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Lambda function URLs are intended to be very lightweight way if invoking Lambda functions. They have limited capabilities, custom domain names is not one of them. Creating an alias in Route53 will not help as the Lambda service will not know which function it needs to invoke.
If you need custom domain names, use API Gateway instead.
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You can use Cloudfront, setting the function url as the origin. Then assign the cloudfront distribution a custom domain as well as certificate if you need https
answered 2 years ago
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You can also consider putting your custom domain name on CloudFront or a public ALB.