Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels
By C. Cascante, J. M. Ortega, Igor Verbitsky. 2003
65 citations recorded by the source corpus.
We study trace inequalities of the type ∥T k f∥ L q (dμ) ≤ C∥f∥ L p (dσ) , f ∈ L p (dσ), in the upper triangle case 1 ≤ q < p for integral operators T k with positive kernels, where dσ and dp are positive Borel measures on R n . Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy E k,σ [μ] = R n(T k [μ]) p' dσ through the L 1 (dμ)-norm of an appropriate nonlinear potential W k,σ [μ] associated with the kernel k and measures dp, dσ. We initially work with a dyadic integral operator with kernel K D (x,y) = Σ Q ∈ D K(Q)X Q (x)X Q (y), where D = {Q} is the family of all dyadic cubes in R n , and K: D → R + . The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.
Subfields: Applied Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/7ce56483c7660726803acabe57471ca63eea0192
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
- N. Phuc
- E. Dynkin
- Igor Verbitsky
- Giuseppe Mingione
- Tuomo Kuusi
- Xu-jia Wang
- Frank Duzaar
- J. A. Pel'aez
In Numerical Analysis
Citations in the corpus
- Superdiffusions and positive solutions of non-linear partial differential equations (2004, methodology): “The case l d 1 can be treated in a slightly dierent way using, for instance, estimates of the energy in terms of nonlinear potentials from [ COIEV04 ].”
- Quasilinear and Hessian equations of Lane-Emden type (2005, methodology): “Very recently, an analogue of Wol’s inequality for general dyadic and radially decreasing kernels was obtained in [ COV ]; some of the tools developed there are employed below.”
- Singular quasilinear and Hessian equations and inequalities (2009, methodology): “In the present paper, we use extensively dyadic models, and various global and local Wol’s potential estimates obtained in [12], [ 13 ], [21], [22], [25], [26], along with fundamental weak continuity theorems for quasilinear and Hessian equations [32]‐[34].”