Hardy-Littlewood-Stein-Weiss type theorems for Riesz potentials and their commutators in Morrey spaces

Canay Aykol, Javanshir J. Hasanov

Abstract


In this paper we consider weighted Morrey spaces \(L_{p,\lambda,|\cdot|^{\gamma}}(\Rn).\) We prove the Hardy-Littlewood-Stein-Weiss type \(L_{p,\lambda,|\cdot|^{\gamma}}(\Rn)\) to \(L_{p,\lambda,|\cdot|^{\mu}}(\Rn)\) theorems for Riesz potential \(I^{\a}\) and its commutators \([b,I^{\alpha}]\) and \(|b,I^{\alpha}|\), where \(0<\alpha<n\), \(0\leq\lambda <n-\alpha\), \(1< p<\frac {n-\lambda}{\alpha}\), \(0\le\gamma<n(p-1)\), \(\mu=\frac{q\gamma}{p}\), \(\frac 1p-\frac 1q = \frac \alpha {n-\lambda}\), \(b\in BMO(\Rn).\) As a result of these we obtain the conditions for the boundedness of the commutator \(|b,I^{\a}|\) from Besov-Morrey spaces \(B_{p,\theta,\lambda,|\cdot|^{\gamma}}^s (\Rn)\) to \(B_{q,\theta,\lambda,|\cdot|^{\mu}}^s (\Rn).\)
\

Furthermore, we consider the Schr\"{o}dinger operator \(-\Delta + V\) on \(\Rn\) and obtain weighted Morrey \(L_{p,\lambda,|\cdot|^{\gamma}}(\Rn)\) estimates for the operators
\(V^{s} (-\Delta+V)^{-\beta}\) and \(V^{s} \nabla (-\Delta+V)^{-\beta}.\) Finally we apply our results to various operators which are estimated from above by Riesz potentials.

Keywords


Riesz potential, commutator, fractional maximal operator, Schr\"{o}dinger operator, Hardy-Littlewood-Stein-Weiss type estimate, Morrey space, BMO space

Full Text:

PDF

References


D.R. Adams, A note on Riesz potentials,

Duke Math., 42 (1975), 765-778.

D.R. Adams, Choquet integrals in potential theory, Publ. Mat. 42 (1998), 3-66.

W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87--130.

P. Auscher and P. Tchamitchian, Square root problem for divergence operators and related topics, Ast'erisque, 249, Soc. Math. France, 1998.

V.I. Burenkov and H.V. Guliyev, Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces, Stud. Math. 163 (2) (2004), 157-176.

L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1990), 253-284.

F. Chiarenza and M. Frasca, Morrey spaces and Hardy--Littlewood maximal function, Rend. Math. 7 (1987), 273-279.

X. T. Duong and L. X. Yan, On commutators of fractional integrals,

Proc. Amer. Math. Soc., 132 (2004), 12, 3549-3557.

G. Di Fazio, D.K. Palagachev and M.A. Ragusa, Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal. 166 (1999), 179-196.

G. Di Fazio and M. A. Ragusa, Commutators and Morrey spaces, Bollettino U.M.I. 7 5-A (1991), 323-332.

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129-206.

M. Giaquinta, Multiple integrals in the calculus of variations and

nonlinear elliptic systems, Princeton Univ. Press, Princeton, NJ,

H.G. Hardy and J.E. Littlewood, Some properties of fractional integrals, I. Math. Z., 27(4):565-606, 1928.

J.J. Hasanov, Hardy-Littlewood-Stein-Weiss inequality in the variable exponent Morrey spaces, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 39 (2013), 47-62.

K-P. Ho, Singular integral operators, John-Nirenberg inequalities and Tribel-Lizorkin type spaces on weighted Lebesgue spaces with variable exponents,

Rev. Un. Mat. Argentina 57(1), 85-101, 2016.

T. Karaman, V. S. Guliyev and A. Serbetci, Boundedness of sublinear operators generatedby Calderon-Zygmund operators on generalized weighted Morrey spaces, An. c{S}tiint. Univ. Al. I. Cuza Iac{s}i. Mat. (N.S.) 60 (2014), no. 1, 227-244.

A. Kufner, O. John and S. Fuc{c}ik, Function spaces, Noordhoff

International Publishing: Leyden, Publishing House Czechoslovak

Academy of Sciences: Prague, 1977.

K.Kurata, S.Nishigaki and S.Sugano, Boundedness of Integral Operators on Generalized Morrey Spaces and Its Application to Schr"odinger operators, Proc. AMS, 1999, 128(4), 1125-1134.

K. Kurata and S. Sugano, A remark on estimates for uniformly elliptic operators on weighted $L_p$ spaces and Morrey spaces, Math. Nachr. 209 (2000), 137-150.

H.Q. Li, Estimations $L_p$ des operateurs de Schr"odinger sur les groupes nilpotents, J. Funct. Anal. 161 (1999), 152-218.

G.Z. Lu, A Fefferman- Phong type inequality for degenerate vector fields and applications, Panamer. Math. J. 6 (1996), 37-57.

Yu Liu, The weighted estimates for the operators $V^{alpha} (-Delta_{G}+V)^{-beta}$ and

$V^{alpha} nabla_{G} (-Delta_{G}+V)^{-beta}$ on the stratified Lie group $mathbb{G}$, J. Math. Anal. Appl. 349 (2009), 235-244.

A.L. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355 (2003), 1297-1364.

A. McIntosh, Operators which have an $H_1$-calculusin, in Proc. Centre Math. Analysis, Vol. 14, Miniconference on Operator Theory and Partial Differential Equations, A. N. U., Canberra, 1986, 210-231.

T. Mizuhara, Boundedness of some classical operators on

generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991), 183-189.

C.B. Morrey, On the solutions of quasi-linear elliptic partial

differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166.

B. Muckenhoupt, Weighted norm inequalities for the Hardy-Littlewood maximal function, Trans. Amer. Math. Soc. 165(1972), 207-226.

E. Nakai, Hardy-Littlewood maximal operator, singular integral

operators and Riesz potentials on generalized Morrey spaces, Math. Nachr.

(1994), 95-103.

J. Peetre, On the theory of ${mathcal L}^{p,lb}$ spaces, J. Funct. Anal. 4 (1969), 71-87.

C. Perez, Two weighted norm inequalities for Riesz potentials and uniform

$L_p$-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), 31-44.

A. Ruiz and L. Vega, Unique continuation for Schr"odinger operators with potential in Morrey spaces, Publ. Mat. 35 (1991), 291-298.

A. Ruiz and L. Vega, On local regularity of Schr"odinger equations, Int. Math. Res. Notices 1993:1 (1993), 13-27.

S. Samko, Hardy-Littlewood-Stein-Weiss inequality in the Lebesgue spaces with variable exponent, Fract. Calc. Appl. Anal., 6(4) (2003), 421-440.

S. Samko, E. Shargorodsky and B. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators,

J. Math. Anal. Appl., 325 (2007), 745-751.

Z.W. Shen, $L_p$ estimates for Schr"odinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995) 513-546.

Z. Shen, The periodic Schr"odinger operators with potentials in the Morrey class, J. Funct. Anal. 193 (2002), 314-345.

Z. Shen, Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains, Amer. J. Math. 125 (2003), 1079-1115.

B. Simon, Maximal and minimal Schr"odinger forms, J. Op. Theory, {1} (1979), 37-47.

E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and

Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

E.M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. and Mech., 7(4):503-514, 1958.

S. Sugano, Estimates for the operators $V^{alpha} (-Delta+V)^{-beta}$ and $V^{alpha} nabla (-Delta+V)^{-beta}$ with certain nonnegative potentials $V$}, Tokyo J. Math. 21 (1998), 441-452.

M.E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407-1456.

J.P. Zhong, Harmonic analysis for some Schr"{o}dinger type operators,

PhD thesis, Princeton University, 1993.




DOI: http://dx.doi.org/10.24193/subbmath.2023.3.11

Refbacks

  • There are currently no refbacks.