〜をよそに
ignoring, unconcerned by
Noun + をよそに
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をよそに means someone carries on regardless of other people's feelings or the situation around them: 親の心配をよそに、息子は一人で旅に出た, "ignoring his parents' worries, the son set off travelling alone".
The noun is usually other people's worry, opposition, criticism or excitement: 心配, 反対, 不安, 期待, 騒ぎ. The second half is what the subject did anyway. It can be critical ("inconsiderately"), or just descriptive: 周りの騒ぎをよそに、赤ちゃんはぐっすり眠っていた, "oblivious to the commotion, the baby slept soundly".
よそ means "somewhere else", so it's literally "treating it as someone else's business". Compare も構わず (N2), "without caring about", and をものともせず, which is about overcoming obstacles bravely.
Examples
親の心配をよそに、息子は一人で旅に出た。
Ignoring his parents' worries, the son set off travelling alone.
周囲の反対をよそに、彼は会社を辞めた。
In spite of everyone's objections, he quit his job.
住民の不安をよそに、工事は進められた。
The construction went ahead, regardless of residents' concerns.
周りの騒ぎをよそに、赤ちゃんはぐっすり眠っていた。
Oblivious to the commotion, the baby slept soundly.
世間の期待をよそに、彼は突然引退を発表した。
Ignoring the public's hopes, he suddenly announced his retirement.