Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Nonlinear Equations and Weighted Norm Inequalities

N. J. Kalton and I. E. Verbitsky
Transactions of the American Mathematical Society
Vol. 351, No. 9 (Sep., 1999), pp. 3441-3497
Stable URL: http://www.jstor.org/stable/117929
Page Count: 57
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Nonlinear Equations and Weighted Norm Inequalities
Preview not available

Abstract

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem -Δ u=vuq+w, u≥ 0 on Ω , u=0 on ∂ Ω , on a regular domain Ω in R{n in the "superlinear case" $q>1$. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω . We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if $v\equiv 1$ and Ω is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary "integration by parts" argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

Page Thumbnails

  • Thumbnail: Page 
3441
    3441
  • Thumbnail: Page 
3442
    3442
  • Thumbnail: Page 
3443
    3443
  • Thumbnail: Page 
3444
    3444
  • Thumbnail: Page 
3445
    3445
  • Thumbnail: Page 
3446
    3446
  • Thumbnail: Page 
3447
    3447
  • Thumbnail: Page 
3448
    3448
  • Thumbnail: Page 
3449
    3449
  • Thumbnail: Page 
3450
    3450
  • Thumbnail: Page 
3451
    3451
  • Thumbnail: Page 
3452
    3452
  • Thumbnail: Page 
3453
    3453
  • Thumbnail: Page 
3454
    3454
  • Thumbnail: Page 
3455
    3455
  • Thumbnail: Page 
3456
    3456
  • Thumbnail: Page 
3457
    3457
  • Thumbnail: Page 
3458
    3458
  • Thumbnail: Page 
3459
    3459
  • Thumbnail: Page 
3460
    3460
  • Thumbnail: Page 
3461
    3461
  • Thumbnail: Page 
3462
    3462
  • Thumbnail: Page 
3463
    3463
  • Thumbnail: Page 
3464
    3464
  • Thumbnail: Page 
3465
    3465
  • Thumbnail: Page 
3466
    3466
  • Thumbnail: Page 
3467
    3467
  • Thumbnail: Page 
3468
    3468
  • Thumbnail: Page 
3469
    3469
  • Thumbnail: Page 
3470
    3470
  • Thumbnail: Page 
3471
    3471
  • Thumbnail: Page 
3472
    3472
  • Thumbnail: Page 
3473
    3473
  • Thumbnail: Page 
3474
    3474
  • Thumbnail: Page 
3475
    3475
  • Thumbnail: Page 
3476
    3476
  • Thumbnail: Page 
3477
    3477
  • Thumbnail: Page 
3478
    3478
  • Thumbnail: Page 
3479
    3479
  • Thumbnail: Page 
3480
    3480
  • Thumbnail: Page 
3481
    3481
  • Thumbnail: Page 
3482
    3482
  • Thumbnail: Page 
3483
    3483
  • Thumbnail: Page 
3484
    3484
  • Thumbnail: Page 
3485
    3485
  • Thumbnail: Page 
3486
    3486
  • Thumbnail: Page 
3487
    3487
  • Thumbnail: Page 
3488
    3488
  • Thumbnail: Page 
3489
    3489
  • Thumbnail: Page 
3490
    3490
  • Thumbnail: Page 
3491
    3491
  • Thumbnail: Page 
3492
    3492
  • Thumbnail: Page 
3493
    3493
  • Thumbnail: Page 
3494
    3494
  • Thumbnail: Page 
3495
    3495
  • Thumbnail: Page 
3496
    3496
  • Thumbnail: Page 
3497
    3497