scholarly journals Approximate solutions of equations defined by complex spherical multiplier operators

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-115
Author(s):  
C. P. Oliveira
Keyword(s):  
Approximate Solutions ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Sobolev Type ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Solutions Of Equations ◽  
Multiplier Operators

This paper studies, in a partial but concise manner, approximate solutions of equations defined by complex spherical multiplier operators. The approximations are from native spaces embedded in Sobolev-type spaces and derived from the use of positive definite functions to perform spherical interpolation.

Sharp Singular Trudinger–Moser Inequalities Under Different Norms

2019 ◽  
Vol 19 (2) ◽  
pp. 239-261 ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu ◽  
Lu Zhang
Keyword(s):  
Exponential Type ◽  
Sobolev Type ◽  
Infinite Volume ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
The Relationship

AbstractThe main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces {D^{N,q}(\mathbb{R}^{N})}, {q\geq 1}, the completion of {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm {\|\nabla u\|_{N}+\|u\|_{q}}. The case {q=N} (i.e., {D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})}) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type {\|\nabla u\|_{N}\leq 1} and full-norm type {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1}, {a>0}, {b>0}. We will show that the Trudinger–Moser-type inequalities hold if and only if {b\leq N}. Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when {b>N}.

scholarly journals Sarason’s Conjecture of Toeplitz Operators on Fock-Sobolev Type Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Xiaofeng Wang ◽  
Jianjun Chen ◽  
Jin Xia
Keyword(s):  
Toeplitz Operators ◽  
Complex Polynomial ◽  
Sobolev Type ◽  
Linear Complex ◽  
Nonzero Constant ◽  
Sobolev Type Spaces ◽  
Type Spaces

In this note, we will solve Sarason’s conjecture on the Fock-Sobolev type spaces and give a well solution that if Toeplitz product TuTv¯, with entire symbols u and v, is bounded if and only if u=eq, v=Ce-q, where q is a linear complex polynomial and C is a nonzero constant.

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