How would a low-cost price leader enforce its leadership through implied threats to a rival?

As a general rule, modern monopoly laws don't allow organizations to coordinate with one another on prices. That means that discount stores and other "price leaders" can't outright threaten competitors with a price war; beyond the unprofessionalism of that conduct, it would likely constitute price-fixing in a court of law.
However, that threat is very valuable and important to stores that rely on discount models to provide competitive advantages. While discount stores typically have production and/or supply line advantages over their competitors, price wars cut into every company's profits, so those discount stores would lose a great deal of money on their way to their inevitable victory.
Therefore, many price leaders will communicate their intentions indirectly, through market signals and actions. The most common way to do this is through communications with customers. For example, you've almost certainly seen fliers and advertisements of stores with price-match guarantees, a published promise that the store will match the price of a product if a competitor lists it for a lower price. These guarantees target customers and are published in customer-facing ads, so they do not typically run afoul of anti-price-fixing statutes, but the public declaration that a company will not be undercut is easily understood by other firms competing with similar products and offerings. Such a guarantee makes clear that there is little competitive advantage to be gained by discounting prices below those of the price leader and therefore that competitors should seek other value propositions than pure pricing in order to distinguish themselves from the price leaders and attract their own market segment.