On the approximation properties of bi-parametric potential-type integral operators

Year 2021, , 1681 - 1691, 14.12.2021

Çağla Sekin Mutlu Güloğlu , İlham Aliyev

https://doi.org/10.15672/hujms.821159

Abstract

In this work we study the approximation properties of the classical Riesz potentials $I^{\alpha }f\equiv (-\Delta )^{-\alpha /2}f$ and the so-called bi-parametric potential-type operators $J_{\beta }^{\alpha }f\equiv(E+(-\Delta )^{\beta /2})^{-\alpha /\beta }f$ as $\alpha \rightarrow \alpha_{0}>0$ where, $\alpha >0$, $\beta >0$, $E$ is the identity operator and $\Delta $ is the laplacian. These potential-type operators generalize the famous Bessel potentials when $\beta =2$ and Flett potentials when $\beta =1$. We show that, if $A^{\alpha}$ is one of operators $J_{\beta }^{\alpha }$ or $I^{\alpha}$, then at every Lebesgue point of $f\in L_{p}(\mathbb{R}^{n})$ the asymptotic equality $(A^{\alpha}f)(x)-(A^{\alpha _{0}}f)(x)=O(1)(\alpha-\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds. Also the asymptotic equality $\left\Vert A^{\alpha }f-A^{\alpha _{0}}f\right\Vert_{p}=O(1)(\alpha -\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds when $A^{\alpha}=J_{\beta }^{\alpha }$.

Keywords

Abel-Poisson semigroup, Gauss-Weierstrass semigroup, Riesz potentials, Bessel potentials, potentials-type operators

References

• [1] İ.A. Aliev, Bi-parametric potentials, relevant function spaces and wavelet-like trans- forms, Integr. Equ. Oper. Theory 65, 151-167, 2009.
• [2] İ.A. Aliev, A.D. Gadjiev and A. Aral, On approximation properties of a family of linear operators at critical value of parameter, J. Approx. Theory 138, 242-253, 2006.
• [3] İ.A. Aliev and B. Rubin, Wavelet-like transforms for admissible semi-groups; inver- sion formulas for potentials and Radon transforms, J. Fourier Anal. Appl. 11 (3), 333-352, 2005.
• [4] İ.A. Aliev, B. Rubin, S. Sezer and S. Uyhan, Composite wavelet transforms: applica- tions and perspectives, Contemp. Math., AMS 464, 1-27, 2008.
• [5] İ.A. Aliev, S. Sezer and M. Eryiğit, An integral transform associated to the Poisson integral and inversion of Flett potentials, J. Math. Anal. Appl. 321 (2), 691-704, 2006.
• [6] N. Aronszajn and K.T. Smith, Theory of Bessel Potentials, I, Ann. Inst. Fourier 11, 385-475, 1961.
• [7] T.M. Flett, Temperatures, Bessel Potentials and Lipschitz spaces, Proc. London. Math. Soc. 22, 385-451, 1971.
• [8] A.D. Gadjiev, A. Aral and İ.A. Aliev, On behaviour of the Riesz and generalized Riesz potentials as order tends to zero, Math. Inequal. Appl. 10 (4), 875-888, 2007.
• [9] S.G. Gal, Approximation by complex potentials generated by the Gamma function, Turkish J. Math. 35, 443-456, 2011.
• [10] R. Johnson, Temperatures, Riesz potentials and the Lipschitz spaces of Herz, Proc. London Math. Soc. 3 (27), 290-316, 1973.
• [11] T. Kurokawa, On the Riesz and Bessel kernels as approximations of the identity, Sci. Rep. Kagoshima Univ. 30, 31-45, 1981.
• [12] B. Rubin, Fractional Integrals and Potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics 82, Longman, Harlow, 1996.
• [13] S.M. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Sci. Publ. London-New York, 1993.
• [14] S. Sezer, On approximation properties of the families of Flett and generalized Flett potentials, Int. J. Math. Anal. 3 (39), 1905-1915, 2009.
• [15] S. Sezer and İ.A. Aliev, A new characterization of the Riesz potential spaces with aid of a composite wavelet transform, J. Math. Anal. Appl. 372, 549-558, 2010.
• [16] E. Stein, The characterization of function arising as potentials, I, Bull. Amer. Math. Soc. 67 (1), 101-104, 1961.
• [17] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970.
• [18] S. Uyhan, A.D. Gadjiev and İ.A. Aliev, On approximation properties of the parabolic potentials, Bull. Aust. Math. Soc. 74 (3), 449-460, 2006.