulopathies[END_REF]. This finding provided the proof of concept that a podocyte antigen could be responsible for human MN, as is the case for megalin in the rat, and laid the foundation for the identification of the M-type receptor for secretory phospholipase A2 (PLA2R1), which was the first podocyte autoantigen shown to be involved in idiopathic MN in humans, soon followed by thrombospondin domain-1 containing 7A [START_REF] Beck Lh | M-type phospholipase A2 receptor as target antigen in idiopathic MN[END_REF][START_REF] Tomas | Thrombospondin type-1 domain-containing 7A in idiopathic membranous nephropathy[END_REF]. Non-podocyte circulating antigens can also be involved as is the case in children for cationic bovine serum albumin (BSA) or enzymes used in enzyme replacement therapy for lysosomal storage disorders, [START_REF] Debiec | Early-childhood membranous nephropathy due to cationic bovine serumalbumin[END_REF] who represent the human counterpart to the rabbit model of "planted" antigen first described by Border et al in the early 1980's [START_REF] Border | Induction of membranous nephropathy in rabbits by administration of an exogenous cationic antigen[END_REF]. In these diseases also, the podocyte is an actor because of defective clearance properties. This review will focus on the role of the podocyte, both as a culprit and a victim. in Heymann nephritis, NEP-related neonatal MN, and other childhood MN.

HEYMANN NEPHRITIS: THE PODOCYTE AS A TARGET OF ANTI-MEGALIN

ANTIBODIES

In 1959, Heymann et al [START_REF] Heymann | Production of nephrotic syndrome in rats by Freund's adjuvants and rat kidney suspension[END_REF] described a rat model of MN, referred to as active Heymann nephritis, which was induced by the immunization of Lewis rats with crude kidney extracts and was reminiscent of the disease in humans. Because the subepithelial deposits were induced by fractions of renal brush-border membrane from the proximal tubule rather than by glomerular extracts, the deposits were initially believed to result from the glomerular trapping of circulating immune complexes composed of brush-border-related antigens and the corresponding antibodies. Subsequently, however, the development of passive Heymann nephritis in rats injected with rabbit anti-rat brush-border antibodies argued against a role for circulating immune complexes [START_REF] Couser | Experimental glomerulonephritis in the isolated perfused rat kidney[END_REF][START_REF] Van Damme | Experimental glomerulonephritis in the rat induced by antibodies directed against tubular antigens. V. Fixed glomerular antigens in the pathogenesis of heterologous immune complex glomerulonephritis[END_REF]. In the passive Heymann nephritis model, immune deposits occur within minutes of antibody injection, which argues against formation of immune compexes in the circulation followed by their deposition in the glomerular capillary wall. By using ex vivo and isolated perfused kidney systems, Van Damme et al [START_REF] Van Damme | Experimental glomerulonephritis in the rat induced by antibodies directed against tubular antigens. V. Fixed glomerular antigens in the pathogenesis of heterologous immune complex glomerulonephritis[END_REF] and Couser et al [START_REF] Couser | Experimental glomerulonephritis in the isolated perfused rat kidney[END_REF] demonstrated that anti-brush-border antibodies bound to an antigenic target located on podocytes, which indicated that the disease was caused by the in situ formation of immune complexes.

The principal autoantigen in both active and passive Heymann nephritis was identified in the early 1980s [START_REF] Kerjaschki | The pathogenic antigen of Heymann nephritis is a membrane glycoprotein of the renal proximal tubule brush border[END_REF][START_REF] Kerjaschki | Immunocytochemical localization of the Heymann nephritis antigen (gp330) in glomerular epithelial cells of normal Lewis rats[END_REF]. This autoantigen is a large podocyte transmembrane protein called megalin (also known as LDL receptor-related protein 2), with a molecular weight of ~600 kDa [START_REF] Saito | Mapping rat megalin: the second cluster of ligand binding repeats contains a 46-amino acid pathogenic epitope involved in the formation of immune deposits in Heymann nephritis[END_REF][START_REF] Yamazaki | All four putative ligand-binding domains in megalin contain pathogenic epitopes capable of inducing passive Heymann nephritis[END_REF]. This polyspecific receptor, which belongs to the LDL receptor superfamily, is present at the sole of podocyte foot processes in clathrin-coated pits. The formation of immune deposits is fueled by the de novo synthesis of megalin [START_REF] Allegri | Polyvalent antigen-antibody interactions are required for the formation of electron-dense immune deposits in passive Heymann's nephritis[END_REF]. Epitope mapping showed that although subepithelial immune deposits could be induced by all four megalin ligand-binding domains [START_REF] Raychowdhury | Induction of Heymann nephritis with a gp330/megalin fusion protein[END_REF][START_REF] Saito | Mapping rat megalin: the second cluster of ligand binding repeats contains a 46-amino acid pathogenic epitope involved in the formation of immune deposits in Heymann nephritis[END_REF][START_REF] Yamazaki | All four putative ligand-binding domains in megalin contain pathogenic epitopes capable of inducing passive Heymann nephritis[END_REF], full-blown disease with proteinuria was associated with a specific epitope located in the first ligand-binding domain of megalin, in a 60 kDa N-terminal fragment [START_REF] Oleinikov | A small N-terminal 60-kD fragment of gp600 (megalin), the major autoantigen of active Heymann nephritis, can induce a full-blown disease[END_REF][START_REF] Tramontano | Nested N-terminal megalin fragments induce high-titer autoantibody and attenuated Heymann nephritis[END_REF][START_REF] Tramontano | Conformation and glycosylation of a megalin fragment correlate with nephritogenicity in Heymann nephritis[END_REF], and with diffusion of immune response to other megalin epitopes, a process named intramolecular epitope spreading [START_REF] Shah | Intramolecular spreading in Heymann nephritis[END_REF].

Heymann nephritis-like MN was also reported in rats and mice [START_REF] Assmann | Membranous glomerulonephritis in the mouse[END_REF][START_REF] Assmann | Comparison of antigenic targets involved in antibody-mediated membranous glomerulonephritis in the mouse and rat[END_REF][START_REF] Assmann | Involvement of an antigen distinct from the Heymann antigen in membranous glomerulonephritis in the mouse[END_REF][START_REF] Ronco | A monoclonal antibody to brush border and passive Heymann nephritis[END_REF], in which the target antigen was dipeptidyl peptidase 4. More recently, a model of passive Heymann nephritis was induced in mice injected with antipodocyte serum [START_REF] Meyer | A new mouse model of immune-mediated podocyte injury[END_REF][START_REF] Meyer-Schwesinger | Nephrotic syndrome and subepithelial deposits in a mouse model of immune-mediated anti-podocyte glomerulonephritis[END_REF]. Although studies of Heymann nephritis have greatly contributed to our understanding of the pathomechanisms of MN (see below), there is no evidence that megalin is involved in human MN.

THE PODOCYTE NEUTRAL ENDOPEPTIDASE AND ALLOIMMUNE MN

This rare form of MN was first identified in a neonate borne with respiratory distress and acute renal failure followed by nephrotic range proteinuria and hypertension [START_REF] Debiec | Role of Truncating mutations in MME gene in feto-maternal alloimmunization and neonatal glomerulopathies[END_REF]. Because of early occurrence of the disease in the last trimester of pregnancy, we hypothesized that the mother became immunized during pregnancy and the pathogenic antibodies were transferred to the fetus (Figure 1). Involvement of NEP antigen was suspected because of interspecies differences in reactivity observed when human, rat and rabbit kidneys were incubated with sera from the mother and her infant. Identification of NEP was confirmed by immunoprecipitation of rat brush-border extracts [START_REF] Debiec | Antenatal membranous glomerulonephritis due to anti-neutral endopeptidase antibodies[END_REF][START_REF] Debiec | Role of Truncating mutations in MME gene in feto-maternal alloimmunization and neonatal glomerulopathies[END_REF]. Transfer experiments, in which a pregnant rabbit was injected with the mother's IgG fraction, and colocalization of NEP, IgG and C5-b9 within the subepithelial immune deposits, established that the disease was caused by anti-NEP antibodies [START_REF] Debiec | Antenatal membranous glomerulonephritis due to anti-neutral endopeptidase antibodies[END_REF][START_REF] Ronco | Molecular pathomechanisms of membranous nephropathy: from Heymann nephritis to alloimmunization[END_REF]. NEP not only is an antigen, it also is an enzyme involved in the degradation of biologically active peptides such as enkephalin (for this reason, it is also called enkephalinase), natriuretic peptides, endothelin, bradykinin and substance P in the vicinity of their receptors. In the kidney, NEP is present at the podocyte surface, in the brush border of the proximal tubule and in vessel walls. It is found in numerous organs but in very specific localizations, and in granulocytes as well. NEP is identical to the common acute lymphoblastic leukaemia antigen (CALLA) and to CD10.

Because the mother did not show any renal manifestation even though she had persistently high titers of anti-NEP antibody, we hypothesized that she might be deficient in NEP. This hypothesis was supported by the lack of reactivity of her granulocyte extracts with a panel of anti-NEP antibodies [START_REF] Debiec | Antenatal membranous glomerulonephritis due to anti-neutral endopeptidase antibodies[END_REF]. Four additional families with materno-fetal alloimmune MN were identified.

All immunized mothers were NEP deficient as a result of the same truncating mutation in exon 7 of the NEP gene [START_REF] Debiec | Role of Truncating mutations in MME gene in feto-maternal alloimmunization and neonatal glomerulopathies[END_REF][START_REF] Vivarelli | Genetic homogeneity but IgG subclassdependent clinical variability of alloimmune membranous nephropathy with anti-neutral endopeptidase antibodies[END_REF], (Figure 1). In the first family, the mother was a compound heterozygote for this mutation and for another truncating mutation in exon 15. A founder effect by a common ancestor is suspected because the same mutation in exon 7 was identified in all families.

Alloimmunization can be triggered by previous spontaneous miscarriages or by the ongoing pregnancy during which the mother's immune system is first exposed to the NEP of paternal origin on syncytiotrophoblastic cells. The NEP-deficient individuals had normal blood pressure, renal functional tests, and lymphocyte phenotype and function [START_REF] Debiec | Role of Truncating mutations in MME gene in feto-maternal alloimmunization and neonatal glomerulopathies[END_REF], whereas NEP null mice showed hypotension, greater sensitivity to septic shock, early onset Alzheimer's disease and prostatic tumors [START_REF] Lu | The control of microvascular per-meability and blood pressure by neutral endopeptidase[END_REF]. Because mutations in the NEP gene are asymptomatic, one can expect a high prevalence in selected populations.

After birth, renal failure and the nephrotic syndrome rapidly improved in all children in parallel with the disappearance from the child's blood of maternal IgG. However, a newborn had persisting renal failure requiring dialysis despite exchange transfusions. This case led to conclude that the anti-NEP IgG1 was responsible for the disease because the mother had high titers of anti-NEP IgG1 with very low titer of anti-NEP IgG4 and other IgG subclasses. This finding is in keeping with the capacity of IgG1 to activate complement whereas IgG4 usually fail to do so. Another patient with the longest follow-up, developed severe chronic renal failure with nephrotic range proteinuria leading to end-stage renal disease requiring transplantation. Although a repeat kidney biopsy could not be performed, the delayed renal manifestations might be induced by advanced MN combined with the postponed consequences of immunologically mediated antenatal nephron loss. Antibodies against idiotypes or allotypes on the maternal IgG might also have contributed to further progression of the disease. These observations suggest that cases of MN or chronic renal failure diagnosed during adolescence or early adulthood might be the late consequence of anti-NEPinduced antenatal renal disease.

Fetomaternal alloimmune glomerulopathy defines a novel organ-specific, fetomaternal disease because of alloimmunization induced by a genetic defect in the mother which leads to the development of MN in her fetus. Rhesus incompatibility was as yet the paradigm of fetomaternal diseases resulting from alloimmunization, which were long considered to be restricted to red blood cells and platelets. Our findings suggest that truncating mutations in genes coding for other podocyte antigens, asymptomatic for the carrier mother, could induce alloimmune glomerulopathy in the neonate. They open new perspectives for neonatal diseases affecting other organs than the kidney.

NON-PODOCYTE ANTIGENS IN CHILDHOOD MN

Altough not being a source of exogenous antigens (per definition), the podocyte also plays a role in MN caused by exogenous antigens which become planted in the glomerular capillary wall because it creates a favorable situation to the growth of immune complexes either by contributing to the synthesis of basement components and cell surface molecules with a negative charge (interacting with positively charged exogenous molecules coming from the blood) or because of its defective clearance properties. These 2 situations are illustrated by BSA-related MN and enzyme replacement therapy (ERT)-caused MN, respectively (Figure 2).

Bovine serum albumin and other dietary antigens

MN is one of the most common causes of nephrotic syndrome in adults but this disease is rarely diagnosed in paediatric populations [7,[START_REF] Chen | Idiopathic membranous nephropathy in pediatric patients: presentation, response to therapy, and long-term outcome[END_REF]. We have identified BSA as a novel antigen in children with MN aged <5 years [START_REF] Debiec | Early-childhood membranous nephropathy due to cationic bovine serumalbumin[END_REF]. These cases that can be considered as the human counterpart to the experimental model in rats and mice induced by cationized BSA [START_REF] Border | Induction of membranous nephropathy in rabbits by administration of an exogenous cationic antigen[END_REF][START_REF] Chen | Mouse model of membranous nephropathy induced by cationic bovine serum albumin: antigen dose-response relations and strain differences[END_REF][START_REF] Koyama | Effect of chemical cationization of antigen on glomerular localization of immune complexes in active models of serum sickness nephritis in rabbits[END_REF], open up new perspectives as to a role of dietary and environmental factors in MN, and more generally in glomerular diseases.

In this model of MN, podocyte antigens do not contribute to the formation of immune deposits which involve an exogenous antigen, BSA, that becomes implanted in the glomerular capillary wall (see Beck et al in this issue). The antigen charge is a key factor for disease induction.

Only the animals that were intravenously given chemically modified cationized BSA shortly after immunization with this antigen developed MN [START_REF] Border | Induction of membranous nephropathy in rabbits by administration of an exogenous cationic antigen[END_REF][START_REF] Chen | Mouse model of membranous nephropathy induced by cationic bovine serum albumin: antigen dose-response relations and strain differences[END_REF][START_REF] Koyama | Effect of chemical cationization of antigen on glomerular localization of immune complexes in active models of serum sickness nephritis in rabbits[END_REF]. When native, slightly acidic BSA was injected, animals did not develop the disease. Positively charged cationic BSA bound to the anionic glomerular capillary wall because of electrostatic interactions instead of being taken up by endocytosis in podocytes, and could then serve as a target for circulating antibodies.

We speculated that the same pathophysiological scenario might occur in humans. High levels of circulating anti-BSA antibodies of IgG1 and IgG4 subclasses were identified in children and adults with MN whereas anti-BSA IgE was not detected. Circulating BSA antigen was identified in the patients with the higher levels of anti-BSA antibodies. BSA immunopurified from children' sera migrated in the basic range of pH whereas in adult patients, it migrated in the neutral region as native BSA. BSA antigen was co-localized with IgG in subepithelial immune deposits only in the children with both circulating cationic BSA and anti-BSA antibodies, but in none of the adult patients with MN, which strongly supported the assumption that both cationic BSA serving as a "planted antigen" and concomitant immunization against this antigen could induce MN.

Additionally, decrease of cationic BSA and antibody levels during MN remission reinforced a direct causal link with the disease. Contrary to the observation made in the majority of adults with MN, neither PLA2R autoantibodies nor PLA2R antigen were detected in the blood and subepithelial immune deposits, respectively, in the children with BSA-related MN [START_REF] Debiec | Early-childhood membranous nephropathy due to cationic bovine serumalbumin[END_REF].

Cow's milk is the major source of BSA in young children. However, the origin of cationic BSA remains obscure. In modern dietary culture, food ingredients are subjected to a variety of processing conditions and contain many additives. Some of these technological processes might induce alterations of food proteins, which could in turn change their physicochemical properties and digestion in the gastrointestinal tract [START_REF] Davis | Protein modification by thermal processing[END_REF][START_REF] Sanchez | Consequences of heat treatment and processing of food on the structure and allergenicity of component proteins[END_REF][START_REF] Sathe | Effects of food processing on the stability of food allergens[END_REF]. On the other hand, it is conceivable that intestinal microbiota and frequent episodes of gastroenteritis as commonly observed in children might lead to pathological modifications of BSA [START_REF] Sreedharan | Gastrointestinal tract[END_REF].

Although very soluble in water and rich in amino-acids recognized by gastrointestinal enzymes, BSA is relatively resistant to digestion. Heat treatment of BSA induces denaturation of the protein and results in reduced proteolysis [START_REF] Alting | Incomplete elimination of the ABBOS epitope of bovine serum albumin under simulated gastrointestinal conditions of infants[END_REF]. at the relatively high pH (3-4) of the infants' stomach compared with that of adults (pH2) [START_REF] Schmidt | Raising the pH of the pepsin-catalysed hydrolysis of bovine whey proteins increases the antigenicity of the hydrolysates[END_REF] In addition, the amount of intact BSA that enters the intestinal circulation is likely higher during infancy before the gastrointestinal tract has matured and its barrier function has been established [START_REF] Sreedharan | Gastrointestinal tract[END_REF][START_REF] Van Elburg | Intestinal permeability in relation to birth weight and gestational and postnatal age[END_REF]. Finally, modification of BSA could directly affect how this protein is cleaved and processed in vivo. Both the predominance of a cationic form and the increased amount of absorbed BSA most likely contribute to the development of MN in young children.

Enzyme replacement therapy as a cause of alloimmune MN

Alloimmune reactions can occur during enzyme replacement therapy [START_REF] Brooks | Immune response to enzyme replacement therapy in lysosomal storage disorder patients and animal models[END_REF][START_REF] Koren | Immune responses to therapeutic proteins in humans--clinical significance, assessment and prediction[END_REF][START_REF] Richards | Immunologic considerations for enzyme replacement therapy in the treatment of lysosomal storage disorders[END_REF]. Because of the absence or very low levels of enzyme in many patients, therapeutic proteins are potential alloantigens that commonly trigger immunization. Allo-antibodies may be without clinical significance or may lead to hypersensitivity reactions, decreased bioavailability, and reduced efficacy of the therapeutic proteins. Only two cases of MN have been reported so far, but the prevalence of ERTinduced renal complications is probably underestimated.

A first case of nephrotic syndrome with mesangial and subepithelial Ig deposits was reported in a highly sensitized patient with Pompe disease treated with recombinant human alpha-glucosidase (rhGAA) [START_REF] Hunley | Nephrotic syndrome complicating alphaglucosidase replacement therapy for Pompe disease[END_REF]. The second case was a patient with mucopolysaccharidosis type VI treated with recommended doses of human recombinant arylsulfatase B (rhASB) who developed typical MN [START_REF] Debiec | Allo-immune membranous nephropathy and recombinant aryl sulfatase replacement therapy: A need for tolerance induction therapy[END_REF]. It is remarkable that this patient developed a nephrotic syndrome of rapid onset just one week after surgery although prior, regular screens for proteinuria were negative . Titers of anti-rhSAB antibody were high but in the same range as in other patients receiving rhASB without renal complication [START_REF] Harmatz | Enzyme replacement therapy for mucopolysaccharidosis VI: A phase 3, randomized, double-blind, placebo-controlled, multinational study of recombinant human N-acetylgalactosamine 4-sulfatase (recombinant human arylsulfatase B or rhASB) and follow-on, open-label extension study[END_REF][START_REF] Hendriksz | Design, baseline characteristics, and early findings of the MPS VI (mucopolysaccharidosis VI) Clinical Surveillance Program (CSP)[END_REF]. It is likely that drugs given during the perioperative period had deleterious effects on podocyte physiological properties. The patient received a cocktail of anesthetic drugs, which were before shown to increase glomerular permeability to proteins and alter podocyte function [START_REF] Giardino | Podocyte glutamatergic signaling contributes to the function of the glomerular filtration barrier[END_REF]. Further, the patient was anesthetized with sevoflurane, which can affect the conductance of Ca 2+ -activated K + channel expressed on podocytes [START_REF] Morton | Human podocytes possess a stretch-sensitive, Ca2+-activated K+ channel: Potential implications for the control of glomerular filtration[END_REF][START_REF] Namba | Inhibition of the human intermediate conductance Ca(2+)-activated K(+) channel, hIK1, by volatile anesthetics[END_REF].

These cases thus add a new cause to the list of MN aetiologies. They point to careful monitoring of proteinuria in all patients receiving ERT. Considering the critical requirement for ERT in patients with lysosomal enzyme deficiencies, it should be stressed that intensive immunosuppressive therapy can allow reintroduction of ERT resulting in a dramatic improvement of the patient's condition without relapse of the renal disease.

MECHANISMS OF PODOCYTE INJURY

In this chapter, we will focus on Heymann nephritis and childhood MN (Figure 2). The reader will find further information on PLA2R-and THSD7A in another chapter devoted to adult MN.

Mechanisms of immune complex formation

Megalin, a member of the LDL-receptor superfamily, is concentrated in clathrin-coated pits.

Epitope-specific IgG arrives from the circulation to form immune complexes with megalin on the podocyte membrane, followed by shedding and rapid immobilization in the glomerular basement membrane, thus preventing clearing of complexes by endocytosis of the podocyte. Large immune deposits form by repeated cycles of this mechanism [START_REF] Kerjaschki | Initial events in the formation of immune deposits in passive Heymann nephritis. gp330-anti-gp330 immune complexes form in epithelial coated pits and rapidly become attached to the glomerular basement membrane[END_REF]. Although megalin is clearly involved in the formation of immune deposits, one does not know whether the complete form of megalin or a proteolytic form is deposited extracellurly.

In the proximal tubule where megalin is strongly expressed, Biemersderfer's laboratory [START_REF] Biemesderfer | Regulated intramembrane proteolysis of megalin: linking urinary protein and gene regulation in proximal tubule?[END_REF] have shown that megalin is subjected to regulated intramembrane proteolysis (RIP), an evolutionarily conserved process linking receptor function with transcriptional regulation. Best characterized by the Notch signaling pathway, RIP involves regulated ectodomain shedding followed by γ-secretase-mediated release of the C-terminal, cytosolic domain. The C-terminus in turn translocates to the nucleus where it interacts with other proteins to regulate expression of specific genes. Megalin is subjected to protein kinase C-regulated, metalloprotease-mediated ectodomain shedding producing a membrane-associated C-terminal fragment (MCTF). The MCTF in turn forms the substrate for γ-secretase. These data implicate megalin as a central element of a Notch-like signaling pathway linking protein reabsorption and gene regulation in proximal tubule.

Megalin processing potentially triggered by anti-megalin antibodies, may also play an important role in podocyte alterations in Heymann nephritis.

Of note, PLA2R1 and THSD7A, the 2 major antigens in primary MN in the adult, are also subjected to proteolytic cleavage, and their involvement in RIP deserves further investigations.

Complement-mediated injury

Complement activation and formation of the C5b-9 attack complex, which is triggered by immune complex deposition, have major roles in sublethal podocyte injury and proteinuria [START_REF] Cunningham | Contrasting roles of complement activation and its regulation in membranous nephropathy[END_REF][START_REF] Cybulsky | Experimental membranous nephropathy redux[END_REF][START_REF] Nangaku | Cellular response to injury in membranous nephropathy[END_REF], (Figure 2). C5b-9, in sublytic quantities that do not kill podocytes, activates various mediators, such as phospholipases, protein kinases, cyclooxygenases, transcription factors and cytokines.

These signals influence podocyte cell metabolic pathways, structure and function of key cytoskeletal proteins, expression and localization of nephrin, the turnover of extracellular matrix, and DNA integrity [START_REF] Cunningham | Contrasting roles of complement activation and its regulation in membranous nephropathy[END_REF][START_REF] Cybulsky | Experimental membranous nephropathy redux[END_REF][START_REF] Nangaku | Cellular response to injury in membranous nephropathy[END_REF]. Evidence also shows that assembly of the sublytic C5b-9 complex on podocytes causes upregulation of NADPH oxidoreductase expression and its translocation to the cell surface. Subsequently, reactive oxygen species are produced locally, which leads to damage of podocyte lipids, membrane proteins and glomerular basement membrane components [START_REF] Cunningham | Contrasting roles of complement activation and its regulation in membranous nephropathy[END_REF][START_REF] Cybulsky | Experimental membranous nephropathy redux[END_REF][START_REF] Kerjaschki | Pathogenetic concepts of membranous glomerulopathy (MGN)[END_REF][START_REF] Nangaku | Cellular response to injury in membranous nephropathy[END_REF]. The injured podocytes also produce new extracellular matrix that is assembled between and around the immune deposits giving rise to the characteristic "spikes" that are well visualized by silver staining. Genetic polymorphisms resulting in reduced activity of circulating or podocyte complement regulatory proteins (CRP) as well as intrinsic or acquired defect of these proteins which control the alternative pathway of complement activation, could contribute as well to pathogenesis of MN and account for its varying severity.

Direct effector role of antibodies

Despite numerous studies supporting a role for complement in the pathogenesis of experimental MN, proteinuria and podocyte damage may also occur without assembly of C5b-9 [START_REF] Leenaerts | Active Heymann nephritis in complement component C6 deficient rats[END_REF][START_REF] Spicer | Induction of passive Heymann nephritis in complement component 6-deficient PVG rats[END_REF]. In addition to forming immune complexes, anti-podocyte antibodies may directly alter podocyte biology. Both megalin and PLA2R are cell surface receptors, and NEP cleaves biologically active peptides at the cell surface. Anti-megalin antibodies were shown to inhibit the uptake of lipoproteins that normally bind the multiligand receptor megalin [START_REF] Kerjaschki | Pathogenic antibodies inhibit the binding of apolipoproteins to megalin/gp330 in passive Heymann nephritis[END_REF]. The accumulation of lipoproteins (apoE and apoB) within the immune deposits together with the production of reactive oxygen species (ROS) potentially favors the formation of lipid peroxidation products.

The dual effect of pathogenic antibodies is well illustrated by anti-NEP antibodies. Mothers producing anti-NEP IgG1 transmitted the disease whereas those producing anti-NEP IgG4 only, failed to do so. These findings point to a critical role of the classical pathway of complement activation as attested by C1q deposition, whereas in adult cases of PLA2R-related autoimmune nephropathy anti-PLA2R IgG4 is prevailing and C1q deposits are rare. On the other hand, NEP also has important enzymatic in the kidney where it is present at the podocyte surface, in the brush border of the proximal tubule, and in vessel walls [START_REF] Dussaule | Characterization of neutral endopeptidase in vascular smooth muscle cells of rabbit renal cortex[END_REF]. NEP-specific IgG1 had a strong inhibitory effect, whereas IgG4 had a very weak effect. Inhibition of NEP activity by anti-NEP IgG1 antibodies may alter metabolism of a number of regulatory peptides and thus have an impact on glomerular hemodynamics, endothelial permeability, tubular function, and podocyte lesions [START_REF] Friedrich | Podocytes are sensitive to fluid shear stress in vitro[END_REF][START_REF] Hu | Anti-neutral endopeptidase, natriuretic peptides disarrangement, and proteinuria onset in membranous nephropathy[END_REF][START_REF] Lewko | Regulation of cGMP synthesis in cultured podocytes by vasoactive hormones[END_REF][START_REF] Lu | The control of microvascular per-meability and blood pressure by neutral endopeptidase[END_REF]. These alterations may be the cause of the unusual, severe ischemic lesions observed in the biopsy and of the disappearance of NEP from the brush border of proximal tubules, as also noted in ischemic and toxic acute kidney injuries [START_REF] Lebeau | Early proximal tubule injury in experimental aristolochic acid nephropathy: functional and histological studies[END_REF][START_REF] Nambi | Down regulation of kidney neutral endopeptidase mRNA, protein and activity during acute renal failure:possible mechanism for ischemiainduced acute renal failure in rats?[END_REF].

It is likely that antibodies directed against PLA2R and THSD7A also behave as bifunctional antibodies activating complement and blocking protein/receptor function, which must be taken into account in the interpretation of the mechanisms of the nephrotic syndrome and of podocytopenia, a major feature of adult MN.

Podocyte recovery and protective mechanisms

In parallel with the various cell damaging signals triggered by C5b-9, other signals are activated to limit complement-induced injury and to promote recovery (Figure 2). Nucleated cells are equipped with several mechanisms that support resistance to complement-dependent cytotoxicity [START_REF] Bohana-Kashtan | Cell signals transduced by complement[END_REF][START_REF] Cunningham | Contrasting roles of complement activation and its regulation in membranous nephropathy[END_REF][START_REF] Tegla | Membrane attack by complement: the assembly and biology of terminal complement complexes[END_REF].

One of the most important protective mechanisms is endocytosis and ectocytosis of the C5b-9. In disease state, C5b-9 inserted into the podocyte membrane is transported intracellularly and extruded into the urinary space, where it is considered as a dynamic marker of ongoing immunologic injury [START_REF] Kerjaschki | Transcellular transport and membrane insertion of the C5b-9 membrane attack complex of complement by glomerular epithelial cells in experimental membranous nephropathy[END_REF][START_REF] Moskovich | Live cell imaging of outward and inward vesiculation induced by the complement c5b-9 complex[END_REF]. In injured podocytes, an accumulation of misfolded proteins in the endoplasmic reticulum (ER) leads to ER stress [START_REF] Cybulsky | Endoplasmic reticulum stress in proteinuric kidney disease[END_REF][START_REF] Kon | Urinary C5b-9 excretion and clinical course in idiopathic human membranous nephropathy[END_REF]. Various studies in animal models and human diseases imply that complement may upregulate the adaptive unfolded protein response together with the ubiquitinproteasome system and autophagy [START_REF] Cybulsky | The intersecting roles of endoplasmic reticulum stress, ubiquitinproteasome system, and autophagy in the pathogenesis of proteinuric kidney disease[END_REF][START_REF] Meyer-Schwesinger | A new role for the neuronal ubiquitin C-terminal hydrolase-L1 (UCH-L1) in podocyte process formation and podocyte injury in human glomerulopathies[END_REF][START_REF] Wang | Autophagy can repair endoplasmic reticulum stress damage of the passive Heymann nephritis model as revealed by proteomics analysis[END_REF]. This response is one of the most potent cytoprotective mechanisms against accumulation of misfolded proteins inside podocytes. However, sustained ER stress may be cytotoxic by causing activation of pro-apoptotic pathways leading to destruction of podocytes [START_REF] Meyer-Schwesinger | A new role for the neuronal ubiquitin C-terminal hydrolase-L1 (UCH-L1) in podocyte process formation and podocyte injury in human glomerulopathies[END_REF]. Mechanisms of immune-mediated podocyte injury, disease progression, and recovery are summarized in Fig. 1 PERSPECTIVES Because the etiology and pathogenesis of MN involve the interplay of a variety of factors acting at different stages of the disease, optimum therapy should be multifaceted being directed at multiple targets and adapted with time (Figure 2). For example at initial stages, a combination of approaches including blocking antibody production, inhibiting complement activation at the site of complement activity, and protecting podocyte from injury triggered by complement activation and C5b-9 assembly should be considered. One major pitfall in MN is the lack of reliable biomarkers with diagnostic, prognostic, and therapeutic values. In addition to the now classical antibodies, the clinical complexity of the disease suggests that a combination of prognostic markers would be the best option for prediction of clinical outcome, recurrence on the graft, and personalized therapy In conclusion, the podocyte is at the center of the pathogenesis of MN either by providing a source of endogenous antigens or by creating an environment favorable to deposition and accumulation of immune complexes containing exogenous (non-podocyte) antigens. The podocyte is also a victim of complement activation and antibody blocking activity against enzymes or receptors. For these reasons, the podocyte should be targeted by innovative drugs aimed at protecting this cell against complement activation and the effects of prolonged ER stress. One should keep in mind that despite successful immunosuppressive treatment, decrease of proteinuria often lags much behind immunological remission, most likely because of irreversible alterations of podocytes.

Figure 1 .

 1 Figure 1. Pathophysiological scenario of fetomaternal alloimmune glomerulopathy (FMAIG) caused by maternal anti-NEP antibodies.

Figure 2 .

 2 Figure 2. Mechanisms of immune mediated podocyte injury and disease progression in MN and pathogenesis based therapeutic approach.

Acknowledgments

The research of the authors is supported by grants from European Research Council (ERC-2012-ADG_20120314 ; Grant Agreement n° 322947), and the 7th Framework Programme of the European Community, contract 2012-305608 « European Consortium for High-Throughput Research in Rare Kidney Diseases (EURenOmics) »


Nathael Gozlan
email: natael.gozlan@parisdescartes.fr

Mokshay Madiman
email: madiman@udel.edu

Cyril Roberto
email: croberto@math.cnrs.fr

Paul-Marie Samson
email: paul-marie.samson@u-pem.fr

Uhlenbeck

DEVIATION INEQUALITIES FOR CONVEX FUNCTIONS MOTIVATED BY THE TALAGRAND CONJECTURE

Keywords: Mathematics Subject Classification. 60E15, 32F32 and 26D10 Ehrhard inequality, Talagrand conjecture, Hypercontractivity, Ornstein-

published or not. The documents may come

Introduction

In the late eighties, Talagrand conjectured that the "convolution by a biased coin", on the hypercube {-1, 1} n , satisfies some refined hypercontractivity property. We refer to Problems 1 and 2 in [17] for precise statements. A continuous version of Talagrand's conjecture for the Ornstein-Uhlenbeck operator has recently attracted some attention [START_REF] Ball | L 1 -smoothing for the Ornstein-Uhlenbeck semigroup[END_REF][START_REF] Eldan | Regularization under diffusion and anti-concentration of temperature[END_REF]11]; in particular, it was resolved by [START_REF] Eldan | Regularization under diffusion and anti-concentration of temperature[END_REF]11] by first proving a deviation inequality for log-semiconvex functions above their means under Gaussian measure. In this paper, we discuss a simpler approach to proving this deviation inequality for the special case of log-convex functions (which is already of interest).

Let us start by presenting the continuous version of Talagrand's conjecture and the history of its resolution. Denote by γ n the standard Gaussian (probability) measure in dimension n, with density x → (2π) -n/2 exp -|x| 2

2

(where |x| denotes the standard Euclidean norm of x ∈ R n ) and, for p ≥ 1, by L p (γ n ) the set of measurable functions f : R n → R such that |f | p is integrable with respect to γ n . Then, given g ∈ L 1 (γ n ), the Ornstein-Ulhenbeck semi-group is defined as (1.1) P t g(x) := g e -t x + 1e -2t y dγ n (y)

x ∈ R n , t ≥ 0.

It is well known that the family (P t ) t≥0 enjoys the so-called hypercontractivity property [13, [START_REF] Nelson | The free Markoff field[END_REF]9] which asserts that, for any p > 1, any t > 0 and q ≤ 1 + (p -1)e 2t , P t g is more regular than g in the sense that, if g ∈ L p (γ n ) then P t g ∈ L q (γ n ) and moreover

P t g q ≤ g p .
However this property is empty when one only assumes that g ∈ L 1 (γ n ). A natural question is therefore to ask if the semi-group has anyway some regularization effect also in this case. Given g : R n → R non-negative with g dγ n = 1, by Markov's inequality and the fact that P s g dγ n = 1 we have

γ n ({P s g ≥ t}) ≤ 1 t ∀t > 0.
The continuous version of Talagrand's conjecture (adapted from [17, Problems 1 and 2]) states that as soon as s > 0,

lim t→∞ sup g≥0, g dγn=1 tγ n ({P s g ≥ t}) = 0.
The most recent paper dealing with this conjecture is due to Lehec [11] who proved that, for any s > 0 there exists a constant α s ∈ (0, ∞) (depending only on s and not on the dimension n) such that for any non-negative function g : R n → R + with g dγ n = 1,

(1.2) γ n ({P s g ≥ t}) ≤ α s t √ log t ∀t > 1
and this bound is optimal in the sense that the factor √ log t cannot be improved. In the first paper dealing with this question [START_REF] Ball | L 1 -smoothing for the Ornstein-Uhlenbeck semigroup[END_REF], Ball, Barthe, Bednorz, Oleszkiewicz and Wolff already obtained a similar bound but with a constant α s depending heavily on the dimension n plus some extra log log t factor in the numerator. Later Eldan and Lee [START_REF] Eldan | Regularization under diffusion and anti-concentration of temperature[END_REF] proved that the above bound holds with a constant α s independent on n but again with the extra log log t factor in the numerator. Finally the conjecture was fully proved by Lehec removing the log log t factor [11] and giving an explicit bound on α s , namely that α s := α max(1, 1 2s ) for some numerical constant α. In both Eldan-Lee and Lehec's papers, the two key ingredients are the following:

(1) for any s > 0, the Ornstein-Uhlenbeck semi-group satisfies, for all non-negative function g ∈ L 1 (γ n ),

Hess (log P s g) ≥ - 1 2s Id,
where Hess denotes the Hessian matrix and Id the identity matrix of R n . This is a somehow standard property easy to prove thanks to the kernel representation (1.1); (2) for any positive function g with Hess (log g) ≥ -βId, for some β > 0, and g dγ n = 1, it holds

γ n ({g ≥ t}) ≤ C β t √ log t ∀t > 1, with C β = α max(1, β).
It will be more convenient to deal with g = e f in the sequel so we move to this setting now. The last inequality can be reformulated as follows: for any f : R n → R with e f dγ n = 1 and Hess (f ) ≥ -βId, it holds

(1.3) γ n ({f ≥ t}) ≤ C β e -t √ t ∀t > 0.
We now describe the two main contributions of this note (which were independently obtained by Ramon van Handel). First, as a warm up, we give in Section 2 a short proof of (1.3) in dimension 1. The main argument of this proof is that due to the semi-convexity of f , the condition (2π) -1/2 e f -1 2 |x| 2 dγ = 1 implies a pointwise comparison between f and the function |x| 2 /2, which then can be turned into a tail comparison.

Then, in dimension n, we give in Section 3 a sharp version of the upper bound (1.3) for convex functions. Our main result states: Theorem 1.4. Suppose that f : R n → R is a convex function such that e f dγ n = 1, then

(1.5) γ n (f ≥ t) ≤ Φ( √ 2t), ∀t ≥ 0,
where

Φ(t) = 1 √ 2π +∞ t e -u 2 /2 du, t ∈ R.
Let us make a few comments on this result. First, using the following classical bound (which is asymptotically optimal)

(1.6) Φ(s) = 1 √ 2π ∞ s e -x 2 /2 dx ≤ 1 √ 2π ∞ s x s e -x 2 /2 dx = e -s 2 /2 √ 2πs , ∀s > 0,
one immediately recovers (1.3) with the constant C ′ 0 = 1/(2 √ π). Furthermore, the bound (1.9) is sharp. Indeed, for a given value of t ≥ 0, Inequality (1.9) becomes an equality for the function

f t (x) = √ 2tx 1 -t, x = (x 1 , . . . , x n ) ∈ R n .
Finally, since the Ornstein-Uhlenbeck semigroup preserves log-convexity (this follows from the fact that any positive combination of log-convex functions remains log-convex, see e.g [START_REF] Marshall | Inequalities: theory of majorization and its applications[END_REF] p. 649), Theorem 1.4 immediately implies the following corollary.

Corollary 1.7. Let g be a log-convex function such that g dγ n = 1, then for any s ≥ 0,

γ n (P s g ≥ t) ≤ Φ( 2 log(t)), ∀t ≥ 1.
In the special case when g is log-convex, Corollary 1.7 is a sharp improvement of Lehec's result (1.2). Note that for log-convex g, the constant α s can be taken independent of s unlike in (1.2), but this already followed from Lehec's inequality (1.3) combined with the preservation of log-convexity by the Ornstein-Uhlenbeck semigroup.

Another consequence of Theorem 1.4 is that a deviation inequality for structured functions also follows for other measures that can be obtained by "nice" pushforwards of Gaussian measure. Indeed, observe that for any coordinate-wise non-decreasing, convex function f on R n , and any convex functions g 1 , . . . , g n : R N → R, the composition f (g 1 (x), . . . , g n (x)) is convex on R N . Hence we immediately have the following corollary.

Corollary 1.8. For a standard Gaussian random vector Z in R N , let the probability measure µ on R n be the joint distribution of (g 1 (Z), . . . , g n (Z)), where g 1 , . . . , g n : R N → R are convex functions. Suppose that f : R n → R is a coordinate-wise non-decreasing, convex function such that R n e f dµ = 1. Then

(1.9) µ(f ≥ t) ≤ Φ( √ 2t), ∀t ≥ 0,
For example, consider the exponential distribution, whose density is e -x on R + = (0, ∞) and which can be realized as

Z 2 1 +Z 2 2 2
with Z 1 , Z 2 i.i.d. standard Gaussian. Clearly a product of exponential distributions on the line is an instance covered by Corollary 1.8, since we can take N = 2n and g i (x) =

x 2 i +x 2 i+1 2
. More generally, Corollary 1.8 applies to a product of χ 2 distributions with arbitrary degrees of freedom, and also to some cases with correlation (consider for example

N = 3, g 1 (x) = x 2 1 +x 2 2 2 and g 2 (x) = x 2 2 +x 2 3
2 ). The proof of Theorem 1.4 is given in Section 3. It relies on the Ehrhard inequality, which we recall now: according to [5, Theorem 3.2], if A, B ⊂ R n are two convex sets, then

(1.10) Φ -1 (γ n (λA + (1 -λ)B)) ≥ λΦ -1 (γ n (A)) + (1 -λ)Φ -1 (γ n (B)), ∀λ ∈ [0, 1],
where λA

+ (1 -λ)B := {λa + (1 -λ)b : a ∈ A, b ∈ B} denotes the usual Minkowski sum and Φ -1 is the inverse of the cumulative distribution function Φ of γ 1 : (1.11) Φ(t) = 1 √ 2π t -∞ e -u 2 /2 du, t ∈ R.
After Ehrhard's pioneer work, Inequality (1.10) was shown to be true if only one set is assumed to be convex by Lata la [10] and finally to arbitrary measurable sets by Borell [4]. See also [START_REF] Barthe | On Gaussian Brunn-Minkowski inequalities[END_REF][START_REF] Van Handel | The Borell-Ehrhard game[END_REF] and the references therein for recent developments on this inequality. Inequality (1.10) (for arbitrary sets A, B) is a very strong statement in the hierarchy of Gaussian geometric and functional inequalities. For instance, it gives back the celebrated Gaussian isoperimetric result of Sudakov-Tsirelson [START_REF] Sudakov | Extremal properties of half-spaces for spherically invariant measures[END_REF] and Borell [START_REF]rue des Saints-Pères 75270 Paris cedex 6[END_REF]. Another elegant consequence of (1.10) due to Kwapień is that if f is a convex function on R n which is integrable with respect to γ n , then the median of f is always less than or equal to the mean of f under γ n . The key ingredient in Kwapień's proof is the observation that the function

α(t) = Φ -1 (γ n (f ≤ t)), t ∈ R
is concave over R; this observation (already made in Ehrhard's original paper) also plays a key role in our proof of Theorem 1.4.

After the completion of this work, we learned that Paouris and Valettas [15] developed in a recent paper similar ideas to derive from (1.10) deviation inequalities for convex functions under their mean.

In Section 4, we give a second proof of Theorem 1.4, and also discuss (following an observation of R. van Handel) the difficulty of its extension to the log-semiconvex case.

Acknowledgement. The results of this note were independently obtained by Ramon van Handel a few months before us, as we learnt after a version of this note was circulated. Although he chose not to publish them, these observations should be considered as due to him. We are also grateful to him for numerous comments on earlier drafts of this note.

The Continuous Talagrand Conjecture in dimension 1

In the next lemma we take advantage of the semi-convexity property Hess (f ) ≥ -βId to derive information on f . More precisely we may compare f to x → |x| 2 /2. The result holds in any dimension, and we give two proofs for completeness. Lemma 2.1. Let f : R n → R and β ≥ 0 be such that e f dγ n = 1, f is smooth and Hess (f ) ≥ -βId. Then,

f (x) ≤ n 2 ln(1 + β) + 1 2 |x| 2 , ∀x ∈ R n . First proof of Lemma 2.1. Let h(x) = f (x) + β 2 |x| 2 .
By assumption on f , the function h is convex on R n and hence

h(x) = sup t∈R n { x, t -h * (t)} , ∀x ∈ R n ,
where

h * (t) := sup x∈R n { t, x -h(x)} , t ∈ R n
is the Legendre transform of h. Now, we have for all

t ∈ R n 1 = e f dγ n = exp h(x) - β 2 |x| 2 dγ n (x) ≥ (2π) -n/2 e -h * (t) exp x, t - 1 + β 2 |x| 2 dx = (1 + β) -n/2 exp -h * (t) + 1 2(1 + β) |t| 2 .
Therefore, for all t ∈ R n it holds

h * (t) ≥ - n 2 ln(1 + β) + 1 2(1 + β) |t| 2 .
In turn

h(x) = sup t { x, t -h * (t)} ≤ n 2 ln(1 + β) + sup t x, t - 1 2(1 + β) |t| 2 = 1 2 n ln(1 + β) + (1 + β)|x| 2
which leads to the desired conclusion.

Second proof of Lemma 2.1. Define h(x) = h(x) + β 2 |x| 2 , x ∈ R n and let γ n,β be the gaussian measure N (0, 1 1+β I), then it holds

1 = e h(x) dγ n (x) = (1 + β) -n/2 e h(x) dγ n,β (x)
For all a ∈ R n , the change of variable formula then gives

1 = (1 + β) -n/2 e -(1+β) 2 |a| 2
e h(y+a)-(1+β)y•a dγ n,β (dy).

The function y → h(y + a) -(1 + β)y • a is convex and the function x → e x is convex and increasing so the function y → exp h(y + a) -(1 + β)y • a is also convex. So applying Jensen inequality yields to

1 ≥ (1 + β) -n/2 e -(1+β) 2 |a| 2 exp h a + y dγ n,β (y) -(1 + β) y • a dγ n,β (y) = e -(1+β) 2 |a| 2 + h(a)
and so h(a) ≤ |a| In principle, one would hope to already get some deviation bound from the above lemma. More precisely, given f as in Lemma 2.1, we have

γ n ({f ≥ t}) ≤ γ n |x| 2 ≥ 2t -n ln(1 + β) ,
thanks to Lemma 2.1, and we are left with a tail estimate for a χ 2 distribution with n degrees of freedom. In dimension n = 1, the tail of the χ 2 distribution behaves like e -t / √ t. Therefore, the above simple argument already gives back the estimate (1.3) and thus provides a quick proof of the continuous Talagrand's conjecture for n = 1, moreover with clean dependence on β, as detailed below.

Theorem 2.3. If f : R → R is smooth and β ≥ 0 are such that e f dγ 1 = 1 and f ′′ ≥ -β pointwise, then

γ 1 ({f ≥ t}) ≤ 1 + β √ 2 e -t √ t ∀t ≥ 1.
Proof. Assume first that t ≥ (1 + β) ln(1 + β)/(2β). Using Inequality (1.6), we get from Lemma 2.1

γ 1 ({f ≥ t}) ≤ γ 1 |x| ≥ 2t -ln(1 + β) ≤ 2(2π) -1/2 exp -t + 1 2 ln(1 + β) 2t -ln(1 + β) = 1 + β π e -t √ t 1 1 -(ln(1 + β)/(2t)) ≤ 1 + β π e -t √ t 1 1 -(β/(1 + β)) = 1 + β √ π e -t √ t .
Now assume that t ≤ (1 + β) ln(1 + β)/(2β). Thanks to Markov's inequality, we have

γ 1 ({f > t}) ≤ e -t ≤ (1 + β) ln(1 + β)/(2β) e -t √ t ≤ √ 1 + β √ 2 e -t √ t ≤ 1 + β √ 2
e -t √ t where, in the third inequality, we used that ln(1 + β) ≤ β. The result follows.

Unfortunately this naive approach of using the pointwise bound from Lemma 2.1 is specific to dimension 1, since in higher dimension the tail of the χ 2 distribution does not have the correct behavior. It should be noticed that Ball et al. [START_REF] Ball | L 1 -smoothing for the Ornstein-Uhlenbeck semigroup[END_REF] also have a quick direct proof of the Talagrand conjecture for n = 1 that also uses a similar tail comparison with the χ 2 distribution, and also noticed that such a tail is not of the correct order for n ≥ 2.

The Deviation Inequality for Log-Convex Functions

Throughout this section f : R n → R is a convex function satisfying e f dγ n = 1 where γ n is the standard Gaussian measure on R n . Given s ∈ R, let A s := {f ≤ s} and ϕ(s) := Φ -1 (γ n (A s )) , where Φ -1 is the inverse of the Gaussian cumulative function Φ defined by (1.11).

The key ingredient in the proof of Theorem 1.4 is the concavity of the function ϕ that, as we shall see in the proof of the next lemma, is a direct consequence of Ehrhard's inequality (1.10). Lemma 3.1. Let f and ϕ be defined as above. Then ϕ is concave, non-decreasing, lim s→∞ ϕ(s) = +∞ and lim s→-∞ ϕ(s) = -∞.

The concavity of ϕ was first observed by Ehrhard in [5]. Below we recall the proof for the reader's convenience.

Proof. That ϕ is non-decreasing and satisfies lim s→∞ ϕ(s) = +∞ and lim s→-∞ ϕ(s) = -∞ is a direct and obvious consequence of the definition. Now we prove that ϕ is concave, using Ehrhard's inequality. Given λ ∈ [0, 1] and s 1 , s 2 ∈ R, we have, by convexity of f ,

A λs 1 +(1-λ)s 2 ⊃ λA s 1 + (1 -λ)A s 2 .
Hence, by monotonicity of Φ -1 , it holds

ϕ(λs 1 + (1 -λ)s 2 ) ≥ Φ -1 (γ n (λA s 1 + (1 -λ)A s 2 )) .
Then, Ehrhard's inequality (1.10) implies that

Φ -1 (γ n (λA s 1 + (1 -λ)A s 2 )) ≥ λΦ -1 (γ n (A s 1 )) + (1 -λ)Φ -1 (γ n (A s 2 )) = λϕ(s 1 ) + (1 -λ)ϕ(s 2 )
from which the concavity of ϕ follows.

Proof of Theorem 1.4. Let f and ϕ be defined as above. Then, it is enough to show that

ϕ(u) ≥ √ 2u, ∀u ≥ 0.
Since -ϕ : R → R∪{+∞} is convex by Lemma 3.1 and lower-semicontinuous, the Fenchel-Moreau Theorem applies and guarantees that

-ϕ(u) = sup t∈R {ut -ψ(t)} , ∀u ∈ R,
where

ψ(t) = (-ϕ) * (t) := sup u∈R {ut + ϕ(u)}
is the Fenchel-Legendre transform of -ϕ. Also we observe that, since lim u→∞ ϕ(u) = +∞, necessarily ψ(t) = +∞ for all t > 0 so that

ϕ(u) = -sup t≤0 {ut -ψ(t)} = inf t≤0 {-ut + ψ(t)} . Now observe that 1 = e f dγ n = ∞ -∞ e u γ n (f ≥ u) du = ∞ -∞ e u (1 -Φ(ϕ(u)) du = ∞ -∞ e u Φ(ϕ(u)) du
where we recall that Φ = 1 -Φ. Using integration by parts and the fact Φ is decreasing, we have for all

t ≤ 0 1 = ∞ -∞ e u Φ(ϕ(u)) du ≥ ∞ -∞ e u Φ(-ut + ψ(t)) du = (-t)e ψ(t) t +∞ -∞ e -v t Φ(v) dv = e ψ(t) t 1 √ 2π +∞ -∞ e -v t e -v 2 /2 dv = exp ψ(t) t + 1 2t 2 .
Therefore, for all t ≤ 0 it holds

ψ(t) ≥ - 1 2t .
In turn,

ϕ(u) = inf t≤0 {-ut + ψ(t)} ≥ inf t≤0 -ut - 1 2t = √ 2u
as expected.

4. Revisiting the deviation inequality, with a discussion of the semi-convex case Suppose that f : R n → R is a function such that e f dγ n = 1. Define µ f the distribution of f under γ n , that it to say

µ f (A) := γ n ({x ∈ R n : f (x) ∈ A}), ∀ Borel A ⊂ R.
Consider the monotone rearrangement transport map T f sending γ 1 onto µ f . It is defined by

T f (u) = F -1 f • Φ(u), ∀u ∈ R,
where F f (t) = µ f ((-∞, t]), t ∈ R, denotes the cumulative distribution function of µ f and

F -1 f (s) = inf{t : F f (t) ≥ s}, s ∈ (0, 1)
its generalized inverse.

The following proposition will yield to a slightly different proof of Theorem 1.4.

Proposition 4.1. With the notation above, if T f is κ-semiconvex, for some κ ≥ 0 i.e

T f ((1 -t)x + ty) ≤ (1 -t)T f (x) + tT f (y) + κ 2 t(1 -t)|x -y| 2 , ∀x, y ∈ R, ∀t ∈ [0, 1], then γ n ({f > u}) ≤ Φ 2u -log(1 + κ) , ∀u ≥ 1 2 log(1 + κ).
Proof. The κ-semiconvexity condition is equivalent to the convexity of the function x → T f (x) + κ x 2 2 . Now observe that 1 = e f dγ n = e y dµ f (y) = e T f (x) dγ 1 (x).

Applying Lemma 2.1 to the function T f in dimension 1, one concludes that

T f (x) ≤ 1 2 x 2 + 1 2 log(1 + κ), ∀x ∈ R.
This is equivalent to

Φ(x) ≤ F f 1 2 x 2 + 1 2 log(1 + κ)
and thus

F f (u) ≥ Φ 2u -log(1 + κ) , ∀u ≥ 1 2 log(1 + κ)
or in other words,

γ n ({f > u}) ≤ Φ 2u -log(1 + κ) , ∀u ≥ 1 2 log(1 + κ)
Second proof of Theorem 1.4. Suppose that f : R n → R is convex and such that e f dγ n = 1. Then according to Lemma 3.1, the function Φ

-1 • F f = T -1 f is concave.
Being also non-decreasing, its inverse T f is convex. Applying Proposition 4.1 with κ = 0 completes the proof.

In view of Proposition 4.1, a natural conjecture would be the following:

Conjecture. There exists a function

κ : [0, ∞) → [0, ∞) such that if f : R n → R is a smooth function such that Hess f ≥ -βId, β ≥ 0, then the map T f is κ(β)-semiconvex on R.
If this conjecture was true, then one would recover completely Eldan-Lee-Lehec result (1.3). Besides the convex case, let us observe that the conjecture is obviously true in dimension 1 for non-decreasing functions f . Indeed, f is clearly a transport map between γ 1 and µ f . Being non-decreasing, f is necessarily the monotone rearrangement map, that is to say : f = T f . Since f is κ-semiconvex, then so is T f .

Unfortunately, this probably too naive conjecture turns out to be false in general. As explained to us by R. van Handel, the presence of local minimizers for f breaks down the semi-convexity of T f . Let us illustrate this in dimension 1. Consider a function f : R → R of class C 1 such that f ′ (x) vanishes only at a finite number of points and such that there is some point

x o ∈ R and η > 0 such that f ′ (x o ) = 0, f ′ (x) < 0 on [x o -η, x o [ and f ′ (x o ) > 0 on ]x o , x o + η]. Denoting by t o = f (x o ), we assume that inf R f < t o ,
that is to say, f only presents a local minimizer at x o . Let us further assume that there are some α o , β o > 0 and some positive integer N such that, for all

t o -α o ≤ t < t o , Card{x ∈ R : f (x) = t} ≤ N and |f ′ (x)| ≥ β o for all x such that t o -α o ≤ f (x) < t o .
Claim. There is no λ ≥ 0 for which the map T := T f is λ-semi-convex.

It is not difficult to exhibit semi-convex functions f enjying the assumptions above, which disclaim the conjecture.

Proof of the Claim. First let us remark that if T was λ-semi-convex for some λ ≥ 0, then the map x → T (x) + λ 2 x 2 would be convex, and so would admit finite left and right derivatives everywhere. Moreover for a convex function the left derivative at some point is always less than or equal to the right derivative at this same point. So the λ-semi-convexity of T would in particular imply that

T ′ -(x) ≤ T ′ + (x), ∀x ∈ R.
We are going to show that T ′ -(u o ) > T ′ + (u o ) for some u o ∈ R which will prove the claim. Since, denoting F := F f ,

T ′ ± (u) = ϕ(u) F ′ ± • T (u)
, at every point u ∈ R where the derivative exists, one conludes that it is enough to show that F ′ -(t o ) < F ′ + (t o ) to have the desired inequality at u o = T -1 (t o ). Note that |T -1 (t o )| < ∞ because µ f ((t o , +∞)) = γ 1 ((T -1 (t o ), +∞)) > 0 and µ f ((∞, t o )) = γ 1 ((-∞, T -1 (t o ))) > 0, as easily follows from our assumptions.

According to the one dimensional general change of variable formula, the probability measure µ f admits the following density It is easily seen that M (h) → 0 as h to 0 + , which implies that F ′ + (t o ) = +∞. Now let us consider the left derivative. Let us note that one can assume without loss of generality that the left derivative exists at t o , since otherwise the function T would clearly not be semi-convex. For any h > 0, it holds

h(t) = x∈{f =t} ϕ(x) |f ′ (x)| , t ∈ R,
F (t o ) -F (t o -h) = to to-h h(t) dt ≤ h N √ 2πβ o
and so F ′ -(t o ) < +∞, which completes the proof of the claim.

where ϕ(x) = 1 √

 1 2π e -x 2 /2 , x ∈ R. Define ε o = max [xo-η,xo+η] ft o > 0 ; then, for h < ε o , it holds F (t o + h) -F (t o ) = to+h to h(t) dt ≥ h m M (h) , where m = inf [xo-η,xo+η] ϕ and M (h) = sup |f ′ (x)| : x ∈ [x oη, x o + η], f (x) ∈ [t o , t o + h] .