「心の奥」は Deep down
または Deep down in (my, your, his, their) heart. という意味です。
例えば: 「日本人は心の奥では反対と思っていても、そう言わない時がある」は Japanese people tend to keep to themselves even though deep down they disagree.
「彼女は心の奥では自分の方が私たちより上だと思っている」は
Deep down, she thinks she’s better than all of us.
「彼は心の奥では本当にあなたのことを気にかけている」は
Deep down in his heart, he really cares about you.
Deep down というフレーズを使うときは主語をはっきりさせる必要があります。
Even though many Japanese people know in their hearts that something is wrong, often they will not say anything.
Even though many Japanese people oppose something deep down, often they will not express that to others.
「心の奥」= in one's heart / deep down
例えば、
I know in my heart...
I know deep down...
(心の奥では知っている。。。)
「日本人は心の奥では反対と思っていても、そう言わないときがある。」=Even though many Japanese people know in their hearts that something is wrong, often they will not say anything. / Even though many Japanese people oppose something deep down, often they will not express that to others.
「心の奥では、自分のほうが上だと思っている」= He/She knows deep down that he is better / He/She knows in his/her heart that she is ....
(この文章の文脈が分かりませんので翻訳しにくいです!)
deep down inside→心の奥では、心の底では
"deep inside"または"deep down"と省略して言うこともあります。
例文)Japanese don't speak up their minds even when they know the answer deep down inside.
「心の奥では答えが分かっていても、日本人は自分の考えをはっきりと言わない。」
「心の奥では、自分のほうが上だと思っている」
→Deep down inside, they think they're greater than others.
「日本人は心の奥では反対と思っていても、そう言わないときがある。」
→Japanese people might disagree with something deep down inside, but they don't necessarily speak up their minds.
★don't necessarily speak up→必ずしもはっきり言うとは限らない
ご参考になれば幸いです。