From Tracing to Writing: the Maps that Thoreau Copied Julien Nègre ENS de Lyon, Université de Lyon -France (IHRIM-CNRS)

There is a place in Concord, Massachusetts, called Beck Stow's swamp. No need to take out your map and look at it: it won't appear on a traditional USGS map and a search on Google Maps will yield no result. Beck Stow's swamp is somewhere out there, but it is located on a different plane of existence. The swamp does appear on some maps, but they belong to a very specific category: those reproduced at the end of certain editions of Thoreau's works. The name of the swamp will indeed be familiar to his readers, as he evokes it several times in his writing, especially his Journal 1 . As he explains, for example, in what is now the "European Cranberry" section of Wild Fruits (167-69), the swamp was a favorite spot of his to search for a very particular variety of cranberryso much so, that editors have felt it necessary to feature it on maps of the area.

Culturally, the swamp has an existence of its own. It appears as a meaningful, noteworthy spot on the paper mapsand mental maps -used by Thoreau's readers. On closer examination, yet, the exact location of the swamp seems to be slightly uncertain. If we compare the map provided by Bradley Dean in his 1993 edition of Faith in a Seed to the one that he included at the end of Wild Fruits, published in 2000, we can notice that Beck Stow's swamp has drifted southward. On the first map (18), which is a copy of the map compiled by Herbert W. Gleason for the 1906 edition of Thoreau's works, the swamp is located to the north east of the village center, near the place where the New Bedford Road meets the Old Bedford Road. In Wild Fruits (410-11), the swamp is placed just south of Lexington Road, along the Mill Brook. If we set aside for a moment the question of where Beck Stow's swamp actually is, what the uncertainty in Dean's maps reveals is the discrepancy between the cultural function of the map and its accuracy. There is a certain amount of disconnection between the spatial configuration of the world and the cultural work that the map accomplishes.

Culturally, what matters in this case is not so much where things are, but rather the fact they are on the map. In the case of Beck Stow's swamp, the place has literally been put "on the map" by Thoreau's texts, but the degree of correspondence between that particular (Thoreauvian) map and the actual map of Concord is only secondary. To use the language of image editing software and Geographic Information Systems, a cartographic "layer" featuring Beck Stow's swamp has been added to the map of Concord. That layer is only visible in certain cultural contexts, but the local map has effectively been altered.

This analogy might be helpful to understand the place and function of a specific category of documents in Thoreau's archives: the tracings of maps that he made throughout his life 2 . Produced by literally placing a layer of thin paper over the surface of the map, Thoreau's tracings are both a copy of their originals and an addition to them, produced by superimposing a new stratum of information. As such, they embody how Thoreau's works can be read as a "supplement:" something that, as is the case with Beck Stow's swamp, is added to the common perception of the world. But what is the nature of this supplement? What does the apparition of Beck Stow's swamp "add" to the local map?

As they are both copies of other documents and hand-drawn originals produced by Thoreau himself, the tracings also imply a logic of appropriation whereby Thoreau made his both the place represented on each map, and the map itself as a situated document. For that reason, these tracings also offer new ways of approaching the issue of Thoreau's complex relationship with mapping and surveying. While maps are objects that anyone familiar with his works readily and almost intuitively associates with Thoreau (think of the ubiquitous maps found on the covers or in the appendices of most recent editions of his texts), cartographic productions also embody many of the things against which Thoreau stood and that he tried to resist. For instance, as Patrick Chura has noted (92-106), Thoreau was highly ambivalent about his work as a surveyor, which allowed him to roam the woods more or less freely but also made him an agent of what one could call territorializationthe inscription and perpetuation of land ownership, but also of a certain conception of space as something that can be stabilized, appropriated and made profitable. Such a process is similar to what Jacques Rancière calls the distribution of the sensible (le partage du sensible, in French). A community, according to Rancière, is based on a certain distribution of time and space (the sensible) that defines which of its members take part in the common, and how (Rancière 12). The distribution is inclusive and exclusive at the same time, the term partage meaning both "sharing" something in common and "dividing" it into several parts. Those who do not "occupy" their time and/or space in a manner that fits the current distribution are left out and their voice is not heard: "Having a particular 'occupation' thereby determines the ability or inability to take charge of what is common to the community; it defines what is visible or not in a common space, endowed with a common language, etc." (12-13). The map is a tool, not only of representation, but also of perpetuation of a certain distribution of the (spatial) sensible: it delineates and inscribes a common space, based on a set of shared assumptions and the use of a common language. If tracing a map is not just an act of reproduction but also a gesture of appropriation, then it becomes the basis of a potentially subversive re-interpretation or rewriting of the distribution of the sensible.

In this paper I explore how Thoreau's tracings can be read as a prototype for what Thoreau accomplishes in his writings. What is at stake here, if we follow Rancière, is to make oneself heard. For that reason, Thoreau's tracings are far more than just reference materials used by him to document his excursions. They are key elements in his writing project and are intimately linked to the act of writing itself.

A Map of Two Rivers: a principle of selection

The earliest map drawn by Thoreau that is available to us is a copy of a map of the Concord and Merrimack rivers, drawn on the occasion of Thoreau's boat trip with his brother in September, 1839a trip that eventually became the topic of his first published book, A Week on the Concord and Merrimack Rivers (1849). The tracing is now among other Thoreau manuscripts in the Albert and Shirley Small Special Collections of the University of Virginia Library in Charlottesville. It was first made public in Stowell's Thoreau Gazetteer.

The map is considered to have been copied "on the occasion" of Thoreau's boat trip, but to this day it remains impossible to date it with precision. The map is drawn in pencil on a sheet of wove paper, watermarked "ALLEN" and countermarked with an image of a bird. It measures 40,8 × 24,2 cm 3 . It is not dated and it does not contain any annotation which might help to date it 4 .

Nevertheless, certain details attest that the map is intimately linked to the boat trip made by the two Thoreau brothers. The lower and upper limits exactly match the starting point of the excursion and its northernmost destination, from Concord, Massachusetts, to New Hampshire's White Mountains. The map also shows the brothers' camping spots, represented by Thoreau with small trianglesor tents. In other words, the map was undoubtedly drawn before, during, or shortly after the voyage up the two rivers. As such, it provides a lot of information about the excursion itself and how Thoreau understood its spatial configuration: for that reason, it was reproduced in the 1980 Princeton Edition of A Week (422), thereby becoming part of the authoritative critical apparatus of the text. Interestingly enough, yet, Thoreau's map has never been described in detail. In the Thoreau Gazetteer, it is simply described as "drawn by Thoreau himself," and its relation to the text of A Week is not explored. This is in keeping with the primary aim of the Gazetteer, which is to "give to its readers an idea of the places Thoreau described in his own books" (xi). More than an idea of the place, yet, the map provides insight into Thoreau's writing project itself.

The map is a tracing of another map, which I have been able to identify as Nathan Hale's Map of the New England States, published in Boston in 1826 5 . Hale's is a large wall map (112 × 95 cm), of which Thoreau copied only a specific portion, located exactly in the middle (see fig. 1). The map provides significant information about how Thoreau perceived his boat tripboth in the details that he copied and in those he decided to leave out. Indeed, the tracing is incomplete and scrupulous at the same time, as Thoreau selected specific categories of cartographic information and copied only what mattered to him. For example, a large part of the Atlantic shoreline is missing on his map: he decided to focus on the Concord and Merrimack rivers only. On the contrary, Thoreau carefully reproduced the geometric shapes of the towns and villages that border the Merrimack River. The left-hand side of the map appears as a complex patchwork of polygons, whose straight lines sharply contrast with the meandering course of the river itself. Thoreau might have copied the map before the excursion, with the purpose of creating a document that he could easily carry with him and refer to to locate the invisible boundary lines separating the villages. As the two brothers were paddling their way upstream, they had no other way to know which township they were crossing. And, according to Thoreau's account in A Week, knowing exactly where they were seems to have a been a major concern for the two brothers. As a narrative, A Week seems to always know exactly where it is going. The carefully organized timeline of the successive chapters (from "Saturday" to "Friday") reflects the way the account itself is spatially "anchored." With unfailing regularity, the narrator informs the reader of the position of the boat and of the names of the villages they are crossing. In "Tuesday," for example, Thoreau writes: "Our course this morning lay between the territories of Merrimack, on the west, and Litchfield, […] on the east" (194). The phrase "the territories of" suggests that Thoreau is not referring here to the center of the village as a visual landmark, but rather to the administrative division of the area into townshipsan invisible network of geometric shapes that is only perceptible on the map he has copied.

If Thoreau insists on defining his position in space, it is because his voyage through the landscape of New England is also envisioned as a voyage through time: the only guidebook the two brothers carry is the New England Gazetteer, which provides them with a compilation of historical anecdotes and economic data about each village (Sattelmeyer 197). When the narrator of A Week enters the territory of a new village, he turns to the Gazetteer and conjures up its ancient spatial arrangement: "Our course this morning lay between the territories of Merrimack, on the west, and Litchfield, once called Brenton's Farm, on the east, which townships were anciently the Indian Naticook. Brenton was a fur trader among the Indians, and these lands were granted to him in 1656." (194). Even though the Gazetteer primarily contains historical facts, Thoreau refers to it as "our Navigator" (90): to him, navigating through space is indissociably linked to the evocation of New England's invisible past. By referring to the exact moment when a piece of land changed status and entered colonial history ("in 1656"), Thoreau also puts into perspective the present-day division of the map into geometric polygons. Just as the coastline fades away on Thoreau's tracing, the administrative boundaries disappear in his text, to make way for a covert, more ancient cartographic layer: the historical past of the Native American tribes who lived in the exact same places before the arrival of the white man, and, beyond that, the even larger dimension of deep time (Dimock 7-22), compared to which the entire human history is but a brief, "wearisome while" (325).

The map that Thoreau traced and possibly carried with him during the excursion is the default layer of the space he is moving through. By revealing the hidden past of the region, the text challenges the stability and permanence of the map's configuration. In the final pages of A Week, Thoreau evokes the existence of another dimension of space, hidden "behind the ordinary," that we are unwilling and unable to perceive. "It is easier to discover another such a new world as Columbus did, than to go within one fold of this which we appear to know so well; the land is lost sight off, the compass vary, and mankind mutiny," Thoreau notes (383). Losing sight of the land is a maritime image of navigation, but it could be read as a new way of looking at the map itself, where contours and boundary lines appear as transient and inconsistent. In that respect, the principle of selection applied by Thoreau on his tracing of the map parallels the way the text puts its configuration into perspective. Tracing the map is only the first step towards its partial erasure and rewriting.

The Maine Woods, "a labyrinth of errors"

In In the first chapter of The Maine Woods, Thoreau explains how, at the start of his first excursion in Maine, he had the opportunity to make a tracing of a copy of Greenleaf's map that was hanging on the wall of a local inn (15). The short description of how Thoreau and his companion oiled a sheet of paper to make it transparent and then "carefully follow[ed] the outlines of the imaginary lakes which that map contains" might be the only instance in his writings where Thoreau evokes his own practice of tracing maps. Thoreau's purpose was to produce a "pocket map" (15) that he would be able to carry with him during his trip. Greenleaf's map is indeed a large wall map (128 × 105,5 cm) but Thoreau probably copied only a small portion of it: on that map, the area where all three of his excursions in Maine took place occupies a 20 × 20 cm square.

Characteristically, Thoreau contrasts his own careful tracing with the inaccuracy of the map itself, which proves to be "a labyrinth of errors" (15). As Thoreau explains in The Maine Woods, the region he is visiting is comparable to a "lesser Oregon" that American westward expansion has left behind, "unexplored" (82). Cartographically speaking, it is an area in transition, comparable to the vaguely known territories that the nation is conquering in the far West. The publication of Greenleaf's map was a landmark in the history of the state of Maine because it was the first consistent attempt at mapping the complex network of the region's waterways after the creation of the new state in 1820 (Ristow 94-96). In effect, yet, the mapping process was still ongoing during Thoreau's lifetime and the maps that were available to travelers were often full of errors. This profoundly shaped the experience of navigating the woods, as one was never completely sure of what stood on the other side of the next lake or at the end of next carry. While Thoreau's tracing of Greenleaf's map has been lost, another document shows the extent to which an excursion in the forests of Maine was also a complex mapping exercise: Thoreau's own copy of Coffin's 1835 map has been preserved to this day and is now in the Concord Free Public Library. A high-resolution picture of the map is available on the library's website; it shows the annotations that Thoreau added to his map in order to correct its errors 7 . Most of these markings appear on a specific area located just north of Moosehead Lake, near the northeast carry that connects the lake itself to the Penobscot River. As Thoreau explains in The Maine Woods (94), the distances between the different topographical features of that zone (the lake, the river, the various streams meeting it and the local ponds) are all overestimated on both Coffin's and Colton's maps. Thoreau's heavily annotated copy indicates the right distances and provides the missing or alternative names of certain ponds and streams.

The experience of reading the map turns here into an experience of writing on the map. The confrontation with an inaccurate, unreliable representation of the region triggers the act of jotting down words, names and figures in order to make sense of the forest's labyrinthine nature. The need to inscribe the surface of the map can be seen as the prototype of the act of writing itself. Throughout The Maine Woods, Thoreau painstakingly takes note of the "the names and distances, for the benefit of future tourists" (45-46). He also tries to take note of Indian names, which encapsulate the radically different knowledge of the terrain that Native Americans possess: "So much geography is there in their names." ( 270). Yet, very often, his notations are marked by a degree of uncertainty, which Thoreau expresses by adding question marks that reveal that he is unable to fully grasp the meaning of what his Indian guides teach him: "Mattawamkeag was a place where two rivers meet. (?)" (141). As a result, Thoreau's text is caught in a tension between his own effort to transcribe Indian words as faithfully as possible, and the unavoidable margin of error and imprecision that separates the two languages and the two cultures.

This narrow but unbridgeable gap mirrors the experience of tracing Greenleaf map, which amounts to producing an accurate and authentic copy, while unwittingly inscribing a series of errors. What is at stake here is the validity of the act of inscribing itselfbe it annotating an inaccurate map, transcribing Indian words, or the very act of producing a text. Characteristically, the passage of "Chesuncook" in which Thoreau evokes the area north of Moosehead Lake also contains the description of the unsettling experience of being faced with uncertain and unreliable inscriptions:

Joe said that they called the chicadee Kecunnilessu in his language. I will not vouch for the spelling of what possibly was never spelt before, but I pronounced after him till he said it would do. We passed close to a woodcock which stood perfectly still on the shore, with feathers puffed up, as if sick. This, Joe said, they called Nipsquecohossus. The king-fisher was Skuscumonsuck, bear was Wassus; Indian Devil lunxus; the mountain ash Upahsis. This was very abundant and beautiful. Moose tracks were not so fresh along this stream, except in a small creek about a mile up it, where a large log had lodged in the spring, marked W-cross-girdle-crowfoot. (97)

The lesson takes a deictic form when Joe can simply point at objects, which ensures that there is no ambiguity about what he is referring to. Yet, Thoreau's transcriptions are by definition asymptotic, as they can only come as close as possible to Joe's pronunciation, without ever matching it. Selfreferentially, the passage points at the uncertain authenticity of its own texture, reflected by the multiple signs that suddenly proliferate around them: moose tracks are visible, but they are "not so fresh," which means that they can't be read easily; in much the same way, the log bears a clear series of signs, but they remain cryptic and beyond Thoreau's reach as they are written in this "alphabet of their own, which only the practised [sic] can read" (42). Reading the map and the narrative together reveals that, in this area of the forest and of the text, inscriptions are uncertain and unstable. Far from coalescing into a fixed and immutable meaning, they remain indefinitely open to interpretation and rewriting. The effect is to put into perspective the narrator's constant effort to take note of accurate measurements, correct distances and exact nameselements that are all part of the basic field work of a professional surveyor. Thoreau appropriates the logic of the map by collecting cartographic data on the terrain, but he also stages his narrative in such a way that the authority of the map is kept at a distance and constantly questioned. Rather than submitting to the deceptive labyrinthine nature of the map, the text remains a flexible repository of inscriptions, free from authoritative interpretation.

Thoreau's maps of Cape Cod: "the thing itself"

A largely similar process is at work in Thoreau's descriptions of Cape Coda place that he depicts as full of optical illusions and deceptions. In 2010, John Hessler showed how Thoreau copied ancient charts of the New England coastline to acquire a diachronic knowledge of how the Cape has been perceived at the beginning of the colonial period ("From Ortelius") 8 . Yet, these ancient documents are not the only maps of Cape Cod that Thoreau used during his excursions. In the first chapter of the book, he explains how he carefully planned his first visit by plotting his trajectory on a map of the Cape: "On studying the map, I saw that there must be an uninterrupted beach on the east or outside of the fore-arm of the Cape" (4). In his Journal, he also explains that, while visiting the Cape, he usually found it easier to determine his own trajectory by looking at his map, rather than by asking the inhabitants for directions: "With my chart and compass I can generally find a shorter way than the inhabitants can tell me." (IX: 428). These notations show that Thoreau was in possession of a modern map of the peninsula. Although it is never identified explicitly in Cape Cod, that map might well be one of the three tracings representing Cape Cod that we find in Thoreau's archives today (Thoreau Gazetteer 22-25). These tracings differ from the documents examined by Hessler in that they are copies of modern maps of the Cape. As such, they reveal how Thoreau apprehended the topographical conformation of the peninsulanot just the evolution of how men have understood it across the centuries.

As was the case with Thoreau's map of the Concord and Merrimack rivers, little is known of the history of these three maps. They are not dated and contain no caption or annotation regarding their purpose or the reason why Thoreau created them in the first place. The first one represents the entire peninsula of Cape Cod and is drawn on a large sheet of paper (36,8 × 24,1 cm), while the other two depict the northern portion of the Cape only, from Orleans to Provincetown. These last two maps have very different sizes: the first one is drawn on a sheet of paper that is comparable to the large map (30 × 23 cm), while the second one is much smaller (20,5 × 13 cm). The Gazetteer treats them as separate documents and assumes that the two maps of the northern portion of the Cape are not copies but originals, entirely produced by Thoreau himself. Under scrutiny, yet, the three documents prove to be closely related.

The map representing the entire peninsula is now in the Concord Free Public Library 9 . The map is extremely detailed and contains dozens of place names, as well as township divisions. By determining the map's scale and comparing certain characteristic features, I was able to identify the source of Thoreau's tracing. Interestingly enough, the map that Thoreau copied is not a map of Cape Cod specifically, but a map of the entire State of Massachusetts, published by Simeon Borden in 1844. Borden's map is a very large wall map (125 × 194 cm) and Cape Cod occupies only a small portion of its surface (the scale is 1:158,400). This is significant because this first map is widely different from a map of the extremity of the Cape published by J. D. Graham in 1836 that Thoreau also mentions in Cape Cod (177-78). The size of Graham's map (140 × 180 cm) is comparable to that of Borden's, but the scale is radically different, as Graham's map only depicts the area around the village of Provincetown (the scale is 1:10,560). What these maps reveal is the profoundly multiscalar nature of the cartographic materials used by Thoreau during his travels, and the special care he put into his preparatory work: his excursions are based on a solid and deliberate effort to acquire as much information about the Cape as possible.

This aspect is also obvious on the second and third maps of the Cape, which are kept in the Huntington Library in San Marino, California. The maps are placed in a binder bearing the title "A Course of Lectures on Cape Cod" that also contains drafts and notes for the conferences on Cape Cod that Thoreau read in the early 1850s 10 . These documents were part of the library of William Keeney Bixby that was purchased by the Huntington in 1918 11 . As noted, the two maps are very different. The first one (see fig. 2) is drawn on a large sheet of very thin tracing paper. It does not include any place name. Its most striking feature is a series of straight lines that radiate from a single point located in the middle. That point does not correspond to any topographical feature in the real world: on the contrary, as surmised by Stowell in the Thoreau Gazetteer (25), these lines are a geometric tool used by Thoreau to maintain proportion on his drawing. However, contrary to what Stowell indicated, the map is not "Thoreau's most sophisticated piece of cartography." Stowell seems to have assumed that the map was entirely created by Thoreau himself, and that the radiating lines were a clever trick used by the skillful surveyor to produce an original survey of the Cape. But Thoreau never attempted to survey the Cape alone. This work was already being done by the Coast Survey and it would have been extremely difficult for Thoreau to perform it all by himself. What Stowell failed to notice is that the map on tracing paper is actually a copy of the larger map that represents the entire peninsula: like the map in the Concord Library, it is a tracing of Borden's map of Massachusetts. As for the radiating lines, they were not used to maintain proportions on this drawing, but only to produce the third map, which is noticeably smaller (see fig. 3). The straight lines drawn by Thoreau enabled him to reduce the dimensions of his first map, possibly by using the principle of similar triangles: with this method, the topographical features inside each triangle can be reduced while maintaining the same dimensional relationships between the different points (Misra 318-19). A draft version of the map of Walden Pond available on the Concord Free Public Library's website shows that this is exactly how Thoreau produced the (literally) "reduced" plan of the pond that appears in Walden (286) 13 .

Figure 3: Map of the northern extremity of Cape Cod drawn by Thoreau by reducing the previous map (see fig. 2). This map is noticeably smaller and it is drawn on thick embossed notepaper. The inscription reads "4 1/6 miles to an inch," which is the scale of the reduced map. Source: Huntington Library.

Nègre 9

In other words, the map with the radiating lines is far from being "Thoreau's most sophisticated piece of cartography." On the contrary, it is merely a draft and an intermediary stage in the process of producing the third map, which is drawn on thicker paper and contains more details, such as place names and boundary lines. Another detail proves that these two maps are closely related. On the map on tracing paper, Thoreau has drawn a mark with his pencil at the point of intersection of each radiating line with the coastline of the Cape (see fig. 4). A close examination of the reduced map on thick paper shows that the exact same points have also been marked on that copy (see fig. 5). These are the points of reference used by Thoreau in the process of reducing the large map.

Figure 4: detail of the map on tracing paper (see fig. 2). The red circle is an example of a point of intersection between Thoreau's radiating straight lines and the coastline of the Cape. Thoreau marked each intersection with his pencil.

Figure 5: detail of Thoreau's reduced map (see fig. 3). The points of intersection marked on figure 4 also appear here, which shows that this map is a reduced copy of the previous one.

The three maps described in the Gazetteer are all different copies of the same original source: Borden's map of Massachusetts. Thoreau first traced the entire peninsula, then he copied the upper part of the Cape's "forearm" on tracing paper, and eventually he used the radiating straight lines to reduce that map and draw the third map on thicker paper. One cannot but notice that this portion of the peninsula exactly corresponds to the long "uninterrupted beach" that Thoreau evokes at the very beginning of his narrative, and on which he intended to walk "straight to Race Point, about twentyeight miles, and not meet with any obstruction." (4) 14 . Significantly enough, the moment in Cape Cod when Thoreau eventually reaches that beach leads to a scene of ecstatic confrontation with the reality of the Cape: "Cape Cod! as it cannot be represented on a map, color it as you will; the thing itself, than which there is nothing more like it, no truer picture or account;" (50). What is at work here is comparable to the "scopic impulse" (pulsion scopique) described by Michel de Certeau about people who went to the top of the World Trade Center to catch hold of the "texturology" of New York City (139-42). Seen from above, de Certeau says, the landscape can be read like a text. The reference to the map is telling: standing at the top of the high bank, Thoreau is able to seize the Cape, mentally and visually, because this moment has been prepared by the panoptic appropriation of the map, which is now brushed aside as a pale copy. The complex syntax ("than which there is nothing more like it") suggests how astounding the correspondence between the mental and the visual images of the Cape is to the observer.

Thoreau's tracing and subsequent reduction of the Cape's map also amounts to an effort to "comprehend" (that is, to understand and to embrace, almost physically) the reality of the Cape while at the same time keeping it at bay and remaining well aware of its deceptive authority and fixity. As Brückner noted in his recent article on the "cartographic turn" in American literary studies, maps possess the "nomothetic capability of positing the lasting existence of named locales or spatial relations," which "enables us (or seduces us) to invent assertive rules" (58). What Thoreau realizes during his excursions is that the shape of the peninsula that he has inscribed on his map as something stable and permanent is actually elusive and transient. The Cape is constantly moving, shifting, collapsing and sinking. For that reason, the dramatic image of the Cape being precariously tethered to the heavens by thousands of stalks of beach-grass (164) can be read as a metaphor of the transient and elusive nature of the map, which works as a layer of fixity superimposed by men on the Cape, even as the peninsula is already being brushed away by the sea and the wind.

Oases of wildness: a new distribution of the sensible

In Wild Fruits, Thoreau notes that he seems to be the only person in Concord who knows about the secret cranberries that grow in Beck Stow's swamp. The place cannot be improved and put to good use; it is not featured on the mental map of the area because it does not fit in the categories used to classify the different kinds of terrain: "I see that all is not garden and cultivated field and copse, that there are square rods in Middlesex County as purely primitive and wild as they were a thousand years ago, which have escaped the plow and the axe and the scythe and the cranberry rakelittle oases of wildness in the desert of our civilization" (168). Not only does the swamp remain invisible to most of the inhabitants, but actually visiting it is seen as a dangerously erratic behavior which can make one invisible, too: any farmer who would "spend an hour thus in this secluded swamp, bare-legged, intent on the sphagnum," Thoreau declares, "would be pronounced insane and have a guardian put over him" (168). To use Rancière's terms, the current distribution of the sensible is such that the place remains unseen, and those who lose their time in it are seen as irresponsible and put on the margin of the community. In his Journal, Thoreau notes: "How little there is on an ordinary map! … Between those lines indicating roads is a plain blank space in the form of a square or triangle or polygon or segment of a circle … . Yet the one, may be covered, in fact, with a primitive oak wood, like that of Boxboro, waving and creaking in the wind … . The waving woods, the dells and glades and green banks and smiling fields, the huge boulders, etc., etc., are not on the map, nor to be inferred from the map. (xxii-xxv). Such seems to be the case with the documents held at the University of Virginia Library, where they are part of the Clifton Waller Barrett Library. A shipping magnate and famous bibliophile, Mr. Barrett (1901-1991) was a major collector of American literary manuscripts and first editions. He purchased the map from an unidentified American bookseller, probably during the 1940s or 1950s, as part of a small but heterogeneous collection of Thoreau manuscripts. Because of that complex history, the date and the function of the map remain mysterious. 5 To my knowledge, the source of Thoreau's hand-drawn copy has never been identified explicitly before. However, the Library of America edition of A Week, Walden, Cape Cod and The Maine Woods, published in 1985, includes a reproduction of a portion of Hale's map. The notes in this volume were written by Robert F. Sayre, who probably also decided to include the map, and who might have been aware that Hale's map had been copied by Thoreau. Yet Thoreau's tracing is not reproduced in that volume and Sayre does not mention it.

Hale's map is available online at http://Hdl.Loc.Gov/Loc.Gmd/G3720.Rr001030. 6 Hale's map is reproduced courtesy of the Library of Congress. Thoreau's tracing is reproduced courtesy of the Albert and Shirley Small Special Collections Library at the University of Virginia Library. Hessler linked his findings to the last chapter of Cape Cod, which contains a long section (178-202) devoted to the question of how the Cape has been represented and named on ancient maps. Thoreau's work of archival excavation of the cartographic past of the Cape is similar to the logic that is at work in A Week. 9 http://www.concordlibrary.org/scollect/Thoreau_surveys/157.htm 10 Information about these maps was provided by Nathalie Russell, curator to of literary manuscripts at the Huntington Library, in emails to the author on July 1, 2013 and March 8, 2016. High-resolution images of the two maps were obtained with the generous support of the LARCA (Laboratoire de Recherche sur les Cultures Anglophones, Université Paris Diderot, France). 11 Howarth mentions Bixby as one of the persons who purchased some of Thoreau's manuscripts from George S. Hellman, who had acquired them from E. H. Russell, who had received them from Harrison Blake, who had received them from Sophia Thoreau (Howarth xxii). Contrary to the map kept in the University of Virginia Library, it is possible to retrace the complete history of those manuscripts. 12 Thoreau's maps of the northern part of Cape Cod are reproduced courtesy of the Huntington Library. 13 http://www.concordlibrary.org/scollect/Thoreau_surveys/133c.htm 14 The inscription "CAPE COD BEACH" that Thoreau carefully inscribed along the outer coastline of the Cape (see fig. 3) does not appear on any map of that area. The implication is that this is a sort of personal toponym coined by Thoreau himself to refer to the long beach he describes at length in Cape Cod. 15 In September, 2014, I contacted Allan H. Schmidt, a specialist of Thoreau's surveys and a resident of Concord, to have his opinion on the question of where Beck Stow's swamp actually is. In just a few days, Allan was able to track down a reference to the swamp in the Herbert Gleason Archives at the Concord Library. He eventually located the swamp on one of Thoreau's land surveys, available on line: http://www.concordlibrary.org/scollect/Thoreau_surveys/7c.htm. The correct location is the one that appears on Bradley Dean's first map, drawn by Gleason himself in 1906.

Figure 1 :

 1 Figure 1: comparison of a portion of Nathan Hale's Map of the New England States (1826) with Thoreau's tracing 6 .

  The Maine Woods, Thoreau evokes three different maps of Maine that he has used during his excursions: Moses Greenleaf's Map of the State of Maine published in 1844 (Maine Woods 15), George W. Coffin's Plan of the Public Land in the State of Maine published in 1835 (Maine Woods 15, 91 and 279) and Joseph H. Colton's Maine published in 1855 (Maine Woods 94 and 279-80).

Figure 2 :

 2 Figure 2: Thoreau's tracing of a map of the northern extremity of Cape Cod. The map is drawn on thin tracing paper 12 .

  7 http://www.concordlibrary.org/scollect/Thoreau_surveys/165.htm 8 Ranging from a 1527 world map by Robert Thorne (copied by Richard Hackluyt in 1582) to Samuel de Champlain's 1632 map of "Nouvelle France," the maps identified by Hessler were copied by Thoreau in the fall of 1855. They cover more than a century of successive efforts to chart the coast of what is now eastern Canada and New England. In a special edition of Cape Cod published in 2010 (Henry David Thoreau on Cape Cod),

  In his texts, Thoreau effectively reclaims the local map by revealing what those blank spaces contain, as evidenced by Bradley Dean's maps featuring Beck Stow's swamp15 . Such an accomplishment is profoundly political, in that it turns the distribution of the sensible on its head and suggests that the ---. A Week on the Concord and Merrimack Rivers, Walden, Cape Cod, The Maine Woods. Library of America, 1985. ---. Wild Fruits: Thoreau's Rediscovered Last Manuscript. Edited by Bradley P. Dean, W. W. Norton & Co., 2000. On the importance of Beck Stow's swamp in Thoreau's writings, see Scheese 142-146 and Howarth, "Reading the Wetlands" 71.2 Most of the documents studied in this paper were made available to the public as soon as 1970, when Robert Stowell published the Thoreau Gazetteer, edited by William Howarth, as a supplementary volume of the authoritative edition of Thoreau's texts published by Princeton University Press.3 Information about the map was provided by David Whitesell, curator at the University of Virginia Library, in an email to the author on March 18, 2013.4 The other Thoreau manuscripts that come with the map are extremely diverse and seem to have no direct relationship to the map itself. As Howarth explained in his study The Literary manuscripts of Henry David Thoreau, most of Thoreau's manuscripts were carelessly dispersed at the end of the 19 th century after having been re-ordered and re-classified by their successive owners, and eventually sold to several different collectors

	" (XIV:
	228-29)

1


Omar El-Fallah
email: elfallah@fsr.ac.ma

LAMA Youssef Elmadani
email: elmadanima@gmail.com

Karim

K Kellay
email: kkellay@math.u-bordeaux.fr

Kernel estimate and capacity in the Dirichlet spaces

Keywords: 2000 Mathematics Subject Classification. Primary 46E22; Secondary 31A05, 31A15, 31A20, 47B32 Dirichlet spaces, reproducing kernel, capacity, polar set

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The reproducing kernel plays an important role in the study of Hilbert spaces of analytic functions. In particular, it allows to determine the rate of growth of functions near the boundary and its tangential behavior; their properties are closely related to embedding theorems, sampling and interpolation sets, and other topics. Here we shall be concerned with Dirichlet spaces D(µ).

Let H 2 denote the classical Hardy space of analytic functions on the unit disc D having square summable Taylor coefficients at the origin. Every function f ∈ H 2 has nontangential limits almost everywhere on the unit circle T = ∂D. We denote by f (ζ) the non-tangential limit of f at ζ ∈ T if it exists. Let µ be a positive finite measure on T, the Dirichlet space D(µ) is the set of analytic functions f ∈ H 2 , such that

D µ (f ) := 1 2π T T |f (ζ) -f (ξ)| 2 |ζ -ξ| 2 |dζ|dµ(ξ) < ∞.
The space D(µ) is endowed with the norm These spaces were introduced by Richter in [START_REF] Richter | Invariant subspaces of the Dirichlet shift[END_REF] by examining the 2-isometries on the Hilbert spaces. A bounded operator T in a Hilbert space H is called 2-isometry operator if

f 2 µ := f 2 H 2 + D µ (f ). If µ = 0, then D(µ) = H
T * 2 T 2 -2T * T -I = 0,
is said to be cyclic if there exists x ∈ H such that span{T n x, n ≥ 0} is dense in H and is called analytic if n≥0 T n H = {0}. Richter in [START_REF] Richter | Invariant subspaces of the Dirichlet shift[END_REF] proved that every cyclic, analytic 2-isometry can be represented as a multiplication by z on a Dirichlet space D(µ) for some positive finite measure µ on T.

As consequence [START_REF] Richter | A representation theorem for cyclic analytic two-isometries[END_REF][START_REF] Richter | Invariant subspaces of the Dirichlet shift[END_REF] Richter gave an analogue of Beurling's theorem for the classical Dirichlet space : every invariant subspace for D is of the form φD(µ φ ) where dµ φ = |φ| 2 dm and φ is an extremal function for D, that is φ µ = 1, and φ, z n φ = 0, n ≥ 1.

We refer to the monograph [START_REF] El-Fallah | A primer on the Dirichlet space[END_REF] for numerous results on the Dirichlet space. By representation theorem for bounded linear functionals on a Hilbert space, to each z ∈ D, there is unique element k µ (•, z) in D(µ), such that

f (z) = f, k µ (•, z) D(µ) , f ∈ D(µ).
The function k µ (ζ, z), with (ζ, z) ∈ D ×D, is called the reproducing kernel for D(µ). Let us Shimorin [START_REF] Shimorin | Complete Nevanlinna-Pick property of Dirichlet spaces[END_REF] proved that all Dirichlet spaces have complete Nevanlinna-Pick reproducing kernels. Let P [µ] be the Poisson integral of the positive finite measure µ on T

P [µ](z) = T 1 -|z| 2 |ζ -z| 2 dµ(ζ), z ∈ D.
In the following theorem, we provide an asymptotic estimate of the reproducing kernel k µ of D(µ) on the diagonal.

Theorem 1. Let µ be a finite positive measure on T. We have

k µ (z, z) ≍ 1 + |z| 0 dr (1 -r)P [µ](rz/|z|) + (1 -r) 2 ,
where the implied constants are absolute.

Our second aim here is to give estimates of capacity associated to the Dirichlet space D(µ) which is closely related to some notions of potential theory. Let D h (µ) be the harmonic version of D(µ) given by

D h (µ) := f ∈ L 2 (T) : f 2 µ := f 2 L 2 (T) + D µ (f ) < ∞ .
D h (µ) is a Dirichlet space in the sense of Beurling-Deny [START_REF] Beurling | Dirichlet spaces[END_REF]. Some aspects of the potential theory associated to the general Dirichlet spaces were studied in several papers (see for instance [START_REF] Fukushima | Dirichlet Forms and Symmetric Markov Processes[END_REF]). In this paper we will focus on the notion of capacity. We recall at first, the definition of capacity in the sense of Beurling-Deny. Let U be an open subset of the unit circle. The c µ -capacity of U is defined by

c µ (U) := inf u 2 µ : u ∈ D h (µ), u ≥ 0 and u ≥ 1 a.e. on U . (1)
As usual we define the c µ -capacity of any subset F ⊂ T by

c µ (F ) = inf{c µ (U) : U open, F ⊂ U}.
Since the L 2 norm dominates the Dirichlet norm, it is completely obvious that sets having c µ -capacity 0 have Lebesgue measure 0. We say that a property holds c µ -quasi-everywhere (c µ -q.e.) if it holds everywhere outside a set of c µ -capacity 0. So, c µ -q.e. implies a.e.. A closed set of capacity zero will be called, throughout this paper, µ-polar set. If µ = m is the Lebesgue measure on T, then c µ = c m is comparable to the logarithmic capacity c, see [START_REF] Meyers | A theory of capacities for potentials of functions in Lebesgue classes[END_REF]Theorem 14] and [START_REF] Adams | Function spaces and potential theory[END_REF]Theorem 2.5.5]. Every function f ∈ D(µ) has non-tangential limits c µ -q.e. Furthermore, if E is a closed subset of T such that c µ (E) = 0, then there exists a cyclic function f ∈ D(µ) continuous on T and vanishing on E, see [START_REF] Elmadani | Cyclicity in the Dirichlet spaces[END_REF]. The generalized Brown-Shields conjecture asserts that an outer function f ∈ D(µ) is cyclic if and only if c µ (Z(f )) = 0, where Z(f ) is the zero set of f . In [6, Theorem 1] we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces see

c µ ({e iθ }) > 0 ⇐⇒ 1 0 dr (1 -r)P [µ](re iθ ) + (1 -r) 2 < ∞. (2)
This allows to construct examples of positive measure µ and a countable closed set E such that µ(E) = 0 and E is uniqueness set for D(µ) [START_REF] Bahajji | Havin-Mazya uniqueness theorem for Dirichlet space[END_REF]Corollary 3.6]. We obtain an estimate of capacity of arcs, more precisely we have Theorem 2. Let I ⊂ T be the arc of length |I| = 1 -ρ with the midpoint at ζ ∈ T. Then

1 c µ (I) ≍ 1 + ρ 0 dr (1 -r)P [µ](rζ) + (1 -r) 2 .
In the sequel we will suppose that E is a closed set which has Lebesgue measure zero and µ is a finite positive measure on T. Now our goal is to give sufficient condition on E to be µ-polar. Let us introduce the local modulus of continuity of µ on E which will play a crucial role in this paper. It is defined by

ρ µ,E (t) := sup{µ(ζe -it , ζe it ) : ζ ∈ E}. (3)
Note that ρ µ,T = ρ µ is the classical modulus of continuity of µ. Let us write

E t := {ζ ∈ T : dist(ζ, E) ≤ t},
where dist denotes the distance with respect to arc-length, and denote by |E t | the Lebesgue measure of E t . We can express the function |E t | in terms of

N E (t) := 2 j 1 {|I j |>2t} ,
where (I j ) are the components of T\E, as follows

t 0 N E (s)ds = |E t |.
In Theorem 5.4, we give a sufficient conditions on a closed subset E, in terms of ρ µ,E and N E , to be µ-polar. To illustrate this theorem we give here some of its corollaries.

(i) If π 0 dt t 0 (ρ µ,E (s)N E (s)/s)ds = +∞, then c µ (E) = 0.
This result can be viewed as an extension of Carleson's Theorem [3, section IV, Theorem 2] which says that if

π 0 dt |E t | = ∞, then c(E) = 0.
In fact if µ = m then ρ µ,E (t) = t and c µ ≍ c (c is the logarithmic capacity).

(ii) Suppose that ρ µ,E (t) = O(t α ) with 1 ≤ α < 2. If π 0 dt t α-1 |E t | = +∞, then c µ (E) = 0. Our proof is different from the one given in [3] (iii) If ρ µ,E (t) = O(t α ) with α > 2, we have c µ (E) = 0.
Note also that if

t α = O(ρ µ,{1} (t)) with α < 1, then by (2), c µ ({1}) > 0.
The plan of the paper is the following. In the next section we recall two formulas of the Dirichlet norm; we also give punctual estimates of some outer functions. In Section 3, we obtain norm estimates of our test functions. In Section 4, we state diagonal asymptotic estimates of reproducing kernel. To prove the lower estimates of k µ (z, z), we establish a sharp norm estimates of some outer functions which peak near z. This allows us to transfer our problem to an estimation of the norm of the kernel of an appropriate weighted Sobolev space. In fact and roughly speaking, Theorem 1 says that k µ (z, z) ≍ K ϕ (1 -|z|, 1 -|z|) where K ϕ is the reproducing kernel of the weighted Sobolev space defined by

W 2 (ϕ) := f ∈ C (0, 2π] : f (x) = f (1) + 2π x g(t)dt, g ∈ L 2 ((0, 2π), ϕdt) , where ϕ(t) = tP [µ]((1 -t)z/|z|) + t 2 .
In section 5, we prove the results on capacities cited above. The proof of Theorem 5.4 uses an idea analogous to the proof of Theorem 1. However, our test functions must peak on the whole set E and the desired weighted Sobolev spaces will depend on µ and E. In fact, we prove that there is no bounded point evaluation at 0 for W 2 (ϕ) (where ϕ depends on ρ µ,E and N E ), then c µ (E) = 0. Note finally, that there is no bounded point evaluation at 0 for W 2 (ϕ) if and only if lim

t→0 + K ϕ (t, t) = ∞.
Throughout the paper, we use the following notations:

• A B means that there is an absolute constant C such that A ≤ CB.

• A ≍ B if both A B and B A hold.

• C(x 1 , . . . , x n ) denote a constant which depends only on variables x 1 , . . . , x n .

Preliminaries and technical lemmas

2.1. Norm formulas. In this subsection we recall some results about norm formulas in Dirichlet spaces which will be used in what follows.

For a finite positive measure µ on T, the harmonic Dirichlet space

D h (µ) consists of functions f ∈ L 2 (T) such that D µ (f ) := T D ξ (f )dµ(ξ) < ∞, where D ξ (f ) is the local Dirichlet integral of f at ξ ∈ T given by D ξ (f ) := T |f (e it ) -f (ξ)| 2 |e it -ξ| 2 dt 2π .
The Douglas' formula, see [START_REF] El-Fallah | A primer on the Dirichlet space[END_REF]Theorem 7.1.3], expresses the Dirichlet integral of a function f in terms of the Poisson transform of µ, namely

D µ (f ) = D |∇P [f ]| 2 P [µ]dA, f ∈ D h (µ),
where dA(z) = dxdy/π stands for the normalized area measure in D. In particular, if

f ∈ D(µ)(= D h (µ) ∩ H 2 ), Douglas' Formula becomes D µ (f ) := D |f ′ (z)| 2 P [µ](z)dA(z) < ∞.
Another useful formula, due to Richter and Sundberg [15, Theorem 3.1], gives the local Dirichlet integral of function f in terms of their zero sequence, their singular measure and the modulus of their radial limit. We will need, throughout this paper, the Richter-Sundberg formula mainly for outer functions. Recall that outer functions are given by

f (z) = exp T ζ + z ζ -z log ϕ(ζ) |dζ| 2π , (z ∈ D),
where ϕ is a positive function such that log ϕ ∈ L 1 (T). Note that the radial limit of f , noted also by f , exists a.e. and |f | = ϕ a.e. on T. We have the following Richer-Sundeberg formula [START_REF] Richter | A formula for the local Dirichlet integral[END_REF] (see also [START_REF] El-Fallah | A primer on the Dirichlet space[END_REF]Theorem 7.4.2])

Theorem 2.1. Let f ∈ H 2 be an outer function such that f (ζ) exists at ζ ∈ T. Then D ζ (f ) = T |f (λ)| 2 -|f (ζ)| 2 -2|f (ζ)| 2 log |f (λ)/f (ζ)| |λ -ζ| 2 |dλ| 2π . (4)
2.2. Punctual estimates of test functions. The result obtained in this subsection will be used in the proof of the lower estimate of the kernel.

Lemma 2.2. Let 1/2 < r = 1 -a < 1 and let I k = [e ia k , e ia k+1 ] with a 0 = 0, a k = 2 k a (k ≥ 1). Let N be the integer such that 2 N a ≤ π < 2 N +1 a, then N -1 k=0 (k + 1)̟(r, I k , D) ≍ 1,
where ̟(r, I k , D) denotes the harmonic measure of I k at r in D.

Proof. Without loss of generality, we may suppose that 2 N a = π. Note that

N -1 k=0 (k + 1)̟(r, I k , D) ≥ N -1 k=0 ̟(r, I k , D) ≍ 1.
For the reverse inequality, let g(z) = log 1/|1 -rz|. Since g is harmonic in the neighbourhood of D,

g(z) = 1 2π π -π 1 -|z| 2 |1 -ze -iθ | 2 log 1 |1 -re iθ | dθ.
So,

g(r) = log 1 1 -r 2 = N -1 k=0 1 π a k+1 a k 1 -r 2 |1 -re -iθ | 2 log 1 |1 -re iθ | dθ. (5)
For k = 0, . . . , N -1, and θ ∈ (a k , a k+1 ), we have

1 |1 -re iθ | ≍ 1 2 k 1 1 -r . By (5), we get log 1 1 -r 2 = log 1 1 -r -log 2 N -1 k=0 k̟(r, I k , D) + O(1)
and our result follows.

Let w : (0, π) → (0, +∞) be a continuous positive function such that log w ∈ L 1 (T). As before f w denote the outer function satisfying

|f w (e it )| = w(|t|) a.e. on (-π, π). ( 6
)
Proposition 2.3. Let w : (-π, π) → (0, +∞) be an even continuous positive decreasing function such that log w ∈ L 1 (T). Suppose that w(x) ≤ 2w(2x). Let f w be an outer function given by [START_REF] El-Fallah | Cyclicity and invariant subspaces in The Dirichlet space[END_REF]. Then

w(1 -r) |f w (r)|, 0 ≤ r < 1.
Proof. Let a , I k and N as in lemma 2.2, and suppose that a N = π. We have

|f w (r)| = exp N -1 k=0 1 π I k 1 -r 2 |1 -re iθ | 2 log w(θ)dθ ≥ exp N -1 k=0 log ω(2 k+1 a) 1 π I k 1 -r 2 |1 -re iθ | 2 dθ ≥ exp N -1 k=0 (log ω(a) -(k + 1) log 2) 1 π I k 1 -r 2 |1 -re iθ | 2 dθ ≥ exp log w(a) N -1 k=0 ̟(r, I k , D) -log 2 N -1 k=0 (k + 1)̟(r, I k , D) .
We obtain from Lemma 2.2, that w(1 -r) |f w (r)|. The case a N < π can be treated in the same way by taking into account the interval [e ia N , e iπ ].

Regularization lemma.

Let µ be a positive finite measure on T, we set dµ(s) = dµ(e is ). Denote by

µ(s) = µ([e -is , e is ]) (0 ≤ s ≤ π) and µ(s) = µ(π) (s > π). Let F µ (x) = π -π x 2 x 2 + s 2 dµ(s) x > 0. ( 7
)
Note that F µ is increasing and F µ (x)/x 2 is decreasing. We extend F µ at the origin by F µ (0) = F µ (0 + ). In the following lemma we collect some elementary properties of F µ which will be used in the sequel.

Lemma 2.4. Let ν be a positive finite measure on T. We have the following

(1) F ν (x) ≍ xP [ν](1 -x), for x > 0, (2) ν(x) F ν (x) for x ≥ 0, (3)
x≤|s|≤π dν(s) s 2 F ν (x) x 2 , for x > 0, (4) If h is a positive monotone function on (0, π). Then a -a h(|x|)dν(x) a 0 h(x) x ν(2x)dx, (5) If ν(2x) ≤ c ν(x) for some constant c < 4, then F ν (x) ν(x) 4 -c . Proof. (1) For x ∈ (0, 1/2) we have obviously x(1 -(1 -x) 2 ) ≍ x 2 and |e iθ -(1 -x)| 2 ≍ x 2 + θ 2 . Hence xP [ν](1 -x) = π -π x(1 -(1 -x) 2 ) |e iθ -(1 -x)| 2 dν(θ) ≍ π -π x 2 x 2 + θ 2 dν(θ) = F ν (x).
(2) Let x ≥ 0, we have

ν(x) = x -x dν(θ) = x -x x 2 x 2 + θ 2 dν(θ) + x -x θ 2 x 2 + θ 2 dν(θ) ≤ 2 x -x x 2 x 2 + θ 2 dν(θ) = 2F µ (x). (3) Fix x > 0, we have x<|s|<π dν(s) s 2 = +∞ n=0 2 n+1 x 2 n x dν(s) s 2 ≍ 1 x 2 +∞ n=0 1 2 2n x 2 2 n+1 x 2 n x dν(s) = 1 x 2 +∞ n=0 1 2 2n 2 n+1 x 2 n x x 2 + s 2 x 2 + s 2 dν(s) ≤ 4 x 2 +∞ n=0 2 n+1 x 2 n x x 2 x 2 + s 2 dν(s) ≤ 4 F ν (x) x 2 .
To prove (4) suppose that h is a decreasing function. Clearly if ν({0}) > 0, then (4) is obvious. So, suppose that ν({0}) = 0. We have

a -a h(|x|)dν(x) = n≥0 2 -n a 2 -(n+1) a h(x)(dν(x) + dν(-x)) m n=0 h(2 -n a) ν(2 -n a) a 0 h(x) x ν(2x)dx.
Analogue argument works if h is increasing. Finally to prove [START_REF] Bahajji | Havin-Mazya uniqueness theorem for Dirichlet space[END_REF], suppose that ν(2x) ≤ c ν(x) with c < 4. We have ν(2 n x) ≤ c n ν(x) and

|t|≥x dν(t) t 2 dt = n≥0 2 n x≤|t|≤2 n+1 x dν(t) t 2 ≤ n≥0 ν(2 n+1 x) 2 2n x 2 ≤ 4c 4 -c ν(x) x 2 . So F ν (x) ≤ ν(x) + |θ|>x x 2 x 2 + θ 2 dν(θ) ≤ ν(x) + x 2 θ≥x dν(x) θ 2 ν(x) 4 -c .

Norm estimate of test functions

3.1. Norm estimate of analytic test functions. The purpose of this subsection is to give an estimate of norms of some outer functions which play an important role in what follows.

The following lemma is the first step to prove Theorem 3.2.

Lemma 3.1. Let w : [0, π] → (0, +∞) be a C 1 decreasing convex function such that w(x) ≤ 2w(2x). Suppose that x 2 |w ′ (x)| is increasing and let f w be the outer function given by [START_REF] El-Fallah | Cyclicity and invariant subspaces in The Dirichlet space[END_REF].

Then D µ (f w ) J 1 + J 2 + J 3 ,
where

J 1 := π x=0 x y=0 |w ′ (y)|w(y) |w ′ (x)| w(x) µ(y)
x dxdy,

J 2 := π s=-π w ′ (s) 2 sdµ(s), J 3 := π x=0 π y=x x|w ′ (y)|w(y) |w ′ (x)| w(x) y≤|s|≤π dµ(s) s 2 dxdy.
Proof. Without loss of generality, we may assume that dµ(s) = dµ(-s). By Richter-Sundberg formula (4) we have Since w(2x) ≍ w(x) and w ′ (2x) ≍ w ′ (x), we have

D e is (f w ) = 8 2π π t=0 t x=s x y=s w ′ (y)w(y) w ′ (x) w(x) dydx dt |e is -e it | 2 .
I 2 = π s=0 2s t=s/2 t x=s x y=s w ′ (y)w ′ (x) w(y) w(x) dydxdt dµ(s) |e is -e it | 2 ≍ π s=0 w ′ (s) 2 2s t=s/2 t x=s x y=s dydxdt dµ(s) (s -t) 2 = 1 2 π s=0 w ′ (s) 2 2s t=s/2 (s -t) 2 dt dµ(s) (s -t) 2 = 3 4 π s=-π w ′ (s) 2 sdµ(s).
Hence [START_REF] El-Fallah | Cyclicity and invariant subspaces in The Dirichlet space[END_REF].

I 2 ≍ π s=-π w ′ (s) 2 sdµ(s). If 0 ≤ t ≤ s/2,
Then D µ (f w ) F µ w ′ ∞ w ∞
, where F µ is given by [START_REF] El-Fallah | A primer on the Dirichlet space[END_REF].

Proof. By (3) of Lemma 2.4, we have µ ≤ F µ . Now Lemma 3.1 gives

J 3 π x=0 π y=x |w ′ (y)|w(y) |w ′ (x)| w(x) F µ (y) y 2 xdydx F µ w ′ ∞ π x=0 π y=x |w ′ (x)| w(y)x w(x)y 2 dydx F µ w ′ ∞ π x=0 π y=x |w ′ (x)| x y 2 dydx F µ w ′ ∞ w ∞ .
Note that x|w ′ (x)| w(x) for all x ∈]0, π]. Indeed, since w(2x) ≍ w(x) and |w ′ (2x)| ≍ |w ′ (x)|, it suffices to prove the inequality for x ∈ [0, π/2]. We have

w(x) ≥ π x t 2 |w ′ (t)| dt t 2 ≥ x 2 |w ′ (x)| 1 x - 1 π ≥ x 2 |w ′ (x)|.
So, again by Lemma 2.4 we get

J 1 F µ w ′ ∞ π x=0 x y=0 |w ′ (x)| w(y) w(x)x dydx F µ w ′ ∞ π x=0 x y=0 π u=y |w ′ (x)||w ′ (u)| w(x)x dudydx + F µ w ′ ∞ π x=0 x y=0 |w ′ (x)| w(π) w(x)x dydx = J 12 + J 22 .
We have

J 12 F µ w ′ ∞ π u=0 π x=0 min(u,x) y=0 |w ′ (x)||w ′ (u)| w(x)x dudydx F µ w ′ ∞ π u=0 u x=0 |w ′ (x)w ′ (u)| w(x) dudx + π u=0 π x=u u|w ′ (x)w ′ (u)| xw(x) dudx F µ w ′ ∞ π x=0 |w ′ (x)|dx + π u=0 π x=u u|w ′ (x)w ′ (u)| xw(x) dudx) F µ w ′ ∞ w ∞ + π u=0 π x=u u|w ′ (u)| x 2 dudx F µ w ′ ∞ w ∞ .
Since w is decreasing, we get

J 22 ≤ F µ w ′ ∞ π x=0 x y=0 |w ′ (x)| dx x dy ≤ F µ w ′ ∞ w ∞ .
Finally, applying (4) of Lemma 2.4 with dν(s) = sdµ(s), we have ν(t) ≤ t µ(t) and

J 2 π 0 w ′ (t) 2 t ν(2t)dt F µ w ′ ∞ w ∞ .
3.2. Norm estimate of test functions in D h (µ). Our goal here is to give an estimate of the norm of some distance functions in D h (µ) (For analytic distance functions see [START_REF] El-Fallah | On the Brown-Shields conjecture for the cyclicity in the Dirichlet space[END_REF]).The result of this subsection will be used in the proof of Theorem 5.4.

Let E be a closed subset of T, µ be a positive finite measure and denote by ρ µ,E the local modulus of continuity of µ on E given by

ρ µ,E (t) = sup ζ∈E µ(ζe -it , ζe -it ).
Note that ρ µ,{1} (t) = µ(t). Recall that

N E (t) := 2 j 1 {|I j |>2t}
where (I j ) are the components of T\E. Note also that for every measurable function

Ω : (0, π] → R + T Ω(dist(ζ, E))|dζ| = π 0 Ω(t)N E (t)dt. ( 8
)
Lemma 3.3. Let Ω : (0, π] → R + be a positive decreasing function, then

T Ω(dist(ζ, E))dµ(ζ) 1 0 Ω(t) ρ µ,E (t) t N E (t)dt.
Proof. Write T \ E = ∪ n I n , where (I n ) n = (e iαn , e iβn ) n are the components of T \ E. Let dµ n (t) = dµ(t + α n ) + dµ(β n -t), By Lemma 2.4

In

Ω(dist(ζ, E))dµ(ζ) ≍ |In|/2 0 Ω(t)dµ n (t) |In|/2 0 Ω(t) µ n (t) t dt |In|/2 0 Ω(t) ρ µ,E (t) t dt.
Summing over all I n , we get

T Ω(dist(ζ, E))dµ(ζ) 1 0 Ω(t) ρ µ,E (t) t N E (t)dt,
and the proof is complete.

I 1 := π 0 π 0 (w(t) -w(t + s)) 2 s 2 ρ µ,E (t) t N E (t)dsdt, I 2 = π 0 w ′ (t) 2 ρ µ,E (2t)N E (t)dt, I 3 := π t=0 π s=t (w(t) -w(t + s)) 2 s 2 ρ µ,E (s) s N E (t)dsdt, Proof. Set δ = dist(ζ, E) and δ ′ = d(ζ ′ , E)
. By Lemma 3.3 we have

J 1 = 1 2π T δ ′ ≤δ (w(δ) -w(δ ′ )) 2 |ζ -ζ ′ | 2 |dζ|dµ(ζ ′ ) T δ ′ ≤δ (w(δ ′ ) -w(δ ′ + |ζ -ζ ′ |)) 2 |ζ -ζ ′ | 2 |dζ|dµ(ζ ′ ) T π s=0 (w(δ ′ ) -w(δ ′ + s)) 2 s 2 dsdµ(ζ ′ ) π 0 π 0 (w(t) -w(t + s)) 2 s 2 ρ µ,E (t) t N E (t)dsdt,
and

J 2 = 1 2π T δ≤δ ′ (w(δ) -w(δ ′ )) 2 |ζ -ζ ′ | 2 dµ(ζ ′ )|dζ| T T (w(δ) -w(δ + |ζ -ζ ′ |)) 2 |ζ -ζ ′ | 2 dµ(ζ ′ )|dζ| ζ∈T |ζ ′ -ζ|≤δ + ζ∈T |ζ ′ -ζ|≥δ = J 21 + J 22 .
Clearly we have

J 21 = T |ζ-ζ ′ |≤δ w ′ (x δ,|ζ-ζ ′ | ) 2 dµ(ζ ′ )|dζ|, where x δ,|ζ-ζ ′ | ∈ (δ, δ + |ζ -ζ ′ |). Since w is convex, then J 21 ≤ T w ′ (2δ) 2 |ζ-ζ ′ |≤δ dµ(ζ ′ )|dζ| ≤ T w ′ (2δ) 2 ρ µ,E (δ)|dζ|.
Hence by ( 8) we get

J 21 T w ′ (δ) 2 ρ µ,E (2δ)|dζ| π 0 w ′ (t) 2 ρ µ,E (2t)N E (t)dt.
With the same calculation, as in Lemma 3.3, we have

T T Ω(dist(ζ ′ , F )dµ(ζ ′ )|dζ|
where Ω(s) = 1 {s≥δ} w(δ) -w(δ + s) s 2 and F = {ζ}. Then

J 22 T π 0 Ω(s) µ(ζe -is , ζe is ) s ds|dζ| = T π 0 w(δ) -w(δ + s) s 2 µ(ζe -is , ζe is ) s ds|dζ|.
Again by ( 8) we get

J 22 π t=0 π s=t (w(t) -w(t + s)) 2 s 2 ρ µ,E (s) s N E (t)dsdt.
Since D µ (w) = J 1 + J 2 , we get our result.

For a positive increasing function ψ such that ψ(0) = 0, we set

M ψ,E (s) = max s 0 ψ(t) t N E (t)dt, ψ(s) s |E s | , (9)
where

E t = {ζ ∈ E : dist(ζ, E) ≤ t}.
If ψ is concave, then ψ(x)/x is decreasing and

ψ(s) s |E s | ≤ ψ(s) s s 0 N E (t)dt ≤ s 0 ψ(t) t N E (t)dt = M ψ,E (s).
And if ψ is convex then ψ(x)/x is increasing, so

s 0 ψ(t) t N E (t)dt ≤ ψ(s) s s 0 N E (t)dt = ψ(s) s |E s | = M ψ,E (s).
The function ψ is called α-admissible if ψ is concave or convex and ψ(s)/s α is decreasing for some α > 0. Now we can state the main result of this subsection Theorem 3.5. Let E be a closed subset of T. Let w be a convex decreasing function and let Ω(ζ) = w(dist(ζ, E)). Suppose that there exists an α-admissible function ψ, with α < 2, such that ρ µ,E (s) ≤ ψ(s). Then

D µ (Ω) ≤ C(α) w ′ M ψ,E ∞ w ∞ .
Proof. We apply lemma 3.4. An analogue calculation, as in the proof of Theorem 3.2, gives

I 1 + I 2 w ′ M ρ µ,E ,E ∞ w ∞ .
Estimates of I 1 : we have

I 1 = π 0 π 0 (w(t) -w(t + s)) 2 s 2 ρ µ,E (t) t N E (t)dsdt, = π 0 π 0 w ′ (x t,s ) 2 ρ µ,E (t) t N E (t)dsdt,
where x t,s ∈ (t, t + s). Since w is convex, we get

I 1 ≤ π t=0 π s=0 w ′ (t + s) 2 ρ µ,E (t) t N E (t)dsdt = π t=0 t s=0 . . . + π t=0 π s=t ≤ π t=0 w ′ (2t) 2 ρ µ,E N E (t)dt + π t=0 π s=t w ′ (2s) 2 ρ µ,E (t) t N E (t)dsdt ≤ π t=0 w ′ (2t) 2 ρ µ,E N E (t)dt + π s=0 w ′ (2s) 2 s t=0 ρ µ,E (t) t N E (t)dtds ≤ 2 π 0 w ′ (t) 2 M ρ µ,E (t)dt ≤ 2 w ′ M ψ,E ∞ w ∞ .
Estimates of I 2 , we have

I 2 = π 0 w ′ (t) 2 ρ µ,E (t)N E (t)dt ≤ 2 π 0 w ′ (t) 2 M ρ µ,E (t)dt ≤ 2 w ′ M ψ,E ∞ w ∞ .
Now we consider the integral I 3 . We have

I 3 = 2 π t=0 π s=t t+s u=t t+s v=t w ′ (u)w ′ (v) ρ µ,E (s) s 3 N E (t)dvdudsdt π t=0 π s=t 2s u=t 2s v=t |w ′ (u)||w ′ (v)| ρ µ,E (s) s 3 N E (t)dvdudsdt π v=0 |w ′ (v)| ψ(v) v 2 v u=0 |w ′ (u)||E u |dudv π v=0 |w ′ (v)| 1 v 2-α v u=0 |w ′ (u)| ψ(u) u α |E u |dudv ≤ C(α) sup u |w ′ (u)| ψ(u) u |E u | π v=0 |w ′ (v)| 1 v 2-α v 2-α dv ≤ C(α) sup u |w ′ (u)| ψ(u) u |E u | w ∞ .
Corollary 3.6. Let E be a closed subset of T. Let w be a convex decreasing function and let

Ω(ζ) = w(dist(ζ, E)). Suppose that ρ µ,E (t) = O(t α ) for some α > 0. Then (1) D µ (Ω) ≤ C(α) sup t≥0 w ′ (t) π t t α-1 N E (t)dt × w ∞ , if 0 < α ≤ 1, (2)
D µ (Ω) ≤ C(α) sup t≥0 w ′ (t)t α-1 |E t | × w ∞ , if 1 ≤ α < 2, (3) D µ (Ω) ≤ C(α) sup t≥0 w ′ (t)| log t||E t | × w ∞ , if α = 2, (4) D µ (Ω) ≤ C(α, h) sup t≥0 w ′ (t)h(t)|E t | × w ∞ , if α > 2,
where h is a positive increasing function such that h(0) = 0 and 0 ds/h(s) < ∞.

Proof. (1) and ( 2) are direct consequences of Theorem 3.5. Now we prove (3). The proof of Theorem 3.5 gives

I 3 π v=0 |w ′ (v)| log 1 v v u=0 |w ′ (u)||E u |dudv,
and we get our estimate. Finally we prove (4). Since |E t | → 0, then there exists a positive increasing function h, h(0) = 0, such that 0 ds/h(s) < ∞. Again, by the proof of Theorem 3.5 we have

I 3 C(α) π v=0 |w ′ (v)| v u=0 h(u) h(u) |w ′ (u)||E u |dudv,
which gives the desired estimate.

kernel estimate

In this section we will prove Theorem 1. The reproducing kernel k µ of D(µ) is defined by,

f (z) = f, k µ (•, z) , f ∈ D(µ), z ∈ D. So, k µ (z, z) = sup{|f (z)| 2 : f ∈ D(µ), f 2 µ ≤ 1}. ( 10
)
It follows obviously that for |z| ≤ 1/2 we have

k µ (z, z) ≍ 1 + |z| 0 dr (1 -r)P [µ](rz/|z|) + (1 -r) 2 .
By Littlewood-Paley identity, we have

f 2 µ = f 2 H 2 + D µ (f ) = |f (0)| 2 + D |f ′ (w)| 2 |[| log |w|| + P [µ](w)]dA(w) ≍ |f (0)| 2 + D |f ′ (w)| 2 |[(1 -|w|) + P [µ](w)]dA(w).
Let f ∈ D(µ)\{0} and let f = I f O f be the inner-outer factorization of f . Then O f ∈ D(µ) and D µ (O f ) ≤ D µ (f ) (see [START_REF] Richter | A formula for the local Dirichlet integral[END_REF]). Thus by [START_REF] Fukushima | Dirichlet Forms and Symmetric Markov Processes[END_REF], we get

k µ (z, z) = sup{|f (z)| 2 : f ∈ D(µ) outer function and f µ ≤ 1}. ( 11
)
This observation will be useful in the proof of the lower estimate.

k µ (ρ, ρ) 1 + 1 1-ρ dx F µ (x) + x 2 . Let f ∈ D(µ), since f ∈ H 2 , |f (iy)| ≤ f H 2 √ 1 -y ≤ √ 2 f H 2 , 0 < y < 1/2.
So, for 0 < y < 1/2, we have

|f (ρ + i(1 -ρ)y)| = f (iy) + ρ 0 f ′ (t + i(1 -t)y)dt ρ 0 |f ′ (t + i(1 -t)y)|dt + f H 2 .
Let ∆ be the triangle with vertices -i/2, 1, i/2 and let ∆ ρ = {x + iy ∈ ∆ : 0 ≤ x ≤ ρ}. By change of variables w = u + iv = t + i(1 -t)y, we get

1 1 -ρ (1-ρ)/2 -(1-ρ)/2 |f (ρ + iη)|dη = 1/2 -1/2 |f (ρ + i(1 -ρ)y)|dy ρ 0 1/2 -1/2 |f ′ (t + i(1 -t)y)|dydt + f H 2 ∆ρ |f ′ (w)| dudv 1 -u + f H 2 D µ (f ) 1/2 ∆ρ dA(w) (1 -u) 2 ((1 -|w|) + P [µ](w)) 1/2 + f H 2 D µ (f ) 1/2 ρ 0 du (1 -u) 2 + (1 -u)P [µ](u) 1/2 + f H 2 D µ (f ) 1/2 1 1-ρ dx F µ (x) + x 2 1/2 + f H 2 . ( 12
)
Denote by D(λ, r) the disc of radius r centered at λ. Since

|f (ρ)| 1 (1 -ρ) 2 ρ+ 1-ρ 4 x=ρ-1-ρ 4 1-x 2 y=-1-x 2 |f (x + iy)dy dx D µ (f ) 1/2 1 5 4 (1-ρ) dx F µ (x) + x 2 1/2 + f H 2 (13)
Now from [START_REF] Fukushima | Dirichlet Forms and Symmetric Markov Processes[END_REF], we get

k µ (ρ, ρ) 1 + 1 1-ρ dx F µ (x) + x 2 ≍ 1 + |z| 0 dr (1 -r)P [µ](rz/|z|) + (1 -r) 2 .
This achieves the proof. 4.2. Lower estimate.

Weighted Sobolev spaces.

In this subsection we introduce weighted Sobolev spaces which will be used in the proof of lower estimate of the norm of the kernel of D(µ) and in the proof of Theorem 5.4. Let ϕ be a nondecreasing continuous function. The Sobolev space associated with ϕ, W ∞ (ϕ), consists of real continuous functions f on ]0, 2π] given by

f (x) = f (2π) + 2π x g(s)ds, 0 < x ≤ 2π, ( 14
)
where g is a measurable function satisfying

||gϕ|| ∞ = sup 0<x |g(x)|ϕ(x) < ∞.
As usual g will be denoted by f ′ . Equipped with the following norm

f W ∞ (ϕ) = f ′ ϕ ∞ , W ∞ (ϕ)
0 dx ϕ(x) < ∞.
We say that f is regular, and write

f ∈ R, if f is a C 1 convex decreasing function on [0, 2π], satisfying f (2t) ≤ 2f (t) and t 2 |f ′ (t)| is increasing.
Our goal is to estimate

γ ϕ (a) = sup f 2 (a) : f ∈ W ∞ (ϕ) ∩ R, f W ∞ (ϕ) f ∞ + f 2 2 ≤ 1 .
First we will examine the Hilbertian case

W 2 (ϕ) = f of the form (14) : f 2 W 2 (ϕ) = f 2 2 + 2π 0 |f ′ (t)| 2 ϕ(t)dt < ∞ .
We also need the following subspace of W 2 (ϕ)

W 2 0 (ϕ) = f of the form (14) : f (2π) = 0, and f 2 W 2 0 (ϕ) = 2π 0 |f ′ (t)| 2 ϕ(t)dt < ∞ .
Clearly, evaluations at points of ]0, 2π] define continuous linear functionals on W 2 (ϕ) and on W 2 0 (ϕ). Let K ϕ , L ϕ be the reproducing kernels of W 2 (ϕ) and of W 2 0 (ϕ) respectively. A simple computation gives the following expression of the reproducing kernel L ϕ of W 2 0 (ϕ). We have

L ϕ (t, s) =        2π t dx ϕ(x) t ≥ s, L ϕ (s, s) t ≤ s. The estimates of K ϕ , on the diagonal is given by K ϕ (a, a) ≍ 1 + L ϕ (a, a) = 1 + 2π a dx ϕ(x) . It means that sup f (a) 2 : f ∈ W 2 (ϕ) , f 2 W 2 (ϕ) ≤ 1 ≍ 1 + 2π a dx ϕ(x) .
We have the following proposition.

Proposition 4.1. Suppose that t 2 /ϕ(t) is increasing and ϕ(t) ≥ t 2 . Let a < 1/2, we have

γ ϕ (a) ≍ K ϕ (a, a). Proof. Since f 2 W 2 (ϕ) ≤ f W ∞ (ϕ) f ∞ , then γ ϕ (a) ≤ K ϕ (a, a). Conversely, let f = f 0 + 1, where f 0 (x) = 2π 2π x+a 2π+a ds ϕ(s) , 0 < x ≤ 2π. Clearly f ∈ R. Since ϕ(2t) ≍ ϕ(t), we have f 0 2 2 = 2π 0 f 0 (t) 2 dt π 0 2π u=t+a 2π v=t+a 1 ϕ(u)ϕ(v) dudvdt a t=0 2π u=a 2π v=a 1 ϕ(u)ϕ(v) dudvdt + π t=a 2π u=t 2π v=t 1 ϕ(u)ϕ(v) dudvdt 2π u=a 2π v=a a ϕ(u)v 2 dudv + 2π u=a 2π v=u u ϕ(u)v 2 dudv f 0 ∞ . Then we obtain K ϕ (a, a) f (a) 2 f W ∞ (ϕ) f ∞ + f 2 2 ≤ γ ϕ (a).
And the proof is complete. Let ϕ(x) = F µ (x) + x 2 . By (11), Proposition 2.3 and Theorem 3.2 we have

k µ (r, r) |f w (r)| 2 f w 2 D(µ) w 2 (1 -r) w W ∞ (Fµ) w ∞ + w 2 2 w 2 (1 -r) w W ∞ (ϕ) w ∞ + w 2 2 . Thus k µ (r, r) γ ϕ (1 -r).
By Proposition 4.1, we obtain the result.

Capacity

Let µ be a positive finite measure on T and let c µ be the associated capacity given by [START_REF] Adams | Function spaces and potential theory[END_REF]. The capacity c µ is a Choquet capacity [START_REF] Carleson | Selected Problems on Exceptional sets[END_REF][START_REF] Guillot | Fine boundary behavior and invariant subspaces of harmonically weighted Dirichlet spaces[END_REF] and so for every borelian set of T we have As consequence of this inequality we have the following properties. (3) The radial limit lim r→1-f (rζ) exists and is finite for every f ∈ D h (µ) if and only if c µ (ζ) > 0.

Proof. See [START_REF] Chacón | Carleson-type measure on Dirichlet spaces[END_REF][START_REF] Guillot | Fine boundary behavior and invariant subspaces of harmonically weighted Dirichlet spaces[END_REF]. Now we will give the proof of the estimate of the capacity of arcs. Now we will give some sufficient conditions on a closed subset of T to be µ-polar. Let E be a closed subset of T. We define n E (ε), the ε-covering number of E, to be the smallest number of the closed arcs of length 2ε that cover E. Note that

εn E (ε) ≤ |E ε | ≤ 4εn E (ε), 0 < ε ≤ π.

2

 2 and if dµ(e it ) = dm(t) = dt/2π, the normalized arc measure on T, then D(µ) is the classical Dirichlet space D.

[ 6 ,

 6 Theorem 2]. The generalized Brown-Shields conjecture is still open. Let ζ ∈ T, if µ = m then c m (ζ) = c(ζ) = 0 (c is the logarithmic capacity) and if µ = δ ζ , the dirac measure on ζ, then c δ ζ (ζ) > 0. In a more general way, since c µ (ζ) ≍ 1/ sup 0<ρ<1 k µ (ρζ, ρζ) (see [6, Lemma 3.2]), we get from Theorem 1

I 3 .

 3 To complete the proof we will estimate each term separately.If 2s ≤ t ≤ π, we have |t -s| ≥ t/2,

Lemma 3 . 4 .

 34 Let E be a closed subset of T. Let w be a convex decreasing function and let Ω(ζ) = w(dist(ζ, E)). Then D µ (Ω) I 1 + I 2 + I 3 , where

4. 1 .

 1 Proof of the upper estimate. Let z = ρ ∈ [1/2, 1). By Lemma 2.4, it suffices to prove that

D(ρ, ( 1 -

 1 ρ)/4) ⊂ {z = x + iy : |x -ρ| ≤ (1 -ρ)/4 and |y| ≤ (1 -x)/4}, by (12) and the subharmonicity of |f | we obtain

4. 2 . 2 .

 22 Proof of the lower estimates. Let z = r ∈]1/2, 1[ and let w ∈ R. We consider the outer function f w given by |f w (e it )| = w(|t|), a.e on [-π, π].

2 µ t 2

 22 c µ (E) = sup{c µ (K) : K compact , K ⊂ E}.Note that c µ satisfies a weak-type inequality. Namely:c µ ({ζ ∈ T : |f (ζ)| ≥ t c µ -q.e.}) ≤ f , f ∈ D h (µ).

Proposition 5 . 1 .

 51 The following properties are satisfied[START_REF] Adams | Function spaces and potential theory[END_REF] Let E ⊂ T be a Borel set and let M µ (E) := {f ∈ D(µ) : g|E = 0 c µ -q.e}. Then the set M µ (E) is closed in D(µ).

( 2 )

 2 If f ∈ D(µ) is cyclic for D(µ) then f is outer function and c µ (Z T (f )) = 0.

Proof Theorem 2 .✷ 5 . 2 . 0 dx ( 1

 25201 Suppose that ζ = 1. Let w ∈ R and let f w be the outer function satisfying |f w (e it )| = w(|t|), a.e on [-π, π]. It's clear that w(|I|) w(x) for |x| ≤ 2|I|. We have γ ϕ (|I|) 1/c µ (I). By proposition 4.1, we obtainc µ (I) 1/k µ (ρ, ρ), (ρ = 1 -|I|).For the reverse inequality note thatc µ (I) = inf{ f 2 µ : f ∈ C 1 , 0 ≤ f ≤ 1 and f = 1 on I}. Consider the function u ∈ D h (µ) such that 0 ≤ u ≤ 1 and u |I = 1. Hence u ∈ D h (µ) ∩ C 1 . We have P [u](1 -|I|) ≍ 1. Let ρ = 1 -|I|,a similar argument, as in the proof of (13), givesP [u](ρ) D µ (u) 1/2 k µ (ρ, ρ) + u L 2 (T) . So k µ (ρ, ρ) -1/2 D µ (u) 1/2, andc µ (I) ≥ 1 k µ (ρ, ρ).As an immediate consequence, we obtainCorollary Let λ ∈ T.c µ ({λ}) > 0 ⇐⇒ 1 -x)P [µ](xλ) + (1 -x) 2 < ∞.

Let κ µ

 µ (r) = inf{k µ (rζ, rζ) : ζ ∈ supp µ}.

It 's easy

  to see that κ µ is unbounded if and only if, for each ζ ∈ T, we have c µ (ζ) = 0. In this case one can prove easily, by the sub-additivity property of capacity, thatCorollary 5.3. Let E be a closed subset of T such that lim r→1 -κ µ (r) = ∞. If n E (ε) = o(κ µ (1 -ε)), ε → 0, then c µ (E) = 0. then c µ (E) = 0. (ii) If µ = mis the Lebesgue measure, then c µ is comparable to the logarithmic capacity and M µ,E (s) = |E s |. Theorem 5.4 says that if 0 dt |E t | = +∞, then c(E) = 0. This result is due to Carleson [3, Theorem 2, p.30]. (iii) Let K be a closed subset of T such that ρ µ,K (s) = O(s 1+β ) for some 0 < β. If β < 1, then every subset of K with Hausdorff dimension less than β is µ-polar. If β > 1, then every subset E of K with |E| = 0 is µ-polar. The following measures dµ(ζ) = dist(ζ, K) β dm(ζ) provide such examples.

  then |t -s| ≥ s/2, Suppose that x 2 |w ′ (x)| is increasing and let f w be the outer function given by

	I 3	π s=0	s t=0	s x=t	s y=x	|w ′ (y)|w(y)	|w ′ (x)| w(x)	dydx	dµ(s) s 2 dt
		π t=0	π x=t	π y=x	|w ′ (y)|w(y)	|w ′ (x)| w(x)	y≤|s|≤π	dµ(s) s 2 dt
		π x=0	π y=x	|w ′ (y)|w(y)	|w ′ (x)| w(x)	π y≤|s|≤π	dµ(s) s 2 xdxdy.
	Theorem 3.2. Let w : [0, π] → (0, +∞) be a C 1 decreasing convex function such that w(x) ≤ 2w(2x).

  is a Banach space. It becomes a topological Banach algebra, if and only if,

Research partially supported by "Hassan II Academy of Science and Technology" for the first and the second authors.

Proof. Let (I i ) 0≤i≤n E (ε) be a ε-coverning of E, then

Recall that the function ψ is called α-admissible if ψ is concave or convex and ψ(s)/s α is decreasing for some α > 0. For a positive increasing function ψ on (0, 2π) such that ψ(0) = 0, we set

Now we can state the main result of this section.

Theorem 5.4. Let E be closed subset of T such that ρ µ,E ≤ ψ where ψ is α-admissible for some α < 2.

Proof. Note that M ψ,E is given by ( 5), so M ψ,E (s) is increasing. Let a > 0. By the definition of capacity and using Theorem 3.5 and Proposition 4.1 we have

.

When a goes to zero, we get

.

And the proof is complete.

Remarks. Now we give some examples:

(i) Let E be a closed subset of T and let