### 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_
"""Adjusts the (still url encoded) URL path"""
        parsed_url = urlparse(path)
        
        
        decoded_query = parse_qs(qs=parsed_url.query, keep_blank_values=True)
        adjusted_path = self._adjust_partition(parsed_url.path, static_partition)
        adjusted_query = self._adjust_partition(decoded_query, static_partition)
        encoded_query = urlencode(adjusted_query, doseq=True)
        
        encoded_query = encoded_query.replace("=&", "&")
        encoded_query = re.sub(r"=$", "", encoded_query)
        return f"{adjusted_path}{('?' + encoded_query) if encoded_query else ''}"
    def _adjust_partition(self, source, static_partition: str = None):
        
        if isinstance(source, dict):
            result = {}
            for k, v in source.items():
                result[k] = self._adjust_partition(v, static_partition)
            return result
        if isinstance(source, list):
            result = []
            for v in source:
                result.append(self._adjust_partition(v, static_partition))
            return result
        elif isinstance(source, bytes):
            try:
                decoded = unquote(to_str(source))