Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sobolev inequalities for products of powers

Authors: A. Eduardo Gatto and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 314 (1989), 727-743
MSC: Primary 26D10; Secondary 46E35
MathSciNet review: 967312
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive weighted Sobolev inequalities of the form $ {\left\Vert f \right\Vert _{L_u^q}} \leq C{\left\Vert {\nabla f} \right\Vert _{L_v^p}}$, $ f \in C_0^\infty ({{\mathbf{R}}^n})$, $ 1 < p \leq q < \infty $, for classes of weight functions $ u$, $ v$ which include $ v$'s that are a finite product of certain power weights times an $ {A_p}$ function.

References [Enhancements On Off] (What's this?)

  • [1] L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259-275. MR 768824 (86c:46028)
  • [2] S. Chanillo and R. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math 107 (1985), 1191-1226. MR 805809 (87f:42045)
  • [3] E. Fabes, C. Kenig, and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116. MR 643158 (84i:35070)
  • [4] A. E. Gatto, C. Gutierrez, and R. Wheeden, Fractional integals on weighted $ {H^p}$ spaces, Trans. Amer. Math. Soc. 289 (1985), 575-589. MR 784004 (86k:42037)
  • [5] F. Gehring, The $ {L^p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. MR 0402038 (53:5861)
  • [6] E. Sawyer and R. Wheeden, Weighted inequalities for fractional and Poisson integrals on Euclidean and homogeneous spaces (to appear).
  • [7] J.-O. Strömberg and R. Wheeden, Fractional integrals on weighted $ {H^p}$ and $ {L^p}$ spaces, Trans. Amer. Math. Soc. 287 (1985), 293-321. MR 766221 (86f:42016)
  • [8] J. Vaisala, Lectures on $ n$-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, 1971. MR 0454009 (56:12260)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26D10, 46E35

Retrieve articles in all journals with MSC: 26D10, 46E35

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society