What does a tilde underneath an inequality mean?

Question

I've recently come across an expression of the form $$\large x \lesssim y$$ What does this expression mean?

Answer 1

One of several possible meanings is a total preorder.

Answer 2

One possible interpretation of $x \lesssim y$ is, $$x < y \ \mbox{ or }\ x \approx y.$$

This is analogous to the way $x \le y$ signifies "$x < y$ or $x = y$."

But from the comments, clearly this is not the only way this symbol might be used. One would hope that prior to its first use in a publication, the same publication would define the symbol.

Answer 3

In the context of partial differential equations and harmonic analysis, this notation is often used to mean

$$ A \lesssim B \iff \exists C > 0 \text{ s.t. } A \leq C B $$

As written this expression is pretty damn useless when $A$ and $B$ are just two real numbers. What's more useful is supposing $A(\lambda), B(\lambda)$ are two families of objects parametrised by $\lambda\in \Lambda$ for some set. Then

$$ A \lesssim B \iff \exists C > 0 \text{ s.t. } \forall \lambda\in \Lambda~,~ A(\lambda) \leq C B(\lambda) $$

Sometimes you will see when $A(\lambda,\pi), B(\lambda,\pi)$ are two families of objects parametrised by $(\lambda,\pi) \in \Lambda\times\Pi$, the notation

$$ A\lesssim_\pi B \iff \forall \pi\in \Pi \exists C = C(\pi)>0 \text{ s.t. } \forall \lambda\in \Lambda ~,~ A(\lambda,\pi) \leq C(\pi) B(\lambda,\pi) $$

that is, the constant in the inequality is universal over $\lambda$ but may depend on $\pi$.

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Similarly one sees the notation

$$ A\approx B \iff A\lesssim B \text{ and } B\lesssim A $$

which also has the $\approx_\pi$ variant in the same way.

Answer 4

Maybe $x \lesssim y$ means $y-x<\epsilon$ fro some small $\epsilon$. In other words, the difference is small but x IS less than y