On weighted estimates for a class of integral operators.(English. Russian original)Zbl 0815.47064

Sib. Math. J. 34, No. 4, 755-766 (1993); translation from Sib. Mat. Zh. 34, No. 4, 184-196 (1993).
Let $$\varphi(t)$$, $$t\in (0,1)$$, be a measurable function satisfying the following conditions:
a) $$\varphi(t) \geq 0$$ and $$\varphi(t)$$ is nonincreasing;
b) $$\varphi (t_ 1+ t_ 2)\leq D(\varphi(t_ 1)+ \varphi(t_ 2))$$, $$0<t_ 1,t_ 2< 1$$,
where $$D$$ is independent of $$t_ 1$$, $$t_ 2$$. For the integral operator $(T_ \varphi f) (x)= \int_ 0^ x \varphi(t/x) f(t) dt$ weighted norm inequalities of the form $\| u T_ \varphi f \|_{L^ q (0,\infty)} \leq c\| vf \|_{L^ p (0,\infty)}$ are studied in the case $$0<q<p <\infty$$, $$p>1$$. Moreover, necessary and sufficient conditions for the operator $$T_ \varphi$$ to be compact are given in the case $$1<p, q<\infty$$.

MSC:

 47G10 Integral operators 47B07 Linear operators defined by compactness properties
Full Text:

References:

 [1] A. Kufner and B. Opic, Hardy-Type Inequalities, Longman-Pittman (1990). · Zbl 0698.26007 [2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987). · Zbl 0617.26004 [3] V. D. Stepanov, ?On weighted Hardy-type inequalities for fractional Riemann-Liouville integrals,? Sibirsk. Mat. Zh.,31, No. 3, 186-197 (1990). [4] V. D. Stepanov, ?Biweighted estimates for Riemann-Liouville integrals,? Izv. Akad. Nauk SSSR Ser. Mat.,54, No. 3, 645-656 (1990). · Zbl 0705.26015 [5] V. D. Stepanov, ?On boundedness and compactness of a class of integral operators,? Dokl. Akad. Nauk SSSR,312, No. 3, 544-546 (1990). · Zbl 0733.47033 [6] P. J. Martin-Reyes and E. T. Sawyer, ?Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater,? Proc. Amer. Math. Soc.,106, 727-733 (1989). · Zbl 0704.42018 [7] R. Oînarov, ?Weighted inequalities for a class of integral operators,? Dokl. Akad. Nauk SSSR,319, No. 5, 1076-1078 (1991). [8] G. Sinnamon, ?Weighted Hardy and Opial type inequalities,? J. Math. Anal. Appl.,160, No. 2, 434-445 (1991). · Zbl 0756.26011 [9] L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1984). · Zbl 0555.46001 [10] K. T. Mynbaev and M. O. Otelbaev, Weighted Functional Spaces and Spectra of Differential Operators [in Russian], Nauka, Moscow (1988). · Zbl 0651.46037 [11] N. Dunford and J. T. Schwartz, Linear Operators. Vol. 1. General Theory [Russian translation], Izdat. Inostr. Lit., Moscow (1962). [12] V. D. Stepanov, ?Weighted Hardy-type inequalities for higher derivatives and their applications,? Trudy Mat. Inst. Steklov. Akad. Nauk SSSR,187, 178-190 (1989). [13] V. G. Maz’ya, Sobolev Spaces [in Russian], Izdat. Leningrad. Univ., Leningrad (1985). [14] T. Ando, ?On compactness of integral operators,? Indag. Math.,24, 235-239 (1962). [15] I. Halperin, ?Function spaces,? Canad. J. Math.,5, 273-288 (1953). · Zbl 0052.11303 [16] E. M. Stein, Singular Integrals and Differentiability Properties of Functions [Russian translation] Mir, Moscow (1973).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.