Abstract
We extend Th. Wolff's inequality to a general class of radially decreasing convolution kernels. As an application we obtain characterizations of nonnegative Borel measures μ on R n such that the trace inequality \(\parallel {\kern 1pt} T_K f{\kern 1pt} \parallel _{L^q ({\text{d}}\mu )} \;\; \leqslant \;\;C\parallel {\kern 1pt} f{\kern 1pt} \parallel _{L^p ({\text{d}}x)} \) holds for every f in L p(dx).
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References
Adams, D.R.: 'Traces of potentials arising from translation invariant operators', Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1971), 203–217.
Adams, D.R.: 'Weighted nonlinear potential theory', Trans. Amer. Math. Soc. 297 (1986), 73–94.
Aikawa, H. and Essén, M.: Potential Theory-Selected Topics, Lecture Notes in Math. 1633, Springer, 1996.
Adams, D.R. and Hedberg, L.I.: Function Spaces and Potential Theory, Springer-Verlag, New York, 1996.
Cascante, C. and Ortega, J.M.: 'Norm inequalities for potential-type operators in homogeneous spaces', to appear in Math. Nachr.
Cascante, C., Ortega, J.M., and Verbitsky, I.E.: 'Trace inequalities of Sobolev type in the upper triangle case', Proc. London Math. Soc. 80 (2000), 391–414.
Fefferman, R.: 'Strong differentiation with respect to measures', Amer. J. Math. 103 (1981), 33–40.
Fefferman, C. and Stein, E.M.: 'Some maximal inequalities', Amer. J. Math. 93 (1971), 107–115.
Hedberg, L.I. and Wolff, Th.H.: 'Thin sets in nonlinear potential theory', Ann. Inst. Fourier (Grenoble) 33 (1983), 161–187.
Kalton, N.J. and Verbitsky, I.E.: 'Nonlinear equations and weighted norm inequalities', Trans. Amer. Math. Soc. 351 (1999), 3441–3497.
Kerman, R. and Sawyer, E.: 'The trace inequality and eigenvalue estimates for Schrödinger operators', Ann. Inst. Fourier (Grenoble) 36 (1986), 207–228.
Maz'ya, V.G.: Sobolev Spaces, Springer-Verlag, New York, 1985.
Maz'ya, V.G. and Netrusov, Y.: 'Some counterexamples for the theory of Sobolev spaces on bad domains', Potential Anal. 4 (1995), 47–65.
Muckenhoupt, B. and Wheeden, R.: 'Weighted norm inequalities for fractional integrals', Trans. Amer. Math. Soc. 192 (1974), 261–274.
Sawyer, E.T.: 'A characterization of a two weight norm inequality for maximal operators', Studia Math. 75 (1982), 1–11.
Sawyer, E.T. and Wheeden, R.L.: 'Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces', Amer. J. Math. 114 (1992), 813–874.
Sawyer, E.T., Wheeden, R.L., and Zhao, S.: 'Weighted norm inequalities for operators of potential type and fractional maximal functions', Potential Anal. 5 (1996), 523–580.
Verbitsky, I.E.: 'Nonlinear potentials and trace inequalities', in: Operator Theory: Advances and Applications 110, 1999, pp. 323–343.
Verbitsky, I.E. and Wheeden, R.L.: 'Weighted norm inequalities for integral operators', Trans. Amer. Math. Soc. 350 (1998), 3371–3391.
Wheeden, R.L. and Zhao, S.: 'Weak type estimates for operators of potential type', Studia Math. 119 (1996), 149–160.
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Cascante, C., Ortega, J.M. & Verbitsky, I.E. Wolff's Inequality for Radially Nonincreasing Kernels and Applications to Trace Inequalities. Potential Analysis 16, 347–372 (2002). https://doi.org/10.1023/A:1014845728367
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DOI: https://doi.org/10.1023/A:1014845728367