Katherine M Davies

PhD Sylvain Bohic
email: bohic@esrf.fr

PhD Asunción Carmona

PhD Richard Ortega

Veronica Cottam

PhD Dominic J Hare

PhD John P M Finberg

Stefanie Reyes

PhD Glenda M Halliday

PhD Julian F B Mercer

PhD Kay L Double
email: kay.double@sydney.edu.au

Copper pathology in vulnerable brain regions in Parkinson's disease

Keywords: copper, copper transporter 1 (Ctr1), human brain, Parkinson's disease, substantia nigra, superoxide dismutase 1 (SOD1)

Synchrotron-based X-ray fluorescence microscopy, immunofluorescence, and western blotting were used to investigate changes in copper (Cu) and Cu-associated pathways in the vulnerable substantia nigra (SN) and locus coeruleus (LC), and non-degenerating brain regions, in cases of Parkinson's disease (PD) and appropriate healthy and disease controls. In PD and incidental Lewy body disease (ILBD), levels of Cu and Cu transporter protein 1 (Ctr1), were significantly reduced in surviving neurons in the SN and LC. Specific activity of the cuproprotein superoxide dismutase 1 (SOD1) was unchanged in the SN in PD but was enhanced in the parkinsonian anterior cingulate cortex (ACC), a region with α-synuclein pathology, normal Cu and limited cell loss. These data suggest that regions affected by αsynuclein pathology may display enhanced vulnerability and cell loss if Cu-dependent protective mechanisms are compromised. Further investigation of copper pathology in PD may identify novel targets for the development of protective therapies for this disorder.

Introduction

Neurodegenerative cascades in Parkinson's disease (PD) involve protein aggregation and oxidative stress, although the triggers for these events are unknown. Changes in biometals have long been suspected to play a role in these cascades. Copper (Cu) is an important biometal in the brain, as exemplified by Menkes and Wilson's diseases; serious neurological disorders of Cu dyshomeostasis [START_REF] De Bie | Molecular pathogenesis of Wilson and Menkes disease: correlation of mutations with molecular defects and disease phenotypes[END_REF][START_REF] Schaefer | Genetic disorders of membrane transport. IV. Wilson's disease and Menkes disease[END_REF]. A significant decrease in total tissue Cu in the degenerating substantia nigra (SN) in PD has been consistently reported over a number of decades [START_REF] Ayton | Ceruloplasmin dysfunction and therapeutic potential for parkinson disease[END_REF][START_REF] Dexter | Increased nigral iron content and alterations in other metal ions occurring in brain in Parkinson's disease[END_REF][START_REF] Loeffler | Increased regional brain concentrations of ceruloplasmin in neurodegenerative disorders[END_REF][START_REF] Popescu | Iron, copper, and zinc distribution of the cerebellum[END_REF][START_REF] Uitti | Regional metal concentrations in Parkinson's disease, other chronic neurological diseases, and control brains[END_REF] and recent evidence suggests that peripheral Cu metabolism is altered in PD [START_REF] Larner | High precision isotope measurements reveal poor control of copper metabolism in Parkinsonism[END_REF]. The complexing of Cu with the PD-associated protein -synuclein increases aggregation and toxicity of this protein [START_REF] Rose | Mechanism of copper(II)-induced misfolding of Parkinson's disease protein[END_REF][START_REF] Wang | Copper binding regulates intracellular alpha-synuclein localisation, aggregation and toxicity[END_REF], possibly via stimulation of free radical production [START_REF] Meloni | Redox activity of alpha-synuclein-Cu is silenced by Zn(7)metallothionein-3[END_REF]. On the other hand, Cu is also a critical cofactor in a range of cuproproteins, including the key protective cellular antioxidant superoxide dismutase 1 (SOD1) [START_REF] Mccord | Superoxide dismutase. An enzymic function for erythrocuprein (hemocuprein)[END_REF]. Studies in model systems demonstrate that Cu depletion is associated with reduced activity of SOD1 and a concomitant increase in free radical production, which can be normalized by Cu supplementation [START_REF] Lombardo | Prolonged copper depletion induces expression of antioxidants and triggers apoptosis in SH-SY5Y neuroblastoma cells[END_REF][START_REF] Prohaska | Comparison of copper metabolism between brindled mice and dietary copper-deficient mice using 67Cu[END_REF]. SOD1 activity is reduced in the plasma of PD patients [START_REF] Torsdottir | Ceruloplasmin and superoxide dismutase (SOD1) in Parkinson's disease: a follow-up study[END_REF] and studies in animal models of PD suggest that overexpression of SOD1 increases neuronal survival [START_REF] Battaglia | Selective blockade of mGlu5 metabotropic glutamate receptors is protective against methamphetamine neurotoxicity[END_REF][START_REF] Botella | Superoxide dismutase overexpression protects dopaminergic neurons in a Drosophila model of Parkinson's disease[END_REF][START_REF] Tripanichkul | 17Beta-estradiol reduces nitrotyrosine immunoreactivity and increases SOD1 and SOD2 immunoreactivity in nigral neurons in male mice following MPTP insult[END_REF]. These data suggest that a reduction in brain Cu in PD may reduce SOD1-mediated antioxidant defense and contribute to neurodegenerative cascades. In the following work, we investigated changes in neuronal Cu levels and transport pathways to determine if changes in Cu are associated with a reduced antioxidant capacity in regions of neurodegeneration in PD.

inserm-00968923, version 1 -1 Apr 2014

Methods

Brain tissue samples

Brain tissue samples were requested and received from the New South Wales Tissue Resource Centre at the University of Sydney and the Sydney Brain Bank at Neuroscience Research Australia. The brain tissue samples were from patients with idiopathic PD and incidental Lewy body disease (ILBD), identified according to the diagnostic criteria of Dickson et al. [START_REF] Dickson | Neuropathological assessment of Parkinson's disease: refining the diagnostic criteria[END_REF], and from Alzheimer's disease (AD) and age-matched controls, identified according to the diagnostic criteria of Montine et al. [START_REF] Montine | National Institute on Aging-Alzheimer's Association guidelines for the neuropathologic assessment of Alzheimer's disease: a practical approach[END_REF] (Supplementary Tables 1, 2 and3). All cases of PD had received anti-parkinsonian medications prior to death, but none of the ILBD, AD, or control cases had received dopamine-replacement therapies. Cases with other neurological or neuropathological conditions were excluded and consent to use autopsy material for research purposes was obtained for all cases.

Brain regions investigated included the SN and locus coeruleus (LC), the brain regions most vulnerable to neuronal loss and Lewy body pathology in PD, the anterior cingulate cortex (ACC) which is affected by Lewy body pathology later in the course of PD and has limited cell loss, and the occipital cortex (OCx), a non-degenerating region in PD [START_REF] Braak | Staging of brain pathology related to sporadic Parkinson's disease[END_REF][START_REF] Harding | Clinical correlates of selective pathology in the amygdala of patients with Parkinson's disease[END_REF]. Samples were collected within 48 hours of death, dissected using polytetrafluoroethylene-coated blades (ProSciTech, Queensland, Australia), and the fresh tissue samples frozen immediately in liquid nitrogen, then stored at -80°C prior to analysis. Fixed tissue samples were stored in formalin prior to cryo-microtome sectioning.

Due to limited tissue availability, not every brain region was used in every experiment.

inserm-00968923, version 1 -1 Apr 2014

Measurement of neuronal and regional Cu levels

Total Cu levels within cells are the sum of different pools of Cu; free Cu and Cu bound to proteins. In the brain the majority of Cu is not free but is tightly regulated by being bound to a range of molecules, including Cu transport proteins and other molecules, such as neuromelanin [START_REF] Bohic | Intracellular chemical imaging of the developmental phases of human neuromelanin using synchrotron X-ray microspectroscopy[END_REF], an intracellular pigment found in vulnerable regions of the PD brain [START_REF] Hirsch | Melanized dopaminergic neurons are differentially susceptible to degeneration in Parkinson's disease[END_REF]. Total Cu levels were quantified within single neurons as well as regions in thin sections (20 µm) cut from fixed or frozen tissue samples using synchrotron radiation X-ray fluorescence microscopy (SRXFM) and particle induced X-ray emission (PIXE) microscopy. Experiments were conducted using the microprobes endstation at ID22 beamline of the European Synchrotron Radiation Facility and at the Diamond Light Source I18 beamline, as previously described [START_REF] Antharam | High field magnetic resonance microscopy of the human hippocampus in Alzheimer's disease: quantitative imaging and correlation with iron[END_REF][START_REF] Bohic | Intracellular chemical imaging of the developmental phases of human neuromelanin using synchrotron X-ray microspectroscopy[END_REF]. Micro-PIXE and micro-backscattering spectrometry (BS) analyses were performed simultaneously using the AIFIRA (Applications Interdisciplinaires des Faisceaux d'Ions en Région Aquitaine) facility to enable quantitative chemical analysis of trace elements in cells and tissues [START_REF] Carmona | Quantitative micro-analysis of metal ions in subcellular compartments of cultured dopaminergic cells by combination of three ion beam techniques[END_REF]. For each case (Supplementary Table 1) two fixed tissue sections of the SN, LC and OCx were investigated: one using SRXFM to raster-scanned individual pigmented neurons (6-10 per section/case) with a step size of 1 µm; and the other using PIXE microprobes to raster-scan over a 683 μm x 683 μm area, corresponding to the larger scan size available for incident protons of 3.0 MeV energy. In the SN and LC, only neurons exhibiting normal pigmented morphology were chosen to avoid measurement of extraneuronal neuromelanin (NM) pigment. Elemental concentrations derived from single neurons analyzed in each section were averaged and only one value from each patient was used in the analysis.

Results from fixed tissue were confirmed in the SN using the same methods on fresh frozen tissue sections from PD and control cases (Supplementary Table 2).

For quantification of whole tissue Cu content, samples of fresh frozen brain (35-60 mg) inserm-00968923, version 1 -1 Apr 2014 from the SN, ACC and OCx were subjected to closed-vessel microwave digestion (Milestone MLS1200) in 4 mL of concentrated nitric acid and 1 mL 35% hydrogen peroxide (both Seastar Chemicals, British Columbia, Canada). Digests were diluted w/w to c.a. 50 g with 1% nitric acid, and total metal concentrations were determined using an Agilent Technologies 7500cs inductively coupled plasma mass spectrometer (ICP-MS) (Forrest Hill, Victoria, Australia), as previously described [START_REF] Davies | Localization of copper and copper transporters in the human brain[END_REF]. Specificity for Cu was confirmed by measuring both m/z 63 and 65. Cu levels in the SN (subsequently referred to as TH-associated Cu levels) were calculated relative to nigral tyrosine hydroxylase determined by Western blot, as an index of dopamine neuron loss.

Preparation of brain tissue proteins for Western blotting and activity assays

Samples from fresh frozen brain (62-84 mg) from the SN, ACC and OCx were homogenized in 17 volumes of buffer solution (10 mM Tris-HCl, 0.25 M sucrose and 1 mM EDTA, pH 7.4) using a hand-held electric homogenizer with a polycarbonate probe (OmniTH, Kelly Scientific, Australia). For assaying SOD activity, homogenates of brain tissue proteins were centrifuged at 1,500 g for 5 min at 4 °C. For Western blot analysis, 1% SDS and 1:50 dilution of protease inhibitor cocktail (Sigma) was added to homogenates of brain tissue proteins before being centrifuged at 10,000 g for 15 min at 4 °C. Supernatants were collected and stored at -80 °C. Pierce ® BCA Protein Assay Kit (Thermo Scientific) was used to determine sample protein concentration.

Primary antibodies used for detection of Cu transport proteins using Western blotting and immunohistochemistry

Primary antibodies used are detailed in Supplementary Table 4. Specificity of each antibody was confirmed by including no primary antibodies and, for Atox1, by preadsorption of the Atox1 antibody with the antigen against which the primary antibody was raised (the antigen inserm-00968923, version 1 -1 Apr 2014

was not available for the other antibodies used). The ATPase antibodies used in this study have been well characterized in previous reports, and their specificity confirmed [START_REF] Cater | Copper binding to the N-terminal metalbinding sites or the CPC motif is not essential for copper-induced trafficking of the human Wilson protein (ATP7B)[END_REF][START_REF] Ke | Alteration of copper physiology in mice overexpressing the human Menkes protein ATP7A[END_REF]. Horseradish peroxidase-conjugated antibodies used for detection of proteins by Western blotting include: mouse anti-goat IgG, goat anti-mouse IgG, and donkey anti-sheep/goat IgG (all Millipore, USA). Fluorescent antibodies used for detection of proteins include: donkey anti-goat Alexa Fluor® 594 IgG (H+L), donkey anti-sheep Alexa Fluor® 594 IgG (H+L), donkey anti-sheep Alexa Fluor® 488 IgG (H+L), donkey anti-mouse Alexa Fluor® 594 IgG (H+L), and donkey anti-rabbit Alexa Fluor® 488 IgG (H+L), all purchased from Invitrogen (Eugene, Oregon, USA).

SDS-PAGE and Western blot analysis of Cu transport proteins

The following protein amounts were loaded onto 4-12% Bis-Tris Criterion pre-cast gels (Bio-Rad); Copper transporter 1 (Ctr1): 40 mg, Atox1: 80 mg, ATP7A: 100 mg, ATP7B: 100 mg, SOD1: 2.5 mg, Tyrosine Hydroxylase (TH): 40 mg, and separated by SDS-PAGE in MES (Atox1) or MOPS (Ctr1, ATP7A, ATP7B, SOD1, TH) buffer (Bio-Rad), according to manufacturer's instructions (Bio-Rad). Separated proteins were transferred to Immobilon-P SQ PVDF (Millipore) (Ctr1, Atox1, SOD1, and TH) or nitrocellulose (Millipore) (ATP7A and ATP7B) membranes. Following transfer, membranes were blocked in either 5% skim milk (Ctr1, ATP7A, ATP7B, SOD1, and TH), or 1% casein (Atox1), in PBS containing 0.1% Tween ® 20 and incubated with appropriate primary antibodies diluted in PBS containing 0.1% Tween ® 20 and 1% skim milk (Ctr1, ATP7A, ATP7B, SOD1, and TH) or 1% casein (Atox1) at the following concentrations: Ctr1 (1:200), Atox1 (1:250), ATP7A (1:750), ATP7B

(1:1500), SOD1 (1:10,000), and TH (1:5000) overnight at 4 °C. Following incubation with horseradish peroxidase-conjugated mouse anti-goat IgG (1:10,000), goat anti-mouse IgG (1:2,000), donkey anti-sheep/goat IgG (1:2,000), or donkey anti-rabbit IgG (1:5000), protein inserm-00968923, version 1 -1 Apr 2014 signals were obtained using an ECL Western blotting detection system (Bio-Rad) as per the manufacturer's instructions, and developed using the Chemi-Doc XRS (Bio-Rad). Signal intensities were quantified by densitometry using Quantity One® software (Bio-Rad) and normalized to -actin (1:10,000) levels. TH levels in the SN were quantified by Western blot to provide an indication of dopaminergic cell loss. As expected, TH levels in the PD SN were significantly reduced by 60% (p=0.013). Levels of the Cu transporters Ctr1 (subsequently referred to as TH-associated Ctr1 levels), Atox1, ATP7A, and ATP7B were calculated relative to TH levels in the SN.

Cu transport protein immunohistochemistry

Formalin fixed slide-mounted paraffin-embedded sections (7µm) were antigen retrieved by either incubation with Tris-EDTA buffer (0.01M Tris, 1 mM EDTA, pH 9.0) at 95 °C for 20 mins (Ctr1), or by microwave irradiation in citrate buffer (pH 6.0) for 15 mins (ATP7A and ATP7B). Non-specific peroxidases were blocked with 1% H 2 O 2 (Fronine Laboratory Supplies) in 50% ethanol for 30 mins at room temperature. Non-specific antigen sites were blocked with a mixture of 1% bovine serum albumin (BSA; Sigma) and 5% normal horse serum (NHS; Australis) (Ctr1), Animal Free Blocker™ (Vector Laboratories) (ATP7A), or 3% BSA (Sigma) (ATP7B), for 20 mins at room temperature. Primary antibodies for immunohistochemistry were used at the following dilutions: Ctr1 (1:100), ATP7A (1:300), ATP7B (1:1,000). Primary antibodies for Ctr1, ATP7A and ATP7B were detected using biotinylated IgG antibodies (Vector Laboratories) (1:200), followed by Vector Elite Kit tertiary antibody complex (Vector Laboratories) (1:100), and visualized using 3'3'diaminobenzidine (DAB; Sigma) with a cresyl violet counterstain (Sigma). Images were taken using a Zeiss AxioCam HRc microscope. 3), formalin fixed slidemounted sections (20 µm; Ctr1; Supplementary Table 1), and formalin fixed slide-mounted paraffin-embedded sections (7 µm; ATP7A, ATP7B; Supplementary Table 3) were antigen retrieved in Tris-EDTA buffer (0.01 M Tris, 1 mM EDTA, pH 9.0) at 95 °C for 25 mins (Ctr1), or citrate buffer (pH 6.0) with microwave irradiation for 15 mins (ATP7A, ATP7B), and permeabilized with 50% ethanol. Non-specific antigen sites were blocked in 10% NHS (Australis; Ctr1), or 0.25% Casein (Sigma; ATP7A, ATP7B) for 1 hour at room temperature and midbrain sections double-stained for the dopaminergic cell marker tyrosine hydroxylase (TH; 1:500) and for Ctr1 (1:25), ATP7A (1:100), or ATP7B (1:400). Primary antibodies were detected using secondary IgG antibodies conjugated to Alexa Fluor® 594 or Alexa Fluor® 488 fluorophores (Invitrogen). Sections were analysed using a Nikon D-Eclipse C1 Si scanning confocal microscope. The cross-reactivity and specificity of the fluorescence reactions were tested by incubating each primary antibody singularly with the secondary antibody solution containing two fluorophores. No cross reactivity was observed.

SOD1 Activity Assay

SOD activity was determined using SOD determination kit (Sigma, Cat. #19160) according to the manufacturer's instructions. Total SOD activity was measured in supernatants, and SOD2 activity was measured by the addition of 5 mM potassium cyanide (Sigma). SOD1 activity (units/μg protein) was measured as the difference between total SOD, and SOD2, activity.

The specific activity of SOD1 was determined by normalizing SOD1 activity (units/μg inserm-00968923, version 1 -1 Apr 2014 protein) to SOD1 protein levels. A single PD case was excluded from analysis due to a medication history of selegiline, demonstrated to increase midbrain SOD1 activity [START_REF] Takahata | Effects of selegiline on antioxidant systems in the nigrostriatum in rat[END_REF].

Statistical Analyses

Statistical analyses were performed using PASW v18 (SPSS inc.). Univariate analysis of variance followed by Bonferroni's t-test was used to test for differences between groups.

Linear regression was used to test for relationships between variables. A p-value of less than 0.05 was accepted as the level of significance. Quantification of iron (Fe) and zinc (Zn) levels by PIXE revealed a 49% increase in Fe (p<0.05) in NM-containing neurons in fresh frozen PD SN compared with fresh frozen control SN, while Zn concentrations were similar between PD and control cases (Supplementary Table 5). These findings were confirmed by SRXFM on an adjacent set of sections (26% increase in Fe, p< 0.05, Supplementary Table 6). In contrast, Fe levels were not altered in fixed tissue within multiple individual pigmented neurons, suggesting that chemical fixation masks changes in Fe in PD. No other changes were found for NM-associated elements investigated across the disease groups in either the SN or LC.

Results

Decreased Cu precedes regional cell loss in PD

Reduced Ctr1 is an early change in regions of cell loss in PD

Increased SOD1 levels relates to reduced Cu levels in regions of cell loss in PD

SOD1 levels and specific activity were not associated with age, PM delay, or tissue pH.

Consistent with our previous findings [START_REF] Davies | Localization of copper and copper transporters in the human brain[END_REF], in the control brain Cu levels were significantly higher in the SN compared with those in the OCx and ACC (p=0.001, p≤0.001, respectively). Confirming the decrease in SN Cu at the neuronal level, total tissue Cu was reduced by 48% in the PD SN compared with control (p=0.032). Regional protein levels of SOD1 in the control SN, OCx, and ACC did not vary significantly, and were not correlated with Cu levels; in contrast in the PD SN there was significantly more SOD1 compared with

Discussion

The current data demonstrate that Cu, and the expression of the Cu transport protein Ctr1, are decreased within the intraneuronal environment of surviving dopaminergic neurons of the SN and LC in ILBD and PD (Fig 1,2,3,and 4). As ILBD is suggested to represent preclinical PD [START_REF] Dickson | Neuropathological assessment of Parkinson's disease: refining the diagnostic criteria[END_REF]) and these cases had not taken anti-parkinsonian medications, this suggests these changes occur early in the disease process and are independent of treatment. As Cu and Ctr1 levels were not altered in the degenerating LC in AD, it is unlikely that the reduction in Cu and Ctr1 observed in these regions in PD results simply from neurodegeneration per se. Further, as Cu and Ctr1 levels were unchanged in regions that do not degenerate in PD, it appears that this early decrease is specific to regions vulnerable to neuronal cell loss in this disorder. show that a decrease in Ctr1 leads to significant neuronal dysfunction and death. Deletion of Ctr1 is embryonic lethal and epithelial cell-specific Ctr1 knockout mice display neonatal defects and severe growth and viability abnormalities [START_REF] Nose | Ctr1 drives intestinal copper absorption and is essential for growth, iron metabolism, and neonatal cardiac function[END_REF]. In addition, organisms lacking Ctr1 are deficient in oxidative stress protection [START_REF] Dancis | The Saccharomyces cerevisiae copper transport protein (Ctr1p). Biochemical characterization, regulation by copper, and physiologic role in copper uptake[END_REF][START_REF] De Freitas | Yeast lacking Cu-Zn superoxide dismutase show altered iron homeostasis. Role of oxidative stress in iron metabolism[END_REF][START_REF] Knight | A widespread transposable element masks expression of a yeast copper transport gene[END_REF]. While our data demonstrate a change in Ctr1 in the parkinsonian brain, the regulation of intracellular Cu is complex and other yet to be characterized Cu transport pathways may also be altered in PD. It has recently been shown, for example, that down-regulation of Ctr1 in a murine model of Wilson's disease is linked to the appearance of a redundant mechanism of Cu transport, the 2 kDa Small Copper Carrier, SCC [START_REF] Gray | Urinary copper elevation in a mouse model of Wilson's disease is a regulated process to specifically decrease the hepatic copper load[END_REF]. A second Ctr protein, structurally related to Ctr1, has also been identified in mammals, designated Copper transporter 2 (Ctr2). This protein has not been identified in human tissue but is thought to play a role in low affinity Cu import [START_REF] Bertinato | Ctr2 is partially localized to the plasma membrane and stimulates copper uptake in COS-7 cells[END_REF] and in intracellular Cu mobilization [START_REF] Rees | Identification of a vacuole-associated metalloreductase and its role in Ctr2-mediated intracellular copper mobilization[END_REF].

To determine the functional consequences of reduced intraneuronal Cu, we assessed a key antioxidant cuproprotein in the brain, SOD1, and identified an associated increase in levels of SOD1 protein in the degenerating PD SN (Fig 6A andB). The relationship we observed between Cu and SOD1 levels is consistent with the hypothesis that cellular Cu levels regulate SOD1 expression [START_REF] Brown | Oxygen and the copper chaperone CCS regulate posttranslational activation of Cu,Zn superoxide dismutase[END_REF][START_REF] Henchcliffe | Mitochondrial biology and oxidative stress in Parkinson disease pathogenesis[END_REF]. Despite this increase in SOD1 protein levels, specific activity of SOD1 was unchanged in the PD SN (Fig 7). In contrast, in the ACC, where Cu and Ctr1 levels are normal SOD1 specific activity was increased above that in the SN of the same brains (Fig 7) and there was a trend for SOD1 specific activity to be higher in the parkinsonian, compared with the control ACC (Fig 7). The increased SOD1 specific activity observed in the ACC where Cu levels are normal, but not in the Cu-deficient SN, suggests that a lack of Cu bioavailability may compromise the function of cuproproteins, such as SOD1. In support of this hypothesis, a profound decrease in the inserm-00968923, version 1 -1 Apr 2014 specific activity of another cuproprotein, ceruloplasmin, was recently reported in the SN in PD [START_REF] Ayton | Ceruloplasmin dysfunction and therapeutic potential for parkinson disease[END_REF], a finding with likely implications for iron accumulation and subsequent oxidative stress in PD.

Genetic alterations causing dysfunctional Cu transport proteins and reduced brain Cu levels result in severe neurodegeneration, as observed in Menkes disease [START_REF] Kaler | ATP7A-related copper transport diseases-emerging concepts and future trends[END_REF][START_REF] Strausak | Copper in disorders with neurological symptoms: Alzheimer's, Menkes, and Wilson diseases[END_REF]. The current understanding of neuronal death in PD supports a role for -synuclein and oxidative stress in these degenerative pathways. We have previously shown that the earliest deposition of -synuclein in the parkinsonian SN occurs on NM [START_REF] Halliday | Alpha-synuclein redistributes to neuromelanin lipid in the substantia nigra early in Parkinson's disease[END_REF]) and both regional and NM-associated Cu levels are significantly higher in the normal SN and LC compared with other brain regions [START_REF] Bohic | Intracellular chemical imaging of the developmental phases of human neuromelanin using synchrotron X-ray microspectroscopy[END_REF][START_REF] Davies | Localization of copper and copper transporters in the human brain[END_REF]. High Cu levels promote the formation of oligomeric (toxic) forms of α-synuclein [START_REF] Binolfi | Bioinorganic chemistry of Parkinson's disease: structural determinants for the copper-mediated amyloid formation of alpha-synuclein[END_REF][START_REF] Rose | Mechanism of copper(II)-induced misfolding of Parkinson's disease protein[END_REF][START_REF] Santner | Metalloproteomics and metal toxicology of alphasynuclein[END_REF][START_REF] Wright | Unique copper-induced oligomers mediate alphasynuclein toxicity[END_REF]. Our current data suggest that the presence of Cu associated with NM in these most vulnerable neurons may underlie the greater propensity of -synuclein to abnormally accumulate in pigmented catecholaminergic neurons, compared with non-pigmented brain regions, such as the ACC. Further studies are required to understand the relationships between NM, Cu and αsynuclein and their roles in neuronal death in the parkinsonian brain.

Our data demonstrate decreased cellular copper levels and disrupted copper pathways in vulnerable brain regions in PD, as well regional changes in SOD1 specific activity that reflect the pattern of neurodegeneration in PD. We also show that neuronal SOD1 protein levels are increased in association with a decrease in Ctr1 and Cu levels prior to cell loss in regions most vulnerable to PD-associated neurodegeneration. These changes are enhanced with increasing disease duration. The reduced activity of copper-dependent SOD1 shown here, and that of ceruloplasmin [START_REF] Ayton | Ceruloplasmin dysfunction and therapeutic potential for parkinson disease[END_REF], suggests that a reduction in cellular Cu compromises the inserm-00968923, version 1 -1 Apr 2014 ability of these neurons to defend themselves against oxidative stress, consistent with the severity of cell loss observed in these vulnerable regions in the PD brain. 5 and6, respectively. Significant differences were determined by multivariate analysis of variance (ANOVA), p<0.05. Alzheimer's disease (AD) patients, compared with normal controls. All tissues were chemically fixed, unless otherwise stated, in which case fresh frozen sections were analysed.

Figure legends

Data reported as mean elemental concentrations, expressed as mg/g dry weight, with standard deviations reported in parentheses.

inserm-00968923, version 1 -1 Apr 2014

  inserm-00968923, version 1 -1 Apr 2014 10 2.7. Cu transport protein immunofluorescence Due to the inherent difficulty of separating positive DAB staining from NM, immunofluorescence staining was performed on midbrain and pons tissue sections to examine the cellular localization of Ctr1, ATP7A, and ATP7B in SN and LC neurons. Formalin fixed free-floating midbrain sections (50 µm; Ctr1; Supplementary Table

  Quantitative analysis of fixed sections by SRXFM (Fig 1A-C) revealed a decrease in NMassociated Cu in the SN from PD and ILBD cases with respect to both age-matched controls (45% decrease in PD, p<0.0001 and 48% decrease in ILBD, p=0.0003) and AD cases (31% decrease in PD, p=0.003 and 34.5% decrease in ILBD, p=0.007) (Fig 1A). Our finding of decreased NM-associated Cu levels in the PD SN was confirmed in fresh frozen tissue sections (65%, p<0.0001; Fig 2). A representative set of selected SRXFM elemental images from typical NM-containing neurons is shown in Fig 2. SRXFM imaging of single pigmented neurons demonstrated that Cu was colocalized with NM and that there was no change in the spatial distribution of Cu between control and PD cases. Similarly, analysis of NM from LC neurons showed a 55% decrease in Cu content in PD compared with controls (p=0.001; Fig 1B) while no differences were seen in the grey matter of the OCx (Fig 1C). This data confirms a specific reduction in Cu within remaining NM-pigmented neurons in PD and inserm-00968923, version 1 -1 Apr 2014 shows that this deficit occurs in cases with ILBD and limited cell loss, suggesting an early deficit that precedes cell death and clinical symptoms. Complementary to SRXFM, we quantified intraneuronal Cu levels in separate tissue sections using micro-PIXE to assess Cu content in randomly sampled regions (683 mm x 683 mm) of the SN, LC, and OCx (Fig 1D-F). Alignment of the carbon map and optical microscopic view allowed independent quantification of melanized and non-melanized tissue within each region. Elemental concentrations obtained using micro-PIXE were within the range of those obtained using SRXFM. Micro-PIXE analysis of NM-containing neurons confirmed a decrease in Cu-associated NM in PD (54%, p=0.0012) and ILBD (60%, p=0.004) compared with controls (Fig 1D). Importantly, the Cu decrease was also observed in NM from neurons of the LC (Fig 1E) but not in the OCx (Fig 1F).

  TH-associated Ctr1 was significantly reduced in the SN of the PD brain (50% reduction in Ctr1, p=0.017, Fig 3A) and was significantly correlated with TH-associated Cu levels (p=0.036; Fig 3B) and disease duration (p=0.008; Fig 3C). We have previously described in the normal SN (Davies et al., 2013) robust Ctr1 staining localized with NM (Fig 4A). In the inserm-00968923, version 1 -1 Apr 2014 PD SN however, consistent with our Western blot data, NM-associated Ctr1 immunoreactivity was markedly reduced (Fig 4B). Interestingly, this was restricted to SN dopaminergic neurons, as robust Ctr1 NM staining was observed in pigmented neurons in the nearby ventral tegmental area (VTA) and retrorubral field (Fig 4B, inset). NM-associated Ctr1 immunoreactivity in the SN of AD cases was equivalent to that in control cases (Fig 4C), in contrast, a marked reduction in NM-associated Ctr1 immunoreactivity was observed in the majority of SN neurons in cases of ILBD (Fig 4D). NM-associated Ctr1 staining in the control LC was more variable compared to the consistently robust staining observed in the control SN. However, while neurons strongly expressing Ctr1 were frequently observed in control (Fig 4E) and AD cases (Fig 4G), strong Ctr1 immunoreactivity was rarely observed in the PD (Fig 4F) and ILBD LC (Fig 4H). We have previously described strong cytoplasmic expression of Ctr1 in neuronal cell bodies in the normal ACC (Davies et al., 2013) (Fig 5A). Interestingly, in contrast to the PD SN and LC, strong cytoplasmic expression of Ctr1 was preserved in the ACC in PD (Fig 5B), consistent with our Western blot data in this brain region (Fig 5C). Immunoreactivity for ATP7A, ATP7B, and Atox1 in SN and ACC did not differ between PD and control cases.

  inserm-00968923, version 1 -1 Apr 2014theOCx and ACC (p=0.002, p=0.007, respectively; Fig 6A and B) and SOD1 levels were correlated with Cu levels (p=0.004; Fig 6C). 3.4. Initial enhancement of SOD1 antioxidant activity in regions affected by late PD pathology Specific activity of SOD1 did not vary between the regions studied in the control brains (p=1.000; Fig 7), but in contrast, in the PD brains SOD1 specific activity was significantly greater in the ACC compared with the SN (p=0.032; Fig 7), and there was a trend for increased SOD1 specific activity in the parkinsonian ACC compared with the control ACC (p=0.076; Fig 7).

  Our data in the PD brain show an association between decreasing Ctr1 and Cu levels selectively in regions degenerating in PD (Fig 4), with the decrease in Ctr1 in PD enhanced in patients with longer disease durations and greater cell loss (Fig 3). Data from animal models inserm-00968923, version 1 -1 Apr 2014

Figure 1 .

 1 Figure 1. Intraneuronal copper (Cu) levels measured in fixed brain tissue by Synchrotron X-

Figure 2 .

 2 Figure 2. Representative map for copper (Cu) obtained from single intact neuromelanin-

Figure 3 .

 3 Figure 3. Tyrosine hydroxylase (TH)-associated Copper transporter 1 (Ctr1) levels in the

Figure 4 .

 4 Figure 4. Representative photomicrographs of Copper transporter 1 (Ctr1) in the control

Figure 5 .Figure 6 .

 56 Figure 5. The cellular localization (A-B) and levels (C) of the copper transport protein

Figure 7 .

 7 Figure 7. Superoxide dismutase 1 (SOD1) specific activity in the substantia nigra (SN),

  Figure 3

  Figure 7

Figure

  Figure(s) Click here to download high resolution image

inserm-00968923, version 1 -1 Apr 2014

Acknowledgements

Tissues were received from the New South Wales Tissue Resource Centre at the University of Sydney, supported by the National Health and Medical Research Council of Australia, Schizophrenia Research Institute and the National Institute of Alcohol Abuse and Alcoholism (NIH (NIAAA) R24AA012725), and from the Sydney Brain Bank, which is supported by Neuroscience Research Australia, the University of New South Wales and the National Health and Medical Research Council of Australia. The European Synchrotron Radiation Facility is acknowledged for provision of synchrotron radiation beam time (exp MD-304, MD-258). The synchrotron Diamond is acknowledged for provision of beam time (exp. SP7238). The authors are grateful to AIFIRA facility for granted access to nuclear microprobe end-station. This work was supported by research grants from the Partenariats Hubert Curien (PHC)-FAST, the Australian Department of Innovation, Industry, Science and Research International Science Linkage FAST programs, the National Health and Medical Research Council of Australia (project grant to KLD and JFBM), Parkinson's New South Wales (seed grant to KLD), the 36th District of Quota International, AMP, and a donation from Greg Connolly on behalf of the AMP Foundation and Charter Financial Planning. KMD was a recipient of an Australian Postgraduate Award and a Research Excellence Award form the University of New South Wales. DJH was a recipient of a Research Fellowship from the Australian Research Council, and KLD and GMH were recipients of Research Fellowships from the National Health and Medical Research Council of Australia. We thank Tina Geraki for assistance at diamond I18 beamline, Vanessa Krapp for her assistance with immunocytochemical preparations and Francine Carew-Jones for tissue preparation.

inserm-00968923, version 1 -1 Apr 2014

Disclosure statement

The authors declare no actual or potential conflicts of interest.

inserm-00968923, version 1 -1 Apr 2014

Supplementary Information

Tissue pH 5.9 -7.0 5.9 -6.8 6.1 -6.7 5.9 - Iron (Fe), copper (Cu) and zinc (Zn) concentrations, determined by Particle-Induced X-Ray Emission (PIXE), in the occipital cortex, substantia nigra pars compacta (SN), and within neuromelanin (NM) of neurons from the substantia nigra pars compacta (SN) and locus inserm-00968923, version 1 -1 Apr 2014


M I Isaev

R G Novikov

Effectivized Hölder-logarithmic stability estimates for the Gel'fand inverse problem

We give effectivized Hölder-logarithmic energy and regularity dependent stability estimates for the Gel'fand inverse boundary value problem in dimension d = 3. This effectivization includes explicit dependance of the estimates on coefficient norms and related parameters. Our new estimates are given in L 2 and L ∞ norms for the coefficient difference and related stability efficiently increases with increasing energy and/or coefficient difference regularity. Comparisons with preceeding results are given.

Introduction and main results

We consider the equation

-∆ψ + v(x)ψ = Eψ, x ∈ D ⊂ R 3 , (1.1)
where

D is an open bounded domain in R 3 , ∂D ∈ C 2 , (1.2)
v ∈ L ∞ (D).

(1.3) Equation (1.1) can be regarded as the stationary Schrödinger equation of quantum mechanics at fixed energy E. Equation (1.1) at fixed E arises also in acoustics and electrodynamics.

As in Section 5 of Gel'fand's work [START_REF] Gelfand | Some problems of functional analysis and algebra[END_REF] we consider an operator establishing a relationship between ψ and ∂ψ/∂ν on ∂D for all sufficiently regular solutions ψ of equation (1.1) in D = D ∪ ∂D at fixed E, where ν is the outward normal to ∂D. As in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF], [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF] (for example) we represent such an operator as the Dirichlet-to-Neumann map Φ(E) defined by the relation

Φ(E)(ψ| ∂D ) = ∂ψ ∂ν | ∂D , (1.4)
where we assume also that E is not a Dirichlet eigenvalue for operator -∆ + v in D.

(1.5)

The map Φ = Φ(E) can be regarded as all possible boundary measurements for the physical model described by equation (1.1) at fixed energy E under assumption (1.5).

We consider the following inverse boundary value problem for equation (1.1):

Problem 1.1. Given Φ for some fixed E, find v.

This problem is known as the Gel'fand inverse boundary value problem for the Schrödinger equation at fixed energy E in three dimensions (see [START_REF] Gelfand | Some problems of functional analysis and algebra[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]). For E = 0 this problem can be regarded also as a generalization of the Calderón problem of the electrical impedance tomography in three dimensions (see [START_REF] Calderón | On an inverse boundary problem[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]). Problem 1.1 can be also considered as an example of ill-posed problem; see [START_REF] Beilina | Approximate global convergence and adaptivity for coefficient inverse problems[END_REF], [START_REF] Lavrentev | Ill-posed problems of mathematical physics and analysis[END_REF] for an introduction to this theory.

Let, for real m ≥ 0,

H m (R 3 ) = w ∈ L 2 (R 3 ) : F -1 (1 + |ξ| 2 ) m 2 F w ∈ L 2 (R 3 ) , ||w|| H m (R 3 ) = F -1 (1 + |ξ| 2 ) m 2 F w L 2 (R 3 ) , (1.6)
where F denote the Fourier transform

F w(ξ) = 1 (2π) 3 R 3 e iξx w(x)dx, ξ ∈ R 3 .
In addition, for real m ≥ 0, we consider the spaces W m (R 3 ) defined by

W m (R 3 ) = w ∈ L 1 (R 3 ) : (1 + |ξ| 2 ) m 2 F w ∈ L ∞ (R 3 ) , ||w|| W m (R 3 ) = (1 + |ξ| 2 ) m 2 F w L ∞ (R 3 ) . (1.7)
We note that for integer m the space W m (R 3 ) contains the standard Sobolev space W m,1 (R 3 ) of m-times smooth functions in L 1 on R 3 .

In the present work we obtain, in particular, the following theorems:

Theorem 1.1. Suppose that D satisfies (1.2) and v 1 , v 2 satisfy (1.3), (1.5) for some real E. Suppose also that:

||v j || L ∞ (D) ≤ N for some N > 0, j = 1, 2; supp(v 2 -v 1 ) ⊂ D, v 2 -v 1 ∈ H m (R 3 ), v 2 -v 1 H m (R 3 ) ≤ N H m for some m > 0 and N H m > 0. Let δ = || Φ2 (E) -Φ1 (E)|| L ∞ (∂D)→L ∞ (∂D) , (1.8)
where Φ1 (E), Φ2 (E) denote the Dirichlet-to-Neumann maps for v 1 , v 2 , respectively. Then, there exist some positive constants A, B, α, β depending on D only such that

||v 2 -v 1 || L 2 (D) ≤ A αE + β(1 -τ ) 2 ln 3 + δ -1 2 1 2 δ τ + +B (1 + N ) 4m 3 N H m αE + β(1 -τ ) 2 ln 3 + δ -1 2 -m 3 (1.9)
for any τ ∈ (0, 1] and E ≥ 0. Besides, estimate (1.9) is also fulfilled for any τ ∈ (0, 1) and E < 0 under the following additional condition:

αE + β(1 -τ ) 2 ln 3 + δ -1 2 > 0.
(1.10) Theorem 1.2. Suppose that D satisfies (1.2) and v 1 , v 2 satisfy (1.3), (1.5) for some real E. Suppose also that:

||v j || L ∞ (D) ≤ N for some N > 0, j = 1, 2; supp(v 2 -v 1 ) ⊂ D, v 2 -v 1 ∈ W m (R 3 ), v 2 -v 1 W m (R 3 ) ≤ N W m for some m > 3 and N W m > 0.
Let δ be defined by (1.8). Then, there exist some positive constants Ã, B, α, β depending on D only such that

||v 2 -v 1 || L ∞ (D) ≤ Ã αE + β(1 -τ ) 2 ln 3 + δ -1 2 1 2 δ τ + + B (1 + N ) 2(m-3) 3 N W m m -3 αE + β(1 -τ ) 2 ln 3 + δ -1 2 -m-3 6 (1.11)
for any τ ∈ (0, 1] and E ≥ 0.

Besides, estimate (1.11) is also fulfilled for any τ ∈ (0, 1) and E < 0 under the following additional condition

αE + β(1 -τ ) 2 ln 3 + δ -1 2 > 0.
(1.12) Theorems 1.1 and 1.2 are proved in Sections 3 and 4, respectively. These proofs are based on Lemmas 2.1, 2.2 and 2.3 given in Section 2. Then these proofs are based on the intermediate estimates (3.7), (4.8) which may be of independent interest. Remark 1.1. The estimates of Theorem 1.2 can be regarded as a significant effectivization of the following estimates of [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF] for the three-dimensional case:

||v 2 -v 1 || L ∞ (D) ≤ C 1 (N m , D, m, E) ln 3 + δ -1 -s1 (1.13) for E ∈ R; ||v 2 -v 1 || L ∞ (D) ≤ C 2 (N m , D, m, τ )(1 + √ E)δ τ + + C 3 (N m , D, m, τ )(1 + √ E) s-s1 ln 3 + δ -1 -s (1.14)
for E ≥ 0, τ ∈ (0, 1) and any s ∈ [0, s 1 ]. Here δ is defined by (1.8) and s 1 = (m -3)/3. In addition, estimates (1.13) and (1.14) were obtained in [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF] under the assumptions that: D satisfies (1.2), v j satisfies (1.3), (1.5)

, supp v j ⊂ D, v j ∈ W m,1 (R 3 ), v j W m,1 (R 3 ) ≤ N m , j = 1, 2,
for some integer m > 3 and N m > 0.

Actually, Theorem 1.2 was obtained in the framework of finding the dependance of C 1 , C 2 , C 3 of (1.13), (1.14) on N m , m and τ . One can see that the estimates of Theorem 1.2 depend explicitely on coefficient norms N , N W m and parameteres m, τ and imply (1.13), (1.14) with some C 1 , C 2 , C 3 explicitely dependent on N m , m, τ as a corollary. Besides, in Theorem 1.2 we do not assume that each of potentials v 1 , v 2 is m-times differentiable and is supported in D (in a similar way with Theorem 2.1 of [START_REF] Novikov | Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy[END_REF]).

By the way we would like to note also that even for E = 0 the reduction of Hölder-logarithmic stability estimates like (1.9), (1.11) to pure logarithmic estimates like (1.13) is not optimal for large m because of the following asymptotic formula:

sup δ∈(0,1] δ τ (ln(3 + δ -1 )) -µ = O µ τ µ e -µ τ as µ → +∞.
In particular, even for E = 0 the Hölder-logarithmic estimates (1.9), (1.11) are much more informative than their possible pure logarithmic reductions.

Remark 1.2. Theorem 1.1 was obtained as an extention of Theorem 1.2 to the L 2 -norm case. In addition, it is important to note that the second ("logarithmic") term of the right-hand side of (1.9) is considerably better than the analogous term of (1.11). In particular,

R = O ln 3 + δ -1 -2m 3 for δ → 0, R = O E -m 3 for E → +∞, whereas R = O ln 3 + δ -1 -m-3 3 for δ → 0, R = O E -m-3 6 for E → +∞,
where R and R denote the second ("logarithmic") terms of the right-hand sides of (1.9) and (1.11), respectively.

Remark 1.3. The estimates of Theorem 1.1 should be compared also with the following estimate of [START_REF] Isakov | Increasing stability of the inverse boundary value problem for the Schrödinger equation[END_REF] for the three-dimensional case:

v 2 -v 1 H -m (R 3 ) ≤ C E 2 δ + √ E + ln δ -1 -(2m-3) , (1.15)
where

C = C (N m , D, supp(v 2 -v 1 ), m) > 0, v j H m (D) ≤ N m (j = 1, 2), supp (v 2 -v 1 ) ⊂ D, m > 3/2,
δ is the distance between the boundary measurements (Cauchy data) for v 1 , v 2 and is, roughly speaking, similar to δ of (1.8) and where δ ≤ 1/e. A principal advantage of (1.9) in comparison with (1.15) consists in estimation v 2 -v 1 in the L 2 -norm instead of the H -m -norm. Besides, estimate (1.9) depends explicitely on coefficient norms N , N W m and parameteres m, τ in contrast with (1.15). In addition, in (1.9) we do not assume that each of v 1 , v 2 belongs to H m . Remark 1.4. In the literature on Problem 1.1 estimates of the form (1.13) are known as global logarithmic stability estimates. The history of these estimates goes back to [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] for the case when s 1 ≤ 1 and to [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF] for the case when s 1 > 1.

In addition, estimates of the form (1.9), (1.11), (1.14), (1.15) are known in the literature as Hölder-logarithmic energy and regularity dependent stability estimates. For the case when τ = 1 in (1.9), (1.11) or when s = 0 in (1.14) the history of such estimates in dimension d = 3 goes back to [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF], [START_REF] Novikov | The ∂-approach to monochromatic inverse scattering in three dimensions[END_REF], where such energy and regularity dependent rapidly convergent approximate stability estimates were given for the inverse scattering problem.

Then for Problem 1.1 energy dependent stability estimates changing from logarithmic type to Hölder type for high energies were given in [START_REF] Isakov | Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map[END_REF]. However, this high energy stability increasing of [START_REF] Isakov | Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map[END_REF] is slow. The studies of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF], [START_REF] Novikov | The ∂-approach to monochromatic inverse scattering in three dimensions[END_REF], [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], [START_REF] Isakov | Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map[END_REF] were continued, in particular, in [START_REF] Nagayasu | Increasing stability in an inverse problem for the acoustic equation[END_REF], [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF], [START_REF] Isakov | Increasing stability of the inverse boundary value problem for the Schrödinger equation[END_REF] and in the present work.

Remark 1.5. In Theorems 1.1, 1.2 we consider the three-dimensional case for simplicity only. Similar results hold in dimension d > 3.

As regards to logarithmic and Hölder-logarithmic stability estimates for Problem 1.1 in dimension d = 2, we refer to [START_REF] Novikov | A global stability estimate for the Gelfand-Calderon inverse problem in two dimensions[END_REF], [START_REF] Santacesaria | Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions[END_REF], [START_REF] Santacesaria | A Holder-logarithmic stability estimate for an inverse problem in two dimensions[END_REF]. In addition, for problems like Problem 1.1 the history of energy and regularity dependent rapidly convergent approximate stability estimates in dimension d = 2 goes back to [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF].

Remark 1.6. In a similar way with results of [START_REF] Isaev | Stability estimates for determination of potential from the impedance boundary map[END_REF], [START_REF] Isaev | Reconstruction of a potential from the impedance boundary map[END_REF] and subsequent studies of [START_REF] Päivärinta | The inverse Robin boundary value problem in a half-space[END_REF], estimates (1.9), (1.11) can be extended to the case when we do not assume that condition (1.5) is fulfiled and consider an appropriate impedance boundary map (Robin-to-Robin map) instead of the Dirichlet-to-Neumann map.

Remark 1.7. Apparently, estimates analogous to estimates of Theorems 1.1 and 1.2 hold if we replace the difference of DtN maps by the difference of corresponding near field scattering data in a similar way with results of [START_REF] Hähner | New stability estimates for the inverse acoustic inhomogeneous medium problem and applications[END_REF], [START_REF] Isaev | Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions[END_REF], [START_REF] Isaev | New global stability estimates for monochromatic inverse acoustic scattering[END_REF].

Remark 1.8. The optimality (in different senses) of estimates like (1.13), (1.14) was proved in [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF], [START_REF] Isaev | Exponential instability in the Gel'fand inverse problem on the energy intervals[END_REF], [START_REF] Isaev | Instability in the Gel'fand inverse problem at high energies[END_REF]. See also [START_REF] Di Cristo | Rondi Examples of exponential instability for inverse inclusion and scattering problems[END_REF], [START_REF] Isaev | Exponential instability in the inverse scattering problem on the energy interval[END_REF] and references therein for the case of inverse scattering problems.

Remark 1.9. Estimates (1.9), (1.11) for τ = 1 are roughly speaking coherent with stability properties of the approximate monochromatic inverse scattering reconstruction of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF], [START_REF] Novikov | The ∂-approach to monochromatic inverse scattering in three dimensions[END_REF], implemented numerically in [START_REF] Alexeenko | Solution of threedimensional acoustical inverse scattering problem,II: modified Novikov algorithm[END_REF]. Estimates (1.9), (1.11) for E = 0 are roughly speaking coherent with stability properties of the reconstruction of [START_REF] Novikov | An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions[END_REF].

In addition, estimates (1.9), (1.11) can be used for the convergence rate analysis for iterative regularized reconstructions for Problem 1.1 in the framework of an effectivization of the approach of [START_REF] Hähner | New stability estimates for the inverse acoustic inhomogeneous medium problem and applications[END_REF] for monochromatic inverse scattering problems.

Lemmas

Let v denote the Fourier transform of v:

v(ξ) = F v(ξ) = 1 (2π) 3 R 3 e iξx v(x)dx, ξ ∈ R 3 .
(2.1)

Lemma 2.1. Suppose that D satisfies (1.2) and v 1 , v 2 satisfy (1.3), (1.5) for some real E. Suppose also that ||v j || L ∞ (D) ≤ N, j = 1, 2, for some N > 0. Let δ be defined by (1.8). Then

|v 2 (ξ) -v1 (ξ)| ≤ c 1 (1 + N ) 2 e 2ρL δ + v 1 -v 2 L 2 (D) E + ρ 2 (2.2)
for any ρ > 0 such that

|ξ| ≤ 2 E + ρ 2 , E + ρ 2 ≥ (1 + N ) 2 r 2 1 , where L = max x∈∂D |x| and constants c 1 , r 1 > 0 depend on D only.
Some version of estimate (2.2) was given in [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF] (see formula (4.13) of [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF]). Lemma 2.1 is proved in Section 6. This proof is based on results presented in Section 5.

Lemma 2.2. Let w ∈ H m (R 3 ), w H m (R 3 ) ≤ N H m
for some real m > 0 and N H m > 0, where the space H m (R 3 ) is defined in (1.6). Then, for any r > 0,

   |ξ|≥r |F w(ξ)| 2 dξ    1/2 ≤ c 2 N H m r -m , (2.3)
where F w is defined according to (2.1) and c 2 = (2π) -3/2 .

Proof of Lemma 2.2. Note that

|ξ|≥r |F w(ξ)| 2 dξ ≤ (1 + |ξ| 2 ) m 2 F w r m 2 L 2 (R 3 )
.

(2.4)

Using (1.6), (2.4) and the Parseval theorem

F w L 2 (R 3 ) = (2π) -3/2 w L 2 (R 3 ) (2.5) for w ≡ F -1 (1 + |ξ| 2 ) m 2 F w, we get estimate (2.3). Lemma 2.3. Let w ∈ W m (R 3 ), w W m (R 3 ) ≤ N W m
for some real m > 3 and N W m > 0, where the space W m (R 3 ) is defined in (1.7). Then, for any r > 0,

|ξ|≥r |F w(ξ)|dξ ≤ c2 N W m m -3 r 3-m , (2.6)
where F w is defined according to (2.1) and c2 = 4π.

Proof of Lemma 2.3. Note that

r m |F w(ξ)| ≤ (1 + |ξ| 2 ) m/2 |F w(ξ)| ≤ N W m for |ξ| ≥ r. (2.7)
Using (2.7), we obtain that

|ξ|≥r |F w(ξ)|dξ ≤ +∞ r N W m t m 4πt 2 dt ≤ 4πN W m m -3 r 3-m . (2.8) 3
Proof of Theorem 1.1

Using the Parseval formula (2.5), we get that

v 2 -v 1 L 2 (D) = (2π) 3/2 v2 -v1 L 2 (R 3 ) ≤ (2π) 3/2 (I 1 (r) + I 2 (r)), (3.1)
for r > 0, where vj is defined according to (2.1) with v j ≡ 0 on R 3 \ D, j = 1, 2,

I 1 (r) =    |ξ|≤r |v 2 (ξ) -v1 (ξ)| 2 dξ    1/2 , I 2 (r) =    |ξ|≥r |v 2 (ξ) -v1 (ξ)| 2 dξ    1/2 . Let r = q(1 + N ) -4/3 (E + ρ 2 ) 1/3 , q = 1 2π 16πc 2 1 3 -1/3 , (3.2)
where c 1 is the constant of Lemma 2.1. Then, using Lemma 2.1 for |ξ| ≤ r, we get that

I 1 (r) ≤   4πr 3 3 c 2 1 (1 + N ) 4 e 2ρL δ + v 1 -v 2 L 2 (D) E + ρ 2 2   1/2 ≤ ≤ (2π) -3/2 E + ρ 2 e 2ρL δ 2 + v 1 -v 2 L 2 (D) 2 (3.3) for q(1 + N ) -4/3 (E + ρ 2 ) 1/3 ≤ 2 E + ρ 2 and E + ρ 2 ≥ (1 + N ) 2 r 2 1 .
In addition, using (2.3), we have that

I 2 (r) ≤ c 2 N H m r -m . (3.4) Let r 2 = r 2 (D) ≥ r 1 be such that E + ρ 2 ≥ r 2 2 =⇒ q(E + ρ 2 ) 1/3 ≤ 2 E + ρ 2 . (3.5) Using (3.1), (3.3)
-(3.5) with r defined in (3.2), we obtain that

v 2 -v 1 L 2 (D) ≤ E + ρ 2 e 2ρL δ 2 + v 1 -v 2 L 2 (D) 2 + +(2π) 3/2 c 2 (1 + N ) 4m 3 q m N H m (E + ρ 2 ) -m 3 , (3.6)
1 2 v 2 -v 1 L 2 (D) ≤ E + ρ 2 e 2ρL δ 2 + + (1 + N ) 4m 3 q m N H m (E + ρ 2 ) -m 3 , (3.7
)

for E + ρ 2 ≥ (1 + N ) 2 r 2 2
, where L, c 2 are the constants of Lemmas 2.1, 2.2 and q, r 2 are the constants of formulas (3.2), (3.5).

Let τ ∈ (0, 1) and

γ = 1 -τ 2L , ρ = γ ln 3 + δ -1 . (3.8)
Due to (3.7), for δ such that

E + γ ln(3 + δ -1 ) 2 ≥ (1 + N ) 2 r 2 2 , (3.9)
the following estimate holds:

1 2 v 1 -v 2 L 2 (D) ≤ ≤ 1 2 E + γ ln 3 + δ -1 2 1/2 3 + δ -1 2γL δ+ + (1 + N ) 4m 3 q m N H m E + γ ln 3 + δ -1 2 -m 3 , (3.10)
where γ is defined in (3.8). Note that

3 + δ -1 2γL δ = (1 + 3δ) 1-τ δ τ ≤ 4δ τ for δ ≤ 1. (3.11)
Combining (3.10), (3.11), we get that

||v 2 -v 1 || L 2 (D) ≤ A 1 λ E + γ 2 ln 3 + δ -1 2 1 2 δ τ + +B 1 (1 + N ) 4m 3 N H m λ E + γ 2 ln 3 + δ -1 2 -m 3 (3.12)
for δ ≤ 1 satisfying (3.9) and some positive constants A 1 , B 1 , λ depending on D only.

In view of definition (1.6), we have that

||v 2 -v 1 || L 2 (D) ≤ ||v 2 -v 1 || H m (R 3 ) ≤ N H m .
Hence, we get that, for 0

< E + γ ln(3 + δ -1 ) 2 ≤ (1 + N ) 2 r 2 2 , ||v 2 -v 1 || L 2 (D) ≤ (1 + N ) 4m 3 N H m E + γ 2 ln 3 + δ -1 2 r 2 2 -m 3 . (3.13)
On other hand, in the case when

E + γ ln(3 + δ -1 ) 2 ≥ (1 + N ) 2 r 2 2 and δ > 1 we have that ||v 2 -v 1 || L 2 (D) ≤ c 3 ||v 2 -v 1 || L ∞ (D) ≤ c 3 2N ≤ ≤ 2c 3 E + γ 2 ln 3 + δ -1 2 r 2 2 1 2 δ τ , (3.14)
where

c 3 =   D 1 dx   1/2 . (3.15)
Combining (3.8), (3.12)-(3.14), we obtain estimate (1.9). This completes the proof of Theorem 1.1.

Proof of Theorem 1.2

Due to the inverse Fourier transform formula

v(x) = R 3 e -iξx v(ξ)dξ, x ∈ R 3 , (4.1)
we have that

v 1 -v 2 L ∞ (D) ≤ sup x∈D R 3 e -iξx (v 2 (ξ) -v1 (ξ)) dξ ≤ Ĩ1 (r) + Ĩ2 (r) (4.2)
for r > 0, where

Ĩ1 (r) = |ξ|≤r |v 2 (ξ) -v1 (ξ)|dξ, Ĩ2 (r) = |ξ|≥r |v 2 (ξ) -v1 (ξ)|dξ. Let r = q(1 + N ) -2/3 (E + ρ 2 ) 1/6 , q = 8πc 1 c 3 3 -1/3 , (4.3)
where c 1 is the constant of Lemma 2.1 and c 3 is defined by (3.15). Then, combining the definition of Ĩ1 , Lemma 2.1 for |ξ| ≤ r and the inequality

||v 2 -v 1 || L 2 (D) ≤ c 3 ||v 2 -v 1 || L ∞ (D) ,
we get that

Ĩ1 (r) ≤ 4πr 3 3 c 1 (1 + N ) 2 e 2ρL δ + c 3 v 1 -v 2 L ∞ (D) E + ρ 2 ≤ ≤ 1 2c 3 E + ρ 2 e 2ρL δ + v 1 -v 2 L ∞ (D) 2 (4.4) for q(1 + N ) -2/3 (E + ρ 2 ) 1/6 ≤ 2 E + ρ 2 and E + ρ 2 ≥ (1 + N ) 2 r 2 1 .
In addition, using (2.6), we get that

Ĩ2 (r) ≤ c2 N W m m -3 r 3-m . (4.5)
Let r2 = r2 (D) ≥ r 1 be such that

E + ρ 2 ≥ r2 2 =⇒ q(E + ρ 2 ) 1/6 ≤ 2 E + ρ 2 . (4.6)
Using (4.2), (4.4)-(4.6) with r defined in (4.3), we obtain that

v 2 -v 1 L ∞ (D) ≤ 1 2c 3 E + ρ 2 e 2ρL δ + v 1 -v 2 L ∞ (D) 2 + +c 2 (1 + N ) 2(m-3) 3 (m -3)q m-3 N W m (E + ρ 2 ) -m-3 6 , (4.7) 1 2 v 2 -v 1 L ∞ (D) ≤ 1 2c 3 E + ρ 2 e 2ρL δ+ + 4π (1 + N ) 2(m-3) 3 (m -3)q m-3 N W m (E + ρ 2 ) -m-3 6 (4.8) for E + ρ 2 ≥ (1 + N ) 2 r2
2 , where L, c2 are the constants of Lemmas 2.1, 2.3 and c 3 , q, r2 are the constants of formulas (3.15), (4.3), (4.6).

Let τ ∈ (0, 1) and

γ = 1 -τ 2L , ρ = γ ln 3 + δ -1 . (4.9)
Due to (4.8), for δ such that

E + γ ln(3 + δ -1 ) 2 ≥ (1 + N ) 2 r2 2 , (4.10)
the following estimate holds:

1 2 v 1 -v 2 L ∞ (D) ≤ ≤ 1 2c 3 E + γ ln 3 + δ -1 2 1/2 3 + δ -1 2γL δ+ +4π (1 + N ) 2(m-3) 3 (m -3)q m N W m E + γ ln 3 + δ -1 2 -m-3 6
, (4.11) where γ is defined in (4.9). Note that

3 + δ -1 2γL δ = (1 + 3δ) 1-τ δ τ ≤ 4δ τ for δ ≤ 1.
(4.12)

Combining (4.11), (4.12), we get that

||v 2 -v 1 || L ∞ (D) ≤ Ã1 λ E + γ 2 ln 3 + δ -1 2 1 2 δ τ + + B1 (1 + N ) 2(m-3) 3 N W m m -3 λ E + γ 2 ln 3 + δ -1 2 -m- 3 6 (4.13)
for δ ≤ 1 satisfying (4.10) and some positive constants Ã1 , B1 , λ depending on D only. Using (1.7) and (4.2), we get that

||v 2 -v 1 || L ∞ (D) ≤ R 3 (1 + |ξ| 2 ) -m/2 ||v 2 -v 1 || W m (R 3 ) dξ ≤ ≤ N W m +∞ 0 4πt 2 (1 + t 2 ) m/2 dt ≤ c 4 e m-3 m -3 N W m (4.14)
for some c 4 > 0. Here we used also that

+∞ 0 4πt 2 (1 + t 2 ) m/2 dt ≤ 1 0 4πt 2 dt + +∞ 1 4πt 2 t m dt ≤ c 4 1 + 1 m -3 ≤ c 4 e m-3 m -3 .
Using (4.14), we get that, for 0

< E + γ ln(3 + δ -1 ) 2 ≤ (1 + N ) 2 r2 2 , ||v 2 -v 1 || L ∞ (D) ≤ ≤ c 4 (1 + N ) 2(m-3) 3 N W m m -3 E + γ 2 ln 3 + δ -1 2 e 6 r2 2 -m-3 6 . (4.15)
On other hand, in the case when E + γ ln δ -1 2 ≥ (1 + N ) 2 r2 2 and δ > 1 we have that

||v 2 -v 1 || L ∞ (D) ≤ 2N ≤ 2 E + γ 2 ln 3 + δ -1 2 r2 2 1 2 δ τ . (4.16)
Combining (4.9), (4.13), (4.15) and (4.16), we obtain estimate (1.11). This completes the proof of Theorem 1.2.

Faddeev functions

Suppose

that v ∈ L ∞ (D), v ≡ 0 on R 3 \ D, (5.1)
where D satisfies (1.2). More generally, one can assume that v is a sufficiently regular function on R 3

with sufficient decay at infinity. (

Under assumptions (5.2), we consider the functions ψ, µ, h:

ψ(x, k) = e ikx µ(x, k), (5.3)
µ(x, k) = 1 + R 3 g(x -y, k)v(y)µ(y, k)dy, g(x, k) = -(2π) -3 R 3 e iξx dξ ξ 2 + 2kξ , (5.4)
where

x ∈ R 3 , k ∈ C 3 , Im k = 0, h(k, l) = (2π) -3 R 3 e i(k-l)x v(x)µ(x, k)dx, (5.5)
where k, l ∈ C 3 , k 2 = l 2 , Im k = Im l = 0. Here, (5.4) at fixed k is considered as a linear integral equation for µ, where µ is sought in L ∞ (R 3 ). The functions ψ, h and G = e ikx g are known as the Faddeev functions, see [START_REF] Faddeev | Growing solutions of the Schrödinger equation[END_REF], [START_REF] Faddeev | The inverse problem in the quantum theory of scattering[END_REF], [START_REF] Henkin | The ∂-equation in the multidimensional inverse scattering problem[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]. These functions were introduced for the first time in [START_REF] Faddeev | Growing solutions of the Schrödinger equation[END_REF], [START_REF] Faddeev | The inverse problem in the quantum theory of scattering[END_REF].

In particular, we have that

(∆ + k 2 )G(x, k) = δ(x), (-∆ + v(x))ψ(x, k) = k 2 ψ(x, k), where x ∈ R 3 , k ∈ C 3 \ R 3 .
We recall also that the Faddeev functions G, ψ, h are some extension to the complex domain of functions of the classical scattering theory for the Schrödinger equation (in particular, h is an extension of the classical scattering amplitude).

Note also that G, ψ, h in their zero energy restriction, that is for k 2 = 0, l 2 = 0, were considered for the first time in [START_REF] Beals | Multidimensional inverse scattering and nonlinear partial differential equations[END_REF]. The Faddeev functions G, ψ, h were, actually, rediscovered in [START_REF] Beals | Multidimensional inverse scattering and nonlinear partial differential equations[END_REF].

For further considerations we will use the following notations:

Σ E = k ∈ C 3 : k 2 = k 2 1 + k 2 2 + k 2 3 = E , Θ E = {k ∈ Σ E , l ∈ Σ E : Im k = Im l} , |k| = (|Re k| 2 + |Im k| 2 ) 1/2 for k ∈ C 3 .
Under assumptions (5.2), we have that:

µ(x, k) → 1 as |k| → ∞, (5.6)
where

x ∈ R 3 , k ∈ Σ E ; v(ξ) = lim (k, l) ∈ ΘE , k -l = ξ |Im k| = |Im l| → ∞ h(k, l) for any ξ ∈ R 3 , (5.7)
where v is defined by (2.1).

Results of the type (5.6) go back to [START_REF] Beals | Multidimensional inverse scattering and nonlinear partial differential equations[END_REF]. Results of the type (5.7) go back to [START_REF] Henkin | The ∂-equation in the multidimensional inverse scattering problem[END_REF]. These results follow, for example, from equation (5.4), formula (5.5) and the following estimates:

g(x, k) = O(|x| -1 ) for x ∈ R 3 , uniformly in k ∈ C 3 \ R 3 , (5.8)
Λ -s g(k)Λ -s L 2 (R 3 )→L 2 (R 3 ) = O(|k| -1 ), for s > 1/2, as |k| → ∞, k ∈ C 3 \ R 3 , (5.9)
where g(x, k) is defined in (5.4), g(k) denotes the integral operator with the Schwartz kernel g(x -y, k) and Λ denotes the multiplication operator by the function (1 + |x| 2 ) 1/2 . Estimate (5.8) was given in [START_REF] Henkin | The ∂-equation in the multidimensional inverse scattering problem[END_REF]. Estimate (5.9) was formulated, first, in [START_REF] Lavine | On the inverse scattering transform of the n-dimensional Schrödinger operator Topics in Soliton Theory and Exactly Solvable Nonlinear Equations[END_REF]. Concerning proof of (5.9), see [START_REF] Weder | Generalized limiting absorption method and multidimensional inverse scattering theory[END_REF]. In addition, estimate (5.9) in its zero energy restriction goes back to [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF].

In the present work we use the following lemma: Lemma 5.1 is proved in Section 6. This proof is based on estimates (5.8) and (5.9).

In addition, we have that (see [START_REF] Novikov | ∂-method with nonzero background potential. Application to inverse scattering for the two-dimensional acoustic equation[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential[END_REF]):

h 2 (k, l) -h 1 (k, l) = (2π) -3 R 3 ψ 1 (x, -l)(v 2 (x) -v 1 (x))ψ 2 (x, k)dx for (k, l) ∈ Θ E , |Im k| = |Im l| = 0, and v 1 , v 2 satisfying (5.2), (5.11)
h 2 (k, l) -h 1 (k, l) = (2π) -3 ∂D ψ 1 (x, -l) Φ2 -Φ1 ψ 2 (•, k) (x)dx for (k, l) ∈ Θ E , |Im k| = |Im l| = 0,
and v 1 , v 2 satisfying (1.5), (5.1), (5.12) where ψ j , h j denote ψ and h of (5.3) and (5.5) for v = v j , and Φj denotes the Dirichlet-to-Neumann map Φ for v = v j in D, where j = 1, 2.

In the present work we also use the following lemma:

Lemma 5.2. Let D satisfy (1.2). Let v j satisfy (5.1), ||v j || L ∞ (D) ≤ N, j = 1, 2, for some N > 0. Then

|v 1 (ξ) -v2 (ξ) -h 1 (k, l) + h 2 (k, l)| ≤ c 6 N (1 + N ) v 1 -v 2 L 2 (D) (E + ρ 2 ) 1/2 for (k, l) ∈ Θ E , ξ = k -l, |Im k| = |Im l| = ρ, E + ρ 2 ≥ r 2 4 (1 + N ) 2 , (5.13)
where E ∈ R, vj is the Fourier transform of v j , h j denotes h of (5.5) for v = v j , (j = 1, 2) and constants c 6 , r 4 > 0 depend on D only.

Some versions of estimate (5.13) were given in [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF], [START_REF] Novikov | ∂-method with nonzero background potential. Application to inverse scattering for the two-dimensional acoustic equation[END_REF], [START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential[END_REF] (see, for example, formula (3.18) of [START_REF] Isaev | Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions[END_REF]). Lemma 5.2 is proved in Section 6.

6 Proofs of Lemmas 2.1, 5.1 and 5.2

Proof of Lemma 5.1. Using (5.1), (5.4) and (5.9), we get that

µ(•, k) -1 L 2 (D) ≤ R 3 g(• -y)v(y)µ(y, k)dy L 2 (D) ≤ ≤ c 7 N |k| µ(•, k) L 2 (D) , (6.1)
µ(•, k) L 2 (D) ≤ c 3 + c 7 N |k| µ(•, k) L 2 (D) , (6.2)
where c 3 is defined by (3.15) and c 7 is some positive constant depending on D only. Hence, we obtain that

µ(•, k) L 2 (D) ≤ 2c 3 for |k| ≥ 2c 7 N. (6.3)
We use also that

D 1 |x -y| 2 dy ≤ D 1 dy + |x-y|≤1 1 |x -y| 2 dy ≤ c 2 8 , x ∈ D, (6.4)
where c 8 = c 8 (D) > 0. Using (5.1), (5.4), (5.8), (6.3), (6.4), we get that Using estimates (6.1), (6.3), (6.5) in (6.7), we get that (6.12)

|µ(x, k)| ≤ 1 + D g(x -y)v(y)µ(y, k)dy ≤ ≤ 1 +   D |g(x -y)| 2 dy   1/2 N µ(•, k) L 2 (R 3 ) ≤ ≤ c 5 (D)(1 + N ) for x ∈ D, |k| ≥ 2c 7 N .
I ∆ ≤ (2π) -3 µ 1 (•, -l) -1 L 2 (D) v 2 -v 1 L 2 (D)
Note that for any ξ ∈ R 3 satisfying |ξ| ≤ 2 E + ρ 2 (where ρ > 0) there exist some pair (k, l) ∈ Θ E such that ξ = k -l and |Im k| = |Im l| = ρ. Therefore, estimates (5.13) and (6.12) imply (2.2).

Lemma 5 . 1 .

 51 Let D satisfy (1.2) and v satisfy (5.1). Let ||v|| L ∞ (D) ≤ N for some N > 0. Then |µ(x, k)| ≤ c 5 (1 + N ) for x ∈ R 3 , |k| ≥ r 3 (1 + N ), (5.10) where µ(x, k) is the Faddeev function of (5.4) and constants c 5 , r 3 > 0 depend on D only.

(6. 5 ) 3 D ψ 1 3 D 3 D(µ 1 3 D µ 1 3 D

 531331313 Proof of Lemma 5.2. Due to (5.1),(5.11), we have thath 2 (k, l) -h 1 (k, l) = (2π) -(x, -l)(v 2 (x) -v 1 (x))ψ 2 (x, k)dx = = (2π) -e i(k-l)x µ 1 (x, -l)(v 2 (x) -v 1 (x))µ 2 (x, k)dx = = v2 (k -l) -v1 (k -l) + I ∆ (6.6)for (k, l) ∈ Θ E , |Im k| = |Im l| = 0, whereI ∆ = (2π) -(x, -l) -1)(v 2 (x) -v 1 (x))µ 2 (x, k)dx+ +(2π) -(x, -l)(v 2 (x) -v 1 (x))(µ 2 (x, k) -1)dx+ +(2π) -(µ 1 (x, -l) -1)(v 2 (x) -v 1 (x))(µ 2 (x, k) -1)dx. (6.7) Note that, for (k, l) ∈ Θ E , E ∈ R, |Im k| = |Im l| = ρ, |k| = |Re k| 2 + |Im k| 2 = k 2 + 2 |Im k| 2 = E + 2ρ 2 = |l|.(6.8)

µ 2 (E + 2ρ 2 ( 6 . 9 )

 2269 •, -l) L ∞ (D) + + µ 1 (•, -l) L ∞ (D) v 2 -v 1 L 2 (D) µ 2 (•, -l) -1 L 2 (D) + + µ 1 (•, -l) -1 L ∞ (D) v 2 -v 1 L 2 (D) µ 2 (•, -l) -1 L 2 (D) ≤ ≤ 2c 3 c 7 N v 1 -v 2 L 2 (D) c 5 (1 + N ) (2π) 3 |k| + c 5 (1 + N ) v 1 -v 2 L 2 (D) 2c 3 c 7 N (2π) 3 |l| + + (1 + c 5 (1 + N )) v 1 -v 2 L 2 (D) 2c 3 c 7 N (2π) 3 |l| ≤ ≤ c 8 (D) N (1 + N ) v 1 -v 2 L 2 (D) for (k, l) ∈ Θ E , |Im k| = |Im l| = ρ and |k| = |l| = E + 2ρ 2 ≥ 2c 7 N .Formula (6.6) and estimate (6.9) imply (5.13).Proof of Lemma 2.1. Due to (5.11), we have that|h 2 (k, l) -h 1 (k, l)| ≤ c 9 ψ 1 (•, -l) L ∞ (∂D) δ ψ 2 (•, k) L ∞ (∂D) , (k, l) ∈ Θ E , |Im k| = |Im l| = 0, (6.10) where c 9 = (2π) -3 ∂D dx.Using formula (5.3) and Lemma 5.1, we find thatψ j (•, k) L ∞ (∂D) ≤ c 5 (1 + N )e |Im k|L , j = 1, 2, for k ∈ Σ E , |k| ≥ r 3 (1 + N ),(6.11)where L = max x∈∂D |x|. Combining (6.8), (6.10) and (6.11), we get that|h 2 (k, l) -h 1 (k, l)| ≤ c 9 c 2 5 (1 + N ) 2 e 2ρL δ, for (k, l) ∈ Θ E , ρ = |Im k| = |Im l|, E + ρ 2 ≥ r 2 3 (1 + N ) 2 .