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Register a new userThe coarea formula for Sobolev mappings
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We extend Federers coarea formula to mappings $f$ belonging to the Sobolev class $W^{1,p}(R^n;R^m)$, $1 le m < n$, $p>m$, and more generally, to mappings with gradient in the Lorentz space $L^{m,1}(R^n)$. This is accomplished by showing that the graph of $f$ in $R^{n+m}$ is a Hausdorff $n$-rectifiable set.
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We consider contact manifolds equipped with Carnot-Caratheodory metrics, and show that the Rumin complex is respected by Sobolev mappings: Pansu pullback induces a chain mapping between the smooth Rumin complex and the distributional Rumin complex. As a consequence, the Rumin flat complex -- the analog of the Whitney flat complex in the setting of contact manifolds -- is bilipschitz invariant. We also show that for Sobolev mappings between general Carnot groups, Pansu pullback induces a chain mapping when restricted to a certain differential ideal of the de Rham complex. Both results are applications of the Pullback Theorem from our previous paper.
This note concerns the general formulation by Preiss and Uher of Kestelmans influential result pertaining the change of variable, or substitution, formula for the Riemann integral.
We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $sigma=(sigma_1,dots,sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $overline{f} colon {mathbb R}^m to V$ such that $f = sigma circ overline{f}$ is of class $C^{d-1,1}$ is locally of Sobolev class $W^{1,p}$ for all $1 le p<d/(d-1)$. In the case $m=1$ there always exists a continuous choice $overline{f}$ for given $fcolon {mathbb R} to sigma(V) subseteq {mathbb C}^n$. We give uniform bounds for the $W^{1,p}$-norm of $overline{f}$ in terms of the $C^{d-1,1}$-norm of $f$. The result is optimal: in general a lifting $overline{f}$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger Holder class.
We study weighted Poincare and Poincare-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ left (frac{1}{w(Q)}int_Q|f-f_Q|^{q}wright )^frac{1}{q}le C_well(Q)left (frac{1}{w(Q)}int_Q | abla f|^p wright )^frac{1}{p}, $$ with different quantitative estimates for both the exponent $q$ and the constant $C_w$. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality $$ frac{1}{|Q|}int_Q |f-f_Q| dmu le a(Q), $$ for all cubes $Qsubsetmathbb{R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent $p^*_w>p$ associated to the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left hand side of the inequality above. For the endpoint case of $A_1$ weights we reach the classical critical Sobolev exponent $p^*=frac{pn}{n-p}$ which is the largest possible and provide different type of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ constant smaller than $1/p$. We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $pgeq1$, Poincare inequality valid for the whole class of $RH_infty$ weights by showing their intimate connection with the failure of Poincare inequalities, $(p,p)$ in the range $0<p<1$.
In this paper, we prove a new integral representation for the Bessel function of the first kind $J_mu(z)$, which holds for any $mu,zinmathbb{C}$.
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