Weighted quantitative isoperimetric inequalities in the Grushin space Rh+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${R}^{h+1}$\end{document} with density |x|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|x|^{p}$\end{document}

In this paper, we prove weighted quantitative isoperimetric inequalities for the set Eα={(x,y)∈Rh+1:|y|<∫arcsin|x|π2sinα+1(t)dt,|x|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{\alpha}= \{(x,y)\in {R}^{h+1}: \vert y \vert <\int_{\arcsin \vert x \vert }^{\frac{\pi}{2}}\sin^{\alpha +1}(t)\,dt, \vert x \vert <1 \}$\end{document} in half-cylinders in the Grushin space Rh+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${R}^{h+1}$\end{document} with density |x|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert x \vert ^{p}$\end{document}, p≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq0$\end{document}.


Introduction
The study of isoperimetric problems in Carnot-Carathéodory spaces has been an active field over the past few decades. Pansu [] first proved an isoperimetric inequality of the type P H (E) ≥ C|E|   (C > ) in the Heisenberg group H  where P H (E) and |E| denote Heisenberg perimeter and Lebesgue volume of E, respectively. In  Pansu [] conjectured that, up to a null set, a left translation and a dilation, the isoperimetric set in the Heisenberg group H  is a bubble set as follows: In the case of α = , the set E α () is just the Pansu sphere in the Heisenberg group.
On the other hand, manifolds with density, a new category in geometry, have been widely studied. They arise naturally in geometry as quotients of Riemannian manifolds, in physics as spaces with different media, in probability as the famous Gauss space and in a number of other places as well (see [, ] Motivated by the nice work mentioned above, in this paper we consider the quantitative isoperimetric inequalities for the set E α in half-cylinders in the Grushin space R h+ with density |x| p , p ≥ . These inequalities show that the weighted volume distance of a set F from the set E α with the same weighted volume is controlled in terms of the difference of the weighted α-perimeter of F and the weighted α-perimeter of E α . We get the following theorem.

Theorem . Let F be any measurable set in the Grushin space R h+ with density e
Here P α,φ (E) = sup{ E div α (|x| p ϕ) dx dy : ϕ ∈ C  c (R h+ ; R h+ ), max |ϕ| ≤ } and V φ (E) = E |x| p dx dy are called the weighted α-perimeter and the weighted volume of E, respectively. Finally ω h denotes the Euclidean volume of the unit ball.
When p =  in Theorem ., we can obtain the quantitative isoperimetric inequalities for the set E α in half-cylinders in Grushin spaces.

Preliminaries
The Grushin space R h+ = {(x, y) : x ∈ R h , y ∈ R} is a Carnot-Carathéodory space with a system of vector fields where α >  is a given real number and |x| is the standard Euclidean norm of x.
The α-perimeter of a measurable set E ⊂ R h+ in an open set A ⊂ R h+ is defined as where the α-divergence of the vector field ϕ : A → R h+ is given by

. on A and the generalized Gauss-Green formula
Here and hereafter, ·, · denotes the standard Euclidean scalar product. The measure μ E is called α-perimeter measure and the function v E is called measure theoretic inner unit α-normal of E. Now we endow the Grushin space R h+ with density e φ and define the weighted αperimeter of a measurable set E ⊂ R h+ in an open set A ⊂ R h+ as By the definition of div α,φ ϕ, () can also be rewritten as where dV φ = e φ dx dy is the weighted volume measure and dμ E,φ = e φ dμ E is called the weighted α-perimeter measure. For any open set Let be a hypersurface in the Grushin space R h+ with density e φ . can be locally given by the zero set of a function u ∈ C  such that Then we define the weighted α-mean curvature of as Remark . Noticing that the α-mean curvature of is defined by H = - h div α v E , then from () we have To prove Theorem ., we need the following lemma.

Lemma . Let the Grushin space R h+ be endowed with density e
arcsin r sin α+ (t) dt. There exists a continuous function u : (iii) s is a hypersurface of class C  with constant weighted α-mean curvature, that is, Proof The profile function of the set E α is the function f : [, ] → R, Its first and second derivatives are We define the function g : [, ] → R, Its derivative is Now we construct a foliation of C ε . In C ε \ E α , the leaves s of the foliation are vertical translations of the top part of the boundary ∂E α . In C ε ∩ E α , the leaves s are constructed as follows: the surface ∂E α is dilated by a factor larger than  where dilation is defined by (x, y) → (λx, λ α+ y) (∀λ > ), and then it is translated downwards in such a way that the surface {y = y ε = f (ε)} is also the leaf at last.
We construct a function u on the set C ε \ E α as Then we have s ≤  and  = ∂E α . From (), we know u ∈ C  (C ε \ E α ) and s≤ s = C ε \ E α . In the following we will define the function u on the set D ε = C ε ∩ E α . Setting r = |x| and r ε = ε, we let F ε : D ε × (, ∞) → R be a function For any (x, y) ∈ D ε we have On the other hand, using () and () we have So there exists a unique s >  such that F ε (x, y, s) =  for any (x, y) ∈ D ε . Furthermore we can define a function u : D ε → R, s = u(x, y) determined by the equation F ε (x, y, s) = . Obviously we have u ∈ C  (C ε ∩ E α ) and C ε ∩ E α = s> s , where s = {(x, y) ∈ C ε ∩ E α : s = u(x, y) is determined by F ε (x, y, s) = }. By (), we find Using (), () and (), we obtain .

(   )
Then we have , and the square length of the α-gradient of u on D ε is Note that |∇ α u| =  if and only if x = . So for any (x, y) ∈ D ε with x = , we have If (x, y) ∈ D ε tends to (x, y) ∈ ∂E α with x =  and y > , then s = u(x, y) converges to . From (), we have where the right hand side is computed by the definition () of u. The above equality shows that ∇ α u |∇ α u| is continuous on C ε \ {x = }. In the case of e φ = |x| p , we get φ = p ln |x| and ∇ α φ = ( p |x|  x  , . . . , p |x|  x h , ) for x = . From (), we know that the inner unit α-normal of s with s ≤  is v s = -x  , . . . , -x h , - -|x|  .
So the weighted α-mean curvature H s ,φ of s with s ≤  is given by From () we know that the inner unit α-normal of s with s >  is So the weighted α-mean curvature H s ,φ of s with s >  is given by Fixing a point x with |x| < ε and for  ≤ y < f (|x|)y ε , we define the function where s ≥  is uniquely determined by (x, f (|x|)y) ∈ s . Then the function y → h x (y) is increasing and h x () = . From () and (), we know , for all  ≤ y < f (|x|)y ε . By (), g is strictly increasing. So h x (y) satisfies .

(   )
On the other hand, for any s >  we have When ε = , we have r ε = . So () turns into By (), we get Integrating () with h x () = , we get Thus we obtain For any point (x, y) ∈ s with s > , we have , without loss of generality we can assume that the boundary ∂F of F is C ∞ .
For δ > , let E δ α = {(x, y) ∈ E α : |x| > δ}. By () and (), we have Letting δ →  + and using the Cauchy-Schwarz inequality, we obtain By a similar computation, we also have On the other hand, we have From (), () and (), we obtain It is equivalent to For any x with |x| < ε, we define the vertical sections E x α = {y : (x, y) ∈ E α } and F x = {y : (x, y) ∈ F}. By the Fubini theorem, we have Letting m(x) = L  (E x α \ F x ), where L  denotes -dimensional Lebesgue measure, then we obtain where h x (y) = u(x, f (|x|)y) is the function introduced in (). So from () and () we have |x| p dy dx.