Arrington Dixon has a small problem in his campaign for D.C. City Council chairman: many city voters do not seem to know who he is.

"Some people call him Arlington," said his campaign manager. And there are others, like the woman at a recent hearing in the Southeast neighborhood where Dixon grew up, who identified him to her friend as "What's-his-name-you-know-the-guy-who-is-running-against-Doug-Moore."

But Dixon's name recognition problems, which would be all but fatal to the chances of most political candidates, are not regarded as serious by his supporters.

They are comfortable in the knowledge that just about everybody in town knows his opponent, the Rev. Douglas E. Moore, and that they know, among other things, about the time Moore bit that tow truck driver.

According to Dixon's campaign manager, one of their strategies is called "leaving Doug to hang himself." There are indicators that there is some wisdom to that idea. A June poll by The Washington Post revealed intense sentiment against Moore among voters in the city and it showed Dixon with an exceedingly comfortable 2 to 1 lead.

So Arrington Dixon is campaigning these days by presenting himself as a serious, steady man. He drops in on a public hearing here, stops by to shake hands at a cocktail party there. He makes few speeches and has yet to issue any campaign position papers outlining his proposals for the City Council if he is elected.

Dixon and his wife Sharon, who is regarded by observers as his principal political adviser, believe their campaign strategy is the best way to appeal to the voters they are focusing on: older, over 40, black middle-class voters.

The Dixon camp also expects to gain substantial support from white voters who fear such Moore rhetoric as: "I hope Georgetown falls off and sinks in the Potomac."

Through his marriage to the daughter of a prominent local judge, his Air Force Academy background and his law degree from George Washington University, Dixon, himself the son of a Southeast Washington janitor, has become a part of the established black Washington that he is counting on so heavily in this campaign.

He and his wife think they know one thing that those settled, over-40 black Washingtonians want out of their government: They want people who won't embarrass them.

The black voter, Sharon Dixon will tell you, knows what whites will whisper when a black D.C. politician slurs a word or acts foolish in front of a TV camera. And after years of being second class through segregation and not having a local government those older, unpoor, unyoung and largely unnoticed blacks feel they have something to prove. They are out to prove that they are first class. And they don't want black city officials who embarrass them.

"That's why Doug will never get anywhere," Sharon Dixon said about her husband's opponent. "Doug embarrassed them and that's one thing you can't do - embarrass them. All of those newspaper stories about Doug biting the tow truck driver made those people burn inside.

"My grandmother," Sharon Dixon added later, "doesn't want anyone representing her who is going to split their verbs or be soft on their d's and t's in front of Queen Elizabeth. She doesn't want anyone who is going to embarrass her."

Dixon has carefully avoided making his campaign for the City Council chairman a Gatling gun recital of Moore's embarrassing moments: bitting the tow truck driver: being accused of slapping a woman, and then ramming his car into her car and finally flinging a rock through her window; or Moore's troubles with the Small Business Administration over an unpaid $21,000 loan.

"We have one rule in this campaign," Dixon joked with his campaign manager recently, "we don't talk about anyone bitting tow truck drivers."

During political forums Dixon has limited his mention of Moore to vague references, calling Moore the "Lone Ranger," while pointing out that the City Council chairman must be able to work with other councilman and cannot be a maverick.

". . . As chairman of the council I plan to work with every segment of the community," Dixon told one Georgetown audience recently. ". . . If the city government does not do that it will be suicidal. Now that's not to speak of my opponent. We all know he will do that."

Night after night Dixon travels around, making appearances. One night recently he dropped by a Southeast hearing concerning the Metro subway to ask questions and shake hands. Fifteen minutes after he arrived he was off to a Georgetown fashion show to shake more hands and introduce himself to more voters. Dixon didn't make a speech or a political statement that night. He was on stage to be seen, according to the campaign plan.

But Dixon's attempt to get his name and credentials round town is being stymied by a shortage of money. His campaign has received only $55,000 of the approximately $200,000 he expected to collect. He is having a hard time paying for the posters, bumper stickers, buttons and radio and television advertisements that could make his name well known. The "Arrington Dixon for Chairman," committee has no TV or radio ads scheduled, but some may be purchased before election day, according to campaign manager Vivien Cunningham.

Meanwhile, Moore has been busy portraying Dixon as "the Board of Trade's bay," and "the property of the real estate people and the speculators who are pushing black people out of our own city."

Moore recently told a forum in Ward 1 that Dixon takes notes from the Board of Trade, realtors and speculators "like the Redskins' quarterback takes notes from the bench." He told the audience that Dixon is the leader of "the sneaky seven," who raised rents when landlords told him to. And in a loud, deliberate voice, Moore told the audience of over 100 that Dixon is the first candidate to be endorsed by the Metropolitan Washington Board of Trade, in that group's 88-year history.

"That's because he's their candidate," Moore said.

Dixon and his main political advisers, his wife and Vivien Cunningham have ignored Moore's charges, believing that they are not believed, particularly by the middle-class constituency.

"Many of them (over-40 blacks) have houses, steady jobs, children," said Cunningham. Dixon's campaign manager, "and they have a stake in this city. They're not going to pay Doug's talk any mind . . ."

"The worst indictment against Doug Moore," added Cunningham, who is consistently introduced by Dixon as the first woman to run a citywide campaign in the District, "is that in Doug many poor people felt they had an advocate on the council who would protect them, protect their interests. Doug didn't do it. For the poor people in this city it is just as if Doug and all his talk hadn't been on the council all this time."

In speeches and in campaign literature Dixon advertises himself as the councilman who sponsored the voter registration-by-postcard law, the modernized divorce law, the law creating the D.C. Commission on the Aging and the Election and Latino Community Development Amendment, among other bills. And repeatedly Dixon mentions that he is from Anacostia - although he now lives with his wife, his two children and his parents, in North Portal Estates on the rim of Rock Creek Park - and suggests to his audiences that because he comes from Anacostia he has an insider's understanding of the problems facing the poor in the city.

An important part of Dixon's campaign is his wife, Sharon Pratt Dixon, the daughter of D.C. Superior Court Judge Carlisle E. Pratt.

". . . Certainly Arrington benefited from his relationship with my father," said Sharon Dixon when she was asked if her husband had risen socially after his marriage. "From my grandmother on down Arrington has been helped with his politics. My father wasn't very political but he was an active lawyer around town and he is well-known. He has lived in this town all his life."

Sharon Dixon maintains a high profile in her husband's political life and she has been labeled by some persons as the power behind the candidate. Recently, when her husband changed campaign managers, the cause for the switch was reportedly that the campaign manager and the candidate's wife could not get along.

"The only reason peope are concerned about me is because I'm a woman and it is assumed that the man is more intelligent than the woman," said Sharon Dixon. "If I was the candidate and Arrington had a hand in my campaign nothing would be said about it."

"I am one of Arrington's primary advisers," she added, "but I'm not his only adviser."

As a District politician Dixon is sharply distinct from the social activists and street revolutionaries of the late 1960s and early 1970s such as Marion Barry or Julius Hobson.

After leaving the Air Force Academy and graduating from Howard University, Dixon married, went to law school and worked as a professor at Washington Technical Institute. His first political venture was an unsuccessful run for the school board in 1968. He has since run successfully twice to represent Ward 4 on the City Council.

Now that he is running for chairman against Moore. Dixon is receiving the support of whites as well as blacks who view Moore as aracist and a dangerous personality to have at the head of the legislative body in a city seek-midst of a renovation boom.

But Dixon denies that he is selling out to business interests and real estates speculators.

"White businessmen in this twon realize that blacks will control this town as long as they have the voting majority," said Dixon. "All the money in the world can't change that."

At a Georgetown fund raiser for Dixon Alan Winner, a dentist, said he doesn't know anything about Dixon but "look who he's running against."

After Dixon spoke to the group, Barbara Gordon, another Dixon supporter, added: "Oh, he's intelligent and speaks well, doesn't he? I think he'll beat Doug Moore, but my dog could beat Doug Moore, couldn't he, I mean . . ."