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We generalize the Mahowald invariant to the   R  ‐motivic and    C  2   ‐equivariant settings. For all    i  >  0    with    i  ≡  2  ,  3    mod    4   , we show that the   R  ‐motivic Mahowald invariant of(  2  +  ρ  η  )   i   ∈   π   0  ,  0   R    (   S   0  ,  0    )contains a lift of a certain element in Adams' classical    v  1   ‐periodic families, and for all    i  >  0   , we show that the   R  ‐motivic Mahowald invariant ofη  i   ∈   π   i  ,  i   R    (   S   0  ,  0    )contains a lift of a certain element in Andrews'   C  ‐motivic    w  1   ‐periodic families. We prove analogous results about the    C  2   ‐equivariant Mahowald invariants of(  2  +  ρ  η  )   i   ∈   π   0  ,  0    C  2(   S   0  ,  0    )andη  i   ∈   π   i  ,  i    C  2(   S   0  ,  0    )by leveraging connections between the classical, motivic, and equivariant stable homotopy categories. The infinite families we construct are some of the first periodic families of their kind studied in the   R  ‐motivic and    C  2   ‐equivariant settings.

Real motivic and C 2 ‐equivariant Mahowald invariants