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We introduce a family of Balakrishnan–Rubin-type hypersingular integrals depending on a parameter ε and generated by the Gauss–Weierstrass semigroup. Then the connection between the order of Lp –smoothness of a Lp –function φ and the rate of Lp -convergence of these families to φ , as ε tends to 0, is obtained.

On the rate of $L_p$-convergence of Balakrishnan--Rubin-type hypersingular integrals associated to the Gauss-Weierstrass semigroup