Weighted Sobolev Inequalities Research Articles

Non-linear operators of divergence form on the Sierpinski gasket

We consider non-linear operators L of divergence type on the Sierpinski gasket. We prove the Harnack inequality for non-negative L-harmonic functions on the Sierpinski gasket. The proof is based on Moser’s method, and we construct a new measure λ by combining a Hausdorff measure and an energy measure, which plays an essential role at key steps. The Harnack inequality is achieved by involving the Caccioppoli type inequality and the weighted Sobolev inequality which give the local boundedness and the Weak Harnack inequality of solutions with respect to the measure λ.

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Hardy's inequality and the isotropic Landau equation

In this manuscript we establish an L∞ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent γ in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincaré inequality, which in turn implies the propagation of some Lp norms of solutions. From here, the L∞ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.

On a sharp weighted Sobolev inequality on the upper half-space and its applications

On a sharp weighted Sobolev inequality on the upper half-space and its applications

Adimensional weighted Sobolev inequalities in PI spaces

We provide a family of global weighted Sobolev inequalities and Hardy inequalities on PI spaces with possibly non-maximal volume growth. Our results apply notably to non-trivial Ahlfors regular spaces like Laakso spaces and Kleiner-Schioppa spaces.

Sobolev inequality with non-uniformly degenerating gradient

In this paper we prove the following weighted Sobolev inequality in a bounded domain Ω ⊂ R n , n ≥ 1 , of a homogeneous space ( R n , ρ , w d x ) , under suitable compatibility conditions on the positive weight functions ( v , w , ω 1 , ω 2 , … , ω n ) and on the quasi-metric ρ , ( ∫ Ω | f | q v w d z ) 1 q ≤ C ∑ i = 1 N ( ∫ Ω | f z i | p ω i M S w d z ) 1 p , f ∈ L i p 0 ( Ω ¯ ) , where q ≥ p > 1 and M S denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.

Divergent operator with degeneracy and related sharp inequalities

In this paper we classify all nonnegative extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves a divergent operator with degeneracy on the boundary. As an application of the results, we derive a sharp Sobolev type inequality involving Baouendi-Grushin operator, and classify certain extremal functions for all τ>0 and m≠2 or n≠1.

Weighted Sobolev inequalities in CD(0,N) spaces

In this note, we prove global weighted Sobolev inequalities on non-compact CD(0,N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result from [V. Minerbe,G.A.F.A.18(2009) 1696–1749] stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0,N) spaces to get a uniform bound of the corresponding weighted heat kernelviaa weighted Nash inequality.

On a Precise Scaling to Caffarelli-Kohn-Nirenberg Inequality

We analyze the general form of Caffarelli-Kohn-Nirenberg inequality. Due to a new introduced parameter, this inequality presents two distinguishable ranges. One of them, the inequality is shown to be the interpolation between Hardy and weighted Sobolev inequalities. The other range, which is no more an interpolation, the positive constant in the inequality is not necessarily bounded for all value of the parameters. In both cases, the precise value of the constants were given.

Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form ∫ E | u ( x ) | q ω ( x ) d x 1 / q ⩽ K 0 ∫ E | ∇ u ( x ) | p σ ( x ) d x 1 / p , u ∈ C 0 ∞ ( R n ) , (WSI)where p ⩾ 1 and q > 0 is the corresponding Sobolev critical exponent. Here E ⊆ R n is an open convex cone, and ω , σ : E → ( 0 , ∞ ) are two homogeneous weights verifying a general concavity-type structural condition. The constant K 0 = K 0 ( n , p , q , ω , σ ) > 0 is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that K 0 is optimal in (WSI) if and only if ω and σ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to partial differential equations are also provided.

Weighted Sobolev inequality in grand mixed norm Lebesgue spaces

We derive necessary and sufficient conditions on weights guaranteeing the one-weight Sobolev-type inequality for strong fractional maximal and multiple fractional integral operators in grand mixed norm Lebesgue spaces. As a consequence, appropriate statements for fractional integral operators defined on regular curves and domains in $${{\mathbb {R}}}^n$$ are obtained.

A sharp Sobolev inequality and its applications to an indefinite elliptic equation with Neumann boundary conditions

In this paper, we prove a sharp weighted Sobolev inequality on the upper half-space. As a consequence of this inequality we derive some existence, nonexistence and multiplicity results to a class of indefinite semilinear elliptic equations with Neumann boundary conditions.

The sharp Gagliardo–Nirenberg–Sobolev inequality in quantitative form

Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert [49], we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo–Nirenberg–Sobolev inequalities due to Del Pino and Dolbeault [19].

Some isolation results for f-harmonic maps on weighted Riemannian manifolds with boundary

In this paper, we study f -harmonic (respectively, ( p , f ) -harmonic) maps on a compact weighted Riemannian manifold of nonempty boundary and of positive Bakry–Émery Ricci curvature. We first establish a Bochner–Reilly formula for such maps and deduce therefrom some immediate isolation results. In addition, using a weighted Sobolev inequality obtained in [12] , we prove that, under some energy level depending on the Bakry–Émery curvature of the initial manifold and on the sectional curvature of the final one, the only f -harmonic (respectively, ( p , f ) -harmonic) maps are the constant ones. A gap property of the energy density of f -harmonic maps from a weighted Riemannian manifold to a unit sphere is also obtained.

Space-time L 2 estimates, regularity and almost global existence for elastic waves

Abstract In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

On Caffarelli-Kohn-Nirenberg Inequalities for Block-Radial Functions

The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. An elementary example of such inequality is the following inequality of Hardy type for functions u = u(r1(x), r2(x)), where \(r_{1}(x) = \sqrt {{{x}_{1}^{2}} + {{x}_{2}^{2}}}\) and \(r_{2}(x) = \sqrt {{{x}_{3}^{2}} + {{x}_{4}^{2}}}\) from the subspace \({\dot {H}}_{(2,2)}^{1,3} (\mathbb {R}^{4})\) of the Sobolev space \(\dot {H}^{1,3}(\mathbb {R}^{4})\), radially symmetric in variables (x1, x2) and in variables (x3, x4): $$ {\int }_{\mathbb {R}^{4}} \frac {u^{2}}{r_{1}(x)r_{2}(x)} dx \le C {\int }_{\mathbb {R}^{4}} | \nabla u|^{2} dx, $$ Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the radial estimate in Strauss (Comm. Math. Phys. 55, 149–162 1977). Furthermore, the “multi-radial” weights are a stronger singularity than radial weights of the same homogeneity, e.g. \(\frac {1}{r_{1}(x)r_{2}(x)} \ge \frac {1}{2|x|^{2}}\).

Weighted Lp‐theory for vector potential operators in three‐dimensional exterior domains

In the present paper, we study the vector potential problem in exterior domains of . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces. Copyright © 2015 John Wiley & Sons, Ltd.

Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half-spaces via mass transport and consequences

By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space R n−1 × R+, n ≥ 1 equipped with the weight !(x) = x a, a ≥ 0. It amounts to work with the fractional dimension na = n + a. The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted inequalities are also extended to the domain R n−m × R m and the weights are !(x,t) = t a1 1 ...t a m m , where n ≥ m, m ≥ 0 and a1,··· ,am ≥ 0. A weighted L p -logarithmic Sobolev inequality is derived from these inequalities.

Random Weighted Sobolev Inequalities and Application to Quantum Ergodicity

This paper is a continuation of Poiret et al. (Ann Henri Poincaré 16:651–689, 2015), where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential V on \({\mathbb{R}^d}\). We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on \({\mathbb{R}^d}\). This construction relies on accurate estimates on the spectral function in a non-compact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all bases of Hermite functions are quantum uniquely ergodic.

An Extension of Hörmander’s Hypoellipticity Theorem

Motivated by applications to stochastic differential equations, an extension of Hormander’s hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-wise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.