### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

APiGateway Model that being created by CDK project stores the JSON schema with a single quotes instead of double quotes which leading to an error in parsing this json object. 

you can refere to this article to see the steps 

https://dev.to/dvddpl/reduce-lambda-invocations-and-code-boilerplate-by-adding-a-json-schema-validator-to-your-api-gateway-15af

and it works fine on AWS account 

also you can look on the attached image how localstack stores the JSON object with a single quotes, I tried to edit it manually through UI but it return it back with single quotes

![Screen Shot 2022-10-18 at 4 28 08 PM](https://user-images.githubusercontent.com/106184681/196769273-0834865a-2fb3-4efc-b3a9-9da6b2ad511d.png)

### Expected Behavior

store the json schema in correct JSON format

### How are you starting LocalStack?

With the `localstack` script

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    localstack start

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

    cdklocal deploy


### Environment

```markdown
- OS:OSX
- LocalStack: 1.0.4
```


### Anything else?

_No response_
self,
        context: RequestContext,
        domain_name: String,
        base_path: String,
        patch_operations: ListOfPatchOperation = None,
    ) -> BasePathMapping:
        region_details = get_apigateway_store()
        mappings_list = region_details.base_path_mappings.get(domain_name) or []
        mapping = ([m for m in mappings_list if m["basePath"] == base_path] or [None])[0]
        if mapping is None:
            raise NotFoundException(
                f"Not found: mapping for domain name {domain_name}, "
                f"base path {base_path} in list {mappings_list}"
            )
        patch_operations = ensure_list(patch_operations)
        for operation in patch_operations:
            if operation["path"] == "/restapiId":
                operation["path"] = "/restApiId"
        result = apply_json_patch_safe(mapping, patch_operations)
        for i in range(len(mappings_list)):
            if mappings_list[i]["basePath"] == base_path:
                mappings_list[i] = result
        result = to_base_mapping_response_json(domain_name, base_path, result)
        return BasePathMapping(**result)
    def delete_base_path_mapping(