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In the present paper, we investigate the Hardy–Littlewood type and the integration by parts result for ψ$$ \psi $$–Riemann–Liouville fractional integrals. Also, we attack the integration by parts for the ψ$$ \psi $$–Riemann–Liouville and ψ$$ \psi $$–Hilfer fractional derivatives. To finish, we investigated Sobolev‐type inequalities involving the ψ$$ \psi $$–Riemann–Liouville and the ψ$$ \psi $$–Hilfer fractional derivatives in weighted space. [ABSTRACT FROM AUTHOR]