Conductor Inequalities and Criteria for Sobolev-lorentz Two-weight Inequalities
نویسنده
چکیده
In this paper we present integral conductor inequalities connecting the Lorentz p, q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p, q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures.
منابع مشابه
On Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces
Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of co...
متن کاملSharp Boundary Trace Inequalities
This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ R . The inequalities bound (semi-)norms of the boundary trace by certain norms of the function and its gradient on the region and two specific constants kρ and kΩ associated with the domain and a weight function. These inequalities are sharp in that there are functions for which eq...
متن کاملSobolev and Isoperimetric Inequalities with Monomial Weights
We consider the monomial weight |x1|1 · · · |xn|n in R, where Ai ≥ 0 is a real number for each i = 1, ..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|1 · · · |xn|ndx, and they contain the best or critical exponent (which depends on A1, ..., An). More i...
متن کاملA Poincaré Inequality on Loop Spaces
We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L2 variance of an admissible function is controlled by the homogeneous H1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homogeneous H1 norm alone may not control the L2 norm and a potential term involving the end value of the...
متن کاملIsoperimetry and Symmetrization for Sobolev Spaces on Metric Spaces
This is a follow up to our recent work [10], where we obtained new symmetrization inequalities for Sobolev functions that compare the rearrangement of a function with the rearrangement of its gradient, and incorporate in their formulation the isoperimetric profile (cf. (2.1) below). These inequalities imply in a straightforward fashion functional inequalities for very general rearrangement inva...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008