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Transactions of the American Mathematical Society

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



Sobolev interpolation inequalities with weights

Authors: Cristian E. Gutiérrez and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 323 (1991), 263-281
MSC: Primary 46E99; Secondary 35B45, 46M35
MathSciNet review: 994166
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Abstract: We study weighted local Sobolev interpolation inequalities of the form

\begin{displaymath}\begin{gathered}\frac{1} {{{w_2}(B)}}{\int\limits_B {\vert u(... ...\vert u(x){\vert^p}v(x)dx} } \right), \hfill \\ \end{gathered} \end{displaymath}

, where $ 1 < p < \infty,h > 1, B$ is a ball in $ {{\mathbf{R}}^n}$, and $ v$ ,$ {w_1}$, and $ {w_2}$ are weight functions. The case $ p = 2$ is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case $ u$ has compact support in $ B$, and we derive global Landau inequalities $ {\left\Vert {\nabla u} \right\Vert _{L_w^q}} \leq c\left\Vert {\nabla u} \righ... ...eft\Vert {{\nabla ^2}u} \right\Vert _{L_v^p}^a,0 < a < 1,1 < p \leq q < \infty $, when $ u$ has compact support.

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  • [1] R. C. Brown and D. B. Hinton, Weighted interpolation inequalities of sum and product form in $ {R^n}$, Proc. London Math. Soc. (3) 56 (1988), 261-280. MR 922656 (89c:26027)
  • [2] C. Carton-Lebrun and H. P. Heinig, Weighted norm inequalities involving gradients (to appear). MR 1014662 (90g:42035)
  • [3] S. Chanillo and R. L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985), 1191-1126. MR 805809 (87f:42045)
  • [4] F. Chiarenza and R. Serapioni, A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9 (1984), 719-749. MR 748366 (86c:35082)
  • [5] -, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985), 179-190. MR 799906 (87a:35110)
  • [6] J. A. Goldstein, M. K. Kwong, and A. Zettl, Weighted Landau inequalities, J. Math. Anal. Appl. 95 (1983), 20-28. MR 710417 (84j:26019)
  • [7] C. E. Gutierrez and G. Nelson, Bounds for the fundamental solution of degenerate parabolic equations, Comm. Partial Differential Equations 13 (1988), 635-649. MR 919445 (90c:35128)
  • [8] C. E. Gutierrez and R. L. Wheeden, Mean value and Harnack inequalities for degenerate parabolic equations, Colloq. Math., vol. dedicated to A. Zygmund (to appear). MR 1096367 (92d:35025)
  • [9] D. Jerison, The Poincaré inequality for vector fields satisfying Hàrmander's condition, Duke Math. J. 53 (1986), 503-523. MR 850547 (87i:35027)
  • [10] R. V. Kohn, New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78 (1982), 131-172. MR 648942 (83f:73018)
  • [11] V. G. Maz'ja, Sobolev spaces, Springer-Verlag, 1985. MR 817985 (87g:46056)
  • [12] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727-740. MR 0288405 (44:5603)
  • [13] F. O. Porper and S. D. Eidel'man, Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications, Russian Math. Surveys 39 (1984), no. 3, 119-178. MR 747792 (86b:35078)
  • [14] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals in Euclidean and homogeneous spaces (to appear).
  • [15] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259-275. MR 768824 (86c:46028)
  • [16] C.-S. Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations 11 (1986), 1515-1538. MR 864416 (88a:46037)

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