可能是由于以下原因导致API网关JWT Authorizer返回401：

1. 未正确设置JWT Authorizer。
2. 用户身份验证失败。
3. 请求没有授权令牌或授权令牌无效。
4. 可能是缓存。

在AWS环境中，可以通过出现以下代码的方式验证API Gateway是否成功调用其JWT Authorizer：

def token_auth(event, context):
    ** # Code to parse and validate JWT goes here **

    # Return the Allow policy for the root resource with an empty principalId to bypass security
    policy = AuthPolicy(principal_id='', aws_account_id='account-id', region='us-west-2', rest_api_id='rest-api-id', stage='stage')
    policy.allow_all_methods()
    return policy.build()

class AuthPolicy(object):
 
    # omitted code…

    def _get_empty_statement(self):
        statement = {
            'Action': 'execute-api:Invoke',
            'Effect': 'Deny',
            'Resource': self._method_arn
        }
        return statement
    
    def _get_template(self):
        policy = {
            'principalId': self._principal_id,
            'policyDocument': {
                'Version': '2012-10-17',
                'Statement': []
            }
        }
        return policy
  
      # omitted code…

    def allow_all_methods(self):
        self.add_method('Allow', '*', '*', '*')
      
      # omitted code…

    def add_method(self, effect, verb, resource, conditions):
        statement = self._get_empty_statement()
        statement['Effect'] = effect
        if conditions is None:
            statement['Condition'] = {}
        else:
            statement['Condition'] = conditions
        statement['Action'] = 'execute-api:Invoke'
        statement['Resource'] = self._method_arn
        if verb != '*':
            statement['Resource'] += '/' + verb
        if resource != '*':
            statement['Resource'] += '/' + resource
        self._statements.append(statement)

如果API Gateway JWT Authorizer测试成功，但调用API时返回401错误，则可以尝试清除API Gateway中的缓存