A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.


Notations and Main Result
For n ≥ 2, let R n denote the n-dimensional positive half-space R n : x x 1 , x , x 1 > 0, x : x 2 , . . ., x n ∈ R n−1 .

1.1
Let σ ∈ C ∞ R be a function such that σ x 1 x 1 close to x 1 0, and σ x 1 1 for x 1 ≥ 1.For j 1, 2, . . ., n, we set Then, for every multi-index α α 1 , . . ., α n ∈ N n , the conormal derivative Z α is defined by For every positive integer m the anisotropic Sobolev space H m * R n is defined as The space H m * * R n , endowed with its norm 1.7 is a Hilbert space.For the sake of convenience we also set where • denotes the integer part except for H m loc R n , all imbeddings are continuous .The anisotropic spaces H m * , H m * * are the natural function spaces for the study of initialboundary-value problems of symmetric hyperbolic systems with characteristic boundary, see 1-6 .In fact, for such problems, the full regularity i.e., solvability in the usual Sobolev spaces H m cannot be expected generally because of the possible loss of derivatives in the normal direction to the characteristic boundary, see 7, 8 .The introduction of the anisotropic Sobolev spaces H m * , H m * * is motivated by the observation that the one-order gain of normal differentiation should be compensated by two-order loss of conormal differentiation.
The equations of ideal magnetohydrodynamics provide an important example of illposedness in Sobolev spaces H m , see 7 .Application to MHD of H m * and H m * * spaces may be found in 9-13 .For an extensive study of such spaces we refer the reader to 2, 3, 14, 15 and references therein.Function spaces of this type have also been considered in 16, 17 .The purpose of this note is the proof of the following Theorems 1.1 and 1.2.These results are an important calculus tool in the use of the anisotropic spaces H m * , H m * * , and accordingly for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.Typically, in such problems one has to deal with terms of the form A∂ 1 U, where A is a real d × d matrix-valued function, and U is a vector function with d components.The matrix A admits the decomposition A I,I 0 0 0 , A 2|x 1 0 0, 1.10 with A I,I invertible in a neighborhood of the boundary {x 1 0}.Hence, one may write Then Proof.For all integers m ≥ 1, the space Hence, without loss of generality, we may assume that u is supported in a small neighborhood of x 1 0 where σ x 1 x 1 .For the proof of the theorem we use an induction argument somehow inspired from 18 .
The case m 2 follows from the classical Hardy inequality, see 19 .Given any x ∈ R n−1 , the Hardy inequality yields Integrating in x and using 1.9 with m 2 we get Let us now assume that inequality 1.13 holds for a given m ≥ 2, and suppose that u ∈ 1.17 From its definition, we see that f 0 for x 1 0. Next, we obtain the identity

International Journal of Differential Equations
We deduce from 1.18 that which by substitution in 1.16 yields the identity 1.20 Given any multi-index α α 1 , . . ., α n ∈ N n , with α 1 0, we also get 1.21 from which it readily follows that From 1.22 and 1.24 we deduce It follows that for every multi-index α α 1 , . . ., α n ∈ N n , with α 1 0, and k ∈ N such that |α| 2k ≤ m − 1.
In order to treat the case α 1 ≥ 1, we use an induction argument.We first invert the position of conormal and normal derivatives in the norm 1.5 to get where the last term comes from the control of the commutator.Then, from the inductive assumption Let us consider the estimate Notice that 1.29 holds true if α 1 0, because of 1.26 .Assume that 1.29 is true for every multi-index α α 1 , . . ., α n ∈ N n and k ∈ N such that |α| 2k ≤ m − 1 and 0 because for the first term we have |α| − 1 2 k 1 ≤ m, and for the second term we can apply estimate 1.13 , true for m by inductive assumption.Hence 1.29 is true also for α 1 β 1 .We deduce that 1.29 holds for every multi-index α α 1 , . . ., α n ∈ N n , and k ∈ N such that |α| 2k ≤ m − 1.
Therefore, from 1.28 and 1.29 we get The proof of Theorem 1.1 is complete.
In the second anisotropic space H m * * Ω we have the following results.
and looks for an estimate of HZ 1 U in H m * , H m * * , as sharp as possible.Given suitable estimates for the product of functions, the problem is then the estimate of H in H m * and H m * * .This motivates the following results.Theorem 1.1.Let m ≥ 2. Let u ∈ H m * R n ∩ H 1 0 R n be a function, and let H be defined by International Journal of Differential EquationsTheorem 1.2.Let u ∈ H m * * R n ∩ H 1 0 R n , for m ≥ 1, and let H be the function defined in 1.12 .1Ifm 1, then H L 2 R n ≤ C u H 1 R n ≤ C u H 1 * * R n .Proof.The proof of 1.32 follows by direct application of Hardy's inequality; then 1.33 follows by applying 1.32 to Zu.In case of m ≥ 3 the proof is similar to that of Theorem 1.1, hence we omit the details.