Eu espero que você melhore logo.

Because esperar que (to hope that…) typically triggers the present subjunctive in Portuguese when you’re expressing a wish, uncertainty, or something not yet realized.

• melhore = present subjunctive of melhorar (that you get better)
• melhora = present indicative (you get better / you are getting better)
  So Eu espero que você melhore is like “I hope that you will get better.”

In form, melhore can look like both:

• Present subjunctive: (que) você melhore
• Formal imperative (for você commands): Melhore! = “Get better!”

In your sentence, the que clause after espero makes it clearly subjunctive, not a command.

In Portuguese, after esperar + a full clause, you normally use que to link the two parts:

• Eu espero que você melhore… = I hope (that) you get better…

English can drop “that,” but Portuguese generally keeps que in this structure.

Yes, very commonly. Portuguese often drops subject pronouns when the verb ending makes the subject clear.

• (Eu) espero que você melhore logo.
  Both are natural; including Eu can sound a bit more explicit/emphatic.

You can absolutely use tu, depending on region and style. With tu, the subjunctive form changes:

• Eu espero que você melhore logo.
• Eu espero que tu melhores logo. (more “grammar-book” agreement)

In many parts of Brazil, people say tu but still conjugate like você (e.g., tu melhore), but that’s regional.

Yes, esperar can mean to wait or to hope, but the structure makes the meaning clear:

• esperar + que + clause → usually to hope: Espero que…
• esperar + direct object / por → often to wait (for): Estou esperando você / Estou esperando por você

So Eu espero que você melhore is naturally understood as “I hope you get better.”

That version is generally not the standard way to say it in Brazilian Portuguese. The most natural/common is:

• Eu espero que você melhore logo.

There is a different construction where Portuguese can use an infinitive after certain verbs, but with esperar meaning “hope,” the que + subjunctive pattern is the go-to.

It’s neutral and very common in everyday Brazilian Portuguese.
Using você is typical in much of Brazil and can be informal-to-neutral. For more formal/polite, you might see:

• Espero que o(a) senhor(a) melhore logo.

Yes, a few very common ones:

• Melhoras! = “Get well soon!” (super common, short)
• Espero que você fique bem logo. (more general: “feel well”)
• Tomara que você melhore logo. (more heartfelt/casual: “I hope…”)
• Desejo melhoras. (more formal)