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

Abstract

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.

1. Notations and Main Result

For 𝑛≥2, let ℝ^𝑛+ denote the 𝑛-dimensional positive half-space ℝ^𝑛+∶=𝑥=𝑥₁,𝑥^,𝑥₁>0,𝑥′∶=𝑥₂,…,𝑥_𝑛∈ℝ^𝑛−1.(1.1) Let 𝜎∈𝐶^∞(ℝ₊) be a function such that 𝜎(𝑥₁)=𝑥₁ close to 𝑥₁=0, and 𝜎(𝑥₁)=1 for 𝑥₁≥1. For 𝑗=1,2,…,𝑛, we set 𝑍₁∶=𝜎𝑥₁𝜕₁,𝑍_𝑗∶=𝜕_𝑗,for𝑗≥2.(1.2) Then, for every multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛, the conormal derivative 𝑍^𝛼 is defined by 𝑍^𝛼∶=𝑍^𝛼1₁⋯𝑍^𝛼𝑛_𝑛.(1.3) For every positive integer 𝑚 the anisotropic Sobolev space 𝐻^𝑚∗(ℝ^𝑛+) is defined as 𝐻^𝑚∗ℝ^𝑛+∶=𝑤∈𝐿²ℝ^𝑛+∶𝑍^𝛼𝜕^𝑘1𝑤∈𝐿²ℝ^𝑛+,|𝛼|+2𝑘≤𝑚.(1.4) In 𝐻^𝑚∗(ℝ^𝑛+) we introduce the norm ‖𝑤‖^2𝐻_{𝑚∗ℝ𝑛+}∶=_{|𝛼|+2𝑘≤𝑚}‖‖𝑍^𝛼𝜕^𝑘1𝑤‖‖^2𝐿_{2ℝ𝑛+}.(1.5) The space 𝐻^𝑚∗(ℝ^𝑛+), endowed with its norm (1.5) is a Hilbert space. We also introduce a second anisotropic Sobolev space. For every positive integer 𝑚, the space 𝐻^𝑚∗∗(ℝ^𝑛+) is defined as 𝐻^𝑚∗∗ℝ^𝑛+∶=𝑤∈𝐿²ℝ^𝑛+∶𝑍^𝛼𝜕^𝑘1𝑤∈𝐿²ℝ^𝑛+,|𝛼|+2𝑘≤𝑚+1,|𝛼|≤𝑚.(1.6) In particular, 𝐻^1∗∗(Ω)=𝐻¹(Ω). In 𝐻^𝑚∗∗(ℝ^𝑛+), we introduce the natural norm ‖𝑤‖^2𝐻_{𝑚∗∗ℝ𝑛+}∶=_{|𝛼|+2𝑘≤𝑚+1,|𝛼|≤𝑚}‖‖𝑍^𝛼𝜕^𝑘1𝑤‖‖^2𝐿_2(ℝ𝑛+).(1.7) The space 𝐻^𝑚∗∗(ℝ^𝑛+), endowed with its norm (1.7) is a Hilbert space. For the sake of convenience we also set 𝐻^0∗(ℝ^𝑛+)=𝐻^0∗∗(ℝ^𝑛+)=𝐿²(ℝ^𝑛+). We observe that 𝐻𝐻^𝑚ℝ^𝑛+↪𝐻^𝑚∗∗ℝ^𝑛+↪𝐻^𝑚∗ℝ^𝑛+⊂𝐻^{𝑚𝑙𝑜𝑐}ℝ^𝑛+,(1.8)_𝑚∗ℝ^𝑛+↪𝐻^[𝑚/2]ℝ^𝑛+,𝐻^𝑚∗∗ℝ^𝑛+↪𝐻^[(𝑚+1)/2]ℝ^𝑛+,(1.9) where [⋅] denotes the integer part (except for 𝐻^{𝑚𝑙𝑜𝑐}(ℝ^𝑛+), all imbeddings are continuous).

The anisotropic spaces 𝐻^𝑚∗,𝐻^𝑚∗∗ are the natural function spaces for the study of initial-boundary-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 ill-posedness 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𝜕₁𝑈, 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=𝐴₁+𝐴₂,𝐴₁∶=⎛⎜⎜⎝𝐴^𝐼,𝐼000⎞⎟⎟⎠,𝐴_2∣𝑥1=0=0,(1.10) with 𝐴^𝐼,𝐼 invertible in a neighborhood of the boundary {𝑥₁=0}. Hence, one may write 𝐴₂𝜕₁𝑈=𝐻𝑍₁𝑈,(1.11) where 𝐻(𝑥)=𝜎(𝑥₁)⁻¹𝐴₂(𝑥), and looks for an estimate of 𝐻𝑍₁𝑈 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+)∩𝐻¹⁰(ℝ^𝑛+) be a function, and let 𝐻 be defined by 𝐻𝑥₁,𝑥′=𝑢𝑥¹,𝑥′𝜎𝑥₁.(1.12) Then ‖𝐻‖_𝐻^∗𝑚−2_(ℝ^𝑛+₎≤𝐶‖𝑢‖𝐻^𝑚∗ℝ^𝑛+.(1.13)

Proof. For all integers 𝑚≥1, the space 𝐶^∞(0)(R^𝑛+) (𝐶^∞(0)(ℝ^𝑛+) denotes the set of restriction to ℝ^𝑛+ of functions in 𝐶^∞0(ℝ^𝑛+)) is dense in 𝐻^𝑚∗(ℝ^𝑛+), see [4]. Hence, without loss of generality, we may assume that 𝑢 is supported in a small neighborhood of 𝑥₁=0 where 𝜎(𝑥₁)=𝑥₁. 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 ^∞0||||𝑢𝑥₁,𝑥^𝑥₁||||²𝑑𝑥₁≤4^∞0||𝜕₁𝑢𝑥₁,𝑥^||²𝑑𝑥₁,∀𝑢∈𝐻¹⁰ℝ^𝑛+.(1.14) Integrating in 𝑥^ and using (1.9) with 𝑚=2 we get ‖‖‖𝑢𝑥₁‖‖‖_{𝐿2(ℝ𝑛+)}≤2‖𝑢‖_𝐻¹_(ℝ^𝑛+₎≤𝐶‖𝑢‖_𝐻^2∗_(ℝ^𝑛+₎.(1.15) Let us now assume that inequality (1.13) holds for a given 𝑚≥2, and suppose that 𝑢∈𝐻^∗𝑚+1(ℝ^𝑛+)∩𝐻¹⁰(ℝ^𝑛+). A simple computation shows that for 𝑘∈ℕ, 𝜕^𝑘1𝑥𝑢₁=𝑥𝑓^1𝑘+1,(1.16) with 𝑓=^𝑘_ℎ=0⎛⎜⎜⎝ℎ𝑘⎞⎟⎟⎠𝜕_1𝑘−ℎ𝑢ℎ!(−1)^ℎ𝑥^1𝑘−ℎ.(1.17) From its definition, we see that 𝑓=0 for 𝑥₁=0. Next, we obtain the identity 𝜕₁𝑓=^𝑘_ℎ=0⎛⎜⎜⎝ℎ𝑘⎞⎟⎟⎠𝜕_{1𝑘+1−ℎ}𝑢ℎ!(−1)^ℎ𝑥^1𝑘−ℎ+^𝑘−1_ℎ=0⎛⎜⎜⎝ℎ𝑘⎞⎟⎟⎠𝜕_1𝑘−ℎ=𝜕_1𝑘+1𝑢ℎ!(−1)^ℎ𝑥^{1𝑘−ℎ−1}(𝑘−ℎ)𝑢𝑥^𝑘1+^𝑘_ℎ=1⎛⎜⎜⎝ℎ𝑘⎞⎟⎟⎠𝜕_{1𝑘+1−ℎ}𝑢ℎ!(−1)^ℎ𝑥^1𝑘−ℎ+^𝑘−1_ℎ=0⎛⎜⎜⎝ℎ+𝑘1⎞⎟⎟⎠𝜕_1𝑘−ℎ𝑢(ℎ+1)!(−1)^ℎ𝑥^{1𝑘−ℎ−1}=𝜕^1𝑘+1𝑢𝑥^𝑘1.(1.18) We deduce from (1.18) that 𝑓𝑥₁,𝑥^=^𝑥1₀𝜕^1𝑘+1𝑢𝑦₁,𝑥^𝑦^𝑘1𝑑𝑦₁,(1.19) which by substitution in (1.16) yields the identity 𝜕^𝑘1𝑥𝑢₁𝑥¹,𝑥^=∫^𝑥10𝜕^1𝑘+1𝑢𝑦₁,𝑥^𝑦^𝑘1𝑑𝑦₁𝑥^1𝑘+1.(1.20) Given any multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛, with 𝛼₁=0, we also get 𝑍^𝛼𝜕^𝑘1𝑥𝑢₁(𝑥)=∫^𝑥10𝑍^𝛼𝜕^1𝑘+1𝑢𝑦₁,𝑥^𝑦^𝑘1𝑑𝑦₁𝑥^1𝑘+1,(1.21) from which it readily follows that ||||𝑍_𝛼𝜕^𝑘1𝑥𝑢₁(𝑥)||||≤∫^𝑥10||𝑍^𝛼𝜕^1𝑘+1𝑢𝑦₁,𝑥^||𝑑𝑦₁𝑥₁.(1.22) Setting 𝑔𝑥₁,𝑥^=^𝑥1₀||𝑍^𝛼𝜕^1𝑘+1𝑢𝑦₁,𝑥^||𝑑𝑦₁(1.23) the Hardy inequality yields ^∞0||||𝑔𝑥₁𝑥₁,𝑥^||||²𝑑𝑥₁≤4^∞0||𝜕₁𝑔𝑥₁,𝑥^||²𝑑𝑥₁.(1.24) From (1.22) and (1.24) we deduce ‖‖‖𝑍_𝛼𝜕^𝑘1𝑥𝑢₁‖‖‖^2𝐿_{2ℝ𝑛+}≤4‖‖𝑍^𝛼𝜕^1𝑘+1𝑢‖‖^2𝐿_{2ℝ𝑛+}.(1.25) It follows that ‖‖‖𝑍_𝛼𝜕^𝑘1𝑥𝑢₁‖‖‖_{𝐿2(ℝ𝑛+)}≤𝐶‖𝑢‖_𝐻^∗𝑚+1_(ℝ^𝑛+₎(1.26) for every multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛, with 𝛼₁=0, and 𝑘∈ℕ such that |𝛼|+2𝑘≤𝑚−1.
In order to treat the case 𝛼₁≥1, we use an induction argument. We first invert the position of conormal and normal derivatives in the norm (1.5) to get ‖‖‖𝑢𝑥₁‖‖‖^2𝐻_{∗𝑚−1(ℝ𝑛+)}≤𝐶_{|𝛼|+2𝑘≤𝑚−1}‖‖‖𝜕_𝑘1𝑍^𝛼𝑥𝑢₁‖‖‖^2𝐿_2(ℝ𝑛+)+𝐶‖‖‖𝑢𝑥₁‖‖‖_{𝐻∗𝑚−2(ℝ𝑛+)},(1.27) where the last term comes from the control of the commutator. Then, from the inductive assumption ‖‖‖𝑢𝑥₁‖‖‖^2𝐻_{∗𝑚−1(ℝ𝑛+)}≤𝐶_{|𝛼|+2𝑘≤𝑚−1}‖‖‖𝜕_𝑘1𝑍^𝛼𝑥𝑢₁‖‖‖^2𝐿_{2ℝ𝑛+}+𝐶‖𝑢‖_𝐻^𝑚∗_(ℝ^𝑛+₎.(1.28) Let us consider the estimate _|𝛼|+2_{𝑘≤𝑚−1}‖‖‖𝜕_𝑘1𝑍^𝛼𝑥𝑢₁‖‖‖_{𝐿2(ℝ𝑛+)}≤𝐶‖𝑢‖_𝐻^∗𝑚+1_(ℝ^𝑛+₎.(1.29) Notice that (1.29) holds true if 𝛼₁=0, because of (1.26). Assume that (1.29) is true for every multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛 and 𝑘∈ℕ such that |𝛼|+2𝑘≤𝑚−1 and 0≤𝛼₁≤𝛽₁−1, for some 1≤𝛽₁≤𝑚−1. We have _{|𝛼|+2𝑘≤𝑚}_{−1,1≤𝛼1≤𝛽1}‖‖‖𝜕_𝑘1𝑍^𝛼𝑥𝑢₁‖‖‖^2𝐿_{2ℝ𝑛+}=_{|𝛼|+2𝑘≤𝑚−1,1≤𝛼1≤𝛽1}‖‖‖𝜕_𝑘1𝑍^𝛼′𝑍^𝛼1−1𝑥₁𝜕₁𝑥𝑢₁‖‖‖^2𝐿_{2ℝ𝑛+}=_{|𝛼|+2𝑘≤𝑚−1,1≤𝛼1≤𝛽1}‖‖‖𝜕_𝑘1𝑍^𝛼′𝑍^𝛼1−1𝜕¹𝑢−𝑢𝑥₁‖‖‖^2𝐿_2(ℝ𝑛+)≤𝐶_{|𝛼|+2𝑘≤𝑚−1,1≤𝛼1≤𝛽1}‖‖𝜕^1𝑘+1𝑍^𝛼′𝑍^𝛼1−1𝑢‖‖^2𝐿_{2ℝ𝑛+}+‖‖‖𝜕^𝑘1𝑍^𝛼′𝑍^𝛼1−1≤𝐶‖𝑢‖_{2𝐻𝑚∗ℝ𝑛+}𝑥𝑢₁‖‖‖^2𝐿_2(ℝ𝑛+)(1.30) 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 𝛼₁=𝛽₁. We deduce that (1.29) holds for every multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛, and 𝑘∈ℕ such that |𝛼|+2𝑘≤𝑚−1.
Therefore, from (1.28) and (1.29) we get ‖‖‖𝑢𝑥₁‖‖‖_{𝐻∗𝑚−1(ℝ𝑛+)}≤𝐶‖𝑢‖_𝐻^∗𝑚+1_(ℝ^𝑛+₎.(1.31) The proof of Theorem 1.1 is complete.

In the second anisotropic space 𝐻^𝑚∗∗(Ω) we have the following results.

Theorem 1.2. Let 𝑢∈𝐻^𝑚∗∗(ℝ^𝑛+)∩𝐻¹⁰(ℝ^𝑛+), for 𝑚≥1, and let 𝐻 be the function defined in (1.12).(1)If 𝑚=1, then ‖𝐻‖_𝐿²_(ℝ^𝑛+₎≤𝐶‖𝑢‖_𝐻¹_(ℝ^𝑛+₎≤𝐶‖𝑢‖_𝐻^1∗∗_(ℝ^𝑛+₎.(1.32)(2)If 𝑚=2, then ‖𝐻‖_𝐻^1∗_(ℝ^𝑛+₎≤𝐶‖𝑢‖_𝐻^2∗∗_(ℝ^𝑛+₎.(1.33)(3)If 𝑚≥3, then ‖𝐻‖_𝐻^{𝑚−2∗∗}_(ℝ^𝑛+₎≤𝐶‖𝑢‖_𝐻^𝑚∗∗_(ℝ^𝑛+₎.(1.34)

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.

Acknowledgment

The work was supported by the National Research Project PRIN 2007 “Equations of Fluid Dynamics of Hyperbolic Type and Conservation Laws.”