Abstract
New inequalities for fractional integrals of a function and its derivative are proved. Lower estimates of weighted norms of the derivative through fractional Riemann–Liouville integrals are obtained.
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K. Adimurthi, S. Filippas, and A. Tertikas, “On the best constant of Hardy–Sobolev inequalities,” Nonlin. Anal., 70, 2826–2833 (2009).
F. G. Avkhadiev, “Hardy-type inequalities on planar and spatial open sets,” Tr. Mat. Inst. Steklova, 255, 8–18 (2006).
F. G. Avkhadiev, “A geometric description of domains whose Hardy constant is equal to 1/4,” Izv. Ross. Akad. Nauk. Ser. Mat., 78, No. 5, 3–26 (2014).
F. G. Avkhadiev, “Integral inequalities in domains of hyperbolic type and their applications,” Mat. Sb., 206, No. 12, 3–28 (2015).
F. G. Avkhadiev, “Hardy–Rellich inequalities in domains of the Euclidean space,” J. Math. Anal. Appl., 442, 469–484 (2016).
F. G. Avkhadiev and R. G. Nasibullin, “Hardy-type inequalities in arbitrary domains with finite inner radius,” Sib. Mat. Zh., 55, No. 2, 239–250 (2014).
F. G. Avkhadiev and K.-J.Wirths, “Unified Poincaré and Hardy inequalities with sharp constants for convex domains,” Z. Angew. Math. Mech., 87, No. 8, 632–642 (2007).
F. G. Avkhadiev and K.-J. Wirths, “Weighted Hardy inequalities with sharp constants,” Lobachevskii J. Math., 31, 1–7 (2010).
F. G. Avkhadiev and K.-J. Wirths, “Sharp Hardy-type inequalities with Lamb’s constants,” Bull. Belg. Math. Soc. Simon Stevin., 18, 723–736 (2011).
F. G. Avkhadiev and K.-J. Wirths, “On the best constants for the Brezis–Marcus inequalities in balls,” J. Math. Anal. Appl., 396, No. 2, 473–480 (2012).
A. A. Balinsky and A. V. Laptev, “Sobolev generalized Hardy inequality for the magnetic Dirichlet forms,” J. Stat. Phys., 116, Nos. 1–4, 507–521 (2004).
D. W. Boyd and J. S. W. Wong, “An extension of Opial’s inequality,” J. Math. Anal. Appl., 19, No. 1, 100–102 (1967).
H. Brezis and M. Marcus, “Hardy’s inequality revisited,” Ann. Scu. Norm. Super. Pisa Cl. Sci. (4), 25, Nos. 1–2, 217–237 (1997).
Yu. A. Dubinskii, “A Hardy-type inequality and its applications,” Tr. Mat. Inst. Steklova, 269, 112–132 (2010).
S. Filippas, V. Maz’ya, A. Tertikas, “Critical Hardy–Sobolev inequalities,” J. Math. Pures Appl., 87, No. 9, 37–56 (2007).
S. Filippas, A. Tertikas, J. Tidblom, “On the structure of Hardy–Sobolev–Maz’ya inequalities,” J. Eur. Math. Soc., 11, No. 6, 1165–1185 (2009).
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, Cambridge (1934).
V. Levin, “Notes on inequalities. II. On a class of integral inequalities,” Mat. Sb., 4 (46), No. 2, 309–324 (1938).
V. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin (1985).
R. G. Nasibullin, “Generalizations of Hardy-type inequalities in the form of Dubinskii,” Mat. Zametki, 95, No. 1, 109–122 (2014).
R. G. Nasibullin, “On a discrete Hardy-type inequality with logarithmic weight,” Vladikavkaz. Mat. Zh., 18, No. 2, 67–75 (2016).
R. G. Nasibullin and A. M. Tukhvatullina, “Hardy-type inequalities with logarithmic and power weights for a special family of nonconvex domains,” Ufim. Mat. Zh., 5, No. 2, 43–55 (2013).
B. G. Pachpatte, “A note on certain inequalities related to Hardy’s inequality,” Indian J. Pure Appl. Math., 23, No. 11, 773–776 (1992).
Y. Pinchover and K. Tintarev, “On the Hardy–Sobolev–Maz’ya inequality and their generalizations,” in: Sobolev Spaces in Mathematics (Maz’ya, V., ed.), Int. Math. Ser., 1 (2009), pp. 281–297.
D. V. Prokhorov and V. D. Stepanov, “On weighted Hardy inequalities in mixed norms,” Tr. Mat. Inst. Steklova, 283, 155–170 (2013).
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).
D. T. Shum, “On a class of new inequalities,” Trans. Am. Math. Soc., 204, 299–341 (1975).
V. D. Stepanov, “Weighted inequalities of Hardy type for Riemann–Liouville fractional integrals,” Sib. Mat. Zh., 31, No. 3, 186–197 (1990).
G. Talenti, “Best constant in Sobolev inequality,” Ann. Mat. Pura Appl., 110, No. 4, 353–372 (1976).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 140, Differential Equations. Mathematical Physics, 2017.
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Nasibullin, R.G. Inequalities Involving Fractional Integrals of a Function and Its Derivative. J Math Sci 241, 448–457 (2019). https://doi.org/10.1007/s10958-019-04436-1
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DOI: https://doi.org/10.1007/s10958-019-04436-1