See

Jonsson, A.; Wallin, Hans, A Whitney extension theorem in (L^p) and Besov spaces, Ann. Inst. Fourier 28, No. 1, 139-192 (1978). ZBL0369.46031.

Proposition 7.3 is exactly what you are looking for (and a bit more).

See

Jonsson, A.; Wallin, Hans, A Whitney extension theorem in (L^p) and Besov spaces, Ann. Inst. Fourier 28, No. 1, 139-192 (1978). ZBL0369.46031.

Proposition 7.1 in Section 7.3 is exactly what you are looking for (and a bit more).