### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

Authorization headers being sent to API gateway are received on lambdas as "empty string"

### Expected Behavior

_No response_

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

Run a API Gateway with lambda integration

### Environment

```markdown
- OS:
- LocalStack:
```


### Anything else?

_No response_
headers = response.headers
        
        for header in ALLOWED_CORS_RESPONSE_HEADERS:
            if headers.get(header) == "":
                del headers[header]
        
        
        if config.DISABLE_CORS_HEADERS or not should_enforce_self_managed_service(context):
            return
        request_headers = context.request.headers
        if ACL_ORIGIN not in headers:
            headers[ACL_ORIGIN] = (
                request_headers["origin"]
                if request_headers.get("origin") and not config.DISABLE_CORS_CHECKS
                else "*"
            )
        if ACL_METHODS not in headers:
            headers[ACL_METHODS] = ",".join(CORS_ALLOWED_METHODS)
        if ACL_ALLOW_HEADERS not in headers:
            requested_headers = headers.get(ACL_REQUEST_HEADERS, "")
            requested_headers = re.split(r"[,\s]+", requested_headers) + CORS_ALLOWED_HEADERS
            headers[ACL_ALLOW_HEADERS] = ",".join([h for h in requested_headers if h])
        if ACL_EXPOSE_HEADERS not in headers:
            headers[ACL_EXPOSE_HEADERS] = ",".join(CORS_EXPOSE_HEADERS)
        if (
            request_headers.get(ACL_REQUEST_PRIVATE_NETWORK) == "true"
            and ACL_ALLOW_PRIVATE_NETWORK not in headers
        ):
            headers[ACL_ALLOW_PRIVATE_NETWORK] = "true"