Abstract
This paper is devoted to weighted Hardy type inequalities. Using the Bessel functions, we prove one-dimensional inequalities and give some remarks on extensions of the one-dimensional inequalities to \(n\)-dimensional convex domains with finite inner radius. Constants in those inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases. We establish the inequalities in \(L^{p}\) spaces, with \(p\in[1,\infty)\). Also new \(L_{2}\)- inequality with sharp constants is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
F. G. Avkhadiev and K.-J. Wirths, ‘‘Weighted Hardy inequalities with sharp constants,’’ Lobachevskii J. Math. 31 (1), 1–7 (2010).
F. G. Avkhadiev, ‘‘Hardy type inequalities in higher dimensions with explicit estimate of constants,’’ Lobachevskii J. Math. 21, 3–31 (2006).
F. G. Avkhadiev, ‘‘Hardy-type inequalities on planar and spatial open sets,’’ Proc. Steklov Inst. Math. 255 (2), 2–12 (2006).
F. G. Avkhadiev and R. G. Nasibullin, ‘‘Hardy-type inequalities in arbitrary domains with finite inner radius,’’ Sib. Math. J. 55, 191–200 (2014).
F. G. Avkhadiev and K.-J. Wirths, ‘‘Unified Poincaré and Hardy inequalities with sharp constants for convex domains,’’ Z. Angew. Math. Mech. 87, 632–642 (2007).
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).
A. A. Balinsky, W. D. Evans, and R. T. Lewis, The Analysis and Geometry of Hardy’s Inequality (Universitext, Springer, Heidelberg, 2015).
P. R. Beesack, ‘‘Hardy’s inequality and its extensions,’’ Pacif. J. Math. 11, 39–61 (1961).
D. W. Boyd, ‘‘Best constants in a class of integral inequalities,’’ Pacif. J. Math. 30, 367–383 (1969).
H. Brezis and M. Marcus, ‘‘Hardy’s inequalities revisited,’’ Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25, 217–237 (1998).
W. D. Evans and R. T. Lewis, ‘‘Hardy and Rellich inequalities with remainders,’’ J. Math. Inequal. 1, 473–490 (2007).
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge Univ. Press, Cambridge, 1973).
M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and A. Laptev, ‘‘A geometrical version of Hardy’s inequality,’’ J. Funct. Anal. 189, 539–548 (2002).
A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type (World Scientific, New Jersey, London, Singapore, Hong Kong, 2003).
V. Levin, ‘‘Notes on inequalities. II. On a class of integral inequalities,’’ Mat. Sb. 4 (46), 309–324 (1938).
L. E. Persson and V. D. Stepanov, ‘‘Weighted integral inequalities with the geometric mean operator,’’ J. Inequal. Appl. 7, 727–746 (2002).
V. G. Maz’ya, Sobolev Spaces (Springer, Berlin, Heidelberg, 2011).
B. Muckenhoupt, ‘‘Hardy’s inequality with weights,’’ Studia Math. XLIV 44, 31–38 (1972).
R. G. Nasibullin, ‘‘Brezis–Marcus type inequalities with Lamb constant,’’ Sib. Elektron. Mat. Izv. 16, 449–464 (2019).
R. G. Nasibullin, ‘‘Sharp Hardy type inequalities with weights depending on Bessel function,’’ Ufa Math. J. 9, 89–97 (2017).
R. G. Nasibullin, ‘‘Multidimensional Hardy Type Inequalities with Remainders,’’ Lobachevskii J. Math. 40, 1383–1396 (2019).
R. G. Nasibullin, ‘‘Hardy type inequalities with weights dependent on the bessel functions,’’ Lobachevskii J. Math. 37 (3), 274–283 (2016).
R. G. Nasibullin, ‘‘A geometrical version of Hardy–Rellich type inequalities,’’ Math. Slov. 69, 785–800 (2019).
D. T. Shum, ‘‘On a class of new inequalities,’’ Trans. Am. Math. Soc. 204, 299–341 (1975).
G. Talenti, ‘‘Osservazioni sopra una classe di disuguaglianze,’’ Rend. Sem. Mat. Fiz. Milano 39, 171–185 (1969).
G. Tomaselli, ‘‘A class of inequalities,’’ Boll. Un. Mat. Ital. 2, 622–631 (1969).
H. Lamb, ‘‘Note on the induction of electric currents in a cylinder placed across the lines of magnetic force,’’ Proc. London. Math. Soc. s1-15, 270–275 (1884).
J. Tidblom, ‘‘A geometrical version of Hardy’s inequality for \(W^{1,p}_{0}(\Omega)\),’’ Proc. Am. Math. Soc. 132, 2265–2271 (2004).
J. Tidblom, ‘‘A Hardy inequality in the half-space,’’ J. Funct. Anal. 221, 482–495 (2005).
G. N. Watson, A Threatise on the Theory of the Bessel Functions (Cambridge Univ. Press, Cambridge, 1966).
Funding
This work was supported by the grant of the President of the Russian Federation no. MK-709.2019.1.
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by F. G. Avkhadiev)
Rights and permissions
About this article
Cite this article
Makarov, R.V., Nasibullin, R.G. & Shaymardanova, G.R. Weighted Hardy Type Inequalities and Parametric Lamb Equation. Lobachevskii J Math 41, 2198–2210 (2020). https://doi.org/10.1134/S199508022011013X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S199508022011013X