### 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, rest_api_id: String, parent_id: String, path_part: String
    ) -> Resource:
        moto_rest_api = get_moto_rest_api(context, rest_api_id)
        parent_moto_resource: MotoResource = moto_rest_api.resources.get(parent_id, None)
        
        
        if not parent_moto_resource:
            raise NotFoundException("Invalid Resource identifier specified")
        parent_path = parent_moto_resource.path_part
        if is_greedy_path(parent_path):
            raise BadRequestException(
                f"Cannot create a child of a resource with a greedy path variable: {parent_path}"
            )
        store = get_apigateway_store(account_id=context.account_id, region=context.region)
        rest_api = store.rest_apis.get(rest_api_id)
        children = rest_api.resource_children.setdefault(parent_id, [])
        if is_variable_path(path_part):
            for sibling in children:
                sibling_resource: MotoResource = moto_rest_api.resources.get(sibling, None)
                if is_variable_path(sibling_resource.path_part):
                    raise BadRequestException(
                        f"A sibling ({sibling_resource.path_part}) of this resource already has a variable path part -- only one is allowed"
                    )
        response: Resource = call_moto(context)
        
        children.append(response["id"])
        return response
    def delete_resource(